ordinary differential equations, boundary value problems, oscillation theory, qualitative theory, partial differential equations, hyperbolic equations

functional differential equations, dynamical systems

nonlinear ordinary differential equations, periodic solutions, boundary value problems, topological methods

ordinary and delay differential equations, Volterra integral equations, fractional differential equations

Ordinary Differential Equations (Boundary Velue Problems, Oscillation Theory)

Boundary value problems for ODEs and elliptic and parabolic PDEs

boundary value problems for ordinary differential equations, boundary value problems for finite difference equations, boundary value problems for fractional differential equations

Nonlinear Evolutionary Equations and stochastic PDES

Qualitative theory of differential and difference equations

linear differential and difference equations and systems, half-linear differential equations

ordinary differential equations, partial differential equations, dynamical systems, bifurcation, chaos

Control Theory, Differential equations with deviated argument, Oscillations, Stability

BVPs for differentilal equations, fractional differential equations

boundary value problems, fractional differential and difference equations, ordinary and delay differential equations

Ordinary and functional differential equations (delay and neutral delay equations), Difference equations, Nonlinear dynamical systems, Mathematical models in Biology, Ecology, Economy, Epidemiology, and Medicine, etc.

Dynamic equations on time scales, dynamic inequalities on time scales, Applications of opial and Wirtinger inequalities on the zeros of Riemann Zeta Function.

Dynamic equations on time scales, dynamic inequalities on time scales, Applications of opial and Wirtinger inequalities on the zeros of Riemann Zeta Function.

functional differential equations; dynamical systems; invariant manifolds; periodic orbits; stability; global attractor; state-dependent delay

Partial Differential Equations and applications.

Stability of Nonlinear Systems; Artificial Neural Networks, Volterra Integro-differential Systems, Planning Algorithms, and Internet Congestion Control (new)

Volterra Integro-differential Systems

These have their roots in biological growth problems, whose origins can be traced from the Malthusian model through the logistic equation, the predator-prey system of Lotka and Volterra and Volterra's own formulation of integral equations regarding age distribution in population.

Artificial Neural Networks

ANNs are crude mathematical models of biological neural systems. They have to be designed in such a way that their synaptic weights, which are the strengths of signals or communications between neurons, could effectively store and

retrieve memories.

Planning Algorithms

In robotics, motion planning is an important component. The focus is on designing algorithms that generate useful motions by processing complicated geometric models.

Swarm Intelligence

Swarming, or aggregations of organisms in groups, can be found in nature in many organisms ranging from simple bacteria to mammals. A relatively new area of research looks into the behavior of swarms, in particular to how a swarm's collective behavior could be mimicked to solve challenging engineering problems.

Internet Congestion Control (new area of interest)

One of the most recent and exciting areas of research in the stability analysis of systems deals with the need to control traffic in the ever-growing Internet in a more systematic, rigorous and efficient manner.

Volterra Integro-differential Systems

These have their roots in biological growth problems, whose origins can be traced from the Malthusian model through the logistic equation, the predator-prey system of Lotka and Volterra and Volterra's own formulation of integral equations regarding age distribution in population.

Artificial Neural Networks

ANNs are crude mathematical models of biological neural systems. They have to be designed in such a way that their synaptic weights, which are the strengths of signals or communications between neurons, could effectively store and

retrieve memories.

Planning Algorithms

In robotics, motion planning is an important component. The focus is on designing algorithms that generate useful motions by processing complicated geometric models.

Swarm Intelligence

Swarming, or aggregations of organisms in groups, can be found in nature in many organisms ranging from simple bacteria to mammals. A relatively new area of research looks into the behavior of swarms, in particular to how a swarm's collective behavior could be mimicked to solve challenging engineering problems.

Internet Congestion Control (new area of interest)

One of the most recent and exciting areas of research in the stability analysis of systems deals with the need to control traffic in the ever-growing Internet in a more systematic, rigorous and efficient manner.

Fractional order differential equations, integral inequalities, impulsive differential equations, quantitative and qualitative properties of solutions

Impulsive differential equations and inclusions, fractional differential equations and inclusions, solutions sets, fixed point theory and applications.

Boundary Value problems for differential equations, Nonlinear Analysis

34K

stochastic stability and stochastic optimal control

Dynamical systems

Partial functional differential equations

Evolution equations

Delay and ordinary differential equations

Partial functional differential equations

Evolution equations

Delay and ordinary differential equations

ordinary differential equations, dynamic equations on time scales

partial differential equations

impulsive ODE, singular ODE

partial differential equations, mathematical biology, population dynamics

Differential Equations

ordinary differential equations

Integral equations; integro-differential equations

fractional differential equations, boundary value problems, stability, difference equations, Volterra integral equations

Boundary value problems

differential inclusions, optimal control

qualitative theory of impulsive and stochastic systems, delay differential systems and neural networks

Differential equations, difference equations, q-difference equations, dynamic equations and inequalities on time scales, discrete dynamical systems, integral inequalities, asymptotic analysis, and boundary value problems.

qualitative behvior of differential equations,and difference equations,

Qualitative Theory of ordinary differential equations, Mathematical Biology

Nonlinear Functional Analysis, Ordinary Differential Equation

Oscillation Theory.

Control Theory, Problems in Banach Space.

Partial differential equations - self similar solutions

Impulsive differential equation, boundary value problem

Qualitative theory of solutions of Difference equations/ Differential equations/ dynamic equaions

numerical methods of differential equations, numerical linear algebra, splitting methods

Nonlinear boundary problems for ordinary differential equations, asymmetric problems, qualitative behavior of solutions of nonlinear differential equations, differential-geometric structures on manifolds

Quadratic integral equations, Functional Equations, Integral and differential equations in Banach space, Fractional Calculus, Coupled System.

Fractional Differential Equations, Applications of Semigroups Theory to Abstract Differential Equations, Inverse Problems and Control Problems.

Evolution equations, Inequality theorey, Fractional Differential equations, Fractional calculus, Numerical analysis and applied amthematics.

differential inclusions, differential equations, optimal control

boundary value problems for ordinary differential equations, nonlinear operators, degree theory

Nonlinear evolution equations; Impulsive differential equations; Fractional differential equations; Optimal controls; Fractional Hermite-Hadamard Inequalities.

The boundary value problems of differential equations and nonlinear analysis.

Partial Differential equations

Oscillation; Nonoscillation; Stability; Differential Equations; Difference Equations; Dynamic Equations; Time Scales

Partial differential equations

Integral Equations, Difference Equations, Stability theory, Fixed point theory, Qualitative properties of differential, difference, and integral equations, dynamic equations on time scales.

Theory of Differntial Equations, Theory of Difference Equation, Calculus on Time Scales, Theory of Fractional Difference Equations

Nonlocal boundary value problems for ODEs.

oscillation theory of differential equations, difference equations, and dynamic equations on time scales

Differential Equations: Oscillation, Stability, Boundedness, Periodicity.

Functional Differential Equations: Oscillation, Positive solutions.

Lyapunov's type Inequalities.

Difference Equations.

Functional Differential Equations: Oscillation, Positive solutions.

Lyapunov's type Inequalities.

Difference Equations.

Complex analysis, Nevanlinna theory, Complex differential equation

Complex analysis, Complex differential equations, Complex difference equations

complex linear differential equation; complex difference equation; value distribution theory,

Impulsive Differential System, Partial Differential Equations,Differential Inclusions, Computational PDE

Stochastic/deterministic Differential Equations/ Inclusions.

Differential Equations, Integral Equations, Dynamical Equations On Time scales

Stability, instability, boundedness, asymptotic behaviors, oscillation, non-oscillation of solutions, Lyapunov functions and functional for Ordinary Differential Equations and Functional Differential Equations.

Applied Mathematics, PDEs

Optimization, Nonsmooth Analysis, Variational Inequalities, Differential Inclusions

Ordinary and Partial Differential Equations, Impulsive Differential Equations,

Functional Differential Equations, Difference Equations, Time Scale Calculus.

Functional Differential Equations, Difference Equations, Time Scale Calculus.

differential equations, difference equation, time scale calculus

oscillation theory of functional differential equations with deviating argument

Nonlinear analysis

Fractional differential equations

Impulsive differential equations

Biomedical applications

Fractional differential equations

Impulsive differential equations

Biomedical applications

RESEARCH INTERESTS:

Major research interests are in the theory of functions, functional analysis and the theory of differential equations:

- Linear and nonlinear integral and matrix operators;

- Weighted inequalities;

- Weighted embedding theorem;

- Spectral theory of operators;

- The qualitative properties of quasilinear differential and difference equations.

Major research interests are in the theory of functions, functional analysis and the theory of differential equations:

- Linear and nonlinear integral and matrix operators;

- Weighted inequalities;

- Weighted embedding theorem;

- Spectral theory of operators;

- The qualitative properties of quasilinear differential and difference equations.

Boundary value problem,Switched system,Biological system,Logical system

bifurcation theory of differential equations with delays

Singular perturbations in ODES

Variational method, Sobolev and Lebesgue spaces with variable exponent, Elliptic and Parabolic equations, p-Laplacian, p(x)-Laplacian, discrete boundary value problem, Hardy type inequalities, Sobolev type inequalities

Fractional Calculus

differential equations and inclusions, critical point theory

Nonlinear Function Analysis, Partial Differential Equations (Elliptic Differential inclusion problem); variable exponent inclusion problem

boundary value problems, reaction-diffusion equations, multivalued analysis and evolution equations

nonlinear functional analysis, critical point theory

Boundary Value Problems.

Optimal control, abstract differential equations with fractional orders

Ordinary differential equations, stability theory, asymptotic behaviour of solutions

Limit cycles, isochronous center, critical period bifurcation,traveling wave solution of nolinear wave equations.

fractional differential equations

stability theory of ordinary differential equations

ordinary differential equations, qualitative theory

ordinary differential equations, functional differential equations

Nonlinear Analysis, Boundary value problems,

Dynamical systems

Qualitative th. of differential eqs., Dynamical systems (Lagrangian dynamic, ergodic theory), differential geometry.

Nonlinear Dynamics, Biological Mathematics

partial differential equation

Partial Differential Equation

Qualitative theory of differential equations, Functional differential equations,

Inequalities, Dynamic Equations on Time Scales.

Inequalities, Dynamic Equations on Time Scales.

Ordinary differential equations, Semigroups of operators, Control Theory

partial differential equations, nonlinear analysis

Differentia equations, Integral equations, Fractional calculus, Dynamical systems

Partial Differential Equations

Nonlinear Analysis; Partial Differential Equations of Elliptic Type

bifurcation theory, variational methods, topological methods

Theories on parabolic and elliptic equations

Singular perturbation theory, Operational Research and Cybernetics

delay differential equations, difference equations, equations of mathematical biology

control theory of partial differential equations

Dynamic systems on time scale, Volterra integro-dynamic systems, set differential equations, Hardy type inequalities.

Impulsive differential equations, functional differential equations and difference equations

Ordinary Differential Equations and Inclusions. Fixed point Theory.

ordinary differential equations

Nonlinear functional Analysis and Differential Equations

Wave equation, Thermoelasticity, viscoelasticity, Timoshenko systems

Oscillation Theory; Lyapunov-type Inequality; Mean Value Theorem.

fractional differential equations

Functional Differential Equations

ordinary differential and difference equations, functional differential equations and systems

functional differential equations, boundary value problems

Topological methods in nonlinear analysis, ordinary differential equations on differentiable manifolds.

ordinary and partial differential equations, boundary value problems, spectral theory, operator theory, nonlinear functional analysis, fractional differential equations

ordinary differential equations, difference equations, boundary value problems

Partial Differential Equation

Fixed point theory and its applications

value distribution, difference equations, differential equations, tropical nevanlinna theory

Nonlinear Functional Analysis

Differential equations, dynamical systems, bifurcation theory, control theory,

integral inequalities, evolution equations

integral inequalities, evolution equations

Operator splitting, partial differential equations, numerical methods

Asymptotic behaviour of solutions of differential equations with fading perturbations.

Partial differential equations

Existence of solutions of differential equations of fractional order

elliptic partial differential equations, ordinary differential equations

medical physics, specifically MRI (magnetic resonance imaging) and also NMR (nuclear magnetic resonance)

Differential equations

theory of functional differential equations, boundary value problems, positivity of solutions, nonoscillation, stability, integro-differential equations, impulsive equations

Differential delay equations

Partial differential equations

delay differential equations, modeling of infectious diseases

qualitative and quantitative properties of solutions of ordinary differential and difference equations

ordinary differential equations, Lienard dynamical system, oscillation theory, qualitative theory, stability theory

Functional Analysis, Partial Differential Equations

Complex differential equations, value distribution theory

Partial Differential Equations

Dynamical Systems,

Mathematical Biology,

Epidemiology

Mathematical Biology,

Epidemiology

Nonlinear Analysis

partial differential equations, evolutions equations (parabolic, hyperbolic), boundary value problems, analysis, applied math.

functional differential equations, boundary value problems

Stability of (impulsive) delay differential equations and applications

Differential equations

nonlinear boundary value problem, semilinear hyperbolic equations

Delay differential equations; Stability theory; Oscillation theory

Nonlinear boundary value problems for ordinary differential equations and difference equations; stability of functional differential equations.

Ordinary differential equations

Differential Equations, Fractional Differential Equations, Impulsive Differential Equations, Method of Lines

spectral theory, operator theory, boundary value problem of ODE, nonlinear analysis, fractional differential equations

ordinary differential equations

PDE

Coincide degree and nonlinear differential equations, Delay Differential Equations with Applications in Population Dynamics

Applied Mathematics, PDE

Boundary value problems, Fixed point theory, Differential equations.

coincide degree and nonlinear differential equations, delay differential equations with applications in population dynamics

nonlinear boundary value problems, variational and topological methods

numerical computation, functional differential equation, optimal control, mathematical control theory, Stefan problem, spectral interpolation problem

differential equations, dynamical systems

ODE and PDE, Fixed Point Theory, Option pricing

fractional differential equations;

Bound value problem

Bound value problem

Nonlinear functional analysis

partial differential equations, oscillation theorem

Fractional Calculus, ODE, PDE

Ordinary differential equations: oscillation theory, asymptotic analysis, qualitative analysis

Mathematica, Differential Equations, Singular Differential Equations

Differential equations

PDE, Degenerate equations, Reaction Diffusion Systems, Control, stabilisation.

Differential equations, Fractional derivatives, complex analysis, functional inequalities.

stochastic differential equation, stability of stochastic systems

partial differential equations, abstract differential equations

abstract evolution equations, partial differential equations, operator theory

functional differential equation, stability theory

fractional differential equations

ordinary differential equations

qualitative theory of evolution equations

periodic solutions of nonlinear differential equations

His research interests in the areas of dynamics of neural networks, and qualitative theory of differential equations and difference equations.

My research interests include nonlinear dynamic systems, neural networks, Functional differential equations; Almost periodic differential equation.

boundedness, periodicity, stability

fractional Calculus, maximum principle, partial differential equations

Ordinary and Partial Differential Equations

monotone dynamic systems, qualitative theory of differential equations and difference equations, computer-aided geometric design

calculus of variations, functional analysis

dynamics of nonlinear differential equations

operator theory, abstract Volterra equations

boundary value problem, spectral theory

Partial Differential Equations

partial differential equations, dynamical systems, operator theory

half-linear differential equations, oscillation and nonoscillation of solutions, comparison, regular variation

partial differential equations, controllability

Nonlinear Evolution Equations, Differential Inclusions, Control Problems, Nonlinear Dynamical Systems

periodic differential equations, nonlinear dynamic systems, neural network dynamics

stabilization and controllability of bilinear/semilinear systems

Nonlinear dynamic systems, neural networks, biomathematics and applied mathematics.

ordinary and delay differential equations, mathematical biology and medicine, control theory, and dynamical systems

ordinary differential equations, boundary value problems, fractional differential equations

differential equations and inclusions, fixed point theory

dynamical systems

Fractional Differential Equations and Inclusions

Dynamical Systems

Control Theory

Dynamical Systems

Control Theory

impulsive differential equations and stochastic impulsive differential equations, dynamical equations on time scales, bifurcation theory for discrete dynamical systems

Neural networks, Stability,Optimization,

Sobolev spaces, Orlicz-Sobolev spaces,PDE's: elliptic equations, unilateral problems, existence, uniqueness, regularity

nonlinear dynamicsystems, neural networks, and qualitative theory of differential equations and difference equations.

variational methods, critical point theory, difference equations, analysis on manifolds

Nonlinear Stability Theory

Ordinary Differential Equations

Ordinary Differential Equations

lattice differential equations

stability and bifurcation

Differential Equations

nonlinear Analysis, functional Analysis, nonlinear analysis

Hamiltonian systems; elliptic partially differential equations

Integral Equations, Numerical Analysis, Fixed Point Theory.

Calculus of Variations, Topology, functional Analysis, fixed point theory, nonlinear integral equations

Theory of Differential and Integral Inequalities and its Applications, Semigroups of Linear Operators and Their Applications to Partial Differential Equations, Theory of Positive Operators and Stability of Difference Schemes, Theory of Interpolation of Operators and its Applications, Difference Schemes for Stochastic Partial Differential Equations, Uniform Difference Schemes for Singular Perturbation Problems, Non-Classical Problems of Mathematical Physics, Well-Posedness of Parabolic and Elliptic Differential and Difference Equations, Stability of the Neutral Delay Differential and Difference Equations, Numerical Solving of Applied Problems, Reading, Taking Part in the Preparation of Secondary School and University Students for Mathematical Olympiads

Partial Differential Equations of Elliptic, Variational Methods, PDE's Systems

Computational fluid dynamics, Convergence analysis of Interface Element method

Nonlinear Analysis

Spectral theory of differential equations, Cryptography

Parabolic partial differential equations.

functional differential equations, difference equations, asymptotic properties of solutions

functional differential equations, difference equations, asymptotic properties of solutions

Nonlinear operators and differential equations: the fixed point theory, operatorial equations and inclusions: existence results, uniqueness and data dependence (monotonicity, continuity, the derivatives with respect to parameters) and stability for the solution of functional-differential equations, differential equations with delayed argument and differential equations with mixed argument, obtained using weakly Picard operators technique.

- Techniques for approximating solutions of delay differential equations: numerical methods using successive approximations sequences and trapezoidal quadrature formula and spline functions method, the step method.

- Techniques for approximating solutions of delay differential equations: numerical methods using successive approximations sequences and trapezoidal quadrature formula and spline functions method, the step method.

continuum mechanics, contact mechanics, EDP

Differential inclusions, control

Regularity estimates for elliptic and parobolic equations in Sobolev spaces or Orlicz spaces; Homogeneous groups; Hormander's vector fields

nonlinear analysis and potential theory

Ordinary differential equations, partial differential equations, finite difference equations

Ordinary Differential Equations in Banach Spaces,

Differential Equations with Impulses,

Functional Analysis, Topology

Differential Equations with Impulses,

Functional Analysis, Topology

Nonlinear functional analysis,singular boundary value problems

Almost periodic equations, KAM thoery and applications

elasticity

Spectral theory and eigenvalue problems. Inverse problems

Spectral theory of differential operators

Ordinary Differential Equations; Numerical Analysis

Partial Differential Equations: Existence, Regularity, Asymptotic of solutions

Qualitative study of Partial Differential Equations of Hyperbolic and Parabolic. type

differential equations, integral equations, fixed point theory

Bifurcation and chaos in dynamics and fractional order differential equations,chaos control.

Asymptotic properties of solutions of difference equations

Partial Differential Equations

Differential Equations and Dynamical Systems

Differential Equations and Dynamical Systems

differential equation

stability and bifurcation theory of ordinary differential equation functional differential equation.

Differential Equations, Integro-differential equations, dynamical systems on Time Scales.

Delay differential, equations;Ordinary differential Equations

Stability;Coupled system

stability for coupled systems of differential equations

My research interests are in the areas of dynamics of neural networks, and qualitative theory of differential equations and difference equations.

differential equations, fixed point theory

nonlinear diffusion equation, image processing

differential equation

Differential equations, variational methods

Variational methods, Fixed point theory

Dynamic contact problems: elastic, viscoelastic, viscoplastic, elastic-

viscoplastic, electro-elastic, ... ect (with or

without friction, damage, adhesion, Normal

compliance, ... )

viscoplastic, electro-elastic, ... ect (with or

without friction, damage, adhesion, Normal

compliance, ... )

Delay Differential Equations with

Applications to the Life Sciences;

Functional Differential Equations;

Almost Periodic Differential Equations;

Delay Differential Equations with Applications in

Population Dynamics

Applications to the Life Sciences;

Functional Differential Equations;

Almost Periodic Differential Equations;

Delay Differential Equations with Applications in

Population Dynamics

Analysis. Differential Equations in finite and infinite dimensional spaces. Difference Equations. Fixed point theory.

PDE, Calculus of Variations, elliptic equations, parabolic equations, nodal solution, maximum principles, comparison principles

PDE

differential equations, variational methods

Applied mathematics, EDP, initial boundary value problems in mechanics

Fractional calculus

Nonlinear elliptic equations: local and Nonlocal

Hamiltonian systems

Nonlinear elliptic equations: local and Nonlocal

Hamiltonian systems

Qualitative theory of functional differential and difference equations

Qualitative behavior in difference and differential equations

Nonlinear elliptic and parabolic equations

This paper is concerned with systems of functional differential equations with either finite or infinite delay. We give conditions on the system and on a Liapunov function to ensure that the zero solution is asymptotically stable. The main result of this paper is that the assumption on boundedness in Marachkov type stability results may be replaced (in both the finite and the infinite delay case) with the condition that $|f(t,\varphi)|\le F(t)$ such that $\int^{\infty} 1/F(t) dt=\infty$.

Our aim is studing the asymptotic behaviour of the solutions of the equation $\dot x(t) = -a(t)x(t)+a(t)x(pt)$ where $0<p<1$ is a constant. This equation is a special case of the so called pantograph equations of the form $\dot x(t) = -a(t)x(t)+b(t)x(p(t))$. First we prove an asymptotic estimate of the solutions of the later equation, then using this result we show the asymptotic behavior of the solutions of the former equation. In particular, we prove that all solutions are asymptotically logarithmically periodic.

Some property which is equivalent to the concept of an asymptotically almost periodic integral for almost periodic processes is introduced. By using this property, it is shown that a precompact integral of almost periodic processes which is uniformly asymptotically stable is an asymptotically almost periodic integral. Results are applied to the existence of almost periodic solutions of some evolution equation.

This paper discusses the existence of almost periodic solutions of neutral functional differential equations. Using a Liapunov function and the Razumikhin's technique, we obtain the existence, uniqueness and stability of almost periodic solutions.

We derive some sufficient conditions for certain classes of ordinary differential inequalities of neutral type with distributed delay not to have eventually positive or negative solutions. These, together with the technique of spatial average, the Green's Theorem and Jensen's inequality, yield some sufficient conditions for all solutions of a class of neutral partial functional differential equations to be oscillatory. An example is given to illustrate the result.

We use a fixed point index theorem in cones to study the existence of positive solutions for boundary value problems of second-order functional differential equations of the form $$\left\{ \begin{array}{ll} y''(x)+r(x)f(y(w(x)))=0,&0<x<1,\\ \alpha y(x)-\beta y'(x)=\xi (x),&a\leq x\leq 0,\\ \gamma y(x)+\delta y'(x)=\eta (x),&1\leq x\leq b; \end{array}\right.$$ where $w(x)$ is a continuous function defined on $[0,1]$ and $r(x)$ is allowed to have singularities on $[0,1]$. The result here is the generalization of a corresponding result for ordinary differential equations.

We prove the existence and uniqueness of a global solution of a damped quasilinear hyperbolic equation. Key point to our proof is the use of the Yosida approximation. Furthermore, we apply a method based on a specific integral inequality to prove that the solution decays exponentially to zero when the time t goes to infinity.

In this paper we give a new definition of the positive-definiteness of the Liapunov functional involved in the stability and asymptotic stability investigation. Using this notion we prove Liapunov type theorems and apply these results to the scalar equation $\dot x(t)=b(t)x(t-r(t))$, where $b(t)$ and $r(t)$ may be unbounded.

This paper is concerned with the boundary exact controllability of the equation $$u''-\Delta u-\int_0^t g(t-\sigma)\Delta u(\sigma) d\sigma=-\nabla p$$ where $Q$ is a finite cilinder $\Omega\times]0,T[$, $\Omega$ is a bounded domain of $R^n$, $u=(u_1(x,t),\ldots,u_n(x,t))$, $x=(x_1,\ldots,x_n)$ are $n$-dimensional vectors and $p$ denotes the pressure term. The result is obtained by applying HUM (Hilbert Uniqueness Method) due to J. L. Lions. The above equation is a simple model of dynamical elasticity equations for incompressible materials with memory.

This paper is concerned with equations of the form: $u'=A(t)u + f(t)$, where $A(t)$ is (unbounded) periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations which says that if $u$ is a bounded uniformly continuous mild solution and $P$ is the monodromy operator, then their spectra satisfy $e^{i sp_{AP(u)}}\subset \sigma(P)\cap S^1$, where $S^1$ is the unit circle. This result is then applied to find almost periodic solutions to the abovementioned equations. In particular, parabolic and functional differential equations are considered. Existence conditions for almost periodic and quasiperiodic solutions are discussed.

In this paper, we obtain some sufficient conditions for the existence of $2\pi$-periodic solutions of some semilinear equations at resonance where the kernel of the linear part has dimension $2n(n\ge 1)$. Our technique essentially bases on the Brouwer degree theory and Mawhin's coincidence degree theory.

In this work, the nonexistence of the global solutions to a class of initial boundary value problems with dissipative terms in the boundary conditions is considered for a quasilinear system of equations. The nonexistence proof is achieved by the use of a lemma due to O. Ladyzhenskaya and V.K. Kalantarov and by the usage of the so called generalized convexity method. In this method one writes down a functional which reflects the properties of dissipative boundary conditions and represents the norm of the solution in some sense, then proves that this functional satisfies the hypotheses of Ladyzhenskaya-Kalantarov lemma. Hence from the conclusion of the lemma one deduces that in a finite time $t_2$, this functional and hence the norm of the solution blows up.

This paper deals with the hysteresis operator coupled to the system of Urysohn-Volterra equations. The local solutions of the system as well as the global solutions have been obtained.

In this paper we consider the differential system (1.1)

$u_i'(t)=f_i\big(t,u_1(\tau_{i1}(t)),u_2(\tau_{i2}(t))\big) (i=1,2)$

with the boundary conditions (1.2)

$\varphi\big(u_1(0),u_2(0)\big)=0, u_1(t)=u_1(a), u_2(t)=0 for t\geq a,$

where $f_i: [0,a]\times \Bbb{R}^2\to \Bbb{R}$ $(i=1,2)$ satisfy the local Carath\'{e}odory conditions, while $\varphi: \Bbb{R}^2\to \Bbb{R}$ and $\tau_{ik}: [0,a]\to [0,+\infty[$ $(i,k=1,2)$ are continuous functions. The optimal, in a certain sense, sufficient conditions are obtained for the existence and uniqueness of a nonnegative solution of the problem (1.1), (1.2).

$u_i'(t)=f_i\big(t,u_1(\tau_{i1}(t)),u_2(\tau_{i2}(t))\big) (i=1,2)$

with the boundary conditions (1.2)

$\varphi\big(u_1(0),u_2(0)\big)=0, u_1(t)=u_1(a), u_2(t)=0 for t\geq a,$

where $f_i: [0,a]\times \Bbb{R}^2\to \Bbb{R}$ $(i=1,2)$ satisfy the local Carath\'{e}odory conditions, while $\varphi: \Bbb{R}^2\to \Bbb{R}$ and $\tau_{ik}: [0,a]\to [0,+\infty[$ $(i,k=1,2)$ are continuous functions. The optimal, in a certain sense, sufficient conditions are obtained for the existence and uniqueness of a nonnegative solution of the problem (1.1), (1.2).

We consider the wave equation with a mild internal dissipation. It is proved that any small dissipation inside the domain is sufficient to uniformly stabilize the solution of this equation by means of a nonlinear feedback of memory type acting on a part of the boundary. This is established without any restriction on the space dimension and without geometrical conditions on the domain or its boundary.

We consider the second order, linear differential equation

$$y''(x) + (\lambda.q(x)) y(x) = 0 \eqno{(1)}$$

where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$ with periodic and semiperiodic boundary conditions. Our approach to regularizing (1) follows that used by Atkinson [1], Everitt and Race [4], and Harris and Race [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semiperiodic eigenvalues of (1) in the case where $q(x) = 1/|1-x|$.

$$y''(x) + (\lambda.q(x)) y(x) = 0 \eqno{(1)}$$

where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$ with periodic and semiperiodic boundary conditions. Our approach to regularizing (1) follows that used by Atkinson [1], Everitt and Race [4], and Harris and Race [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semiperiodic eigenvalues of (1) in the case where $q(x) = 1/|1-x|$.

We study exact multiplicity of positive solutions for a class of Dirichlet problems on a ball. We consider nonlinearities generalizing cubic, allowing both $f(0)=0$ and non-positone cases. We use bifurcation approach. We first prove our results for a special case, and then show that the global picture persists as we vary the roots.

We prove the solvability of the parabolic problem

$$\partial_t u-\sum_{i,j=1}^N \partial_{x_i}(a_{i,j}(x,t)\partial_{x_j}u)+\sum_{i=1}^N b_i(x,t)\partial_{x_i}u=f(x,t,u)\hbox{ in }\Omega\times R$$

$$u(x,t)=0\hbox{ on }\partial\Omega\times R$$

$$u(x,t)=u(x,t+T)\hbox{ in }\Omega\times R$$

assuming certain conditions on the ratio $2\int_0^s f(x,t,\sigma) d\sigma/s^2$ with respect to the principal eigenvalue of the associated linear problem. The method of proof, whcih is based on the construction of upper and lower solutions, also yields information on the localization and the stability of the solution.

$$\partial_t u-\sum_{i,j=1}^N \partial_{x_i}(a_{i,j}(x,t)\partial_{x_j}u)+\sum_{i=1}^N b_i(x,t)\partial_{x_i}u=f(x,t,u)\hbox{ in }\Omega\times R$$

$$u(x,t)=0\hbox{ on }\partial\Omega\times R$$

$$u(x,t)=u(x,t+T)\hbox{ in }\Omega\times R$$

assuming certain conditions on the ratio $2\int_0^s f(x,t,\sigma) d\sigma/s^2$ with respect to the principal eigenvalue of the associated linear problem. The method of proof, whcih is based on the construction of upper and lower solutions, also yields information on the localization and the stability of the solution.

In this paper on the segment $I=[a,b]$ we will consider the system of linear functional differential equations

\begin{equation}\label{1}

x'_i(t)=\sum\limits_{k=1}^n\ell_{ik}(x_k)(t)+q_i(t)\qquad (i=1,\dots,n)

\end{equation}

and its particular case

$$x'_i(t)=\sum\limits_{k=1}^n p_{ik}(t)x_k(\tau_{ik}(t))+q_i(t)\qquad (i=1,\dots,n)\eqno{(1')}$$

with the boundary conditions

\begin{equation}\label{2}

\int_a^b x_i(t)d\varphi_i(t)=c_i\qquad (i=1,\dots,n).

\end{equation}

Here $\ell_{ik}:C(I;\Bbb R)\to L(I;\Bbb R)$ are linear bounded operators, $p_{ik}$ and $q_i\in L(I;\Bbb R)$, $c_i\in\Bbb R$ $(i,k=1,\dots,n)$, $\varphi_i:I\to\Bbb R$ $(i=1,\dots,n)$ are the functions with bounded variations, and $\tau_{ik}:I\to I$ $(i,k=1,\dots,n)$ are measurable functions. The optimal, in some sense, conditions of unique solvability of the problems $(\ref{1})$, $(\ref{2})$ and $(1')$, $(\ref{2})$ are established.

\begin{equation}\label{1}

x'_i(t)=\sum\limits_{k=1}^n\ell_{ik}(x_k)(t)+q_i(t)\qquad (i=1,\dots,n)

\end{equation}

and its particular case

$$x'_i(t)=\sum\limits_{k=1}^n p_{ik}(t)x_k(\tau_{ik}(t))+q_i(t)\qquad (i=1,\dots,n)\eqno{(1')}$$

with the boundary conditions

\begin{equation}\label{2}

\int_a^b x_i(t)d\varphi_i(t)=c_i\qquad (i=1,\dots,n).

\end{equation}

Here $\ell_{ik}:C(I;\Bbb R)\to L(I;\Bbb R)$ are linear bounded operators, $p_{ik}$ and $q_i\in L(I;\Bbb R)$, $c_i\in\Bbb R$ $(i,k=1,\dots,n)$, $\varphi_i:I\to\Bbb R$ $(i=1,\dots,n)$ are the functions with bounded variations, and $\tau_{ik}:I\to I$ $(i,k=1,\dots,n)$ are measurable functions. The optimal, in some sense, conditions of unique solvability of the problems $(\ref{1})$, $(\ref{2})$ and $(1')$, $(\ref{2})$ are established.

We study the existence and asymptotic behavior of the global solutions of the nonlinear equation

$$u_tt-\Delta_p u+(-\Delta)^\alpha u_t+g(u)=f$$

where $0<alpha\leq 1$ and $g$ does not satisfy the sign condition $g(u)u \geq 0$.

$$u_tt-\Delta_p u+(-\Delta)^\alpha u_t+g(u)=f$$

where $0<alpha\leq 1$ and $g$ does not satisfy the sign condition $g(u)u \geq 0$.

We consider the $n$'th order ordinary differential equation $(-1)^{n-k} y^{(n)}=\lambda a(t) f(y)$, $t\in[0,1]$, $n\geq 3$ together with the boundary condition $y^{(i)}(0)=0$, $0\leq i\leq k-1$ and $y^{(l)}=0$, $j\leq l\leq j+n-k-1$, for $1\leq j\leq k-1$ fixed. Values of $\lambda$ are characterized so that the boundary value problem has a positive solution.

In this paper we consider a functional differential equation of the form

$$x'=F(t,x,\int_0^t C(at-s) x(s)\,ds)$$

where $a$ is a constant satisfying $0<a<\infty$. Thus, the integral represents the memory of past positions of the solution $x$. We make the assumption that $\int_0^\infty |C(t)|\, dt<\infty$ so that this is a fading memory problem and we are interested in studying the effects of that memory over all those values of $a$. Very different properties of solutions emerge as we vary $a$ and we are interested in developing an approach which handles them in a unified way.

$$x'=F(t,x,\int_0^t C(at-s) x(s)\,ds)$$

where $a$ is a constant satisfying $0<a<\infty$. Thus, the integral represents the memory of past positions of the solution $x$. We make the assumption that $\int_0^\infty |C(t)|\, dt<\infty$ so that this is a fading memory problem and we are interested in studying the effects of that memory over all those values of $a$. Very different properties of solutions emerge as we vary $a$ and we are interested in developing an approach which handles them in a unified way.

See also an addendum to this paper: EJQTDE, No. 3. (2001)

We establish new oscillation theorems for the nonlinear differential equation

$$[a(t)\psi(x(t))|x'(t)|^{\alpha-1}x'(t)]'+q(t)f(x(t))=0, \alpha>0$$

where $a,q:[t0,\infty)\rightarrow R, \psi,f:R\rightarrow R$ are continuous, $a(t)>0$ and $\psi(x)>0$, $xf(x)>0$ for $x\not=0$. These criteria involve the use of averaging functions.

$$[a(t)\psi(x(t))|x'(t)|^{\alpha-1}x'(t)]'+q(t)f(x(t))=0, \alpha>0$$

where $a,q:[t0,\infty)\rightarrow R, \psi,f:R\rightarrow R$ are continuous, $a(t)>0$ and $\psi(x)>0$, $xf(x)>0$ for $x\not=0$. These criteria involve the use of averaging functions.

In this paper, we consider the Lidstone boundary value problem $y^{(2m)}(t) = \lambda a(t)f(y(t), ..., y^{(2j)}(t), ... y^{(2(m-1))}(t), 0 < t < 1, y^{(2i)}(0) = 0 = y^{(2i)}(1), i = 0, ..., m - 1$, where $(-1)^m f > 0$ and $a$ is nonnegative. Growth conditions are imposed on $f$ and inequalities involving an associated Green's function are employed which enable us to apply a well-known cone theoretic fixed point theorem. This in turn yields a $\lambda$ interval on which there exists a nontrivial solution in a cone for each $\lambda$ in that interval. The methods of the paper are known. The emphasis here is that $f$ depends upon higher order derivatives. Applications are made to problems that exhibit superlinear or sublinear type growth.

Geometric properties of self--similar solutions to the equation $ u_t = u_{xx} + \gamma(u^q)_x,\ x > 0,\ t > 0 $ are studied, $ q $ is positive and $ \gamma\in \mathbb{R}\setminus\{0\}$. Two critical values of $ q $ (namely 1 and 2) appear the corresponding shapes are of very different nature.

The even order neutral differential equation

$$\frac{d^n}{dt^n} [ x(t) + \lambda x(t-\tau) ] + f(t,x(g(t))) = 0\leqno{(1.1)}$$

is considered under the following conditions: $n\ge 2$ is even; $\lambda>0$; $\tau>0$; $g \in C[t_0,\infty)$, $\lim_{t\to\infty} g(t) = \infty$; $f \in C([t_0,\infty) \times {\bf R})$, $u f(t,u) \ge 0$ for $(t,u) \in [t_0,\infty) \times {\bf R}$, and $f(t,u)$ is nondecreasing in $u \in {\bf R}$ for each fixed $t\ge t_0$. It is shown that equation (1.1) is oscillatory if and only if the non-neutral differential equation

$$x^{(n)}(t) + \frac{1}{1+\lambda} f(t,x(g(t))) = 0$$

is oscillatory.

$$\frac{d^n}{dt^n} [ x(t) + \lambda x(t-\tau) ] + f(t,x(g(t))) = 0\leqno{(1.1)}$$

is considered under the following conditions: $n\ge 2$ is even; $\lambda>0$; $\tau>0$; $g \in C[t_0,\infty)$, $\lim_{t\to\infty} g(t) = \infty$; $f \in C([t_0,\infty) \times {\bf R})$, $u f(t,u) \ge 0$ for $(t,u) \in [t_0,\infty) \times {\bf R}$, and $f(t,u)$ is nondecreasing in $u \in {\bf R}$ for each fixed $t\ge t_0$. It is shown that equation (1.1) is oscillatory if and only if the non-neutral differential equation

$$x^{(n)}(t) + \frac{1}{1+\lambda} f(t,x(g(t))) = 0$$

is oscillatory.

We consider the neutral delay Duffing Equations of the form

$$ax''(t)+bx'(t)+cx(t)+g(x(t-\tau_1), \ x'(t-\tau_2), x''(t-\tau_3))=p(t)=p(t+2\pi).$$

and establish a sufficient coudition for the existence of $2\pi$-periodic solution of above equations.

$$ax''(t)+bx'(t)+cx(t)+g(x(t-\tau_1), \ x'(t-\tau_2), x''(t-\tau_3))=p(t)=p(t+2\pi).$$

and establish a sufficient coudition for the existence of $2\pi$-periodic solution of above equations.

In this paper we will study the asymptotic behaviour of positive solutions to the system

$$\left|

\begin{array}{lcr}

x_1^{\prime}(t)={{A(t)}\over {1+x_2^n(t)}}-x_1(t)\\

x_2^{\prime}(t)={{B(t)}\over {1+x_1^n(t)}}-x_2(t),

\end{array}

\right.\leqno{(1)}$$

where $A$ and $B$ belong to ${\cal C}_+$ and ${\cal C}_+$ is the set of continuous functions $g:{\cal R}\longrightarrow {\cal R}$, which are bounded above and below by positive constants. $n$ is fixed natural number. The system (1) describes cell differentiation, more precisely - its passes from one regime of work to other without loss of genetic information. The variables $x_1$ and $x_2$ make sense of concentration of specific metabolits. The parameters $A$ and $B$ reflect degree of development of base metabolism. The parameter $n$ reflects the highest row of the repression's reactions.

$$\left|

\begin{array}{lcr}

x_1^{\prime}(t)={{A(t)}\over {1+x_2^n(t)}}-x_1(t)\\

x_2^{\prime}(t)={{B(t)}\over {1+x_1^n(t)}}-x_2(t),

\end{array}

\right.\leqno{(1)}$$

where $A$ and $B$ belong to ${\cal C}_+$ and ${\cal C}_+$ is the set of continuous functions $g:{\cal R}\longrightarrow {\cal R}$, which are bounded above and below by positive constants. $n$ is fixed natural number. The system (1) describes cell differentiation, more precisely - its passes from one regime of work to other without loss of genetic information. The variables $x_1$ and $x_2$ make sense of concentration of specific metabolits. The parameters $A$ and $B$ reflect degree of development of base metabolism. The parameter $n$ reflects the highest row of the repression's reactions.

Let $L:\hbox{dom} L\subset L^2(\Omega;R^N)\rightarrow L^2(\Omega;R^N)$ be a linear operator, $\Omega$ being open and bounded in $R^M$. The aim of this paper is to study the Fu\v c\'\i k spectrum for vector problems of the form $Lu=\alpha Au^+ -\beta Au^-$, where $A$ is an $N\times N$ matrix, $\alpha, \beta$ are real numbers, $u^+$ a vector defined componentwise by $(u^+)_i=\max\{u_i,0\}$, $u^-$ being defined similarly. With $\lambda^*$ an eigenvalue for the problem $Lu=\lambda Au$, we describe (locally) curves in the Fu\v c\'\i k spectrum passing through the point $(\lambda^*,\lambda^*)$, distinguishing different cases illustrated by examples, for which Fu\v c\'\i k curves have been computed numerically.

If $A$ is a $\omega$-periodic matrix Floquet's theorem states that the differential equation $y'=A y$ has a fundamental matrix $P(x)\exp(J x)$ where $J$ is constant and $P$ is $\omega$-periodic, i.e., $P(x)=P^*(\e^{2\pi ix/\omega})$. We prove here that $P^*$ is rational if $A$ is bounded at the ends of the period strip and if all solutions of $y'=A y$ are meromorphic. This version of Floquet's theorem is important in the study of certain integrable systems.

We investigate the controlled harmonic oscillator

\begin{equation}\label{eq3.1}

\ddot{\xi}+\xi=u,

\end{equation}

where an external force (the control function) $u$ depends on the coordinate $\xi$, only. It can be shown that no ordinary (even nonlinear) feedback controls of the form $u=f(\xi(t))$ can asymptotically stabilize the solutions of the system (\ref{eq3.1}). However, one is able to make the system (\ref{eq3.1}) asymptotically stable if one designs a special feedback control $u$ depending on $\xi(\cdot)$ which is called {\it a hybrid feedback control}. We demonstrate in the paper that the dynamics of a typical linear system of ordinary differential equations equipped with a linear hybrid feedback control possesses some irregular properties that dynamical systems without delay do not have. For example, solutions with different initial conditions may cross or even partly coincide. This proves that the hybrid dynamics cannot, in general, be described by a system of ordinary differential equations, neither linear, nor nonlinear, so that time-delays have to be incorporated into the system.

\begin{equation}\label{eq3.1}

\ddot{\xi}+\xi=u,

\end{equation}

where an external force (the control function) $u$ depends on the coordinate $\xi$, only. It can be shown that no ordinary (even nonlinear) feedback controls of the form $u=f(\xi(t))$ can asymptotically stabilize the solutions of the system (\ref{eq3.1}). However, one is able to make the system (\ref{eq3.1}) asymptotically stable if one designs a special feedback control $u$ depending on $\xi(\cdot)$ which is called {\it a hybrid feedback control}. We demonstrate in the paper that the dynamics of a typical linear system of ordinary differential equations equipped with a linear hybrid feedback control possesses some irregular properties that dynamical systems without delay do not have. For example, solutions with different initial conditions may cross or even partly coincide. This proves that the hybrid dynamics cannot, in general, be described by a system of ordinary differential equations, neither linear, nor nonlinear, so that time-delays have to be incorporated into the system.

In this paper we are concerned with a class of nonlinear differential equations and obtaining the sufficient conditions for the uniqueness of the periodic solution by using Brouwer's fixed point theory and the Sturm Theorem.

In this paper the concepts of lower mild and upper mild solutions combined with a fixed point theorem for condensing maps and the semigroup theory are used to investigate the existence of mild solutions for first order impulsive semilinear evolution inclusions.

In this paper we study the radial symmetric solutions of the two-dimensional Cahn-Hilliard equation with degenerate mobility. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence and the nonnegativity of weak solutions. Keywords. Cahn-Hilliard equation, radial solution, degenerate mobility, nonnegativity.

This addendum concerns the paper of the above title found in EJQTDE No. 13 (1999). Throughout that paper was the tacit assumption that the coefficient functions $h(t)$, $b(t)$, and $C(t)$ are all continuous on their respective domains. Every result, as well as the existence result stated at the end of the first section, depended on those functions being continuous. A search of the paper indicates that we failed to state this. We regret any inconvenience which this may have caused any reader.

See also: EJQTDE, No. 13. (1999)

We study the existence, multiplicity, and stability of positive solutions to:

$$\eqalign{- u''(x) &= \lambda f(u(x)) \ \text{for} \ x \in (-1, 1), \lambda > 0, \cr

u(-1)&= 0\ = u(1) ,}$$

where $f : [0, \infty) \to \Bbb R$ is semipositone ($f(0)<0$) and superlinear ($\lim_{t \to \infty} f(t) /t = \infty)$. We consider the case when the nonlinearity $f$ is of concave-convex type having exactly one inflection point. We establish that $f$ should be appropriately concave (by establishing conditions on $f$) to allow multiple positive solutions. For any $\lambda > 0$, we obtain the exact number of positive solutions as a function of $f(t)/t$ and establish how the positive solution curves to the above problem change. Also, we give examples where our results apply. This work extends the work in [1] by giving a complete classification of positive solutions for concave-convex type nonlinearities.

$$\eqalign{- u''(x) &= \lambda f(u(x)) \ \text{for} \ x \in (-1, 1), \lambda > 0, \cr

u(-1)&= 0\ = u(1) ,}$$

where $f : [0, \infty) \to \Bbb R$ is semipositone ($f(0)<0$) and superlinear ($\lim_{t \to \infty} f(t) /t = \infty)$. We consider the case when the nonlinearity $f$ is of concave-convex type having exactly one inflection point. We establish that $f$ should be appropriately concave (by establishing conditions on $f$) to allow multiple positive solutions. For any $\lambda > 0$, we obtain the exact number of positive solutions as a function of $f(t)/t$ and establish how the positive solution curves to the above problem change. Also, we give examples where our results apply. This work extends the work in [1] by giving a complete classification of positive solutions for concave-convex type nonlinearities.

We prove existence and asymptotic behaviour results for weak solutions of a mixed problem (S). We also obtain the existence of the global attractor and the regularity for this attractor in $\left[H^{2}(\Omega )\right] ^{2}$ and we derive estimates of its Haussdorf and fractal dimensions.

We investigate an algebraic structure of the space of solutions of autonomous nonlinear differential equations of certain type. It is shown that for these equations infinitely many binary algebraic laws of addition of solutions exist. We extract commutative and conjugate commutative groups which lead to the conjugate differential equations. Besides one is being able to write down particular form of extended Fourier series for these equations. It is shown that in space with a moving field, there always exist metrics geodesics of which are the solutions of a given differential equation and its conjugate equation. Connection between the invariant group and algebraic structure of solution space has also been studied.

The purpose of this paper is to prove uniform boundedness and so global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients satisfying a balance law. Our technics are based on invariant regions and Lyapunov functional methods. The nonlinearity of the reaction term which we take positive in an invariant region has been supposed to be polynomial.

We prove existence of solutions for parabolic initial value problems $\partial_t u = \Delta u + f(u)$ on $R^N$ , where $f : R \rightarrow R$ is a bounded, but possibly discontinuous function.

In this paper, we obtain some results on the nonoscillatory behaviour of the system (1), which contains as particular cases, some well known systems. By negation, oscillation criteria are derived for these systems. In the last section we present some examples and remarks, and various well known oscillation criteria are obtained.

In this article, we generalize the results obtained in [16] concerning uniform bounds and so global existence of solutions for reaction-diffusion systems with a full matrix of diffusion coefficients satisfying a balance law and with homogeneous Neumann boundary conditions. Our techniques are based on invariant regions and Lyapunov functional methods. We demonstrate that our results remain valid for nonhomogeneous boundary conditions and with out balance law's condition. The nonlinear reaction term has been supposed to be of polynomial growth.

Using interesting techniques, an existence result for the problem $\ddot{x}+2f\left( t\right) \dot{x}+x+g\left( t,x\right) =0,$ $\lim\limits_{t\rightarrow +\infty }x\left( t\right) =\lim\limits_{t\rightarrow +\infty }\dot{x}\left( t\right) =0,$ is given in [2]. This note treates the same problem via Schauder-Tychonoff and Banach theorems.

Consider the Lienard system $u''+f(u) u'+g(u)=0$ with an isolated periodic solution. This paper concerns the behavior of periodic solutions of Lienard system under small periodic perturbations.

We investigate the growth of solutions of the differential equation $f^{\left( n\right) }+A_{n-1}\left( z\right) f^{\left( n-1\right) }+...+A_{1}\left( z\right) f^{^{\prime }}+A_{0}\left( z\right) f=0,$ where $A_{0}\left( z\right) ,...,A_{n-1}\left( z\right) $\ \ are\ entire functions with $A_{0}\left( z\right) \not\equiv 0$. We estimate the hyper-order with respect to the conditions of $A_{0}\left( z\right) ,...,A_{n-1}\left( z\right) $\ if\ $f\not\equiv 0$ has infinite order.

In this paper a variant of a fixed point theorem to Krasnoselskii-Schaefer type is proved and it is further applied to certain nonlinear integral equation of mixed type for proving the existence of the solution.

In this paper, we study the stability of solutions for semilinear wave equations whoseboundary condition includes a integral that represents the memory effect. We show that the dissipation is strong enough to produce exponential decay of the solution, provided the relaxation function also decays exponentially. When the relaxation function decays polinomially, we show that the solution decays polynomially and with the same rate.

The cooperating two-species Lotka-Volterra model is discussed. We study the blowup properties of solutions to a parabolic system with homogeneous Dirichlet boundary conditions. The upper and lower bounds of blowup rate are obtained.

The problem of existence of the solutions for ordinary differential equations vanishing at $\pm \infty $ is considered.

The method of quasilinearization for nonlinear impulsive differential equations with linear boundary conditions is studied. The boundary conditions include periodic boundary conditions. It is proved the convergence is quadratic.

The small-amplitude motion of a thin elastic membrane is investigated in $n$-dimensional bounded and unbounded domains, with $n\in N$. Existence and uniqueness of solutions are established. Asymptotic of solutions is proved too.

Using Lyapunov and Lyapunov-like functionals, we study the stability and boundedness of the solutions of a system of Volterra integrodifferential equations. Our results, also extending some of the more well-known criteria, give new sufficient conditions for stability of the zero solution of the nonperturbed system, and prove that the same conditions for the perturbed system yield boundedness when the perturbation is $L^2$.

In this paper, the authors investigate the existence of solutions for nonresonance impulsive higher order functional differential inclusions in Banach spaces with nonconvex valued right hand side. They present two results. In the first one, they rely on a fixed point theorem for contraction multivalued maps due to Covitz and Nadler, and for the second one, they use Schaefer's fixed point theorem combined with lower semi-continuous multivalued operators with decomposable values.

An integral equation, $x(t)=a(t)-\int^t_{-\infty} D(t,s)g(x(s))ds$ with $a(t)$ bounded, is studied by means of a Liapunov functional. There results an {\it{a}} {\it{priori}} bound on solutions. This gives rise to an interplay between continuity and compactness and leads us to a fixed point theorem of Schaefer type. It is a very flexible fixed point theorem which enables us to show that the solution inherits properties of $a(t)$, including periodic or almost periodic solutions in a Banach space.

In this paper we apply a cone theoretic fixed point theorem and obtain conditions for the existence of positive solutions to the three-point nonlinear second order boundary value problem

$$ u''(t)+\lambda a(t)f(u(t)) = 0, \;\;\;t\in(0,1)$$

$$u(0)=0,\;\;\;\; \alpha u(\eta)=u(1),$$

where $0<\eta<1$ and $0<\alpha <\frac{1}{\eta}.$

$$ u''(t)+\lambda a(t)f(u(t)) = 0, \;\;\;t\in(0,1)$$

$$u(0)=0,\;\;\;\; \alpha u(\eta)=u(1),$$

where $0<\eta<1$ and $0<\alpha <\frac{1}{\eta}.$

In this article we consider the second order quasilinear elliptic system of the form

$$\Delta_{p_i} u_i=H_i(|x|)u_{i+1}^{\alpha_i}, x\in R^N, i=1,2,...,m$$

with nonnegative continuous function $H_i$. Sufficient conditions are given to have nonnegative nontrivial radial entire solutions. When $H_i$, $i = 1, 2, ..., m$, behave like constant multiples of $|x|^\lambda$, $\lambda\in R$, we can completely characterize the existence property of nonnegative nontrivial radial entire solutions.

$$\Delta_{p_i} u_i=H_i(|x|)u_{i+1}^{\alpha_i}, x\in R^N, i=1,2,...,m$$

with nonnegative continuous function $H_i$. Sufficient conditions are given to have nonnegative nontrivial radial entire solutions. When $H_i$, $i = 1, 2, ..., m$, behave like constant multiples of $|x|^\lambda$, $\lambda\in R$, we can completely characterize the existence property of nonnegative nontrivial radial entire solutions.

This paper is concerned with the nonlinear boundary eigenvalue problem

$$-(|u'|^{p-2}u')'=\lambda m|u|^{p-2}u\qquad u \in I=]a,b[,\quad u(a)=u(b)=0,$$

where $p>1$, $\lambda$ is a real parameter, $m$ is an indefinite weight, and $a$, $b$ are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and verifies the strict monotonicity property with respect to the weight $m$ and the domain $I$, the k-th eigenfunction, corresponding to the $k$-th eigenvalue, has exactly $k-1$ zeros in $(a,b)$. At the end, we give a simple variational formulation of eigenvalues.

$$-(|u'|^{p-2}u')'=\lambda m|u|^{p-2}u\qquad u \in I=]a,b[,\quad u(a)=u(b)=0,$$

where $p>1$, $\lambda$ is a real parameter, $m$ is an indefinite weight, and $a$, $b$ are real numbers. We prove there exists a unique sequence of eigenvalues for this problem. Each eigenvalue is simple and verifies the strict monotonicity property with respect to the weight $m$ and the domain $I$, the k-th eigenfunction, corresponding to the $k$-th eigenvalue, has exactly $k-1$ zeros in $(a,b)$. At the end, we give a simple variational formulation of eigenvalues.

The existence of radially symmetric solutions $u(x;a)$ to the Dirichlet problems

$$\Delta u(x)+f(|x|,u(x),|\nabla u(x)|)=0\qquad x\in B,\ u|_\Gamma=a\in{\R}\ (\Gamma:=\partial B)$$

is proved, where $B$ is the unit ball in ${\R}^n$ centered at the origin $(n\ge2)$, $a$ is arbitrary $(a>a_0\ge-\infty);f$ is positive, continuous and bounded. It is shown that these solutions belong to $C^2(\ov{B})$. Moreover, in the case $f\in C^1$ a sufficient condition (near necessary) for the smoothness property $u(x;a)\in C^3(\ov{B})\quad\forall a>a_0$ is also obtained.

$$\Delta u(x)+f(|x|,u(x),|\nabla u(x)|)=0\qquad x\in B,\ u|_\Gamma=a\in{\R}\ (\Gamma:=\partial B)$$

is proved, where $B$ is the unit ball in ${\R}^n$ centered at the origin $(n\ge2)$, $a$ is arbitrary $(a>a_0\ge-\infty);f$ is positive, continuous and bounded. It is shown that these solutions belong to $C^2(\ov{B})$. Moreover, in the case $f\in C^1$ a sufficient condition (near necessary) for the smoothness property $u(x;a)\in C^3(\ov{B})\quad\forall a>a_0$ is also obtained.

We study the stability property of a simple periodic solution of an autonomous neutral functional differential equation (NFDE) of the form

$${d\over dt} D(x_t) = f (x_t).$$

A new proof based on local integral manifold theory and the implicit function theorem is given for the classical result that a simple periodic orbit of the equation above is asymptotically orbitally stable with asymptotic phase. The technique used overcomes the difficulty that the solution operator of a NFDE does not smooth as $t$ increases.

$${d\over dt} D(x_t) = f (x_t).$$

A new proof based on local integral manifold theory and the implicit function theorem is given for the classical result that a simple periodic orbit of the equation above is asymptotically orbitally stable with asymptotic phase. The technique used overcomes the difficulty that the solution operator of a NFDE does not smooth as $t$ increases.

It was shown in [1] that for a wide class of differential equations there exist infinitely many binary laws of addition of solutions such that every binary law has its conjugate. From this set of operations we extract commutative algebraic object that is a pair of two alternative to each other fields with common identity elements.

The goal of the present paper is to detect those mathematical constructions that are related to the existence of alternative fields dictated by differential equations. With this in mind we investigate differential and integral calculus based on the commutative algebra that is generated by a given differential equation. It turns out that along with the standard differential and integral calculus there always exists an isomorphic alternative calculus. Moreover, every system of differential equations generates its own calculus that is isomorphic (or homomorphic) to the standard one. The given system written in its own calculus appears to be linear.

It is also shown that there always exist two alternative to each other geometries, and matrix algebra has its alternative isomorphic to the classical one.

The goal of the present paper is to detect those mathematical constructions that are related to the existence of alternative fields dictated by differential equations. With this in mind we investigate differential and integral calculus based on the commutative algebra that is generated by a given differential equation. It turns out that along with the standard differential and integral calculus there always exists an isomorphic alternative calculus. Moreover, every system of differential equations generates its own calculus that is isomorphic (or homomorphic) to the standard one. The given system written in its own calculus appears to be linear.

It is also shown that there always exist two alternative to each other geometries, and matrix algebra has its alternative isomorphic to the classical one.

See also an addendum to this paper: EJQTDE, No. 7. (2004)

In this paper we investigate the existence of solutions for initial and boundary value problems for second order impulsive functional differential inclusions. We shall rely on a fixed point theorem for contraction multivalued maps due to Covitz and Nadler.

In this paper we generalize a result obtained in [15] concerning uniform boundedness and so global existence of solutions for reaction-diffusion systems with a general full matrix of diffusion coefficients satisfying a balance law. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinearity of the reaction term which we take positive in an invariant region has been supposed to be polynomial or of weak exponential growth.

Using a particular locally convex space and Schaefer's theorem, a generalization of Krasnoselskii's fixed point Theorem is proved. This result is further applied to certain nonlinear integral equation proving the existence of a solution on $\R_{+}=[0,+\infty).$

In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial first integrals for a general nonlinear systems of ordinary differential equations $\dot x = f(x)$, $x \in R^n$ with $f(0) = 0$. We show that if the eigenvalues of the Jacobi matrix of the vector field $f(x)$ are $Z$-independent, then the system has no nontrivial Laurent polynomial integrals.

We investigate the coexistence of positive steady-state solutions to a parabolic system, which models a single species on two growth-limiting, non-reproducing resources in an un-stirred chemostat with diffusion. We establish the existence of a positive steady-state solution for a range of the parameter $(m,n)$, the bifurcation solutions and the stability of bifurcation solutions. The proof depends on the maximum principle, bifurcation theorem and perturbation theorem.

In this paper existence and uniqueness results for an abstract measure delay differential equation are proved, by using Leray-Schauder nonlinear alternative, under Carathéodory conditions.

We give a complete description of the set of solutions to the boundary value problem

\[

\left\{

\begin{array}{c}

-\left( \varphi \left( u^{\prime }\right) \right) ^{\prime }=f\left(

u\right) \text{ in }\left( 0,1\right) \\

u\left( 0\right) =u\left( 1\right) =0

\end{array}

\right.

\]

where $\varphi $ is an odd increasing homeomorphism of $\Bbb{R}$ concave on $\Bbb{R}^{+}$ and $f$ $\in C\left( \Bbb{R}\text{, }\Bbb{R}\right) $is odd and superlinear.

\[

\left\{

\begin{array}{c}

-\left( \varphi \left( u^{\prime }\right) \right) ^{\prime }=f\left(

u\right) \text{ in }\left( 0,1\right) \\

u\left( 0\right) =u\left( 1\right) =0

\end{array}

\right.

\]

where $\varphi $ is an odd increasing homeomorphism of $\Bbb{R}$ concave on $\Bbb{R}^{+}$ and $f$ $\in C\left( \Bbb{R}\text{, }\Bbb{R}\right) $is odd and superlinear.

In this paper, we study the global existence of classical solutions for the Cahn-Hilliard equation with terms of lower order and non-constant mobility. Based on the Schauder type estimates, under some assumptions on the mobility and terms of lower order, we establish the global existence of classical solutions.

In this paper we investigate the existence of mild solutions for first and second order impulsive semilinear evolution inclusions in separable Banach spaces. By using suitable fixed point theorems, we study the case when the multivalued map has convex and nonconvex values.

We develop the harmonic analysis approach for parabolic operator with one order term in the parabolic Kato class on $C^{1,1}$-cylindrical domain $\Omega$. We study the boundary behaviour of nonnegative solutions. Using these results, we prove the integral representation theorem and the existence of nontangential limits on the boundary of $\Omega$ for nonnegative solutions. These results extend some first ones proved for less general parabolic operators.

Let $f:\R\times \R^{N}\rightarrow \R^{N}$ be a continuous function and let $h:\R\rightarrow \R$ be a continuous and strictly positive function. A sufficient condition such that the equation $\dot{x}=f\left( t,x\right) $ admits solutions $x:\R\rightarrow \R^{N}$ satisfying the inequality $\left| x\left( t\right) \right| \leq k\cdot h\left( t\right) ,$ $t\in \R,$ $k>0$, where $\left| \cdot \right| $ is the euclidean norm in $\R^{N},$ is given. The proof of this result is based on the use of a special function of Lyapunov type, which is often called guiding function. In the particular case $h\equiv 1$, one obtains known results regarding the existence of bounded solutions.

In this paper the existence of extremal solutions of a functional integral inclusion involving Carathéodory is proved under certain monotonicity conditions. Applications are given to some initial and boundary value problems of ordinary differential inclusion for proving the existence of extremal solutions. Our results generalize the results of Dhage [8] under weaker conditions and complement the results of O'Regan [16].

The linear stability of the travelling wave solutions of a general reaction-diffusion system is investigated. The spectrum of the corresponding second order differential operator $L$ is studied. The problem is reduced to an asymptotically autonomous first order linear system. The relation between the spectrum of $L$ and the corresponding first order system is dealt with in detail. The first order system is investigated using exponential dichotomies. A self-contained short presentation is shown for the study of the spectrum, with elementary proofs. An algorithm is given for the determination of the exact position of the essential spectrum. The Evans function method for determining the isolated eigenvalues of $L$ is also presented. The theory is illustrated by three examples: a single travelling wave equation, a three variable combustion model and the generalized KdV equation.

We investigate in this article the null conrollability for the semilinear heat operator $u' - \Delta u +f(u)$ in a domain which boundary is moving with the time $t$.

In this paper we prove the uniqueness of bounded solutions to a viscous diffusion equation based on approximate Holmgren's approach.

The aim of this paper is to analyse the properties of the solution map to the Cauchy problem for the wave map equation with a source term, when the target is the hyperboloid ${\cal H}^2$ that is embedded in ${\cal R}^3$. The initial data are in ${\dot H}^1\times L^2$. We prove that the solution map is not uniformly continuous.

The generalized method of quasilinearization is applied to obtain a monotone sequence of iterates converging uniformly and rapidly to a solution of second order nonlinear boundary value problem with nonlinear integral boundary conditions.

We focus our attention on a class of perturbed integral equations in modular spaces, by using fixed point Theorem I.1 (see [1]).

We study the following bifurcation problem in a bounded domain $\Omega$ in $\RR^N$:

$$\left\{\begin{array}{lll}

-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,+ f(x,u,v,\lambda)&

\mbox{in} \ \Omega\\

-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \, + g(x,u,v,\lambda) &

\mbox{in} \ \Omega\\

(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega). & \

\end{array}

\right.

$$

We prove that the principal eigenvalue $\lambda_1$ of the following eigenvalue problem

$$\left\{\begin{array}{lll}

-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,& \mbox{in} \ \Omega\\

-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \,& \mbox{in} \ \Omega\\

(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)& \

\end{array}

\right.$$

is simple and isolated and we prove that $(\lambda_1,0,0)$ is a bifurcation point of the system mentioned above.

$$\left\{\begin{array}{lll}

-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,+ f(x,u,v,\lambda)&

\mbox{in} \ \Omega\\

-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \, + g(x,u,v,\lambda) &

\mbox{in} \ \Omega\\

(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega). & \

\end{array}

\right.

$$

We prove that the principal eigenvalue $\lambda_1$ of the following eigenvalue problem

$$\left\{\begin{array}{lll}

-\Delta_p u=&\lambda |u|^{\alpha}|v|^{\beta}v \,& \mbox{in} \ \Omega\\

-\Delta_q v=&\lambda |u|^{\alpha}|v|^{\beta}u \,& \mbox{in} \ \Omega\\

(u,v)\in & W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)& \

\end{array}

\right.$$

is simple and isolated and we prove that $(\lambda_1,0,0)$ is a bifurcation point of the system mentioned above.

The author proves the $W^{1,p}$ convergence of the symmetric minimizers

$u_{\varepsilon}=(u_{\varepsilon 1},u_{\varepsilon 2},u_{\varepsilon 3})$ of a p-energy functional as $\varepsilon \to 0$, and the zeros of $u_{\varepsilon 1}^2+u_{\varepsilon 2}^2$ are located roughly. In addition,the estimates of the convergent rate of $u_{\varepsilon 3}^2$ (to $0$) are presented. At last, based on researching the Euler-Lagrange equation of symmetric solutions and establishing its $C^{1,\alpha}$ estimate, the author obtains the $C^{1,\alpha}$ convergence of some symmetric minimizer.

$u_{\varepsilon}=(u_{\varepsilon 1},u_{\varepsilon 2},u_{\varepsilon 3})$ of a p-energy functional as $\varepsilon \to 0$, and the zeros of $u_{\varepsilon 1}^2+u_{\varepsilon 2}^2$ are located roughly. In addition,the estimates of the convergent rate of $u_{\varepsilon 3}^2$ (to $0$) are presented. At last, based on researching the Euler-Lagrange equation of symmetric solutions and establishing its $C^{1,\alpha}$ estimate, the author obtains the $C^{1,\alpha}$ convergence of some symmetric minimizer.

In this paper the existence of a solution of a general nonlinear functional two point boundary value problem is proved under mixed generalized Lipschitz and Carath\'eodory conditions. An existence theorem for extremal solutions is also proved under certain monotonicity and weaker continuity conditions. Examples are provided to illustrate the theory developed in this paper.

Let $\mathbb{T}$ be a time scale such that $0, T \in \mathbb{T}$. We us a cone theoretic fixed point theorem to obtain intervals for $\lambda$ for which the second order dynamic equation on a time scale,

\begin{gather*}

u^{\Delta\nabla}(t) + \lambda a(t)f(u(t)) = 0, \quad t \in (0,T) \cap \mathbb{T},\\

u(0) = 0, \quad \alpha u(\eta) = u(T),

\end{gather*}

where $\eta \in (0, \rho(T)) \cap \mathbb{T}$, and $0 < \alpha <T/\eta$, has a positive solution.

\begin{gather*}

u^{\Delta\nabla}(t) + \lambda a(t)f(u(t)) = 0, \quad t \in (0,T) \cap \mathbb{T},\\

u(0) = 0, \quad \alpha u(\eta) = u(T),

\end{gather*}

where $\eta \in (0, \rho(T)) \cap \mathbb{T}$, and $0 < \alpha <T/\eta$, has a positive solution.

Let $X$ be an arbitrary (real or complex) Banach space, endowed with the norm $\left| \cdot \right| .$ Consider the space of the continuous functions $C\left( \left[ 0,T\right] ,X\right) $ $\left( T>0\right) $, endowed with the usual topology, and let $M$ be a closed subset of it. One proves that each operator $A:M\rightarrow M$ fulfilling for all $x,y\in M$ and for all $t\in \left[ 0,T\right] $ the condition

\begin{eqnarray*}

\left| \left( Ax\right) \left( t\right) -\left( Ay\right) \left( t\right)

\right| &\leq &\beta \left| x\left( \nu \left( t\right) \right) -y\left( \nu

\left( t\right) \right) \right| + \\

&&+\frac{k}{t^{\alpha }}\int_{0}^{t}\left| x\left( \sigma \left( s\right)

\right) -y\left( \sigma \left( s\right) \right) \right| ds,

\end{eqnarray*}

(where $\alpha ,$ $\beta \in \lbrack 0,1)$, $k\geq 0$, and $\nu ,$ $\sigma :\left[ 0,T\right] \rightarrow \left[ 0,T\right] $ are continuous functions such that $\nu \left( t\right) \leq t,$ $\sigma \left( t\right)\leq t,$ $\forall t\in \left[ 0,T\right] $) has exactly one fixed point in $M $. Then the result is extended in $C\left( \R_{+},X\right) ,$ where $\R_{+}:=[0,\infty ).$

\begin{eqnarray*}

\left| \left( Ax\right) \left( t\right) -\left( Ay\right) \left( t\right)

\right| &\leq &\beta \left| x\left( \nu \left( t\right) \right) -y\left( \nu

\left( t\right) \right) \right| + \\

&&+\frac{k}{t^{\alpha }}\int_{0}^{t}\left| x\left( \sigma \left( s\right)

\right) -y\left( \sigma \left( s\right) \right) \right| ds,

\end{eqnarray*}

(where $\alpha ,$ $\beta \in \lbrack 0,1)$, $k\geq 0$, and $\nu ,$ $\sigma :\left[ 0,T\right] \rightarrow \left[ 0,T\right] $ are continuous functions such that $\nu \left( t\right) \leq t,$ $\sigma \left( t\right)\leq t,$ $\forall t\in \left[ 0,T\right] $) has exactly one fixed point in $M $. Then the result is extended in $C\left( \R_{+},X\right) ,$ where $\R_{+}:=[0,\infty ).$

The main object of this paper is to give a representation of the covariance operator associated to the mild solutions of time-varying,linear, stochastic equations in Hilbert spaces. We use this representation to obtain a characterization of the uniform exponential stability of linear stochastic equations with periodic coefficients.

We describe the complete polynomial vector fields and their fixed points in a finite-dimensional simplex and we apply the results to differential equations of genetical evolution models.

In this work we give a new criteria for the existence of periodic and almost periodic solutions for some differential equation in a Banach space. The linear part is nondensely defined and satisfies the Hille-Yosida condition. We prove the existence of periodic and almost periodic solutions with condition that is more general than the known exponential dichotomy. We apply the new criteria for the existence of periodic and almost periodic solutions for some partial functional differential equation whose linear part is nondensely defined.

This addendum concerns the paper of the above title found in EJQTDE No. 2 (2003 ). On page 3, before (2.2), instead of "Applying the mean value theorem" it should read "By simple calculations". It does not change any results presented in the paper, but excludes the mistake of applying the mean value theorem for a complex valued function of a real variable. The mean value theorem is not needed.

See also: EJQTDE, No. 2. (2003)

In this paper we study a boundary value problem for a first order functional differential equation on an infinite interval. Using fixed point theorems on appropriate cones in Banach spaces, we derive multiple positive solutions for our boundary value problem.

This paper is concerned with the bifurcation result of nonlinear Neumann problem

$$

\left\{\begin{array}{lll}

-\Delta_p u=& \lambda m(x)|u|^{p-2}u + f(\lambda,x,u)& \mbox{in} \ \Omega\\

\frac{\partial u}{\partial \nu}\hspace*{0.55cm}= & 0 & \mbox{on}

\ \partial\Omega.

\end{array}

\right.

$$

We prove that the principal eigenvalue $\lambda_1$ of the corresponding eigenvalue problem with $f\equiv 0,$ is a bifurcation point by using a generalized degree type of Rabinowitz.

$$

\left\{\begin{array}{lll}

-\Delta_p u=& \lambda m(x)|u|^{p-2}u + f(\lambda,x,u)& \mbox{in} \ \Omega\\

\frac{\partial u}{\partial \nu}\hspace*{0.55cm}= & 0 & \mbox{on}

\ \partial\Omega.

\end{array}

\right.

$$

We prove that the principal eigenvalue $\lambda_1$ of the corresponding eigenvalue problem with $f\equiv 0,$ is a bifurcation point by using a generalized degree type of Rabinowitz.

New oscillation criteria for the second order nonlinear neutral delay differential equation

$[y(t)+p(t)y(t-\tau )]^{^{\prime \prime}}+q(t)\,f(y(g(t)))=0$, $t\geq t_{0}$

are given. The relevance of our theorems becomes clear due to a carefully selected example.

$[y(t)+p(t)y(t-\tau )]^{^{\prime \prime}}+q(t)\,f(y(g(t)))=0$, $t\geq t_{0}$

are given. The relevance of our theorems becomes clear due to a carefully selected example.

Cooke and Yorke developed a theory of biological growth and epidemics based on an equation $x'(t)=g(x(t))-g(x(t-L))$ with the fundamental property that $g$ is an arbitrary locally Lipschitz function. They proved that each solution either approaches a constant or $\pm \infty$ on its maximal right-interval of definition. They also raised a number of interesting questions and conjectures concerning the determination of the limit set, periodic solutions, parallel results for more general delays, and stability of solutions. Although their paper motivated many subsequent investigations, the basic questions raised seem to remain unanswered.

We study such equations with more general delays by means of two successive applications of contraction mappings. Given the initial function, we explicitly locate the constant to which the solution converges, show that the solution is stable, and show that its limit function is a type of "selective global attractor." In the last section we examine a problem of Minorsky in the guidance of a large ship. Knowledge of that constant to which solutions converge is critical for guidance and control.

We study such equations with more general delays by means of two successive applications of contraction mappings. Given the initial function, we explicitly locate the constant to which the solution converges, show that the solution is stable, and show that its limit function is a type of "selective global attractor." In the last section we examine a problem of Minorsky in the guidance of a large ship. Knowledge of that constant to which solutions converge is critical for guidance and control.

We apply fixed point theorems to obtain sufficient conditions for existence of infinitely many solutions of a nonlinear fourth order boundary value problem

$$\displaylines{ u^{(4)}(t) = a(t)f(u(t)), \quad 0 < t < 1, \cr u(0) = u(1) = u'(0) = u'(1) = 0, }$$

where $a(t)$ is $L^p$-integrable and $f$ satisfies certain growth conditions.

$$\displaylines{ u^{(4)}(t) = a(t)f(u(t)), \quad 0 < t < 1, \cr u(0) = u(1) = u'(0) = u'(1) = 0, }$$

where $a(t)$ is $L^p$-integrable and $f$ satisfies certain growth conditions.

In this work, we obtain algebraic conditions which assure the Fredholm solvability of an abstract differential equation of elliptic type. In this respect, our work can be considered as an extension of Yakubov's results to the case of boundary conditions containing a linear operator. Although essential technical, this extension is not straight forward as we show it below. The obtained abstract result is applied to a non regular boundary value problem for a second order partial differential equation of an elliptic type in a cylindrical domain. It is interesting to note that the problems considered in cylindrical domains are not coercive.

We introduce a structure condition of parabolic type, which allows for the generalization to quasilinear parabolic systems of the known results of integrability, and boundedness of local solutions to singular and degenerate quasilinear parabolic equations.

In this paper the existence of a solution of general nonlinear functional differential equations is proved under mixed generalized Lipschitz and Carath\'eodory condition. An existence theorem for the extremal solutions is also proved under certain monotonicity and weaker continuity conditions. Examples are provided to illustrate the abstract theory developed in this paper.

The author studies the asymptotic behavior of minimizers $u_{\varepsilon}$ of a p-energy functional with penalization as $\varepsilon \to 0$. Several kinds of convergence for the minimizer to the p-harmonic map are presented under different assumptions.

A generalization of the Leray-Schauder principle for multivalued mappings is given. Using this result, an existence theorem for an integral inclusion is obtained.

In this paper, we study the instability of the traveling waves of a generalized diffusion model in population problems. We prove that some traveling wave solutions are nonlinear unstable under $H^2$ perturbations. These traveling wave solutions converge to a constant as $x\to\infty$.

In this paper, we study a non-local initial boundary-value problem arising in Ohmic heating. By using a dynamical systems approach, some existence and uniqueness results are proved and the existence of a compact attractor is shown.

In this paper we analyze the global existence and asymptotic behavior of a reaction diffusion system with degenerate diffusion arising in modeling the spatial spread of an epidemic disease.

We consider a boundary value problem for the beam equation, in which the boundary conditions mean that the beam is embedded at one end and free at the other end. Some new estimates to the positive solutions to the boundary value problem are obtained. Some sufficient conditions for the existence of at least one positive solution for the boundary value problem are established. An example is given at the end of the paper to illustrate the main results.

The work presented in this article shows that the viscous/inviscid coupled problem (VIC) has a unique solution when interfacial data are imposed. Domain decomposition techniques and non-uniform relaxation parameters were used to characterize the solution of the new system. Finally, some exact solutions for the VIC problem are provided. These type of solutions are an improvement over those found in recent literatures.

We study the positive solutions of equation

\begin{equation*}

\div(\norm{\nabla u}^{p-2}\nabla u)+\ss{\vec b(x)}{\norm{\nabla

u}^{p-2}\nabla u}+c(x)|u|^{q-2}u=0,

\end{equation*}

via the Riccati technique and prove an integral sufficient condition on the potential function $c(x)$ and the damping $\vec b(x)$ which ensures that no positive solution of the equation satisfies a lower (if $p>q$) or upper (if $q>p$) bound eventually.

\begin{equation*}

\div(\norm{\nabla u}^{p-2}\nabla u)+\ss{\vec b(x)}{\norm{\nabla

u}^{p-2}\nabla u}+c(x)|u|^{q-2}u=0,

\end{equation*}

via the Riccati technique and prove an integral sufficient condition on the potential function $c(x)$ and the damping $\vec b(x)$ which ensures that no positive solution of the equation satisfies a lower (if $p>q$) or upper (if $q>p$) bound eventually.

In this paper, we investigate a mathematical model for a nonlinear coupled system of Kirchhoff type of beam equations with nonlocal boundary conditions. We establish existence, regularity and uniqueness of strong solutions. Furthermore, we prove the uniform rate of exponential decay. The uniform rate of polynomial decay is considered.

It is proved that if a linear operator $\ell:C([a,b];\R)\rightarrow L([a,b];\R)$ is nonpositive and for the initial value problem $$u''(t)=\ell(u)(t)+q(t),\quad u(a)=c_1,\quad u'(a)=c_2 $$ the theorem on {\it differential inequalities} is valid, then $\ell$ is an $a-$Volterra operator.

Some oscillation criteria are established for the nonlinear damped elliptic differential equation of second order

$$

\sum_{i,\,j=1}^{N}D_i[\,a_{ij}(x)D_jy\,]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0,

\eqno{(E)}

$$

which are different from most known ones in the sense that they are based on a new weighted function $H(r,s,l)$ defined in the sequel. Both the cases when $D_ib_i(x)$ exists for all $i$ and when it does not exist for some $i$ are considered.

$$

\sum_{i,\,j=1}^{N}D_i[\,a_{ij}(x)D_jy\,]+\sum_{i=1}^{N}b_i(x)D_iy+p(x)f(y)=0,

\eqno{(E)}

$$

which are different from most known ones in the sense that they are based on a new weighted function $H(r,s,l)$ defined in the sequel. Both the cases when $D_ib_i(x)$ exists for all $i$ and when it does not exist for some $i$ are considered.

We consider a system of perturbed Volterra integro-differential equations for which the solution approaches a nontrivial limit and the difference between the solution and its limit is integrable. Under the condition that the second moment of the kernel is integrable we show that the solution decays exponentially to its limit if and only if the kernel is exponentially integrable and the tail of the perturbation decays exponentially.

The paper deals with the impulsive nonlinear boundary value problem

\begin{displaymath}

u''(t) = f(t,u(t),u'(t)) \quad\mbox{for a.~e.}\ t \in [0,T],

\end{displaymath}

\begin{displaymath}

u(t_j+) = J_j(u(t_j)),\quad u'(t_j+) = M_j(u'(t_j)),\quad j = 1,\ldots,m,

\end{displaymath}

\begin{displaymath}

g_1(u(0),u(T)) = 0, \quad g_2(u'(0),u'(T)) = 0,

\end{displaymath}

where $f \in Car([0,T]\times\rr^{2})$, $g_1$, $g_2 \in C(\rr^2)$, $J_j$, $M_j \in C(\rr)$. An existence theorem is proved for non--ordered lower and upper functions. Proofs are based on the Leray--Schauder degree and on the method of a~priori estimates.

\begin{displaymath}

u''(t) = f(t,u(t),u'(t)) \quad\mbox{for a.~e.}\ t \in [0,T],

\end{displaymath}

\begin{displaymath}

u(t_j+) = J_j(u(t_j)),\quad u'(t_j+) = M_j(u'(t_j)),\quad j = 1,\ldots,m,

\end{displaymath}

\begin{displaymath}

g_1(u(0),u(T)) = 0, \quad g_2(u'(0),u'(T)) = 0,

\end{displaymath}

where $f \in Car([0,T]\times\rr^{2})$, $g_1$, $g_2 \in C(\rr^2)$, $J_j$, $M_j \in C(\rr)$. An existence theorem is proved for non--ordered lower and upper functions. Proofs are based on the Leray--Schauder degree and on the method of a~priori estimates.

We investigate the existence of local approximate and strong solutions for a time-dependent subdifferential evolution inclusion with a nonconvex upper-semicontinuous perturbation.

In this paper, we investigate the existence of solutions and extremal solutions for a first order impulsive dynamic inclusion on time scales. By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values.

Oscillatory properties of even order self-adjoint linear differential equations in the form

$$

\sum_{k=0}^{n}

(-1)^k\nu_k\left(\frac{y^{(k)}}{t^{2n-2k-\alpha}}\right)^{(k)}

=(-1)^m\left(q_m(t)y^{(m)}\right)^{(m)},\; \nu_n:=1,

$$

where $m \in \{0, 1\}$, $\alpha \not\in \{1, 3, \dots , 2n-1\}$ and $\nu_0, \dots, \nu_{n-1},$ are real constants satisfying certain conditions, are investigated. In particular, the case when $q_m(t)=\frac{\beta}{t^{2n-2m-\alpha}\ln^2 t }$ is studied.

$$

\sum_{k=0}^{n}

(-1)^k\nu_k\left(\frac{y^{(k)}}{t^{2n-2k-\alpha}}\right)^{(k)}

=(-1)^m\left(q_m(t)y^{(m)}\right)^{(m)},\; \nu_n:=1,

$$

where $m \in \{0, 1\}$, $\alpha \not\in \{1, 3, \dots , 2n-1\}$ and $\nu_0, \dots, \nu_{n-1},$ are real constants satisfying certain conditions, are investigated. In particular, the case when $q_m(t)=\frac{\beta}{t^{2n-2m-\alpha}\ln^2 t }$ is studied.

We solve the following problem: If $(u,\,v)=(u(x,\,t),\,v(x,\,t))$ is a solution of the Dispersive Coupled System with $t_{1}<t_{2}$ which are sufficiently smooth and such that:

$\,\mbox{supp}\;u(\,.\,,\,t_{j})\subset (a,\,b)\,$

and

$\,\mbox{supp}\;v(\,.\,,\, t_{j})\subset (a,\,b),\,-\,\infty<a<b<\infty ,\,$ $j=1,\,2.\,$

Then $u\equiv 0$ and $v\equiv 0.$

$\,\mbox{supp}\;u(\,.\,,\,t_{j})\subset (a,\,b)\,$

and

$\,\mbox{supp}\;v(\,.\,,\, t_{j})\subset (a,\,b),\,-\,\infty<a<b<\infty ,\,$ $j=1,\,2.\,$

Then $u\equiv 0$ and $v\equiv 0.$

We establish the existence of a smallest eigenvalue for the fourth-order four-point boundary value problem

$\left (\phi_p(u''(t)) \right )'' = \lambda h(t) u(t), \, u'(0) = 0, \,

\beta_0 u(\eta_0) = u(1), \, \phi_p'(u''(0)) = 0, \,

\beta_1\phi_p(u''(\eta_1)) = \phi_p(u''(1))$, $p > 2$,

$0 < \eta_1,\eta_0 < 1, 0 < \beta_1, \beta_0 < 1$,

using the theory of u$_0$-positive operators with respect to a cone in a Banach space. We then obtain a comparison theorem for the smallest positive eigenvalues, $\lambda_1$ and $\lambda_2$, for the differential equations

$\left ( \phi_p(u''(t)) \right )'' = \lambda_1 f(t) u(t)$ and

$\left ( \phi_p(u''(t)) \right )'' =\lambda_2 g(t) u(t)$ where $0 \leq f(t) \leq g(t), t \in [0,1]$.

$\left (\phi_p(u''(t)) \right )'' = \lambda h(t) u(t), \, u'(0) = 0, \,

\beta_0 u(\eta_0) = u(1), \, \phi_p'(u''(0)) = 0, \,

\beta_1\phi_p(u''(\eta_1)) = \phi_p(u''(1))$, $p > 2$,

$0 < \eta_1,\eta_0 < 1, 0 < \beta_1, \beta_0 < 1$,

using the theory of u$_0$-positive operators with respect to a cone in a Banach space. We then obtain a comparison theorem for the smallest positive eigenvalues, $\lambda_1$ and $\lambda_2$, for the differential equations

$\left ( \phi_p(u''(t)) \right )'' = \lambda_1 f(t) u(t)$ and

$\left ( \phi_p(u''(t)) \right )'' =\lambda_2 g(t) u(t)$ where $0 \leq f(t) \leq g(t), t \in [0,1]$.

Sufficient conditions for oscillation of all solutions of a class of neutral impulsive differential-difference equations of first order with deviating argument and fixed moments of impulse effect are found.

In this paper, a recent Frigon nonlinear alternative for contractive multivalued maps in Fr\'echet spaces, combined with semigroup theory, is used to investigate the existence of integral solutions for first order semilinear functional differential inclusions. An application to a control problem is studied. We assume that the linear part of the differential inclusion is a nondensely defined operator and satisfies the Hille-Yosida condition.

In this article, we study the existence of solutions for nonlocal variational elliptic inequality

$$ -M(\|u\|^2)\Delta u \geq f(x,u) $$

Making use of the penalized method and Galerkin approximations, we establish existence theorems for both cases when $M$ is continuous and when $M$ is discontinuous.

$$ -M(\|u\|^2)\Delta u \geq f(x,u) $$

Making use of the penalized method and Galerkin approximations, we establish existence theorems for both cases when $M$ is continuous and when $M$ is discontinuous.

We deal with the variational study of some type of nonlinear inhomogeneous elliptic problems arising in models of solar flares on the halfplane $\R$.

Let $\H$ denote the Hilbert transform and $\eta \ge 0$. We show that if the initial data of the following problems

$ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \,

u(\cdot , 0) = \phi (\cdot)$ and

$ v_t + \frac{1}{2} (v_x)^2 + v_{xxx} + \eta(\mathcal{H} v_x + \mathcal{H} v_{xxx}) = 0, \,

v(\cdot , 0) = \psi (\cdot)$

has an analytic continuation to a strip containing the real axis, then the solution has the same property, although the width of the strip might diminish with time. When $\eta>0$ and the initial data is complex-valued we prove local well-posedness of the two problems above in spaces of analytic functions, which implies the constancy over time of the radius of the strip of analyticity in the complex plane around the real axis.

$ u_t + u u_x + u_{xxx} + \eta(\mathcal{H} u_x + \mathcal{H} u_{xxx}) = 0, \,

u(\cdot , 0) = \phi (\cdot)$ and

$ v_t + \frac{1}{2} (v_x)^2 + v_{xxx} + \eta(\mathcal{H} v_x + \mathcal{H} v_{xxx}) = 0, \,

v(\cdot , 0) = \psi (\cdot)$

has an analytic continuation to a strip containing the real axis, then the solution has the same property, although the width of the strip might diminish with time. When $\eta>0$ and the initial data is complex-valued we prove local well-posedness of the two problems above in spaces of analytic functions, which implies the constancy over time of the radius of the strip of analyticity in the complex plane around the real axis.

In this paper some existence theorems for the first order differential equations in Banach algebras is proved under the mixed generalized Lipschitz, Carath\'eodory and monotonicity conditions.

This paper proves the existence of solutions for a third order initial value nonconvex differential inclusion. We start with an upper semicontinuous compact valued multifunction \emph{F} which is contained in a lower semicontinuous convex function $\partial V$ and show that,

$x^{(3)}(t) \in F(x(t),x'(t),x''(t)),$ $ \: x(0)=x_{0}, \: x'(0)=y_{0}, \: x''(0)=z_{0}$.

$x^{(3)}(t) \in F(x(t),x'(t),x''(t)),$ $ \: x(0)=x_{0}, \: x'(0)=y_{0}, \: x''(0)=z_{0}$.

In this paper, we discuss the oscillatory behavior of the second-order forced nonlinear dynamic equation

\begin{equation*}

\left( a(t)x^{\Delta }(t)\right) ^{\Delta }+p(t)f(x^{\sigma })=r(t),

\end{equation*}%

on a time scale ${\mathbb{T}}$ when $a(t)>0$. We establish some sufficient conditions which ensure that every solution oscillates or satisfies $\lim \inf_{t\rightarrow \infty }\left\vert x(t)\right\vert =0.$ Our oscillation results when $r(t)=0$ improve the oscillation results for dynamic equations on time scales that has been established by Erbe and Peterson [Proc. Amer. Math. Soc \ 132 (2004), 735-744], Bohner, Erbe and Peterson [J. Math. Anal. Appl. 301 (2005), 491--507] since our results do not require $% \int_{t_{0}}^{\infty }q(t)\Delta t>0$ and $\int_{\pm t_{0}}^{\pm \infty }% \frac{du}{f(u)}<\infty .$ Also, as a special case when ${\mathbb{T=R}}$, and $r(t)=0$ our results improve some oscillation results for differential equations. Some examples are given to illustrate the main results.

\begin{equation*}

\left( a(t)x^{\Delta }(t)\right) ^{\Delta }+p(t)f(x^{\sigma })=r(t),

\end{equation*}%

on a time scale ${\mathbb{T}}$ when $a(t)>0$. We establish some sufficient conditions which ensure that every solution oscillates or satisfies $\lim \inf_{t\rightarrow \infty }\left\vert x(t)\right\vert =0.$ Our oscillation results when $r(t)=0$ improve the oscillation results for dynamic equations on time scales that has been established by Erbe and Peterson [Proc. Amer. Math. Soc \ 132 (2004), 735-744], Bohner, Erbe and Peterson [J. Math. Anal. Appl. 301 (2005), 491--507] since our results do not require $% \int_{t_{0}}^{\infty }q(t)\Delta t>0$ and $\int_{\pm t_{0}}^{\pm \infty }% \frac{du}{f(u)}<\infty .$ Also, as a special case when ${\mathbb{T=R}}$, and $r(t)=0$ our results improve some oscillation results for differential equations. Some examples are given to illustrate the main results.

We consider a system of reaction-diffusion equations for which the uniform boundedness of the solutions can not be derived by existing methods. The system may represent, in particular, an epidemic model describing the spread of an infection disease within a population. We present an $L^{p}$ argument allowing to establish the global existence and the uniform boundedness of the solutions of the considered system.

In the present Note an existence result of asymptotically stable solutions for the integral equation

$$

x\left( t\right) =q\left( t\right) +\int_{0}^{t}K\left( t,s,x\left( s\right) \right) ds

+\int_{0}^{\infty }G\left( t,s,x\left( s\right) \right) ds

$$

is presented.

$$

x\left( t\right) =q\left( t\right) +\int_{0}^{t}K\left( t,s,x\left( s\right) \right) ds

+\int_{0}^{\infty }G\left( t,s,x\left( s\right) \right) ds

$$

is presented.

In this paper we study stability and boundedness of the nonlinear difference equation

\begin{equation} x(t+1)=a(t)x(t)+c(t)\Delta x(t-g(t))+q(x(t),x(t-g(t))\big).\nonumber \end{equation}

In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis.

\begin{equation} x(t+1)=a(t)x(t)+c(t)\Delta x(t-g(t))+q(x(t),x(t-g(t))\big).\nonumber \end{equation}

In particular we study equi-boundedness of solutions and the stability of the zero solution of this equation. Fixed point theorems are used in the analysis.

In this paper, we establish the existence and boundedness of solutions of a doubly nonlinear parabolic system. We also obtain the existence of a global attractor and the regularity property for this attractor in $\left[ L^{\infty }(\Omega )\right] ^{2}$ and

${\displaystyle \prod_{i=1}^{2}}{B_{\infty }^{1+\sigma _{i},p_{i}}( \Omega )} $.

${\displaystyle \prod_{i=1}^{2}}{B_{\infty }^{1+\sigma _{i},p_{i}}( \Omega )} $.

This paper concerns several variants of an integral equation

$$ x(t)=a(t)-\int^t_0 C(t,s) x(s)ds $$, a resolvent $$ R(t,s) $$, and a variation-of-parameters formula

$$ x(t)=a(t)-\int^t_0 R(t,s) a(s)ds $$ with special accent on the case in which $a(t)$ is unbounded. We use contraction mappings to establish close relations between $a(t)$ and $\int^t_0R(t,s) a(s)ds$.

$$ x(t)=a(t)-\int^t_0 C(t,s) x(s)ds $$, a resolvent $$ R(t,s) $$, and a variation-of-parameters formula

$$ x(t)=a(t)-\int^t_0 R(t,s) a(s)ds $$ with special accent on the case in which $a(t)$ is unbounded. We use contraction mappings to establish close relations between $a(t)$ and $\int^t_0R(t,s) a(s)ds$.

We study the existence of solutions for a periodic fourth order problem. We prove an associated uniform antimaximum principle and develop a method of upper and lower solutions in reversed order. Furthermore, by the quasilinearization method we construct an iterative sequence that converges quadratically to a solution.

In this work, we are concerned with a boundary value problem associated with a generalized Fisher-like equation. This equation involves an eigenvalue and a parameter which may be viewed as a wave speed. According to the behavior of the nonlinear source term, existence results of bounded solutions, positive solutions, classical as well as weak solutions are provided. We mainly use fixed point arguments.

In this paper, we consider the initial-boundary value problem of a nonlinear parabolic equation with double degeneracy, and establish the existence and uniqueness theorems of renormalized solutions which are stronger than $BV$ solutions.

Within the last decade, there has been growing interest in the study of multiple solutions of two- and multi-point boundary value problems of nonlinear ordinary differential equations as fixed points of a cone mapping. Undeniably many good results have emerged. The purpose of this paper is to point out that, in the special case of second-order equations, the shooting method can be an effective tool, sometimes yielding better results than those obtainable via fixed point techniques.

In this paper we investigate the existence of positive solutions of two-point boundary value problems for nonlinear second order differential equations of the form $(py^{\prime})^{\prime}(t)+q(t)y(t)=f(t,y(t),y^{\prime}(t))$, where $f$ \ is a Carath\'{e}odory function, which may change sign, with respect to its second argument, infinitely many times.

In the paper, sufficient conditions are given under which all nontrivial solutions of $(g(a(t)y'))' + r(t) f (y) = 0$ are proper where $a>0, r>0, f(x) x>0, g(x) x>0$ for $x\ne0$ and $g$ is increasing on $R$. A sufficient condition for the existence of a singular solution of the second kind is given.

In this paper, we investigate the iterated order of entire solutions of homogeneous and non-homogeneous linear differential equations with entire coefficients.

We present several new criteria for the oscillation of the second-order linear equation $ y''(t)+q(t)y(t)=0 $, in which the coefficient $ q $ may or may not change signs. The criteria involve the integral $ \int t^\gamma q(t)\, dt $ for some $ \gamma >0 $. The special case $ \gamma =2 $ is then studied in greater details.

The aim of this paper is to present new results on existence theory for perturbed BVPs for first order ordinary differential systems. A nonlinear alternative for the sum of a contraction and a compact mapping is used.

We find an infinite number of smooth, localized, radial solutions of $\Delta_{p} u + f(u) = 0$ in ${\Bbb R}^{N}$ - one with each prescribed number of zeros - where $\Delta_{p}u$ is the $p$-Laplacian of the function $u$.

In this paper, some nonexistence, existence and multiplicity of positive solutions are established for a class of singular boundary value problem. The authors also obtain the relation between the existence of the solutions and the parameter $\lambda$. The arguments are based upon the fixed point index theory and the upper and lower solutions method.

We define the principal matrix solution $Z(t,s)$ of the linear Volterra vector integro-differential equation

\[ x'(t) = A(t)x(t) + \int_s^t B(t,u)x(u)\,du \]

in the same way that it is defined for $x' = A(t)x$ and prove that it is the unique matrix solution of

\[ \frac{\partial}{\partial{t}}Z(t,s) = A(t)Z(t,s) + \int_{s}^t B(t,u)Z(u,s)\,du, \quad Z(s,s) = I. \]

Furthermore, we prove that the solution of

\[ x'(t) = A(t)x(t) + \int_{\tau}^t B(t,u)x(u)\,du + f(t), \quad x(\tau) = x_0\]

is unique and given by the variation of parameters formula

\[ x(t) = Z(t,\tau)x_0 + \int_{\tau}^t Z(t,s)f(s)\,ds.\]

We also define the principal matrix solution $R(t,s)$ of the adjoint equation

\[ r'(s) = -r(s)A(s) - \int_s^t r(u)B(u,s)\,du \]

and prove that it is identical to Grossman and Miller's resolvent, which is the unique matrix solution of

\[ \frac{\partial}{\partial{s}}R(t,s) = -R(t,s)A(s) - \int_{s}^t R(t,u)B(u,s)\,du, \quad R(t,t) = I. \]

Finally, we prove that despite the difference in their definitions $R(t,s)$ and $Z(t,s)$ are in fact identical.

\[ x'(t) = A(t)x(t) + \int_s^t B(t,u)x(u)\,du \]

in the same way that it is defined for $x' = A(t)x$ and prove that it is the unique matrix solution of

\[ \frac{\partial}{\partial{t}}Z(t,s) = A(t)Z(t,s) + \int_{s}^t B(t,u)Z(u,s)\,du, \quad Z(s,s) = I. \]

Furthermore, we prove that the solution of

\[ x'(t) = A(t)x(t) + \int_{\tau}^t B(t,u)x(u)\,du + f(t), \quad x(\tau) = x_0\]

is unique and given by the variation of parameters formula

\[ x(t) = Z(t,\tau)x_0 + \int_{\tau}^t Z(t,s)f(s)\,ds.\]

We also define the principal matrix solution $R(t,s)$ of the adjoint equation

\[ r'(s) = -r(s)A(s) - \int_s^t r(u)B(u,s)\,du \]

and prove that it is identical to Grossman and Miller's resolvent, which is the unique matrix solution of

\[ \frac{\partial}{\partial{s}}R(t,s) = -R(t,s)A(s) - \int_{s}^t R(t,u)B(u,s)\,du, \quad R(t,t) = I. \]

Finally, we prove that despite the difference in their definitions $R(t,s)$ and $Z(t,s)$ are in fact identical.

In this paper, some random fixed point theorems for monotone increasing, condensing and closed multi-valued random operators are proved. They are then applied to first order ordinary nonconvex random differential inclusions for proving the existence of solutions as well as the existence of extremal solutions under certain monotonicity conditions.

In this paper we consider the semilinear differential equation with deviated argument in a Fr\'echet space $x^{\prime}(t) = A x(t) + f(t, x(t), x[\alpha(x(t),t)]),$ $t \in {\mathbb{R}}$, where $A$ is the infinitesimal (bounded) generator of a $C_{0}$-semigroup satisfying some conditions of exponential stability. Under suitable conditions on the functions $f$ and $\alpha$ we prove the existence and uniqueness of an almost automorphic mild solution to the equation.

Substituting the usual growth condition by an assumption that a specific initial value problem has a maximal solution, we obtain existence results for functional nonlinear integral equations with variable delay. Application of the technique to initial value problems for differential equations as well as to integrodifferential equations are given.

We study resonances of the semi-classical Schr\"odinger operator $H = -h^2 \Delta + V$ on $L^2( \R^N)$. We consider the case where the potential $V$ have an absolute degenerate maximum. Then we prove that $H$ has resonances with energies $E = V_0 + e^{-i {\pi \over \sigma +1}} h^{ 2 \sigma \over \sigma +1} k_j + {\cal O}( h^{ 2 \sigma +1 \over \sigma +1} ),$ where $k_j $ is in the spectrum of some quartic oscillator.

In this paper we study the existence of solutions for the generated boundary value problem, with initial datum being an element of $L^1(\Omega)+W^{-1, p'}(\Omega, w^{*})$

$$-{\rm div}a(x, u, \nabla u) + g(x, u, \nabla u) = f-{\rm div}F $$

where $a(.)$ is a Carath\'eodory function satisfying the classical condition of type Leray-Lions hypothesis, while $g(x, s, \xi)$ is a non-linear term which has a growth condition with respect to $\xi$ and no growth with respect to $s$, but it satisfies a sign condition on $s$.

$$-{\rm div}a(x, u, \nabla u) + g(x, u, \nabla u) = f-{\rm div}F $$

where $a(.)$ is a Carath\'eodory function satisfying the classical condition of type Leray-Lions hypothesis, while $g(x, s, \xi)$ is a non-linear term which has a growth condition with respect to $\xi$ and no growth with respect to $s$, but it satisfies a sign condition on $s$.

We study the asymptotic stability of the zero solution of the Volterra difference delay equation

\begin{equation}

x(n+1)=a(n)x(n)+c(n)\Delta x(n-g(n))+\sum^{n-1}_{s=n-g(n)}k(n,s)h(x(s)).\nonumber

\end{equation}

A Krasnoselskii fixed point theorem is used in the analysis.

\begin{equation}

x(n+1)=a(n)x(n)+c(n)\Delta x(n-g(n))+\sum^{n-1}_{s=n-g(n)}k(n,s)h(x(s)).\nonumber

\end{equation}

A Krasnoselskii fixed point theorem is used in the analysis.

In this paper, we discuss an extended form of generalized quasilinearization technique for first order nonlinear impulsive differential equations with a nonlinear three-point boundary condition. In fact, we obtain monotone sequences of upper and lower solutions converging uniformly and quadratically to the unique solution of the problem.

A theorem on existence of mild solutions for partial neutral functional differential inclusions with unbounded linear part generating a noncompact semigroup in Banach space is established.

The paper examines the existence of pseudo almost periodic solutions to some classes of partial hyperbolic evolution equations. Namely, some sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those classes of hyperbolic evolution equations are given. As an application, we consider the existence of pseudo almost periodic solutions to the heat equations with delay.

The existence of a positive solution is obtained for the $n^{th}$ order right focal boundary value problem $y^{(n)}=f(x,y)$, $0 < x \leq 1$, $y^{(i)}(0)=y^{(n-2)}(p)=y^{(n-1)}(1)=0, i=0,\cdots, n-3$, where $\frac{1}{2}<p<1$ is fixed and where $f(x,y)$ is singular at $x=0, y=0$, and possibly at $y=\infty$. The method applies a fixed-point theorem for mappings that are decreasing with respect to a cone.

In the paper, the nonlinear differential equation $(a(t)|y^{\prime}|^{p-1}y^{\prime})^{\prime}+b(t)g(y^{\prime})+r(t)f(y)=e(t)$ is studied. Sufficient conditions for the nonexistence of singular solutions of the first and second kind are given. Hence, sufficient conditions for all nontrivial solutions to be proper are derived. Sufficient conditions for the nonexistence of weakly oscillatory solutions are given.

The fractional calculus (integration and differentiation of fractional-order) is a one of the singular integral and integro-differential operators. In this work a class of fractional-order non-autonomous systems will be considered. The stability (and some other properties concerning the existence and uniqueness) of the solution will be proved.

This paper deals with the existence result of viable solutions of the differential inclusion \begin{center}$\dot{x}(t) \in f(t,x(t)) + F(x(t))$\\$x(t) \in K$ on $[0,T],$ \end{center} where $K$ is a locally compact subset in separable Hilbert space $H,$ $(f(s,\cdot))_s$ is an equicontinuous family of measurable functions with respect to $s$ and $F$ is an upper semi-continuous set-valued mapping with compact values contained in the Clarke subdifferential $\partial_{c} V(x)$ of an uniformly regular function $V.$

We prove a Filippov type existence theorem for solutions of a higher order differential inclusion in Banach spaces with nonconvex valued right hand side by applying the contraction principle in the space of the derivatives of solutions instead of the space of solutions.

In this paper we prove the exponential decay in the case $n>2$, as time goes to infinity, of regular solutions for a nonlinear coupled system of beam equations of Kirchhoff type with memory and weak damping

\begin{eqnarray*}

&&u_{tt}+\Delta^2 u-M(||\nabla u||^2_{L^2(\Omega_t)}+||\nabla

v||^2_{L^2(\Omega_t)})\Delta u\\

&&+\int^{t}_{0}g_1(t-s)\Delta u(s)ds

+\alpha u_{t}+h(u-v)=0 \quad \mbox{in} \quad \hat{Q},\\

&&v_{tt}+\Delta^2 v-M(||\nabla u||^2_{L^2(\Omega_t)}+||\nabla

v||^2_{L^2(\Omega_t)})\Delta v \\

&&+\int^{t}_{0}g_2(t-s)\Delta v(s)ds + \alpha v_{t}-h(u-v)=0 \quad

\mbox{in} \quad \hat{Q}

\end{eqnarray*}

in a non cylindrical domain of $\R^{n+1}$ $(n\ge1)$ under suitable hypothesis on the scalar functions $M$, $h$, $g_1$ and $g_2$, and where $\alpha$ is a positive constant. We show that such dissipation is strong enough to produce uniform rate of decay. Besides, the coupling is nonlinear which brings up some additional difficulties, which plays the problem interesting. We establish existence and uniqueness of regular solutions for any $n\ge 1$.

\begin{eqnarray*}

&&u_{tt}+\Delta^2 u-M(||\nabla u||^2_{L^2(\Omega_t)}+||\nabla

v||^2_{L^2(\Omega_t)})\Delta u\\

&&+\int^{t}_{0}g_1(t-s)\Delta u(s)ds

+\alpha u_{t}+h(u-v)=0 \quad \mbox{in} \quad \hat{Q},\\

&&v_{tt}+\Delta^2 v-M(||\nabla u||^2_{L^2(\Omega_t)}+||\nabla

v||^2_{L^2(\Omega_t)})\Delta v \\

&&+\int^{t}_{0}g_2(t-s)\Delta v(s)ds + \alpha v_{t}-h(u-v)=0 \quad

\mbox{in} \quad \hat{Q}

\end{eqnarray*}

in a non cylindrical domain of $\R^{n+1}$ $(n\ge1)$ under suitable hypothesis on the scalar functions $M$, $h$, $g_1$ and $g_2$, and where $\alpha$ is a positive constant. We show that such dissipation is strong enough to produce uniform rate of decay. Besides, the coupling is nonlinear which brings up some additional difficulties, which plays the problem interesting. We establish existence and uniqueness of regular solutions for any $n\ge 1$.

In this paper, the global exponential stability of dynamical systems with distributed delays and impulsive effect is investigated. By establishing an impulsive differential-integro inequality, we obtain some sufficient conditions ensuring the global exponential stability of the dynamical system. Three examples are given to illustrate the effectiveness of our theoretical results.

In this study, linear second-order matrix $q$-difference equations are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary conditions. A generalized Wronskian is introduced and a Lagrange identity and Abel's formula are established. Two reduction-of-order theorems are given. The analysis and characterization of $q$-dominant and $q$-recessive solutions at infinity are presented, emphasizing the case when the quantum system is disconjugate.

In this paper we study the properties of the solutions to the Cauchy problem

$$

(u_{tt}-\Delta u)_{g_s}=f(u)+g(|x|),\quad t\in [0, 1], x\in {\cal R}^3,

\leqno{(1)}

$$

$$

u(1, x)=u_0\in {\dot H}^1({\cal R}^3),\quad

u_t(1, x)=u_1\in L^2({\cal R}^3),

\leqno{(2)}

$$

where $g_s$ is the Reissner-Nordstr${\ddot o}$m metric (see [2]); $f\in {\cal C}^1({\cal R}^1)$, $f(0)=0$, $a|u|\leq f'(u)\leq b|u|$, $g\in {\cal C}({\cal R}^+)$, $g(|x|)\geq 0$, $g(|x|)=0$ for $|x|\geq r_1$, $a$ and $b$ are positive constants, $r_1>0$ is suitable chosen. When $g(r)\equiv 0$ we prove that the Cauchy problem $(1)$, $(2)$ has a nontrivial solution $u(t, r)$ in the form $u(t, r)=v(t)\omega(r)\in {\cal C}((0, 1]{\dot H}^1({\cal R}^+))$, where $r=|x|$, and the solution map is not uniformly continuous. When $g(r)\ne 0$ we prove that the Cauchy problem $(1)$, $(2)$ has a nontrivial solution $u(t, r)$ in the form $u(t, r)=v(t)\omega(r)\in {\cal C}((0, 1]{\dot H}^1({\cal R}^+))$, where $r=|x|$, and the solution map is not uniformly continuous.

$$

(u_{tt}-\Delta u)_{g_s}=f(u)+g(|x|),\quad t\in [0, 1], x\in {\cal R}^3,

\leqno{(1)}

$$

$$

u(1, x)=u_0\in {\dot H}^1({\cal R}^3),\quad

u_t(1, x)=u_1\in L^2({\cal R}^3),

\leqno{(2)}

$$

where $g_s$ is the Reissner-Nordstr${\ddot o}$m metric (see [2]); $f\in {\cal C}^1({\cal R}^1)$, $f(0)=0$, $a|u|\leq f'(u)\leq b|u|$, $g\in {\cal C}({\cal R}^+)$, $g(|x|)\geq 0$, $g(|x|)=0$ for $|x|\geq r_1$, $a$ and $b$ are positive constants, $r_1>0$ is suitable chosen. When $g(r)\equiv 0$ we prove that the Cauchy problem $(1)$, $(2)$ has a nontrivial solution $u(t, r)$ in the form $u(t, r)=v(t)\omega(r)\in {\cal C}((0, 1]{\dot H}^1({\cal R}^+))$, where $r=|x|$, and the solution map is not uniformly continuous. When $g(r)\ne 0$ we prove that the Cauchy problem $(1)$, $(2)$ has a nontrivial solution $u(t, r)$ in the form $u(t, r)=v(t)\omega(r)\in {\cal C}((0, 1]{\dot H}^1({\cal R}^+))$, where $r=|x|$, and the solution map is not uniformly continuous.

We express radial solutions of semilinear elliptic equations on $R^n$ as convergent power series in $r$, and then use Pade approximants to compute both ground state solutions, and solutions to Dirichlet problem. Using a similar approach we have discovered existence of singular solutions for a class of subcritical problems. We prove convergence of the power series by modifying the classical method of majorants.

In this paper, we deal with a transmission problem for a class of quasilinear pseudoparabolic equations. Existence, uniqueness and continuous dependence of the solution upon the data are obtained via the Rothe method. Moreover, the convergence of the method and an error estimate of the approximations are established.

In this paper, we investigate the existence of solutions for a class of second order functional differential inclusions with integral boundary conditions. By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values.

We use Krasnoselskii's fixed point theorem to show that the nonlinear neutral differential equation with delay

\begin{equation}

\frac{d}{dt}[x(t) - ax(t-\tau)]= r(t)x(t)- f(t, x(t-\tau))

\end{equation}

has a positive periodic solution. An example will be provided as an application to our theorems.

\begin{equation}

\frac{d}{dt}[x(t) - ax(t-\tau)]= r(t)x(t)- f(t, x(t-\tau))

\end{equation}

has a positive periodic solution. An example will be provided as an application to our theorems.

This article focuses on the approximation of conventional second order derivative via the combined (diamond-$\alpha$) dynamic derivative on time scales with necessary smoothness conditions embedded. We will show the constraints under which the second order dynamic derivative provides a consistent approximation to the conventional second derivative; the cases where the dynamic derivative approximates the derivative only via a proper modification of the existing formula; and the situations in which the dynamic derivative can never approximate consistently even with the help of available structure correction methods. Constructive error analysis will be given via asymptotic expansions for practical hybrid modeling and computational applications.

Intervals of the parameter $\lambda$ are determined for which there exist positive solutions for the system of nonlinear differential equations, $u^{(n)} + \lambda a(t) f(v) = 0, \ v^{(n)} +\lambda b(t) g(u) = 0, $ for $0 < t <1$, and satisfying three-point nonlocal boundary conditions, $u(0) = 0, u'(0) = 0, \ldots, u^{(n-2)}(0) = 0, \ u(1)=\alpha u(\eta), v(0) = 0, v'(0) = 0, \ldots, v^{(n-2)}(0) = 0, \ v(1)=\alpha v(\eta)$. A Guo-Krasnosel'skii fixed point theorem is applied.

We shall establish a necessary and sufficient condition under which we have the null controllability of some first order impulsive evolution equation in a Hilbert space.

We study the oscillatory behavior of solutions of the second order linear differential equation of Euler type: $(E)\ y'' + \lambda x^{-\alpha} y = 0, \ x \in (0, 1]$, where $\lambda > 0$ and $\alpha> 2$. Theorem (a) For $2 \le \alpha < 4$, all solution curves of $(E)$ have finite arc length; (b) For $\alpha \ge 4$, all solution curves of $(E)$ have infinite arc length. This answers an open problem posed by M. Pasic [8]

We prove an existence theorem for a quadratic functional-integral equation of mixed type. The functional-integral equation studied below contains as special cases numerous integral equations encountered in nonlinear analysis. With help of a suitable measure of noncompactness, we show that the functional integral equation of mixed type has solutions being continuous and bounded on the interval $[0,\infty)$ and those solutions are globally attractive.

Existence of solutions to some nonlinear integral equations with variable delays are obtained by the use of a fixed point theorem due to Dhage. As applications of the main results, existence results to some initial value problems concerning differential equations of higher order as well as integro-differential equations are derived. The case of Lipschitz-type conditions is also considered. Our results improve and generalize, in several ways, existence results already appeared in the literature.

By using coincidence degree theory of Mawhin, existence results for some higher order resonance multipoint boundary value problems with one dimensional p-Laplacian operator are obtained.

An existence result of a solution for a class of nonlinear parabolic systems is established. The data belong to $L^1$ and no growth assumption is made on the nonlinearities.

We present new oscillation criteria for the second order nonlinear differential equation with damping term of the form

\begin{equation*}

\left( r(t)\psi (x)f(\dot{x})\right) ^{\cdot }+p(t)\varphi \left(g(x),r(t)\psi(x)f(\dot{x})\right)+q(t)g(x)=0,

\end{equation*}

where $p$, $q$, $r:[t_{o},\infty )\rightarrow \mathbf{R}$ and\ $\psi $, $g $, $f:\mathbf{R}\rightarrow \mathbf{R}$ are continuous, $r(t)>0$,\ $p(t)\geq 0$ and $\psi (x)>0$, $xg(x)>0$ for $x\neq 0$, $uf(u)>0$ for $u\neq0 $. Our results generalize and extend some known oscillation criteria in the literature. The relevance of our results is illustrated with a number of examples.

\begin{equation*}

\left( r(t)\psi (x)f(\dot{x})\right) ^{\cdot }+p(t)\varphi \left(g(x),r(t)\psi(x)f(\dot{x})\right)+q(t)g(x)=0,

\end{equation*}

where $p$, $q$, $r:[t_{o},\infty )\rightarrow \mathbf{R}$ and\ $\psi $, $g $, $f:\mathbf{R}\rightarrow \mathbf{R}$ are continuous, $r(t)>0$,\ $p(t)\geq 0$ and $\psi (x)>0$, $xg(x)>0$ for $x\neq 0$, $uf(u)>0$ for $u\neq0 $. Our results generalize and extend some known oscillation criteria in the literature. The relevance of our results is illustrated with a number of examples.

The existence of positive solutions are established for the multi-point boundary value problems

$$

\left\{ \begin{array}{ll}

(-1)^nu^{(2n)}(x)=\lambda p(x)f(u(x)), \hspace*{.2in} 0<x<1 \\

u^{(2i)}(0)=\sum_{j=1}^{m}a_ju^{(2i)}(\eta _j), \quad

u^{(2i+1)}(1)=\sum_{j=1}^{m}b_ju^{(2i+1)}(\eta _j), \quad i=0, 1,

\ldots , n-1

\end{array} \right.

$$

where $a_j,b_j\in[0,\infty), \ j=1, 2, \ldots, m,$ with $0<\sum_{j=1}^{m}a_j<1, \ 0<\sum_{j=1}^{m}b_j<1,$ and $ \eta_j \in(0,1)$ with $0<\eta_1<\eta_2<\ldots <\eta_m<1,$ under certain conditions on $f$ and $p$ using the Krasnosel'skii fixed point theorem for certain values of $\l$. We use the positivity of the Green's function and cone theory to prove our results.

$$

\left\{ \begin{array}{ll}

(-1)^nu^{(2n)}(x)=\lambda p(x)f(u(x)), \hspace*{.2in} 0<x<1 \\

u^{(2i)}(0)=\sum_{j=1}^{m}a_ju^{(2i)}(\eta _j), \quad

u^{(2i+1)}(1)=\sum_{j=1}^{m}b_ju^{(2i+1)}(\eta _j), \quad i=0, 1,

\ldots , n-1

\end{array} \right.

$$

where $a_j,b_j\in[0,\infty), \ j=1, 2, \ldots, m,$ with $0<\sum_{j=1}^{m}a_j<1, \ 0<\sum_{j=1}^{m}b_j<1,$ and $ \eta_j \in(0,1)$ with $0<\eta_1<\eta_2<\ldots <\eta_m<1,$ under certain conditions on $f$ and $p$ using the Krasnosel'skii fixed point theorem for certain values of $\l$. We use the positivity of the Green's function and cone theory to prove our results.

In this paper, we study existence, uniqueness and stability questions for the nonlinear parabolic equation:

$$ \frac{\partial u}{\partial t}-\triangle_{p}u+\alpha(u)=f in ]0,\ T[\times\Omega, $$ with Neumann-type boundary conditions and initial data in $L^1$. Our approach is based essentially on the time discretization technique by Euler forward scheme.

$$ \frac{\partial u}{\partial t}-\triangle_{p}u+\alpha(u)=f in ]0,\ T[\times\Omega, $$ with Neumann-type boundary conditions and initial data in $L^1$. Our approach is based essentially on the time discretization technique by Euler forward scheme.

We consider planar vector fields depending on a real parameter. It is assumed that this vector field has a family of limit cycles which can be described by means of the limit cycles function $l$. We prove a relationship between the multiplicity of a limit cycle of this family and the order of a zero of the limit cycles function. Moreover, we present a procedure to approximate $l(x)$, which is based on the Newton scheme applied to the Poincar\'e function and represents a continuation method. Finally, we demonstrate the effectiveness of the proposed procedure by means of a Li\'enard system.

The system under consideration is

$$

-\Delta u+a_uu=u^3-\beta uv^2, \quad u=u(x),

$$

$$

-\Delta v+a_vv=v^3-\beta u^2v, \quad v=v(x), \ x\in \mathbb {R}^3,

$$

$$

u\big| _{|x|\to \infty }=v\big| _{|x|\to \infty }=0,

$$

where $a_u,a_v$ and $\beta $ are positive constants. We prove the existence of a component-wise positive smooth radially symmetric solution of this system. This result is a part of the results presented in the recent paper by Sirakov [1]; in our opinion, our method allows one to treat the problem simpler and shorter.

$$

-\Delta u+a_uu=u^3-\beta uv^2, \quad u=u(x),

$$

$$

-\Delta v+a_vv=v^3-\beta u^2v, \quad v=v(x), \ x\in \mathbb {R}^3,

$$

$$

u\big| _{|x|\to \infty }=v\big| _{|x|\to \infty }=0,

$$

where $a_u,a_v$ and $\beta $ are positive constants. We prove the existence of a component-wise positive smooth radially symmetric solution of this system. This result is a part of the results presented in the recent paper by Sirakov [1]; in our opinion, our method allows one to treat the problem simpler and shorter.

In this work, we are concerned with a nonlinear weighted Cauchy type problem of a differ-integral equation of fractional order. We will prove some local and global existence theorems for this problem, also we will study the uniqueness and stability of its solution.

In this paper, we investigate the problem of existence and nonexistence of positive solutions for the nonlinear boundary value problem:

\begin{eqnarray*}

u^{(n)}(t)+\lambda a(t)f(u(t))=0,\,\,\, 0<t<1,

\end{eqnarray*}

satisfying three kinds of different boundary value conditions. Our analysis relies on Krasnoselskii's fixed point theorem of cone. An example is also given to illustrate the main results.

\begin{eqnarray*}

u^{(n)}(t)+\lambda a(t)f(u(t))=0,\,\,\, 0<t<1,

\end{eqnarray*}

satisfying three kinds of different boundary value conditions. Our analysis relies on Krasnoselskii's fixed point theorem of cone. An example is also given to illustrate the main results.

In this paper we study integral equations of the form $x(t)=a(t)-\int^t_0 C(t,s)x(s)ds$ with sharply contrasting kernels typified by $C^*(t,s)=\ln (e+(t-s))$ and $D^*(t,s)=[1+(t-s)]^{-1}$. The kernel assigns a weight to $x(s)$ and these kernels have exactly opposite effects of weighting. Each type is well represented in the literature. Our first project is to show that for $a\in L^2[0,\infty)$, then solutions are largely indistinguishable regardless of which kernel is used. This is a surprise and it leads us to study the essential differences. In fact, those differences become large as the magnitude of $a(t)$ increases.

The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t)$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient.

The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t)$ as the sum of a bounded, but badly behaved function, and a large well behaved function.

The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t)$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient.

The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t)$ as the sum of a bounded, but badly behaved function, and a large well behaved function.

In this paper we give sufficient conditions for the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problem

\begin{eqnarray*}

&&D^{\alpha}u + a(t) f(u) = 0, \quad 0<t<1, 1<\alpha\leq2,\\

&&u(0) = 0 ,u'(1)= 0,

\end{eqnarray*}

where $ D^{\alpha}$ is the Riemann-Liouville differential operator of order $\alpha $, $f: [0,\infty)\rightarrow [0,\infty)$ is a given continuous function and $a$ is a positive and continuous function on $[0,1]$.

\begin{eqnarray*}

&&D^{\alpha}u + a(t) f(u) = 0, \quad 0<t<1, 1<\alpha\leq2,\\

&&u(0) = 0 ,u'(1)= 0,

\end{eqnarray*}

where $ D^{\alpha}$ is the Riemann-Liouville differential operator of order $\alpha $, $f: [0,\infty)\rightarrow [0,\infty)$ is a given continuous function and $a$ is a positive and continuous function on $[0,1]$.

In this work we consider a partial integro-differential equation. We reformulate it a functional integro-differential equation in a suitable Hilbert space. We apply the method of lines to establish the existence and uniqueness of a strong solution.

In this paper the Fountain theorem is employed to establish infinitely many solutions for the class of quasilinear Schr\"{o}dinger equations $-L_pu+ V(x)|u|^{p-2}u=\lambda|u|^{q-2}u+\mu |u|^{r-2}u$ in $\mathbb{R}$, where $L_pu=(|u'|^{p-2}u')'+ (|(u^2)'|^{p-2}(u^2)')'u$, $\lambda, \mu$ are real parameters, $1 < p < \infty$, $1<q<p$, $r>2p$ and the potential $V(x)$ is nonnegative and satisfies a suitable integrability condition.

In this paper we study the oscillatory and global asymptotic stability of a single neuron model with two delays and a general activation function. New sufficient conditions for the oscillation and nonoscillation of the model are given. We obtain both delay-dependent and delay-independent global asymptotic stability criteria. Some of our results are new even for models with one delay.

In this paper, we investigate the existence of positive solutions for a class of nonlinear semipositone $n$th-order boundary value problems. Our approach relies on the Krasnosel'skii fixed point theorem. The result of this paper complement and extend previously known result.

In this paper, we investigate the existence of positive solutions for singular $n$th-order boundary value problem $u^{(n)}(t)+a(t)f(t,u(t))=0,\quad 0\le t\le1,$ $u^{(i)}(0)=u^{(n-2)}(1)=0,\quad 0\le i\le n-2,$ where $n\ge2$, $a\in C((0,1),[0,+\infty))$ may be singular at $t=0$ and (or) $t=1$ and the nonlinear term $f$ is continuous and is allowed to change sign. Our proofs are based on the method of lower solution and topology degree theorem.

In this paper we prove the existence of mild solutions for semilinear neutral functional differential inclusions with unbounded linear part generating a noncompact semigroup in a Banach space. This work generalizes the result given in [4].

In the present study, using the characterizations of measures of noncompactness we prove a theorem on the existence and local asymptotic stability of solutions for a quadratic functional integral equation via a fixed point theorem of Darbo. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. An example is indicated to demonstrate the natural realizations of abstract result presented in the paper.

In this paper sufficient conditions are obtained for oscillation of all solutions of a class of nonlinear neutral delay difference equations of the form $\Delta^2(y(n)+p(n)y(n-m))+q(n)G(y(n-k))=0$ under various ranges of $p(n)$. The nonlinear function $G, G \in C (R, R)$ is either sublinear or superlinear.

In this article, the contraction mapping principle and Liapunov's method are used to study qualitative properties of nonlinear Volterra equations of the form $x(t) = a(t) -\int^{t}_{0}C(t,s)g(s,x(s))\;ds,t\geq0.$ In particular, the existence of bounded solutions and solutions with various $L^p$ properties are studied under suitable conditions on the functions involved with this equation.

In this work we will prove the existence uniqueness and asymptotic behavior of weak solutions for the system (*) involving the pseudo Laplacian operator and the condition $\displaystyle\frac{\partial u}{\partial t} + \sum_{i=1}^n \big|\frac{\partial u}{\partial x_i}\big|^{p-2}\frac{\partial u}{\partial x_i}\nu_i + |u|^{\rho}u=f$ on $\Sigma_1$, where $\Sigma_1$ is part of the lateral boundary of the cylinder $Q=\Omega \times (0,T)$ and $f$ is a given function defined on $\Sigma_1$.

Consider the three-point boundary value problem for the 3$^{rd}$ order differential equation:

\begin{equation*}\left\{ \begin{tabular}{l}

$x^{^{\prime \prime \prime }}(t)=\alpha \left( t\right) f(t,x(t),x^{\prime}\left( t\right) ,x^{\prime \prime }\left( t\right) ),\;\;\;0<t<1,$ \\

$x\left( 0\right) =x^{\prime }\left( \eta \right) =x^{\prime \prime }\left(1\right) =0,$

\end{tabular}\right.\end{equation*}

under positivity of the nonlinearity. Existence results for a positive and concave solution $x\left( t\right) ,\ 0\leq t\leq 1$ are given, for any $1/2<\eta <1.\ $ In addition, without any monotonicity assumption on the nonlinearity, we prove the existence of a sequence of such solutions with \begin{equation*} \lim_{n\rightarrow \infty }||x_{n}||=0. \end{equation*} Our principal tool is \emph{a very simple applications on a new cone of the plane} of the well-known Krasnosel'ski\u{\i}'s fixed point theorem. The main feature of this aproach is that, we do not use at all the associated Green's function, the necessary positivity of which yields the restriction $\eta \in \left( 1/2,1\right) $. Our method still guarantees that the solution we obtain is positive.

\begin{equation*}\left\{ \begin{tabular}{l}

$x^{^{\prime \prime \prime }}(t)=\alpha \left( t\right) f(t,x(t),x^{\prime}\left( t\right) ,x^{\prime \prime }\left( t\right) ),\;\;\;0<t<1,$ \\

$x\left( 0\right) =x^{\prime }\left( \eta \right) =x^{\prime \prime }\left(1\right) =0,$

\end{tabular}\right.\end{equation*}

under positivity of the nonlinearity. Existence results for a positive and concave solution $x\left( t\right) ,\ 0\leq t\leq 1$ are given, for any $1/2<\eta <1.\ $ In addition, without any monotonicity assumption on the nonlinearity, we prove the existence of a sequence of such solutions with \begin{equation*} \lim_{n\rightarrow \infty }||x_{n}||=0. \end{equation*} Our principal tool is \emph{a very simple applications on a new cone of the plane} of the well-known Krasnosel'ski\u{\i}'s fixed point theorem. The main feature of this aproach is that, we do not use at all the associated Green's function, the necessary positivity of which yields the restriction $\eta \in \left( 1/2,1\right) $. Our method still guarantees that the solution we obtain is positive.

We suggest finding exact solutions of equation:

\begin{equation*}

\frac{\partial u}{\partial t}=(\frac{\partial ^{m}}{\partial x^{m}}u)^{p},

t\geq 0, x\in \mathbb{R}, m, p\in \mathbb{N}, p>1,

\end{equation*}

by a new method that we call the travelling profiles method. This method allows us to find several forms of exact solutions including the classical forms such as travelling-wave and self-similar solutions.

\begin{equation*}

\frac{\partial u}{\partial t}=(\frac{\partial ^{m}}{\partial x^{m}}u)^{p},

t\geq 0, x\in \mathbb{R}, m, p\in \mathbb{N}, p>1,

\end{equation*}

by a new method that we call the travelling profiles method. This method allows us to find several forms of exact solutions including the classical forms such as travelling-wave and self-similar solutions.

We are concerned here with the existence of monotonic and uniformly asymptotically stable solution of an initial-value problem of non-autonomous delay differential equations of arbitrary (fractional) orders.

The "freezing" method for ordinary differential equations is extended to the Volterra integral equations in a Banach space of the type $$ x(t)- \int_0^t K(t, t-s)x(s)ds =f(t)\;(t\geq 0),$$ where $K(t,s)$ is an operator valued function "slowly" varying in the first argument. Besides, sharp explicit stability conditions are derived.

In this paper, we study the existence of positive solutions for a nonlinear four-point boundary value problem with a $p$-Laplacian operator. By using a three functionals fixed point theorem in a cone, the existence of double positive solutions for the nonlinear four-point boundary value problem with a $p$-Laplacian operator is obtained. This is different than previous results.

In this paper, we obtain sufficient conditions so that every solution of

$$

\big(y(t)- \sum_{i=1}^n p_i(t) y(\delta_i(t))\big)'+\sum_{i=1}^m q_i(t) y(\sigma_i(t)) = f(t)

$$

oscillates or tends to zero as $t \to \infty$. Here the coefficients $p_i(t), q_i(t)$ and the forcing term $f(t)$ are allowed to oscillate; such oscillation condition in all coefficients is very rare in the literature. Furthermore, this paper provides an answer to the open problem 2.8.3 in [7, p. 57]. Suitable examples are included to illustrate our results.

$$

\big(y(t)- \sum_{i=1}^n p_i(t) y(\delta_i(t))\big)'+\sum_{i=1}^m q_i(t) y(\sigma_i(t)) = f(t)

$$

oscillates or tends to zero as $t \to \infty$. Here the coefficients $p_i(t), q_i(t)$ and the forcing term $f(t)$ are allowed to oscillate; such oscillation condition in all coefficients is very rare in the literature. Furthermore, this paper provides an answer to the open problem 2.8.3 in [7, p. 57]. Suitable examples are included to illustrate our results.

We establish the existence of radial solutions to the p-Laplacian equation $ \Delta_p u + f(u)=0 $ in $\mathbb {R^N}$, where $f$ behaves like $|u|^{q-1}u$ when $u$ is large and $f(u) < 0$ for small positive $u$. We show that for each nonnegative integer $n$, there is a localized solution $u$ which has exactly $n$ zeros.

We study some properties of bounded and $C^{(1)}$-almost automorphic solutions of the following Li\'enard equation: $$x'' + f(x)x' + g(x) = p(t) $$, where $p : {\bf R} \longrightarrow {\bf R}$ is an almost automorphic function, $f$, $g : (a,b) \longrightarrow {\bf R}$ are continuous functions and $g$ is strictly decreasing.

In this paper, we prove the existence of solutions of initial value problems for nth order nonlinear impulsive integro-differential equations of mixed type on an infinite interval with an infinite number of impulsive times in Banach spaces. Our results are obtained by introducing a suitable measure of noncompactness.

We demonstrate how the method of auxiliary ('reference') equations, also known as N. V. Azbelev's W-transform method, can be used to derive efficient conditions for the exponential Lyapunov stability of linear delay equations driven by a vector-valued Wiener process. For the sake of convenience the W-method is briefly outlined in the paper, its justification is however omitted. The paper contains a general stability result, which is specified in the last section in the form of seven corollaries providing sufficient stability conditions for some important classes of It\^{o} equations with delay.

In this paper, we investigate the existence of three positive solutions for the nonlinear fractional boundary value problem

$D_{0+}^{\alpha} u(t) + a(t)f(t,u(t), u^{\prime \prime}(t))=0, \quad 0 < t < 1, \quad 3 < \alpha \leq 4,$

$u(0) = u^{\prime}(0) = u^{\prime \prime}(0)= u^{\prime \prime}(1)=0 $,

where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative. The method involves applications of a new fixed-point theorem due to Bai and Ge. The interesting point lies in the fact that the nonlinear term is allowed to depend on the second order derivative $u^{\prime \prime}$.

$D_{0+}^{\alpha} u(t) + a(t)f(t,u(t), u^{\prime \prime}(t))=0, \quad 0 < t < 1, \quad 3 < \alpha \leq 4,$

$u(0) = u^{\prime}(0) = u^{\prime \prime}(0)= u^{\prime \prime}(1)=0 $,

where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative. The method involves applications of a new fixed-point theorem due to Bai and Ge. The interesting point lies in the fact that the nonlinear term is allowed to depend on the second order derivative $u^{\prime \prime}$.

We are concerned here with a nonlinear quadratic integral equation of Volterra type. The existence of at least one $L_1-$ positive solution will be proved under the Carath\`{e}odory condition. Secondly we will make a link between Peano condition and Carath\`{e}odory condition to prove the existence of at least one positive continuous solution. Finally the existence of the maximal and minimal solutions will be proved.

See also an addendum to this paper: EJQTDE, No. 51. 2009.

In this paper, we give a necessary and sufficient condition for the existence of $\Psi -$ bounded solutions for the nonhomogeneous linear difference equation x(n + 1) = A(n)x(n) + f(n) on $\mathbb{Z}$. In addition, we give a result in connection with the asymptotic behavior of the $\Psi -$ bounded solutions of this equation.

In this paper we shall study a neutral differential equation with deviated argument in an arbitrary Banach space $X.$ With the help of the analytic semigroups theory and fixed point method we establish the existence and uniqueness of solutions of the given problem. Finally, we give examples to illustrate the applications of the abstract results.

In this paper, existence criteria of three positive solutions to the followimg $p$-Laplacian functional dynamic equation on time scales

\[\left\{\begin{array}{l}

\left[ \Phi _p(u^{\bigtriangleup }(t))\right] ^{\bigtriangledown}+a(t)f(u(t),u(\mu (t)))=0,t\in \left(0,T\right),\\

u_0(t)=\varphi (t), t\in \left[ -r,0\right], u(0)-B_0(u^{\bigtriangleup }(\eta ))=0, u^{\bigtriangleup }(T)=0,

\end{array}\right.\]

are established by using the well-known Five Functionals Fixed Point Theorem.

\[\left\{\begin{array}{l}

\left[ \Phi _p(u^{\bigtriangleup }(t))\right] ^{\bigtriangledown}+a(t)f(u(t),u(\mu (t)))=0,t\in \left(0,T\right),\\

u_0(t)=\varphi (t), t\in \left[ -r,0\right], u(0)-B_0(u^{\bigtriangleup }(\eta ))=0, u^{\bigtriangleup }(T)=0,

\end{array}\right.\]

are established by using the well-known Five Functionals Fixed Point Theorem.

The topic of fractional calculus (integration and differentiation of fractional-order), which concerns singular integral and integro-differential operators, is enjoying interest among mathematicians, physicists and engineers. In this work, we investigate initial value problem of fractional-order differential equation with nonlocal condition. The stability (and some other properties concerning the existence and uniqueness) of the solution will be proved.

In this paper, by introducing the fractional derivative in the sense of Caputo, we apply the Adomian decomposition method for the foam drainage equation with time- and space-fractional derivative. As a result, numerical solutions are obtained in a form of rapidly convergent series with easily computable components.

In this paper, we present an impulsive version of Filippov's Theorem for the first-order nonresonance impulsive differential inclusion

$$

\begin{array}{rlll}

y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash

\{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $J=[0,b]$ and $F: J \times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m.$). Then the relaxed problem is considered and a Filippov-Wasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion

$$

\begin{array}{rlll}

y'(t) &\in& \varphi(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash

\{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $\varphi: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincar\'e operator.

$$

\begin{array}{rlll}

y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash

\{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $J=[0,b]$ and $F: J \times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m.$). Then the relaxed problem is considered and a Filippov-Wasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion

$$

\begin{array}{rlll}

y'(t) &\in& \varphi(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash

\{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $\varphi: J\times \mathbb{R}^n\to{\cal P}(\mathbb{R}^n)$ is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincar\'e operator.

This paper is concerned with the oscillatory behavior of the fourth-order nonlinear differential equation

$$

\bigl(p(t)|x^{\prime\prime}|^{\alpha-1}\,x^{\prime\prime}\bigr)^{\prime\prime}

+q(t)|x|^{\beta-1}x=0\,,\leqno{\rm(E)}

$$

where $\alpha>0$, $\beta>0$ are constants and $p,q:[a,\infty)\to(0,\infty)$ are continuous functions satisfying conditions

$$

\int_a^{\infty}\left( \frac{t}{p(t)}\right)^{\frac{1}{\alpha}}\,dt<\infty,

\int_a^{\infty}\frac{t}{\left(p(t)\right)^{\frac{1}{\alpha}}}\,dt<\infty .

$$

We will establish necessary and sufficient condition for oscillation of all solutions of the sub-half-linear equation (E) (for $\beta<\alpha$) as well as of the super-half-linear equation (E) (for $\beta>\alpha$).

$$

\bigl(p(t)|x^{\prime\prime}|^{\alpha-1}\,x^{\prime\prime}\bigr)^{\prime\prime}

+q(t)|x|^{\beta-1}x=0\,,\leqno{\rm(E)}

$$

where $\alpha>0$, $\beta>0$ are constants and $p,q:[a,\infty)\to(0,\infty)$ are continuous functions satisfying conditions

$$

\int_a^{\infty}\left( \frac{t}{p(t)}\right)^{\frac{1}{\alpha}}\,dt<\infty,

\int_a^{\infty}\frac{t}{\left(p(t)\right)^{\frac{1}{\alpha}}}\,dt<\infty .

$$

We will establish necessary and sufficient condition for oscillation of all solutions of the sub-half-linear equation (E) (for $\beta<\alpha$) as well as of the super-half-linear equation (E) (for $\beta>\alpha$).

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for two classes of first order semilinear functional and neutral functional differential evolution inclusions with infinite delay using a recent nonlinear alternative for contractive multivalued maps in Fr\'echet spaces due to Frigon, combined with semigroup theory.

We consider dynamics of the one-dimensional Mindlin-Timoshenko model for beams with a nonlinear external forces and a boundary damping mechanism. We investigate existence and uniqueness of strong and weak solution. We also study the boundary stabilization of the solution, i.e., we prove that the energy of every solution decays exponentially as $t\rightarrow\infty$.

The paper studies the notion of Stepanov almost periodicity (or $S^2$-almost periodicity) for stochastic processes, which is weaker than the notion of quadratic-mean almost periodicity. Next, we make extensive use of the so-called Acquistapace and Terreni conditions to prove the existence and uniqueness of a Stepanov (quadratic-mean) almost periodic solution to a class of nonautonomous stochastic evolution equations on a separable real Hilbert space. Our abstract results will then be applied to study Stepanov (quadratic-mean) almost periodic solutions to a class of $n$-dimensional stochastic parabolic partial differential equations.

By means of the Kartsatos technique and generalized Riccati transformation techniques, we establish some new oscillation criteria for a second order nonlinear dynamic equations with forced term on time scales in terms of the coefficients.

We prove a theorem on the existence of solutions of a second order differential inclusion governed by a class of nonconvex sweeping process with a mixed semicontinuous perturbation.

We look for radial solutions of a superlinear problem in a ball. We show that for if $n$ is a sufficiently large nonnegative integer, then there is a solution $u$ which has exactly $n$ interior zeros. In this paper we give an alternate proof to that which was given by Castro and Kurepa.

In this paper, we investigate the higher-order linear differential equations with meromorphic coefficients. We improve and extend a result of M.S. Liu and C.L. Yuan, by using the estimates for the logarithmic derivative of a transcendental meromorphic function due to Gundersen, and the extended Winman-Valiron theory which proved by J. Wang and H.X. Yi. In addition, we also consider the nonhomogeneous linear differential equations.

Generalized approximation technique for a solution of one-dimensional steady state heat transfer problem in a slab made of a material with temperature dependent thermal conductivity, is developed. The results obtained by the generalized approximation method (GAM) are compared with those studied via homotopy perturbation method (HPM). For this problem, the results obtained by the GAM are more accurate as compared to the HPM. Moreover, our (GAM) generate a sequence of solutions of linear problems that converges monotonically and rapidly to a solution of the original nonlinear problem. Each approximate solution is obtained as the solution of a linear problem. We present numerical simulations to illustrate and confirm the theoretical results.

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form

$$

\left\{\aligned &u''=-f(t,v), \ \ 0< t< 1,\\&v''=-g(t,u), \ \ 0< t< 1\\&u(0)=v(0)=0,\varsigma u(\zeta)=u(1),\varsigma v(\zeta)=v(1),\endaligned\right.

$$

where $f:(0,1)\times [0,+\infty)\to [0,+\infty),g:[0,1]\times [0,+\infty)\to [0,+\infty),0<\zeta<1, \varsigma>0,$ and $\varsigma\zeta< 1,f$ may be singular at $t = 0$ and/or $t = 1.$ Under some rather simple conditions, by means of monotone iterative technique, a necessary and sufficient condition for the existence of positive solutions is established, a result on the existence and uniqueness of the positive solution and the iterative sequence of solution is given.

$$

\left\{\aligned &u''=-f(t,v), \ \ 0< t< 1,\\&v''=-g(t,u), \ \ 0< t< 1\\&u(0)=v(0)=0,\varsigma u(\zeta)=u(1),\varsigma v(\zeta)=v(1),\endaligned\right.

$$

where $f:(0,1)\times [0,+\infty)\to [0,+\infty),g:[0,1]\times [0,+\infty)\to [0,+\infty),0<\zeta<1, \varsigma>0,$ and $\varsigma\zeta< 1,f$ may be singular at $t = 0$ and/or $t = 1.$ Under some rather simple conditions, by means of monotone iterative technique, a necessary and sufficient condition for the existence of positive solutions is established, a result on the existence and uniqueness of the positive solution and the iterative sequence of solution is given.

This paper deals with a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space. Using impulsive periodic evolution operator given by us, the suitable $T_{0}$-periodic $PC$-mild solution is introduced and $Poincar\acute{e}$ operator is constructed. Showing the compactness of $Poincar\acute{e}$ operator and using a new generalized Gronwall's inequality with impulse, mixed type integral operators and $B$-norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of $T_{0}$-periodic $PC$-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration.

In this paper we prove the existence of positive solutions for a class of second order semipositone singular three-point boundary value problems. The results are obtained by the use of a Guo-Krasnoselskii's fixed point theorem in cones.

The existence of solutions of a boundary value problem for a third order differential inclusion is investigated. New results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equations have been studied by J. Banas. Here we are concerned with a singular quadratic functional integral equations. The existence of positive monotonic solutions $x \in L_1[0,1]$ will be proved. The fractional order nonlinear functional differential equation will be given as a special case.

In this paper we study the stability of the zero solution of difference equations with variable delays. In particular we consider the scalar delay equation

\begin{equation}

\Delta x(n)=-a(n)x(n-\tau(n))

\end{equation}

and its generalization

\begin{equation}

\Delta x(n)=-\sum^{N}_{j=1}a_j(n)x(n-\tau_j(n)).

\end{equation}

Fixed point theorems are used in the analysis.

\begin{equation}

\Delta x(n)=-a(n)x(n-\tau(n))

\end{equation}

and its generalization

\begin{equation}

\Delta x(n)=-\sum^{N}_{j=1}a_j(n)x(n-\tau_j(n)).

\end{equation}

Fixed point theorems are used in the analysis.

We consider the existence of positive solutions and multiple positive solutions for the third order nonlinear differential equation subject to the general two-point boundary conditions using different fixed point theorems.

In this work, we are concerned with a class of elliptic problems with both absorption terms and critical growth in the gradient. We suppose that the data belong to $L^{m}(\Omega)$ with $m>n/2$ and we prove the existence of bounded weak solutions via $L^{\infty}$-estimates. A priori estimates and Stampacchia's $L^{\infty}$-regularity are our main ingredient.

In this paper, we are concerned with the existence of solutions for second order impulsive anti-periodic boundary value problem

${ \left \{\begin{array} {l}

u''(t) + f(t,u(t),u'(t))=0, \quad t \not= t_k, \ t \in [0, T], \\

\triangle u(t_k) = I_k(u(t_k)), \quad k = 1, \cdots , m, \\

\triangle u'(t_k) = I_k^*(u(t_k)), \quad k = 1, \cdots , m, \\

u(0) + u(T) = 0, \ u'(0) + u'(T) = 0.

\end{array}\right.} $

New criteria are established based on Schaefer's fixed-point theorem.

${ \left \{\begin{array} {l}

u''(t) + f(t,u(t),u'(t))=0, \quad t \not= t_k, \ t \in [0, T], \\

\triangle u(t_k) = I_k(u(t_k)), \quad k = 1, \cdots , m, \\

\triangle u'(t_k) = I_k^*(u(t_k)), \quad k = 1, \cdots , m, \\

u(0) + u(T) = 0, \ u'(0) + u'(T) = 0.

\end{array}\right.} $

New criteria are established based on Schaefer's fixed-point theorem.

We prove the existence of solutions for the differential inclusion $\dot x(t)\in F(t,x(t)) + f(t,x(t))$ for a multifunction $F$ upper semicontinuous with compact values contained in the generalized Clarke gradient of a regular locally Lipschitz function and $f$ a Carath\'{e}odory function.

In this note we apply Avery-Peterson multiple fixed point theorem to investigate the existence of multiple positive periodic solutions to the following nonlinear non-autonomous functional differential system with feedback control

\[\left\{\begin{array}{l}

\frac{dx}{dt}=-r(t)x(t)+F(t,x_t,u(t-\delta(t))),\\

\frac{du}{dt}=-h(t)u(t)+g(t)x(t-\sigma(t)).\\

\end{array}\right.\]

We prove the system above admits at least three positive periodic solutions under certain growth conditions imposed on $F$.

\[\left\{\begin{array}{l}

\frac{dx}{dt}=-r(t)x(t)+F(t,x_t,u(t-\delta(t))),\\

\frac{du}{dt}=-h(t)u(t)+g(t)x(t-\sigma(t)).\\

\end{array}\right.\]

We prove the system above admits at least three positive periodic solutions under certain growth conditions imposed on $F$.

In this paper, we consider the existence of triple positive solutions to the boundary value problem of nonlinear delay differential equation

$$

\left\{ \begin{array}{lll}

(\phi(x'(t)))^{\prime} + a(t)f(t,x(t),x'(t),x_{t})=0, \ \ 0 < t<1,\\

x_{0}=0,\\

x(1)=0,

\end{array}\right.

$$

where $\phi: \R \rightarrow \R$ is an increasing homeomorphism and positive homomorphism with $\phi(0)=0,$ and $x_t$ is a function in $C([-\tau,0],\R)$ defined by $x_{t}(\sigma)=x(t+\sigma)$ for $ -\tau \leq \sigma\leq 0.$ By using a fixed-point theorem in a cone introduced by Avery and Peterson, we provide sufficient conditions for the existence of triple positive solutions to the above boundary value problem. An example is also presented to demonstrate our result. The conclusions in this paper essentially extend and improve the known results.

$$

\left\{ \begin{array}{lll}

(\phi(x'(t)))^{\prime} + a(t)f(t,x(t),x'(t),x_{t})=0, \ \ 0 < t<1,\\

x_{0}=0,\\

x(1)=0,

\end{array}\right.

$$

where $\phi: \R \rightarrow \R$ is an increasing homeomorphism and positive homomorphism with $\phi(0)=0,$ and $x_t$ is a function in $C([-\tau,0],\R)$ defined by $x_{t}(\sigma)=x(t+\sigma)$ for $ -\tau \leq \sigma\leq 0.$ By using a fixed-point theorem in a cone introduced by Avery and Peterson, we provide sufficient conditions for the existence of triple positive solutions to the above boundary value problem. An example is also presented to demonstrate our result. The conclusions in this paper essentially extend and improve the known results.

In this paper, we prove the existence of solutions for the strongly nonlinear equation of the type $$ Au+g(x,u)=f $$ where $A$ is an elliptic operator of infinite order from a functional space of Sobolev type to its dual. $g(x,s)$ is a lower order term satisfying essentially a sign condition on $s$ and the second term $f$ belongs to $L^1(\Omega).$

In this study, the behavior of solutions to certain second order quantum ($q$-difference) equations with maxima are considered. In particular, the asymptotic behavior of non-oscillatory solutions is described, and sufficient conditions for oscillation of all solutions are obtained.

It is proved a necessary and sufficient condition for the existence of at least one $Psi-$bounded solution of a linear nonhomogeneous Lyapunov matrix differential equation.

In this paper we study the generalized Fucik type eigenvalue for the boundary value problem of one dimensional $p-$Laplace type differential equations

\begin{displaymath}\left\{

\begin{array}{lll} - (\varphi( u')) ' = \psi(u), \quad -T< x < T; \\

\quad u(-T)=0, \quad u(T)=0 \\

\end{array} \right.\eqno(*)

\end{displaymath}

where $\varphi (s) = \alpha s_+^{p-1} -\beta s_-^{p-1}, \psi (s) = \lambda s_+^{p-1} -\mu s_-^{p-1}, p >1.$ We obtain a explicit characterization of Fucik spectrum $(\alpha, \beta, \lambda, \mu),$ i.e., for which the (*) has a nontrivial solution.

\begin{displaymath}\left\{

\begin{array}{lll} - (\varphi( u')) ' = \psi(u), \quad -T< x < T; \\

\quad u(-T)=0, \quad u(T)=0 \\

\end{array} \right.\eqno(*)

\end{displaymath}

where $\varphi (s) = \alpha s_+^{p-1} -\beta s_-^{p-1}, \psi (s) = \lambda s_+^{p-1} -\mu s_-^{p-1}, p >1.$ We obtain a explicit characterization of Fucik spectrum $(\alpha, \beta, \lambda, \mu),$ i.e., for which the (*) has a nontrivial solution.

This paper investigates the solvability of the second-order boundary value problems with the one-dimensional $p$-Laplacian at resonance on a half-line

$$

\left\{\begin{array}{llll}

(c(t)\phi_{p}(x'(t)))'=f(t,x(t),x'(t)),~~~~0<t<\infty,\\

x(0)=\sum\limits_{i=1}\limits^{n}\mu_ix(\xi_{i}),

~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0

\end{array}\right.

$$

and

$$\left\{\begin{array}{llll}

(c(t)\phi_{p}(x'(t)))'+g(t)h(t,x(t),x'(t))=0,~~~~0<t<\infty,\\

x(0)=\int_{0}^{\infty}g(s)x(s)ds,~~\lim\limits_{t\rightarrow

+\infty}c(t)\phi_{p}(x'(t))=0

\end{array}\right.

$$

with multi-point and integral boundary conditions, respectively, where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results.

$$

\left\{\begin{array}{llll}

(c(t)\phi_{p}(x'(t)))'=f(t,x(t),x'(t)),~~~~0<t<\infty,\\

x(0)=\sum\limits_{i=1}\limits^{n}\mu_ix(\xi_{i}),

~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0

\end{array}\right.

$$

and

$$\left\{\begin{array}{llll}

(c(t)\phi_{p}(x'(t)))'+g(t)h(t,x(t),x'(t))=0,~~~~0<t<\infty,\\

x(0)=\int_{0}^{\infty}g(s)x(s)ds,~~\lim\limits_{t\rightarrow

+\infty}c(t)\phi_{p}(x'(t))=0

\end{array}\right.

$$

with multi-point and integral boundary conditions, respectively, where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results.

In this paper we investigate the formulation of a class of boundary value problems of fractional order with the Riemann-Liouville fractional derivative and integral-type boundary conditions. The existence of solutions is established by applying a fixed point theorem of Krasnosel'skii and Zabreiko for asymptotically linear mappings.

In this paper, an existence and the existence of extremal random solutions are proved for a periodic boundary value problem of second order ordinary random differential equations. Our investigations have been placed in the space of real-valued functions defined and continuous on closed and bounded intervals of real line together with the applications of the random version of a nonlinear alternative of Leray-Schauder type and an algebraic random fixed point theorem of Dhage. An example is also indicated for demonstrating the realizations of the abstract theory developed in this paper.

In this paper, we investigate the relationship between small functions and differential polynomials $g_{f}\left( z\right)=d_{2}f^{^{\prime \prime }} + d_{1}f^{^{\prime }}+d_{0}f$, where $d_{0}\left(z\right), d_{1}\left( z\right), d_{2}\left( z\right) $ are meromorphic functions that are not all equal to zero with finite order generated by solutions of the second order linear differential equation

\begin{equation*}

f^{^{\prime \prime }}+Af^{^{\prime }}+Bf=F,

\end{equation*}

where $A,$ $B,$ $F\not\equiv 0$ are finite order meromorphic functions having only finitely many poles.

\begin{equation*}

f^{^{\prime \prime }}+Af^{^{\prime }}+Bf=F,

\end{equation*}

where $A,$ $B,$ $F\not\equiv 0$ are finite order meromorphic functions having only finitely many poles.

In this paper, lower bounds for the spacing $(b - a)$ of the zeros of the solutions and the zeros of the derivative of the solutions of third order differential equations of the form \[y''' + q(t) y' + p(t)y = 0 (*)\] are derived under the some assumptions on $p$ and $q$. The concept of disfocality is introduced for third order differential equations (*). This helps to improve the Liapunov-type inequality, when y(t) is a solution of (*) with (i) $ y(a) = 0 = y'(b)$ or $ y'(a) = 0 = y(b) $ with $ y(t) \ne 0, t \in (a,b) $ or (ii) $ y(a) = 0 = y'(a), y(b) = 0 = y'(b)$ with $ y(t)\ne 0, t\in (a,b)$. If y(t) is a solution of (*) with $ y(t_{i}) = 0, 1 \le i \le n, n\ge 4, (t_{1} <t_{2} < ...< t_{n} )$ and $ y(t) \neq 0, t \in \bigcup_{i = 1}^{i =n - 1} (t_{i}, t_{i+1})$, then lower bound for spacing $(t_{n}-t_{1})$ is obtained. A new criteria for disconjugacy is obtained for (*) in $[a,b]$.This papers improves many known bounds in the literature.

In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions

$$

\begin{array}{rlll}

y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in

J\backslash \{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1, 2, \ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $J=[0,b]$ and $F: J\times \R^n\to{\cal P}(\R^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, 2, \ldots,m$). The topological structure of solution sets as well as some of their geometric properties (contractibility and $R_\delta$-sets) are studied. A continuous version of Filippov's theorem is also proved.

$$

\begin{array}{rlll}

y'(t)-\lambda y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in

J\backslash \{t_{1},\ldots,t_{m}\},\\

y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1, 2, \ldots,m,\\

y(0)&=&y(b),

\end{array}

$$

where $J=[0,b]$ and $F: J\times \R^n\to{\cal P}(\R^n)$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1, 2, \ldots,m$). The topological structure of solution sets as well as some of their geometric properties (contractibility and $R_\delta$-sets) are studied. A continuous version of Filippov's theorem is also proved.

We carry out spectral analysis of one class of integral operators associated with fractional order differential equations that arises in mechanics. We establish a connection between the eigenvalues of these operators and the zeros of Mittag-Leffler type functions. We give sufficient conditions for complete nonselfadjointness and completeness the systems of the eigenvalues.

In this article, we study the existence of mild solutions for a class of impulsive abstract partial neutral functional differential equations with state-dependent delay. The results are obtained by using Leray-Schauder Alternative fixed point theorem. Example is provided to illustrate the main result.

In this paper, we are interested in the study of the nonexistence of nontrivial solutions for a class of partial differential equations, in unbounded domains. This leads us to extend these results to systems of m equations. The method used is based on energy type identities.

In this paper, we study the existence of solutions for a two-point boundary value problem of fractional semilinear evolution equations in a Banach space. Our results are based on the contraction mapping principle and Krasnoselskii's fixed point theorem.

In this paper we study the existence and regularity of mild solutions for a class of abstract partial neutral integro-differential equations with unbounded delay.

The method of lower and upper solutions for fractional differential equation $D^\delta u(t)+g(t,u(t))=0, t\in (0,1), 1<\delta\leq 2$, with Dirichlet boundary condition $u(0)=a, u(1)=b$ is used to give sufficient conditions for the existence of at least one solution.

See also: EJQTDE, No. 55. (2012)

In this paper, we study the existence of positive solutions of boundary value problems for systems of second-order differential equations with integral boundary condition on the half-line. By using the fixed-point theorem in cones, we show the existence of at least one positive solution with suitable growth conditions imposed on the nonlinear terms. Moreover, the associated integral kernels for the boundary value problems are given.

In this paper, we establish the existence of at least three positive solutions for the system of higher order boundary value problems on time scales by using the well-known Leggett-Williams fixed point theorem. And then, we prove the existence of at least 2k-1 positive solutions for arbitrary positive integer k.

In this paper we prove the global bifurcation theorem for the nonlinear Picard problem. The right-hand side function $\varphi$ is a Caratheodory map, not differentiable at zero, but behaving in the neighbourhood of zero as specified in details below. We prove that in some interval $[a,b]\subset\real$ the Leray\--Schauder degree changes, hence there exists the global bifurcation branch. Later, by means of some approximation techniques, we prove that there exist at least two such branches.

In this paper, we present a criterion on the oscillation of unbounded solutions for higher-order dynamic equations of the following form:

\begin{equation}

\big[x(t)+A(t)x(\alpha(t))\big]^{\Delta^{n}}+B(t)F(x(\beta(t)))=\varphi(t)\qquad\text{for}\ t\in[t_{0},\infty)_{\T},\label{asbeq1}\tag{$\star$}

\end{equation}

where $n\in[2,\infty)_{\Z}$, $t_{0}\in\T$, $\sup\{\T\}=\infty$, $A\in\crd([t_{0},\infty)_{\T},\R)$ is allowed to alternate in sign infinitely many times, $B\in\crd([t_{0},\infty)_{\T},\R^{+})$, $F\in\crd(\R,\R)$ is nondecreasing, and $\alpha,\beta\in\crd([t_{0},\infty)_{\T},\T)$ are unbounded increasing functions satisfying $\alpha(t),\beta(t)\leq t$ for all sufficiently large $t$. We give change of order formula for double(iterated) integrals to prove our main result. Some simple examples are given to illustrate the applicability of our results too. In the literature, almost all of the results for \eqref{asbeq1} with $\T=\R$ and $\T=\Z$ hold for bounded solutions. Our results are new and not stated in the literature even for the particular cases $\T=\R$ and/or $\T=\Z$.

\begin{equation}

\big[x(t)+A(t)x(\alpha(t))\big]^{\Delta^{n}}+B(t)F(x(\beta(t)))=\varphi(t)\qquad\text{for}\ t\in[t_{0},\infty)_{\T},\label{asbeq1}\tag{$\star$}

\end{equation}

where $n\in[2,\infty)_{\Z}$, $t_{0}\in\T$, $\sup\{\T\}=\infty$, $A\in\crd([t_{0},\infty)_{\T},\R)$ is allowed to alternate in sign infinitely many times, $B\in\crd([t_{0},\infty)_{\T},\R^{+})$, $F\in\crd(\R,\R)$ is nondecreasing, and $\alpha,\beta\in\crd([t_{0},\infty)_{\T},\T)$ are unbounded increasing functions satisfying $\alpha(t),\beta(t)\leq t$ for all sufficiently large $t$. We give change of order formula for double(iterated) integrals to prove our main result. Some simple examples are given to illustrate the applicability of our results too. In the literature, almost all of the results for \eqref{asbeq1} with $\T=\R$ and $\T=\Z$ hold for bounded solutions. Our results are new and not stated in the literature even for the particular cases $\T=\R$ and/or $\T=\Z$.

In this paper, we study the existence of triple positive solutions for second-order four-point boundary value problem with sign changing nonlinearities. We first study the associated Green's function and obtain some useful properties. Our main tool is the fixed point theorem due to Avery and Peterson. The results of this paper are new and extent previously known results.

We establish new efficient conditions for the unique solvability of a non-local boundary value problem for first-order linear functional differential equations. Differential equations with argument deviations are also considered in which case further results are obtained. The results obtained reduce to those well-known for the ordinary differential equations.

The time dependent $2$-periodic system

\begin{equation*}

\dot x{(t)} = A(t)x{(t)} , \ t\in \mathbb{R}, \ \ x(t) \in\mathbb{C}^{n}\eqno{(A(t))}

\end{equation*}

is uniformly exponentially stable if and only if for each real number $\mu$ and each $2$-periodic, $\mathbb{C}^{n}$-valued function $f,$ the solution of the Cauchy Problem

\begin{equation*}

\left\{\begin{split}

\dot y{(t)} &= A(t) y{(t)} + e^{i \mu t}f(t),\ \ t\in \mathbb{R}_+, \ y(t) \in \mathbb{C}^{n} \\

y(0) &= 0

\end{split}\right.

\end{equation*}

is bounded. In this note we prove a result that has the above result as an immediate corollary. Some new characterizations for uniform exponential stability of $(A(t))$ in terms of the Datko type theorems are also obtained as corollaries.

\begin{equation*}

\dot x{(t)} = A(t)x{(t)} , \ t\in \mathbb{R}, \ \ x(t) \in\mathbb{C}^{n}\eqno{(A(t))}

\end{equation*}

is uniformly exponentially stable if and only if for each real number $\mu$ and each $2$-periodic, $\mathbb{C}^{n}$-valued function $f,$ the solution of the Cauchy Problem

\begin{equation*}

\left\{\begin{split}

\dot y{(t)} &= A(t) y{(t)} + e^{i \mu t}f(t),\ \ t\in \mathbb{R}_+, \ y(t) \in \mathbb{C}^{n} \\

y(0) &= 0

\end{split}\right.

\end{equation*}

is bounded. In this note we prove a result that has the above result as an immediate corollary. Some new characterizations for uniform exponential stability of $(A(t))$ in terms of the Datko type theorems are also obtained as corollaries.

In this paper, we investigate the existence of positive solutions for a class of singular $n$th-order three-point boundary value problem. The associated Green's function for the boundary value problem is given at first, and some useful properties of the Green's function are obtained. The main tool is fixed-point index theory. The results obtained in this paper essentially improve and generalize some well-known results.

We consider a wave equation in a bounded domain with nonlinear dissipation and nonlinear source term. Characterizations with respect to qualitative properties of the solution: globality, boundedness, blow-up, convergence up to a subsequence towards the equilibria and exponential stability are given in this article.

A multi-point boundary value problem involving the one dimensional $p$-Laplacian and depending on a parameter is studied in this paper and existence of positive solutions is established by means of a fixed point theorem for operators defined on Banach spaces with cones.

In this paper, we are concerned with the solvability for a class of second order nonlinear impulsive boundary value problem. New criteria are established based on Schaefer's fixed-point theorem. An example is presented to illustrate our main result. Our results essentially extend and complement some previous known results.

In this paper, we deal with the following singular four-point boundary value problem with $p$-Laplacian

$$

\left\{\begin{aligned}

&(\phi_{p}(u'(t)))'+q(t)f(t,u(t))=0,\ t\in(0,1),\\

&u(0)-\alpha u'(\xi)=0,\ u(1)+\beta u'(\eta)=0,

\end{aligned}\right.

$$

where $f(t,u)$ may be singular at $u=0$ and $q(t)$ may be singular at $t=0$ or $1$. By imposing some suitable conditions on the nonlinear term $f$, existence results of at least two positive solutions are obtained. The proof is based upon theory of Leray-Schauder degree and Krasnosel'skii's fixed point theorem.

$$

\left\{\begin{aligned}

&(\phi_{p}(u'(t)))'+q(t)f(t,u(t))=0,\ t\in(0,1),\\

&u(0)-\alpha u'(\xi)=0,\ u(1)+\beta u'(\eta)=0,

\end{aligned}\right.

$$

where $f(t,u)$ may be singular at $u=0$ and $q(t)$ may be singular at $t=0$ or $1$. By imposing some suitable conditions on the nonlinear term $f$, existence results of at least two positive solutions are obtained. The proof is based upon theory of Leray-Schauder degree and Krasnosel'skii's fixed point theorem.

Using the theory of fixed point theorem in cone, this paper presents the existence of positive solutions for the singular $m$-point boundary value problem

$$

\left\{\begin{array}{ll}

x''(t)+a(t)f(t,x(t),x'(t))=0,0<t<1,\\

x'(0)=0,\ \ x(1)=\dis \sum_{i=1}^{m-2}\alpha_{i}x(\xi_{i}),

\end{array}\right.

$$

where $0<\xi_{1}<\xi_{2}<\cdots<\xi_{m-2}<1, \alpha_{i}\in [0,1)$, $i = 1, 2, \cdots m-2$ , with $0<\sum_{i=1}^{m-2}\alpha_{i}<1$ and $f$ may change sign and may be singular at $x=0$ and $x'=0$.

$$

\left\{\begin{array}{ll}

x''(t)+a(t)f(t,x(t),x'(t))=0,0<t<1,\\

x'(0)=0,\ \ x(1)=\dis \sum_{i=1}^{m-2}\alpha_{i}x(\xi_{i}),

\end{array}\right.

$$

where $0<\xi_{1}<\xi_{2}<\cdots<\xi_{m-2}<1, \alpha_{i}\in [0,1)$, $i = 1, 2, \cdots m-2$ , with $0<\sum_{i=1}^{m-2}\alpha_{i}<1$ and $f$ may change sign and may be singular at $x=0$ and $x'=0$.

In this study, we investigate the question of nonexistence of nontrivial solutions of the Robin problem

\begin{equation}

\left\vert\begin{array}{l}

-\dfrac{\partial ^{2}u}{\partial x^{2}}-\sum\limits_{s=1}^{n}\dfrac{\partial}{\partial y_{s}}a_{s}(y,\frac{\partial u}{\partial y_{s}})+f(y,u)=0\text{in }\Omega =\mathbb{R}\times D, \\ \\

u+\varepsilon \dfrac{\partial u}{\partial n}=0\text{ on }\mathbb{R}\times \partial D.

\end{array}\right. \tag*{$\left( P\right) $}

\end{equation}

where $a_{s}:D\times \mathbb{R}\rightarrow \mathbb{R}$ are $H^{1}$-functions with constant sign such that

\begin{equation}\begin{array}{c}

2\int\limits_{0}^{\xi _{s}}a_{s}(y,t_{s})dt_{s}-\xi _{s}a_{s}(y,\xi_{s})\leq 0,s=1,...,n

\end{array}\tag*{$\left( H_{1}\right) $}\end{equation}

and $f:D\times \mathbb{R}\rightarrow \mathbb{R}$ is a real continuous locally Liptschitz function such that

\begin{equation} 2F(y,u)-uf(y,u)\leq 0, \tag*{$\left( H_{2}\right) $} \end{equation}.

We show that the function \begin{equation*} E(x)=\int\limits_{D}\left\vert u(x,y)\right\vert ^{2}dy \end{equation*} is convex on $\mathbb{R}$ . Our proof is based on energy (integral) identities.

\begin{equation}

\left\vert\begin{array}{l}

-\dfrac{\partial ^{2}u}{\partial x^{2}}-\sum\limits_{s=1}^{n}\dfrac{\partial}{\partial y_{s}}a_{s}(y,\frac{\partial u}{\partial y_{s}})+f(y,u)=0\text{in }\Omega =\mathbb{R}\times D, \\ \\

u+\varepsilon \dfrac{\partial u}{\partial n}=0\text{ on }\mathbb{R}\times \partial D.

\end{array}\right. \tag*{$\left( P\right) $}

\end{equation}

where $a_{s}:D\times \mathbb{R}\rightarrow \mathbb{R}$ are $H^{1}$-functions with constant sign such that

\begin{equation}\begin{array}{c}

2\int\limits_{0}^{\xi _{s}}a_{s}(y,t_{s})dt_{s}-\xi _{s}a_{s}(y,\xi_{s})\leq 0,s=1,...,n

\end{array}\tag*{$\left( H_{1}\right) $}\end{equation}

and $f:D\times \mathbb{R}\rightarrow \mathbb{R}$ is a real continuous locally Liptschitz function such that

\begin{equation} 2F(y,u)-uf(y,u)\leq 0, \tag*{$\left( H_{2}\right) $} \end{equation}.

We show that the function \begin{equation*} E(x)=\int\limits_{D}\left\vert u(x,y)\right\vert ^{2}dy \end{equation*} is convex on $\mathbb{R}$ . Our proof is based on energy (integral) identities.

In this paper, a new approach to the existence of time optimal controls of system governed by nonlinear equations on Banach spaces is provided. A sequence of Meyer problems is constructed to approach a class of time optimal control problems. A deep relationship between time optimal control problems and Meyer problems is presented. The method is much different from standard methods.

By using the concept of integrable dichotomy, the fixed point theory, functional analysis methods and some new technique of analysis, we obtain new criteria for the existence and uniqueness of bounded and periodic solutions of general and periodic systems of nonlinear integro-differential equations with infinite delay.

This paper is concerned with semilinear differential equations with nonlocal conditions in Banach spaces. Using the tools involving the measure of noncompactness and fixed point theory, existence of mild solutions is obtained without the assumption of compactness or equicontinuity on the associated linear semigroup.

In this paper we consider the question of the existence of fixed points of the derivatives of solutions of complex linear differential equations in the unit disc. This work improves some very recent results of T.-B. Cao.

This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: $u_{1t}=f_1(u_2)(\Delta u_1+a_1u_1),\cdots, u_{(n-1)t}=f_{n-1}(u_n)(\Delta u_{n-1}+a_{n-1} u_{n-1}),\ u_{nt}=f_n(u_1)(\Delta u_n+a_nu_n)$ with homogeneous Dirichlet boundary condition and positive initial condition, where $a_i\ (i=1,2,\cdots,n)$ are positive constants and $f_i\(i=1,2,\cdots,n)$ satisfy some conditions. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that: (i) when $\min\{a_1,\cdots,\ a_n\}\leq\lambda_1$ then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm; (ii) when $\min\{a_1,\cdots,\ a_n\}>\lambda_1$, and the initial datum $(u_{10},\cdots,\ u_{n0})$ satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where $\lambda_1$ is the first eigenvalue of $-\Delta$ in $\Omega$ with homogeneous Dirichlet boundary condition.

In this paper we study the existence of global solutions for a class of abstract functional differential equation with nonlocal conditions. An application is considered.

This addendum concerns the paper of the above title found in EJQTDE No. 25 (2008). There are some misprints in that paper:

(i) Page 3, line 5 should be $k:[0,1] \times[0,1]\rightarrow R_+$ satisfies Carath\'{e}odory condition (i.e. measurable in $t$ for all $s \in [0,1]$ and continuous in $s$ for all $t\in [0,1]$) such that $\int_0^1 k(t,s) m_2(s)ds$ is bounded $\forall t\in[0,1].$

(ii) Page 6, line 6 should be $k:[0,1] \times [0,1]\rightarrow R_+$ satisfies Carath\'{e}odory condition (i.e. measurable in $s$ for all $t \in~[0,1]$ and continuous in $t$ for all $s \in [0,1] $) such that $k(t,s)m_2(s)\in L_1 \forall t\in[0,1].$

(i) Page 3, line 5 should be $k:[0,1] \times[0,1]\rightarrow R_+$ satisfies Carath\'{e}odory condition (i.e. measurable in $t$ for all $s \in [0,1]$ and continuous in $s$ for all $t\in [0,1]$) such that $\int_0^1 k(t,s) m_2(s)ds$ is bounded $\forall t\in[0,1].$

(ii) Page 6, line 6 should be $k:[0,1] \times [0,1]\rightarrow R_+$ satisfies Carath\'{e}odory condition (i.e. measurable in $s$ for all $t \in~[0,1]$ and continuous in $t$ for all $s \in [0,1] $) such that $k(t,s)m_2(s)\in L_1 \forall t\in[0,1].$

See also: EJQTDE, No. 25. (2008)

We develop a generalized approximation method (GAM) to obtain solution of a steady state one-dimensional nonlinear convective-radiative-conduction equation. The GAM generates a bounded monotone sequence of solutions of linear problems. The sequence of approximants converges monotonically and rapidly to a solution of the original problem. We present some numerical simulation to illustrate and confirm our results.

We provide an existence result for a Neumann nonlinear boundary value problem posed on the half-line. Our main tool is the multi-valued version of the Miranda Theorem.

We prove necessary and sufficient conditions for $\Psi -$ (uniform) stability of the trivial solution of a nonlinear Lyapunov matrix differential equation.

This paper examines a three point, nonlocal boundary value problem for a second order ordinary differential equation. We make use of a generalization of the fixed point theorem of compression and expansion of functional type to obtain the existence of positive solutions.

In this work, we are concerned with the existence of positive solutions for a $\phi$ Laplacian boundary value problem on the half-line. The results are proved using the fixed point index theory on cones of Banach spaces and the upper and lower solution technique. The nonlinearity may exhibit a singularity at the origin with respect to the solution. This singularity is treated by regularization and approximation together with compactness and sequential arguments.

Problem of the type $-\Delta_{p}u=f(u)+h(x) \textrm{ in } (a, b) $ with $u=0$ on $ \{a,b\} $ is solved under nonresonance conditions stated with respect to the first eigenvalue and the first curve in the Fu\v{c}ik spectrum of $(-\Delta_{p},W_{0}^{1,p}(a,b))$, only on a primitive of $f$.

We consider the nonlinear Sturm-Liouville problem

$$

-u''(t) + f(u(t), u'(t)) = \lambda u(t),

\enskip u(t) > 0,

\quad t \in I := (-1/2, 1/2), \enskip u(\pm 1/2) = 0,

$$

where $f(x, y) = \vert x\vert^{p-1}x - \vert y\vert^m$, $p > 1, 1 \le m < 2$ are constants and $\lambda > 0$ is an eigenvalue parameter. To understand well the global structure of the bifurcation branch of positive solutions in $\mbox{\bf R}_+ \times L^q(I)$ ($1 \le q < \infty$) from a viewpoint of inverse problems, we establish the precise asymptotic formulas for the eigenvalue $\lambda = \lambda_q(\alpha)$ as $\alpha :=\Vert u_\lambda\Vert_q \to \infty$, where $u_\lambda$ is a solution associated with given $\lambda > \pi^2$.

$$

-u''(t) + f(u(t), u'(t)) = \lambda u(t),

\enskip u(t) > 0,

\quad t \in I := (-1/2, 1/2), \enskip u(\pm 1/2) = 0,

$$

where $f(x, y) = \vert x\vert^{p-1}x - \vert y\vert^m$, $p > 1, 1 \le m < 2$ are constants and $\lambda > 0$ is an eigenvalue parameter. To understand well the global structure of the bifurcation branch of positive solutions in $\mbox{\bf R}_+ \times L^q(I)$ ($1 \le q < \infty$) from a viewpoint of inverse problems, we establish the precise asymptotic formulas for the eigenvalue $\lambda = \lambda_q(\alpha)$ as $\alpha :=\Vert u_\lambda\Vert_q \to \infty$, where $u_\lambda$ is a solution associated with given $\lambda > \pi^2$.

Unimprovable effective efficient conditions are established for the unique solvability of the periodic problem

$$

\begin{aligned}

u'_i (t)&=\sum\limits_{j=2}^{i+1} \ell_{i,j}(u_{j})(t) + q_i(t) \qquad \text{for} \quad 1 \leq i\leq n-1,\\

u'_n (t)&=\sum\limits_{j=1}^{n} \ell_{n,j}(u_{j} )(t) + q_n(t),\\

u_j (0)& = u_j (\omega) \qquad \text{for} \quad 1 \leq j\leq n,

\end{aligned}

$$

where $\omega >0$, $\ell_{ij}:C([0,\omega])\to L([0,\omega])$ are linear bounded operators, and

$q_i \in L([0,\omega])$.

$$

\begin{aligned}

u'_i (t)&=\sum\limits_{j=2}^{i+1} \ell_{i,j}(u_{j})(t) + q_i(t) \qquad \text{for} \quad 1 \leq i\leq n-1,\\

u'_n (t)&=\sum\limits_{j=1}^{n} \ell_{n,j}(u_{j} )(t) + q_n(t),\\

u_j (0)& = u_j (\omega) \qquad \text{for} \quad 1 \leq j\leq n,

\end{aligned}

$$

where $\omega >0$, $\ell_{ij}:C([0,\omega])\to L([0,\omega])$ are linear bounded operators, and

$q_i \in L([0,\omega])$.

In this paper, by defining a class of functions, we establish some oscillation criteria for the second order nonlinear dynamic equations with forced term $$ x^{\Delta\Delta}(t)+a(t)f(x(q(t)))=e(t) $$ on a time scale $\mathbb{T}.$ Our results unify the oscillation of the second order forced differential equation and the second order forced difference equation. An example is considered to illustrate the main results.

In this paper, by using the generalized Riccati technique and the integral averaging technique, some new oscillation criteria for certain second order retarded differential equation of the form

\begin{equation*}

\left( r\left( t\right) \left\vert u^{\prime }\left( t\right) \right\vert

^{\alpha -1}u^{\prime }\left( t\right) \right) ^{\prime }+p\left( t\right)

f\left( u\left( \tau \left( t\right) \right) \right) =0

\end{equation*}

are established. The results obtained essentially improve known results in the literature and can be applied to the well known half-linear and Emden-Fowler type equations.

\begin{equation*}

\left( r\left( t\right) \left\vert u^{\prime }\left( t\right) \right\vert

^{\alpha -1}u^{\prime }\left( t\right) \right) ^{\prime }+p\left( t\right)

f\left( u\left( \tau \left( t\right) \right) \right) =0

\end{equation*}

are established. The results obtained essentially improve known results in the literature and can be applied to the well known half-linear and Emden-Fowler type equations.

In this paper we provide necesssary and sufficient conditions for the existence of at least one $\Psi$-bounded solution on $\mathbb{R}$ for the system $X'=A(t)X +XB(t)+F(t)$, where $F(t)$ is a Lebesgue $\Psi$-integrable matrix valued function on $\mathbb{R}$. Further, we prove a result relating to the asymptotic behavior of the $\Psi$-bounded solutions of this system.

In this paper, we deal with the order of growth and the hyper order of solutions of higher order linear differential equations $$f^{(k)}+B_{k-1}f^{(k-1)}+\cdots+B_1f'+B_0f=F$$ where $B_j(z) (j=0,1,\ldots,k-1)$ and $F$ are entire functions or polynomials. Some results are obtained which improve and extend previous results given by Z.-X. Chen, J. Wang, T.-B. Cao and C.-H. Li.

The authors consider the fourth order quasilinear difference equation $$\Delta^{2}\left(p_{n}|\Delta^{2}x_n|^{\alpha-1}\Delta^{2}x_n\right)+q_{n}|x_{n+3}|^{\beta -1}x_{n+3}=0,$$ where $\alpha$ and $\beta$ are positive constants, and ${\{p_{n}\}}$ and ${\{q_{n}\}}$ are positive real sequences. They obtain sufficient conditions for oscillation of all solutions when $\sum\limits_{n=n_{0}}^{\infty}\left(\frac{n}{p_{n}}\right)^\frac{1}{\alpha}<\infty $ and $\sum\limits_{n=n_{0}}^{\infty}\left(\frac{n}{{p_{n}}^{\frac{1}{\alpha}}}\right)<\infty.$ The results are illustrated with examples.

We consider the Cauchy problem on the positive half-line for the differential-delay equation

$$

\ddot u(t)+2c_0(t)\dot u(t)+c_1(t)\dot u(t-h)+d_0(t)u(t)+d_1(t)u(t-h)+d_2(t)u(t-2h)=0

$$

where $c_k(t), d_j(t) (t\geq 0; k=0,1; j=0,1,2)$ are continuous functions. Conditions providing the positivity of the Green function and a lower bound for that function are derived. Our results are new even in the case of ordinary differential equations. Applications of the obtained results to equations with nonlinear causal mappings are also discussed. Equations with causal mappings include ordinary differential and integro-differential equations. In addition, we establish positivity conditions for solutions of functional differential equations with variable and distributed delays.

$$

\ddot u(t)+2c_0(t)\dot u(t)+c_1(t)\dot u(t-h)+d_0(t)u(t)+d_1(t)u(t-h)+d_2(t)u(t-2h)=0

$$

where $c_k(t), d_j(t) (t\geq 0; k=0,1; j=0,1,2)$ are continuous functions. Conditions providing the positivity of the Green function and a lower bound for that function are derived. Our results are new even in the case of ordinary differential equations. Applications of the obtained results to equations with nonlinear causal mappings are also discussed. Equations with causal mappings include ordinary differential and integro-differential equations. In addition, we establish positivity conditions for solutions of functional differential equations with variable and distributed delays.

In this paper, we investigate the growth of solutions of complex high order linear differential equations with entire or meromorphic coefficients of finite iterated order and we obtain some results which improve and extend some previous results of Z. X. Chen and L. Kinnunen.

This article presents the results on existence, uniqueness and stability of mild solutions of impulsive stochastic semilinear neutral functional differential equations without a Lipschitz condition and with a Lipschitz condition. The results are obtained by using the method of successive approximations.

In this work, we study the existence of periodic solutions for the evolution of p-Laplacian system and we show that these periodic solutions belong to $L^{\infty}(\omega, W^{1,\infty}(\Omega))$ and give a bound of $\left \Vert \nabla u_{i}(t)\right \Vert_{\infty}$ under certain geometric conditions on $\partial \Omega$.

This paper is devoted to the existence and uniqueness results of solutions for fractional differential equations with integral boundary conditions.

$$

\left\{

\begin{array}{l}

^C\hspace{-0.2em}D^\alpha x(t)+f(t,x(t),x'(t))=0,\quad t\in(0,1),\\

x(0)=\int^1_0 g_0(s,x(s))\mathrm{d}s ,\\

x(1)=\int^1_0 g_1(s,x(s))\mathrm{d}s ,\\

x^{(k)}(0)=0,\,\ k=2,3,\cdots, [\alpha]-1.

\end{array} \right.

$$

By means of the Banach contraction mapping principle, some new results on the existence and uniqueness are obtained. It is interesting to note that the sufficient conditions for the existence and uniqueness of solutions are dependent on the order $\alpha$.

$$

\left\{

\begin{array}{l}

^C\hspace{-0.2em}D^\alpha x(t)+f(t,x(t),x'(t))=0,\quad t\in(0,1),\\

x(0)=\int^1_0 g_0(s,x(s))\mathrm{d}s ,\\

x(1)=\int^1_0 g_1(s,x(s))\mathrm{d}s ,\\

x^{(k)}(0)=0,\,\ k=2,3,\cdots, [\alpha]-1.

\end{array} \right.

$$

By means of the Banach contraction mapping principle, some new results on the existence and uniqueness are obtained. It is interesting to note that the sufficient conditions for the existence and uniqueness of solutions are dependent on the order $\alpha$.

In this paper, we investigate the problem of existence and asymptotic behavior of solutions for the nonlinear boundary value problem

\begin{eqnarray*}

\epsilon y''+ky=f(t,y),\quad t\in\langle a,b \rangle, \quad k<0,\quad 0<\epsilon<<1

\end{eqnarray*}

satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions.

\begin{eqnarray*}

\epsilon y''+ky=f(t,y),\quad t\in\langle a,b \rangle, \quad k<0,\quad 0<\epsilon<<1

\end{eqnarray*}

satisfying three point boundary conditions. Our analysis relies on the method of lower and upper solutions.

Let $D^\alpha$ denote the Riemann-Liouville fractional differential operator of order $\alpha$. Let $1 < \alpha < 2$ and $0 < \beta < \alpha$. Define the operator $L$ by $L = D^\alpha - a D^\beta$ where $a \in \mathbb{R}$. We give sufficient conditions for the existence of solutions of the nonlinear fractional boundary value problem

\begin{eqnarray*}

&&Lu(t) + f(t, u(t)) = 0, \quad 0 < t < 1,\\

&&u(0) = 0, \, u(1)= 0.

\end{eqnarray*}

\begin{eqnarray*}

&&Lu(t) + f(t, u(t)) = 0, \quad 0 < t < 1,\\

&&u(0) = 0, \, u(1)= 0.

\end{eqnarray*}

The main objective of the present paper is to study some basic qualitative properties of solutions of a nonstandard Volterra type dynamic integral equation on time scales. The tools employed in the analysis are based on the applications of the Banach fixed point theorem and a certain integral inequality with explicit estimate on time scales.

We consider the question of global existence and uniform boundedness of nonnegative solutions of a system of reaction-diffusion equations with exponential nonlinearity using Lyapunov function techniques.

In this article, the author studies the stability and boundedness of solutions for the non-autonomous third order differential equation with a deviating argument, $r$:

\begin{equation*}

\begin{array}{c}

x^{\prime \prime \prime }(t)+a(t)x^{\prime \prime }(t)+b(t)g_{1}(x^{\prime}(t-r))+g_{2}(x^{\prime}(t))+h(x(t-r)) \\

=p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime }(t-r),x^{\prime \prime }(t)),

\end{array}

\end{equation*}

where $r>0$ is a constant. Sufficient conditions are obtained; a stability result in the literature is improved and extended to the preceding equation for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))=0,$ and a new boundedness result is also established for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))\neq 0.$

\begin{equation*}

\begin{array}{c}

x^{\prime \prime \prime }(t)+a(t)x^{\prime \prime }(t)+b(t)g_{1}(x^{\prime}(t-r))+g_{2}(x^{\prime}(t))+h(x(t-r)) \\

=p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime }(t-r),x^{\prime \prime }(t)),

\end{array}

\end{equation*}

where $r>0$ is a constant. Sufficient conditions are obtained; a stability result in the literature is improved and extended to the preceding equation for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))=0,$ and a new boundedness result is also established for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))\neq 0.$

An existence result of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces is proved. No growth assumption is made on the nonlinearities.

In this paper, we prove the existence of extremal positive, concave and pseudo-symmetric solutions for a general three-point second order p-Laplacian integro-differential boundary value problem by using an abstract monotone iterative technique.

In this paper we prove the existence of solutions of fractional impulsive semilinear evolution equations in Banach spaces. A nonlocal Cauchy problem is discussed for the evolution equations. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the theory.

In this paper we apply the recent extension of the Leggett-Williams Fixed Point Theorem which requires neither of the functional boundaries to be invariant to the second order right focal boundary value problem. We demonstrate a technique that can be used to deal with a singularity and provide a non-trivial example.

In this paper, we study a class of nonlinear Duffing equations with a deviating argument and establish some sufficient conditions for the existence of positive almost periodic solutions of the equation. These conditions are new and complement to previously known results.

Introducing shift operators on time scales we construct the integro-dynamic equation corresponding to the convolution type Volterra differential and difference equations in particular cases $\mathbb{T}=\mathbb{R}$ and

$\mathbb{T}=\mathbb{Z}$. Extending the scope of time scale variant of Gronwall's inequality we determine function bounds for the solutions of the integro dynamic equation.

$\mathbb{T}=\mathbb{Z}$. Extending the scope of time scale variant of Gronwall's inequality we determine function bounds for the solutions of the integro dynamic equation.

In this paper, we study a class of third-order nonlinear differential equations with a deviating argument and establish some sufficient conditions for the existence and exponential stability of anti-periodic solutions of the equation. These conditions are new and complement to previously known results.

In this paper, we investigate the problem of existence and asymptotic behavior of the solutions for the nonlinear boundary value problem

\begin{eqnarray*}

\epsilon y''+ky=f(t,y),\quad t\in\langle a,b \rangle, \quad k>0,\quad 0<\epsilon<<1

\end{eqnarray*}

satisfying Neumann boundary conditions and where critical manifold is not normally hyperbolic. Our analysis relies on the method upper and lower solutions.

\begin{eqnarray*}

\epsilon y''+ky=f(t,y),\quad t\in\langle a,b \rangle, \quad k>0,\quad 0<\epsilon<<1

\end{eqnarray*}

satisfying Neumann boundary conditions and where critical manifold is not normally hyperbolic. Our analysis relies on the method upper and lower solutions.

In this note we consider a scalar integral equation $x(t)= a(t)-\int^t_0 C(t,s)x(s)ds$, together with its resolvent equation, $R(t,s)= C(t,s)-\int^t_s C(t,u) R(u,s)du$, where $C$ is convex. Using a Liapunov functional we show that for fixed $s$ then $|R(t,s) - C(t,s)| \to 0$ as $t \to \infty$ and $\int^{\infty}_s (R(t,s)-C(t,s))^2 dt < \infty$. We then show that the variation of parameters formula $x(t)=a(t)-\int^t_0 R(t,s) a(s)ds$ can be replaced by $X(t)=a(t)-\int^t_0 C(t,s)a(s)ds$ when $a \in L^1[0,\infty)$ and that $|X(t) - x(t)|\to 0$ as $t \to \infty$ and $\int^{\infty}_0 (x(t)-X(t))^2 dt < \infty$. A mild nonlinear extension is given.

The purpose of this paper is to present the fundamental concepts of the basic theory for linear impulsive systems on time scales. First, we introduce the transition matrix for linear impulsive dynamic systems on time scales and we establish some properties of them. Second, we prove the existence and uniqueness of solutions for linear impulsive dynamic systems on time scales. Also we give some sufficient conditions for the stability of linear impulsive dynamic systems on time scales.

Sufficiency criteria are established to ensure the asymptotic stability and boundedness of solutions to third-order nonlinear delay differential equations of the form

\begin{equation*}

\begin{array}{c}

\dddot{x}(t)+e(x(t),\dot{x}(t),\ddot{x}(t))\ddot{x}(t)+g(x(t-r),\dot{x}

(t-r))+\psi (x(t-r)) \\

=p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime }(t-r),x^{\prime \prime }(t)).

\end{array}

\end{equation*}

By using Lyapunov's functional approach, we obtain two new results on the subject, which include and improve some related results in the relevant literature. Two examples are also given to illustrate the importance of results obtained.

\begin{equation*}

\begin{array}{c}

\dddot{x}(t)+e(x(t),\dot{x}(t),\ddot{x}(t))\ddot{x}(t)+g(x(t-r),\dot{x}

(t-r))+\psi (x(t-r)) \\

=p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime }(t-r),x^{\prime \prime }(t)).

\end{array}

\end{equation*}

By using Lyapunov's functional approach, we obtain two new results on the subject, which include and improve some related results in the relevant literature. Two examples are also given to illustrate the importance of results obtained.

In this paper, we study the singular behavior of solutions of a boundary value problem with mixed conditions in a neighborhood of an edge. The considered problem is defined in a nonhomogeneous body of $\mathbb{R}^{3}$, this is done in the general framework of weighted Sobolev spaces. Using the results of Benseridi-Dilmi, Grisvard and Aksentian, we show that the study of solutions' singularities in the spatial case becomes a study of two problems: a problem of plane deformation and the other is of normal plane deformation.

Some existence criteria are established for a class of fourth-order $m$-point boundary value problem by using the upper and lower solution method and the Leray-Schauder continuation principle.

We investigate extinction properties of solutions for the homogeneous Dirichlet boundary value problem of the nonlocal reaction-diffusion equation $u_t-d\Delta u+k u^p=\int_\Omega u^q(x,t)\,dx$ with $p, q\in (0, 1)$ and $k, d >0$. We show that $q=p$ is the critical extinction exponent. Moreover, the precise decay estimates of solutions before the occurrence of the extinction are derived.

By applying the monotone iterative technique, we obtain the existence and uniqueness of $C^1[0,1]$ positive solutions in some set for singular boundary value problems of second order ordinary differential equations with integral boundary conditions.

In this paper, we study the existence of positive solutions for boundary value problems of second-order differential equations system with integral boundary condition on the half-line. By using a three functionals fixed point theorem in a cone and a fixed point theorem in a cone due to Avery-Peterson, we show the existence of at least two and three monotone increasing positive solutions with suitable growth conditions imposed on the nonlinear terms.

In this paper we investigate the existence and uniqueness of global solutions, and a rate stability for the energy related with a Cauchy problem to the viscous Burgers equation in unbounded domain $\re\times(0,\infty)$. Some aspects associated with a Cauchy problem are presented in order to employ the approximations of Faedo-Galerkin in whole real line $\re$. This becomes possible due to the introduction of weight Sobolev spaces which allow us to use arguments of compactness in the Sobolev spaces.

In this paper, we study an impulsively controlled predator-prey model with Monod-Haldane functional response. By using the Floquet theory, we prove that there exists a stable prey-free solution when the impulsive period is less than some critical value, and give the condition for the permanence of the system. In addition, we show the existence and stability of a positive periodic solution by using bifurcation theory.

Let $\mathbb{T}$ be a time scale. In this paper, we study the existence of multiple positive solutions for the following nonlinear singular $m$-point boundary value problem dynamic equations with sign changing coefficients on time scales

$$\left\{\begin{array}{lll}

u^{\triangle\nabla}(t)+ a(t)f(u(t))=0, (0,T)_{\mathbb{T}},

\cr\

u^{\triangle}(0)=\sum_{i=1}^{m-2}a_{i}u^{\triangle}(\xi_i),

\cr

u(T)=\sum_{i=1}^{k}b_{i}u(\xi_i)-\sum_{i=k+1}^{s}b_{i}u(\xi_i)-\sum_{i=s+1}^{m-2}b_{i}u^{\triangle}(\xi_i),

\end{array}\right.$$

where $1\leq k\leq s\leq m-2, a_i, b_i\in(0,+\infty)$ with $0<\sum_{i=1}^{k}b_{i}-\sum_{i=k+1}^{s}b_{i}<1,

0<\sum_{i=1}^{m-2}a_{i}<1, 0<\xi_1<\xi_2<\cdots<\xi_{m-2}<\rho(T)$, $f\in C( [0,+\infty),[0,+\infty))$, $a(t)$ may be singular at $t=0$. We show that there exist two positive solutions by using two different fixed point theorems respectively. As an application, some examples are included to illustrate the main results. In particular, our criteria extend and improve some known results.

$$\left\{\begin{array}{lll}

u^{\triangle\nabla}(t)+ a(t)f(u(t))=0, (0,T)_{\mathbb{T}},

\cr\

u^{\triangle}(0)=\sum_{i=1}^{m-2}a_{i}u^{\triangle}(\xi_i),

\cr

u(T)=\sum_{i=1}^{k}b_{i}u(\xi_i)-\sum_{i=k+1}^{s}b_{i}u(\xi_i)-\sum_{i=s+1}^{m-2}b_{i}u^{\triangle}(\xi_i),

\end{array}\right.$$

where $1\leq k\leq s\leq m-2, a_i, b_i\in(0,+\infty)$ with $0<\sum_{i=1}^{k}b_{i}-\sum_{i=k+1}^{s}b_{i}<1,

0<\sum_{i=1}^{m-2}a_{i}<1, 0<\xi_1<\xi_2<\cdots<\xi_{m-2}<\rho(T)$, $f\in C( [0,+\infty),[0,+\infty))$, $a(t)$ may be singular at $t=0$. We show that there exist two positive solutions by using two different fixed point theorems respectively. As an application, some examples are included to illustrate the main results. In particular, our criteria extend and improve some known results.

In this paper we prove the existence of solutions to the following third order differential inclusion:

$$\left\{

\begin{array}{ll} x^{(3)}(t)\in

F(t,x(t),\dot{x}(t),\ddot{x}(t))+G(x(t),\dot{x}(t),\ddot{x}(t)),

\mbox{ a.e. on } [0,T]\cr x(0)=x_0, \dot x(0)=u_0, \ddot

x(0)=v_0, \mbox{ and }\ddot{x}(t)\in S, \forall t\in [0,T],

\end{array}\right.

$$

where $F:[0,T]\times \mathbb{H}\times \mathbb{H} \times \mathbb{H}\rightarrow \mathbb{H}$ is a continuous set-valued mapping, $G:\mathbb{H}\times \mathbb{H} \times \mathbb{H}\rightarrow \mathbb{H}$ is an upper semi-continuous set-valued mapping with $G(x,y,z)\subset \partial^C g(z)$ where $g: \mathbb{H}\rightarrow \mathbb{R}$ is a uniformly regular function over $S$ and locally Lipschitz and $S$ is a ball compact subset of a separable Hilbert space $\mathbb{H}$.

$$\left\{

\begin{array}{ll} x^{(3)}(t)\in

F(t,x(t),\dot{x}(t),\ddot{x}(t))+G(x(t),\dot{x}(t),\ddot{x}(t)),

\mbox{ a.e. on } [0,T]\cr x(0)=x_0, \dot x(0)=u_0, \ddot

x(0)=v_0, \mbox{ and }\ddot{x}(t)\in S, \forall t\in [0,T],

\end{array}\right.

$$

where $F:[0,T]\times \mathbb{H}\times \mathbb{H} \times \mathbb{H}\rightarrow \mathbb{H}$ is a continuous set-valued mapping, $G:\mathbb{H}\times \mathbb{H} \times \mathbb{H}\rightarrow \mathbb{H}$ is an upper semi-continuous set-valued mapping with $G(x,y,z)\subset \partial^C g(z)$ where $g: \mathbb{H}\rightarrow \mathbb{R}$ is a uniformly regular function over $S$ and locally Lipschitz and $S$ is a ball compact subset of a separable Hilbert space $\mathbb{H}$.

In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential equations with Stepanov almost automorphic forcing terms. A few illustrative examples are discussed at the very end of the paper.

In the paper we investigate a non-local contact-boundary value problem for a system of second order hyperbolic equations with discontinuous solutions. Under some conditions on input data, a priori estimate is obtained for the solution of this problem.

This paper is concerned with the following retarded Li\'{e}nard equation

$$x''(t)+f_1(x(t))(x'(t))^2+f_2(x(t))x'(t)+g_1(x(t))+g_2(x(t-\tau(t)))=e(t).$$

We prove a new theorem which ensures that all solutions of the above Li\'{e}nard equation satisfying given initial conditions are bounded. As one will see, our results improve some earlier results even in the case of $f_1(x)\equiv 0$.

$$x''(t)+f_1(x(t))(x'(t))^2+f_2(x(t))x'(t)+g_1(x(t))+g_2(x(t-\tau(t)))=e(t).$$

We prove a new theorem which ensures that all solutions of the above Li\'{e}nard equation satisfying given initial conditions are bounded. As one will see, our results improve some earlier results even in the case of $f_1(x)\equiv 0$.

Sufficient conditions for the oscillation of the nonlinear second order differential equation $(a(t)x^{\prime })^{\prime }+Q(t,x^{\prime})=P(t,x,x^{\prime })$ are established where the coefficients are continuous and $a(t)$ is nonnegative.

This paper is concerned with the following nonlinear third-order three-point boundary value problem

\[\left\{

\begin{array}{l}

u^{\prime \prime \prime }(t)+f\left( t,u\left( t\right) ,u^{\prime}\left(t\right) \right) =0,\, t\in \left[ 0,1\right], \\

u\left( 0\right) =u^{\prime }\left( 0\right) =0,\, u^{\prime}\left( 1\right) =\alpha u^{\prime }\left( \eta \right),\label{1.1}

\end{array}

\right.\]

where $0<\eta <1$ and $0\leq \alpha <1.$ A new maximum principle is established and some existence criteria are obtained for the above problem by using the upper and lower solution method.

\[\left\{

\begin{array}{l}

u^{\prime \prime \prime }(t)+f\left( t,u\left( t\right) ,u^{\prime}\left(t\right) \right) =0,\, t\in \left[ 0,1\right], \\

u\left( 0\right) =u^{\prime }\left( 0\right) =0,\, u^{\prime}\left( 1\right) =\alpha u^{\prime }\left( \eta \right),\label{1.1}

\end{array}

\right.\]

where $0<\eta <1$ and $0\leq \alpha <1.$ A new maximum principle is established and some existence criteria are obtained for the above problem by using the upper and lower solution method.

This paper is concerned with the multiplicity of positive solutions of boundary value problem for the fourth-order quasilinear singular differential equation

$$

(|u''|^{p-2}u'')''=\lambda g(t)f(u),\quad 0<t<1,

$$

where $p>1$, $\lambda>0$. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold $\lambda^*<+\infty$, such that if $0<\lambda\leq\lambda^*$, then the problem admits at least one positive solution; while if $\lambda \lambda^*$, then the problem has no positive solution. In particular, there exist at least two positive solutions for $0<\lambda<\lambda^*$.

$$

(|u''|^{p-2}u'')''=\lambda g(t)f(u),\quad 0<t<1,

$$

where $p>1$, $\lambda>0$. We apply the fixed point index theory and the upper and lower solutions method to investigate the multiplicity of positive solutions. We have found a threshold $\lambda^*<+\infty$, such that if $0<\lambda\leq\lambda^*$, then the problem admits at least one positive solution; while if $\lambda \lambda^*$, then the problem has no positive solution. In particular, there exist at least two positive solutions for $0<\lambda<\lambda^*$.

We study classes of $n$th order boundary value problems consisting of an equation having a sign-changing nonlinearity $f(t,x)$ together with several different sets of nonhomogeneous multi-point boundary conditions. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. Conditions are determined by the behavior of $f(t,x)/x$ near $0$ and $\pm\infty$ when compared to the smallest positive characteristic values of some associated linear integral operators. This work improves and extends a number of recent results in the literature on this topic. The results are illustrated with examples.

In this paper we study the existence of mild solutions for a class of abstract partial neutral integro-differential equations with state-dependent delay.

This paper deals with the existence of positive solutions for a boundary value problem involving a nonlinear functional differential equation of fractional order $\alpha$ given by $ D^{\alpha} u(t) + f(t, u_t) = 0$, $t \in (0, 1)$, $2 < \alpha \le 3$, $ u^{\prime}(0) = 0$, $u^{\prime}(1) = b u^{\prime}(\eta)$, $u_0 = \phi$. Our results are based on the nonlinear alternative of Leray-Schauder type and Krasnosel'skii fixed point theorem.

We study the existence of homoclinic solutions for the second order Hamiltonian system $\ddot{u}+V_{u}(t,u)=f(t)$. Let $V(t,u)=-K(t,u)+W(t,u)\in C^{1}(\mathbb{R}\times\mathbb{R}^{n}, \mathbb{R})$ be $T$-periodic in $t$, where $K$ is a quadratic growth function and $W$ may be asymptotically quadratic or super-quadratic at infinity. One homoclinic solution is obtained as a limit of solutions of a sequence of periodic second order differential equations.

In this paper, we investigate the iterated order of solutions of higher order homogeneous linear differential equations with entire coefficients. We improve and extend some results of Belaidi and Hamouda by using the concept of the iterated order. We also consider nonhomogeneous linear differential equations.

In this paper, by using the Avery and Peterson fixed point theorem, we establish the existence of multiple positive solutions for the following higher order multi-point nonhomogeneous boundary value problem

$ u^{(n)}(t) + f(t,u(t),u'(t),\ldots,u^{(n-2)}(t)) = 0, t\in (0,1)$,

$ u(0)= u'(0)=\cdots=u^{(n-3)}(0)=u^{(n-2)}(0)=0, u^{(n-2)}(1)-\sum_{i=1}^{m} a_i u^{(n-2)}(\xi_i)=\lambda$,

where $n\ge3$ and $m\ge1$ are integers, $0<\xi_1<\xi_2<\cdots<\xi_m<1$ are constants, $\lambda\in [0,\infty)$ is a parameter, $a_i>0$ for $1\le i\le m$ and $\sum_{i=1}^{m} a_i\xi_i<1$, $f(t,u,u',\cdots,u^{(n-2)})\in C([0,1]\times[0,\infty)^{n-1}, [0,\infty))$. We give an example to illustrate our result.

$ u^{(n)}(t) + f(t,u(t),u'(t),\ldots,u^{(n-2)}(t)) = 0, t\in (0,1)$,

$ u(0)= u'(0)=\cdots=u^{(n-3)}(0)=u^{(n-2)}(0)=0, u^{(n-2)}(1)-\sum_{i=1}^{m} a_i u^{(n-2)}(\xi_i)=\lambda$,

where $n\ge3$ and $m\ge1$ are integers, $0<\xi_1<\xi_2<\cdots<\xi_m<1$ are constants, $\lambda\in [0,\infty)$ is a parameter, $a_i>0$ for $1\le i\le m$ and $\sum_{i=1}^{m} a_i\xi_i<1$, $f(t,u,u',\cdots,u^{(n-2)})\in C([0,1]\times[0,\infty)^{n-1}, [0,\infty))$. We give an example to illustrate our result.

The aim of this paper is to show that any second order nonoscillatory linear differential equation can be converted into an oscillating system by applying a sufficiently large nonlinear perturbation. This can be achieved through a detailed analysis of possible nonoscillatory solutions of the perturbed differential equation which may exist when the perturbation is sufficiently small. As a consequence the class of oscillation-generating perturbations is determined precisely with respect to the original nonoscillatory linear equation.

A Picone type formula for second order linear non-selfadjoint impulsive differential equations with discontinuous solutions having fixed moments of impulse actions is derived. Applying the formula, Leighton and Sturm-Picone type comparison theorems as well as several oscillation criteria for impulsive differential equations are obtained.

In this paper, we consider (n-1, 1)-type conjugate boundary value problem for the nonlinear fractional differential equation

\begin{gather*}\begin{array}{ll}

\mathbf{D}_{0+}^\alpha u(t)+\lambda f(t,u(t))=0,\quad 0<t<1, \lambda >0,\\

u^{(j)}(0)=0, 0\leq j\leq n-2,\\

u(1)=0,

\end{array}\end{gather*}

where $\lambda$ is a parameter, $\alpha\in(n-1, n]$ is a real number and $n\geq 3$, and $\mathbf{D}_{0+}^\alpha$ is the Riemann-Liouville's fractional derivative, and $f$ is continuous and semipositone. We give properties of Green's function of the boundary value problems, and derive an interval of $\lambda$ such that any $\lambda$ lying in this interval, the semipositone boundary value problem has multiple positive solutions.

\begin{gather*}\begin{array}{ll}

\mathbf{D}_{0+}^\alpha u(t)+\lambda f(t,u(t))=0,\quad 0<t<1, \lambda >0,\\

u^{(j)}(0)=0, 0\leq j\leq n-2,\\

u(1)=0,

\end{array}\end{gather*}

where $\lambda$ is a parameter, $\alpha\in(n-1, n]$ is a real number and $n\geq 3$, and $\mathbf{D}_{0+}^\alpha$ is the Riemann-Liouville's fractional derivative, and $f$ is continuous and semipositone. We give properties of Green's function of the boundary value problems, and derive an interval of $\lambda$ such that any $\lambda$ lying in this interval, the semipositone boundary value problem has multiple positive solutions.

In this paper, the following fractional order ordinary differential equation boundary value problem:

\begin{gather*}

D_{0+}^\alpha u(t) =f(t,u(t),D_{0+}^{\alpha-1}u(t))+e(t), 0<t<1,\\

I_{0+}^{2-\alpha}u(t)\mid_{t=0}=0, D_{0+}^{\alpha-1}u(1)=\sum_{i=1}^{m-2}\beta_i D_{0+}^{\alpha-1}u(\eta_i),

\end{gather*}

is considered, where $1< \alpha \leq 2,$ is a real number, $D_{0+}^\alpha$ and $I_{0+}^{\alpha}$ are the standard Riemann-Liouville differentiation and integration, and $f:[0,1]\times R^2 \to R$ is continuous and $e \in L^1[0,1]$, and $\eta_i \in (0, 1), \beta_i \in R, i=1,2, \cdots, m-2$, are given constants such that $\sum_{i=1}^{m-2}\beta_i=1$. By using the coincidence degree theory, some existence results of solutions are established.

\begin{gather*}

D_{0+}^\alpha u(t) =f(t,u(t),D_{0+}^{\alpha-1}u(t))+e(t), 0<t<1,\\

I_{0+}^{2-\alpha}u(t)\mid_{t=0}=0, D_{0+}^{\alpha-1}u(1)=\sum_{i=1}^{m-2}\beta_i D_{0+}^{\alpha-1}u(\eta_i),

\end{gather*}

is considered, where $1< \alpha \leq 2,$ is a real number, $D_{0+}^\alpha$ and $I_{0+}^{\alpha}$ are the standard Riemann-Liouville differentiation and integration, and $f:[0,1]\times R^2 \to R$ is continuous and $e \in L^1[0,1]$, and $\eta_i \in (0, 1), \beta_i \in R, i=1,2, \cdots, m-2$, are given constants such that $\sum_{i=1}^{m-2}\beta_i=1$. By using the coincidence degree theory, some existence results of solutions are established.

By means of the fixed point index theory of strict set contraction operators, we establish new existence theorems on multiple positive solutions to a boundary value problem for second-order impulsive integro-differential equations with integral boundary conditions in a Banach space. Moreover, an application is given to illustrate the main result.

In this paper, we consider the existence of multiple positive solutions for the 2n-th order $m$-point boundary value problems:

$$\left\{\begin{array}{ll} x^{(2n)}(t)=f(t,x(t),x^{''}(t),\cdots ,x^{(2(n-1))}(t)), 0\leq t\leq 1,\\

x^{(2i+1)}(0)=\sum\limits_{j=1}^{m-2}\alpha_{ij}x^{(2i+1)}(\xi_j),\quad

x^{(2i)}(1)=\sum\limits_{j=1}^{m-2}\beta_{ij}x^{(2i)}(\xi_j), 0\leq i\leq n-1,\\

\end{array}\right.$$

where $\alpha_{ij}, \beta_{ij} \ (0\leq i\leq n-1,1\leq j\leq m-2) \in [0,\infty)$, $\sum\limits_{j=1}^{m-2}\alpha_{ij},\sum\limits_{j=1}^{m-2}\beta_{ij}\in (0,1)$, $0<\xi_1<\xi_2<\ldots<\xi_{m-2}<1$. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem.

$$\left\{\begin{array}{ll} x^{(2n)}(t)=f(t,x(t),x^{''}(t),\cdots ,x^{(2(n-1))}(t)), 0\leq t\leq 1,\\

x^{(2i+1)}(0)=\sum\limits_{j=1}^{m-2}\alpha_{ij}x^{(2i+1)}(\xi_j),\quad

x^{(2i)}(1)=\sum\limits_{j=1}^{m-2}\beta_{ij}x^{(2i)}(\xi_j), 0\leq i\leq n-1,\\

\end{array}\right.$$

where $\alpha_{ij}, \beta_{ij} \ (0\leq i\leq n-1,1\leq j\leq m-2) \in [0,\infty)$, $\sum\limits_{j=1}^{m-2}\alpha_{ij},\sum\limits_{j=1}^{m-2}\beta_{ij}\in (0,1)$, $0<\xi_1<\xi_2<\ldots<\xi_{m-2}<1$. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem.

In this paper, we study the initial boundary value problem for a class of metaparabolic equations. We establish the existence of solutions by the energy techniques. Some results on the regularity, blow-up and existence of global attractor are obtained.

We are interested in the existence of nontrivial solutions for the second order nonlinear differential equation (E): $y'' (t) = f\big(t, y (t)\big) = 0, 0 < t < 1$ subject to multi-point boundary conditions at $t=1$ and either Dirichlet or Neumann conditions at $t=0$. Assume that $f(t, y)$ satisfies $|f(t, y)| \le k(t) |y| + h(t)$ for non-negative functions $k, h \in L^1 (0, 1)$ for all $(t, y) \in (0, 1) \times \Bbb R$ and $f(t, 0) \not\equiv 0$ for $t\in (0, 1)$. We show without any additional assumption on $h(t)$ that if $\|k\|_1$ is sufficiently small where $\|\cdot \|_1$ denotes the norm of $L^1(0, 1)$ then there exists at least one non-trivial solution for such boundary value problems. Our results reduce to that of Sun and Liu and Sun for the three point problem with Neumann boundary condition at $t=0$.

In this paper, we give a necessary and sufficient condition for the existence of at least one $\Psi$-bounded solution of a linear nonhomogeneous Lyapunov matrix differential equation on $\mathbb{R}$. In addition, we give a result in connection with the asymptotic behavior of the $\Psi$-bounded solution of this equation.

The aim of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear delay differential equation

\begin{equation*}\label{E0}

\left[a(t)\left[x''(t)\right]^{\gamma}\right]'+q(t)f(x\left[\tau(t)\right])=0.

\tag{$E$}

\end{equation*}

Applying suitable comparison theorems we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of {(\ref{E0})}. Obtained results essentially improve and complement earlier ones. Various examples are considered to illustrate the main results.

\begin{equation*}\label{E0}

\left[a(t)\left[x''(t)\right]^{\gamma}\right]'+q(t)f(x\left[\tau(t)\right])=0.

\tag{$E$}

\end{equation*}

Applying suitable comparison theorems we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of {(\ref{E0})}. Obtained results essentially improve and complement earlier ones. Various examples are considered to illustrate the main results.

In this paper, by employing fixed point index theory and Leray-Schauder degree theory, we obtain the existence and multiplicity of sign-changing solutions for nonlinear second-order differential equations with integral boundary value conditions.

This paper deals with non-simultaneous blow-up for a reaction-diffusion system with absorption and nonlinear boundary flux. We establish necessary and sufficient conditions for the occurrence of non-simultaneous blow-up with proper initial data.

In this paper we investigate the existence of the periodic solutions of a quasilinear differential equation with piecewise constant argument of generalized type. By using some fixed point theorems and some new analysis technique, sufficient conditions are obtained for the existence and uniqueness of periodic solutions of these systems. A new Gronwall type lemma is proved. Some examples concerning biological models as Lasota-Wazewska, Nicholson's blowflies and logistic models are treated.

In this paper, we are focused upon the global uniqueness results for a stochastic integro-differential equation in Fréchet spaces. The main results are proved by using the resolvent operators combined with a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to Frigon and Granas. As an application, a controllability result with one parameter is given to illustrate the theory.

In this paper oscillatory and asymptotic behaviour of solutions of a class of nonlinear second order neutral difference equations with positive and negative coefficients of the form

(E) $ \Delta (r(n) \Delta (y(n) + p(n) y(n-m))) + f(n) H_1(y(n-k_1))-g(n) H_2 (y(n-k_2)) = q(n) $ \\ and

$ \Delta (r(n) \Delta (y(n) + p(n) y(n-m))) + f(n) H_1(y(n-k_1))-g(n) H_2 (y(n-k_2)) = 0 $ \\ \\

are studied under the assumptions

\begin{eqnarray}\sum\limits_{n=0}^{\infty} \frac{1}{r(n)} < \infty \nonumber \end{eqnarray} and

\begin{eqnarray}\sum\limits_{n=0}^{\infty} \frac{1}{r(n)} = \infty \nonumber \end{eqnarray} for various ranges of $p(n)$. Using discrete Krasnoselskii's fixed point theorem sufficient conditions are obtained for existence of positive bounded solutions of (E).

(E) $ \Delta (r(n) \Delta (y(n) + p(n) y(n-m))) + f(n) H_1(y(n-k_1))-g(n) H_2 (y(n-k_2)) = q(n) $ \\ and

$ \Delta (r(n) \Delta (y(n) + p(n) y(n-m))) + f(n) H_1(y(n-k_1))-g(n) H_2 (y(n-k_2)) = 0 $ \\ \\

are studied under the assumptions

\begin{eqnarray}\sum\limits_{n=0}^{\infty} \frac{1}{r(n)} < \infty \nonumber \end{eqnarray} and

\begin{eqnarray}\sum\limits_{n=0}^{\infty} \frac{1}{r(n)} = \infty \nonumber \end{eqnarray} for various ranges of $p(n)$. Using discrete Krasnoselskii's fixed point theorem sufficient conditions are obtained for existence of positive bounded solutions of (E).

In this paper we investigate the properties of nonoscillation for the equation $$(-1)^{n}(\rho(t)|y^{(n)}|^{p-2}y^{(n)})^{(n)}-v(t)|y|^{p-2}y=0,$$ where $1<p<\infty$ and ${v}$ is a non-negative continuous function and ${\rho}$ is a positive $n$-times continuously differentiable function on the half-line $[0,\infty)$. When the principle of reciprocity is used for the linear equation ($p=2$) we suppose that the functions ${v}$ and ${\rho}$ are positive and $n$-times continuously differentiable on the half-line $[0,\infty)$.

In this paper, we prove the existence of positive solutions for Floquet boundary value problem concerning fractional functional differential equations with bounded delay. The results are obtained by using two fixed point theorems on appropriate cones.

In this paper, we consider a semilinear parabolic equation

$$u_t=\Delta u+u^q\int_0^tu^p(x,s)ds,\quad x\in \Omega,\quad t>0$$

with nonlocal nonlinear boundary condition $u|_{\partial\Omega\times(0,+\infty)}=\int_\Omeg\varphi(x,y) u^l(y,t)dy$ and nonnegative initial data, where $p$, $q\geq 0$ and $l>0$. The blow-up criteria and the blow-up rate are obtained.

$$u_t=\Delta u+u^q\int_0^tu^p(x,s)ds,\quad x\in \Omega,\quad t>0$$

with nonlocal nonlinear boundary condition $u|_{\partial\Omega\times(0,+\infty)}=\int_\Omeg\varphi(x,y) u^l(y,t)dy$ and nonnegative initial data, where $p$, $q\geq 0$ and $l>0$. The blow-up criteria and the blow-up rate are obtained.

In this paper, we study one-dimensional Newtonian filtration equation including unbounded sources with multiple delays. The existence of nonnegative non-trivial time periodic solutions will be established by the Leray-Schauder fixed point theorem based on some suitable Lyapunov functionals and some a priori estimates for all possible periodic solutions.

We consider the $(n-1,1)$ conjugate boundary value problem. Some upper estimates to positive solutions for the problem are obtained. We also establish some explicit sufficient conditions for the existence and nonexistence of positive solutions of the problem.

The aim of this paper is to investigate a class of boundary value problem for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness.

Using a fixed point theorem in ordered Banach spaces with lattice structure founded by Liu and Sun, this paper investigates the multiplicity of nontrivial solutions for fourth order $m$-point boundary value problems with sign-changing nonlinearity. Our results are new and improve on those in the literature.

In this paper, a class of third-order three-point boundary value problem on time scales is considered. Using monotone iterative technique and cone expansion and compression fixed point theorem of norm type, we do not only obtain the existence and uniqueness of positive solutions of the problem, but also establish the iterative schemes for approximating the solutions.

In this note, we obtain a new multiplicative perturbation theorem for local $C$-regularized cosine function with a nondensely defined generator $A$. An application to an integrodifferential equation is given.

This paper deals with the existence of a mild solution for some fractional semilinear differential equations with non local conditions. Using a more appropriate definition of a mild solution than the one given in [12], we prove the existence and uniqueness of such solutions, assuming that the linear part is the infinitesimal generator of an analytic semigroup that is compact for $t > 0$ and the nonlinear part is a Lipschitz continuous function with respect to the norm of a certain interpolation space. An example is provided.

This article presents conditions for the oscillation of solutions to neutral partial differential equations. The order of these equations can be even or odd, and the deviating arguments can be distributed over an interval. We also extend our results to a nonlinear equation and to a system of equations.

We study the complexity of solutions for a class of completely integrable, nonlinear integro-differential Schrödinger initial-boundary value problems on a bounded domain, depending on a real bifurcation parameter. The considered Schrödinger problem is a natural extension of the classical Hopf bifurcation model for planar systems into an infinite-dimensional phase space. Namely, the change in the sign of the bifurcation parameter has a consequence that an attracting (or repelling) invariant subset of the sphere in $L^2(\Omega)$ is born. We measure the complexity of trajectories near the origin by considering the Minkowski content and the box dimension of their finite-dimensional projections. Moreover we consider the compactness and rectifiability of trajectories, and box dimension of multiple spirals and spiral chirps. Finally, we are able to obtain the box dimension of trajectories of some nonintegrable Schrödinger evolution problems using their reformulation in terms of the corresponding (not explicitly solvable) dynamical systems in $\Rb^n$.

In this paper we establish some oscillation theorems for second order neutral dynamic equations with distributed deviating arguments. We use the Riccati transformation technique to obtain sufficient conditions for the oscillation of all solutions. Further, some examples are provided to illustrate the results.

In this paper we investigate the step-type contrast structure of a second-order semilinear differential equation with integral boundary conditions. The asymptotic solution is constructed by the boundary function method, and the uniform validity of the formal solution is proved by the theory of differential equalities.

In this paper, regarding the delay as parameter, we investigate the effect of delay on the dynamics of a Rössler system with multiple delayed feedback proposed by Ghosh and Chowdhury. At first we consider the stability of equilibrium and the existence of Hopf bifurcations. Then an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, we give a numerical simulation example which indicates that chaotic oscillation is converted into a stable steady state or a stable periodic orbit when the delay passes through certain critical values.

The aim of this paper is to establish the existence of solutions and some properties of set solutions for a Cauchy problem with causal operator in a separable Banach space.

We discuss the problem

\begin{equation*}

\left\{

\begin{array}{ll}

-\func{div}\left( \left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right)

=\lambda (a\left( x\right) \left\vert u\right\vert ^{q(x)-2}u+b(x)\left\vert u\right\vert ^{h(x)-2}u)\text{,} & \text{for }x\in \Omega , \\ u=0\text{,} & \text{for }x\in \partial \Omega .

\end{array}

\right.

\end{equation*}

where $\Omega $ is a bounded domain with smooth boundary in $\mathbb{R}^{N}$ $\left( N\geq 2\right)$ and $p$ is Lipschitz continuous, $q$ and $h$ are continuous functions on $\overline{\Omega }$ such that $1<q(x)<p(x)<h(x)<p^{\ast }(x)$ and $p(x)<N$. We show the existence of at least one nontrivial weak solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem.

\begin{equation*}

\left\{

\begin{array}{ll}

-\func{div}\left( \left\vert \nabla u\right\vert ^{p(x)-2}\nabla u\right)

=\lambda (a\left( x\right) \left\vert u\right\vert ^{q(x)-2}u+b(x)\left\vert u\right\vert ^{h(x)-2}u)\text{,} & \text{for }x\in \Omega , \\ u=0\text{,} & \text{for }x\in \partial \Omega .

\end{array}

\right.

\end{equation*}

where $\Omega $ is a bounded domain with smooth boundary in $\mathbb{R}^{N}$ $\left( N\geq 2\right)$ and $p$ is Lipschitz continuous, $q$ and $h$ are continuous functions on $\overline{\Omega }$ such that $1<q(x)<p(x)<h(x)<p^{\ast }(x)$ and $p(x)<N$. We show the existence of at least one nontrivial weak solution. Our approach relies on the variable exponent theory of Lebesgue and Sobolev spaces combined with adequate variational methods and the Mountain Pass Theorem.

The quasilinearization method coupled with the method of upper and lower solutions is used for a class of nonlinear boundary value problems with integral boundary conditions. We obtain some less restrictive sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem. An example is also included to illustrate the main result.

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria for a certain class of third order nonlinear delay difference equations. Our results extend and improve some previously obtained ones. An example is worked out to demonstrate the validity of the proposed results.

The main purpose of this paper is to consider the oscillation theory on meromorphic solutions of second order linear differential equations of the form $f^{''}+A(z)f=0$ where $A$ is meromorphic in the complex plane. We improve and extend some oscillation results due to Bank and Laine, Kinnunen, Liang and Liu, and others.

We show the existence of a polynomial system of degree four having three real invariant straight lines forming a triangle with at least three small amplitude limit cycles in the interior. Also, we obtain the necessary and sufficient conditions for the critical point at the interior of the bounded region to be a center.

In this paper, we investigate the existence of nontrivial solutions to the nonlinear $q$-fractional boundary value problem

\begin{align*}

&(D_q^\alpha y)(x)=-f(x,y(x)),\quad 0<x<1,\\

&y(0)=0=y(1),

\end{align*}

by applying a fixed point theorem in cones.

\begin{align*}

&(D_q^\alpha y)(x)=-f(x,y(x)),\quad 0<x<1,\\

&y(0)=0=y(1),

\end{align*}

by applying a fixed point theorem in cones.

In this paper, we study the existence of solutions for a boundary value problem of differential inclusions of order $q \in (1,2]$ with non-separated boundary conditions involving convex and non-convex multivalued maps. Our results are based on the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.

By using the Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order nonlinear difference equation $\Delta \left[p(t)\Delta u(t-1)\right] +f(t,u(t))=0$ has at least one homoclinic orbit, where $t\in \Z,\ u\in \R$.

In this paper, we study a kind of anisotropic p-Laplacian equations in $R^n$. Nontrivial solutions are obtained using mountain pass theorem given by Ambrosetti-Rabinowitz.

We investigate the existence of positive solutions with respect to a cone for a higher-order nonlinear differential system, subject to some boundary conditions which involve $m$ points.

Along the lines of Atkinson, a spectral theorem is proved for the boundary value problem

$$

\left\{\begin{array}{l}

Jz' + f(t) J z + P(t) z= \lambda B(t) z \\

x(0) = x(T) =0, \\

\end{array}\right.

t \in [0, T], z=(x, y) \in \mathbb{R}^N \times \mathbb{R}^N,

$$

where $f(t)$ is real-valued and $P(t), B(t)$ are symmetric matrices, with $B(t)$ positive definite. A suitable rotation index associated to the system is used to highlight the connections between the eigenvalues and the nodal properties of the corresponding eigenfunctions.

$$

\left\{\begin{array}{l}

Jz' + f(t) J z + P(t) z= \lambda B(t) z \\

x(0) = x(T) =0, \\

\end{array}\right.

t \in [0, T], z=(x, y) \in \mathbb{R}^N \times \mathbb{R}^N,

$$

where $f(t)$ is real-valued and $P(t), B(t)$ are symmetric matrices, with $B(t)$ positive definite. A suitable rotation index associated to the system is used to highlight the connections between the eigenvalues and the nodal properties of the corresponding eigenfunctions.

This work is concerned with the existence and uniqueness of positive solutions for Neumann boundary value problems of second order impulsive differential equations. The result is obtained by using a fixed point theorem of generalized concave operators.

In this paper, we study the second-order nonlinear singular Sturm-Liouville boundary value problems with Riemann-Stieltjes integral boundary conditions

\begin{equation*}\begin{cases}

-(p(t)u'(t))'+q(t)u(t)=f(t,u(t)),\; 0<t<1,\\

\alpha_{1}u(0)-\beta_{1}u'(0)=\int_{0}^{1}u(\tau)\mathrm{d}\alpha(\tau),\\

\alpha_{2}u(1)+\beta_{2}u'(1)=\int_{0}^{1}u(\tau)\mathrm{d}\beta(\tau),

\end{cases}\end{equation*}

where $f(t,u)$ is allowed to be singular at $t=0,1$ and $u=0$. Some new results for the existence of positive solutions of the boundary value problems are obtained. Our results extend some known results from the nonsingular case to the singular case, and we also improve and extend some results of the singular cases.

\begin{equation*}\begin{cases}

-(p(t)u'(t))'+q(t)u(t)=f(t,u(t)),\; 0<t<1,\\

\alpha_{1}u(0)-\beta_{1}u'(0)=\int_{0}^{1}u(\tau)\mathrm{d}\alpha(\tau),\\

\alpha_{2}u(1)+\beta_{2}u'(1)=\int_{0}^{1}u(\tau)\mathrm{d}\beta(\tau),

\end{cases}\end{equation*}

where $f(t,u)$ is allowed to be singular at $t=0,1$ and $u=0$. Some new results for the existence of positive solutions of the boundary value problems are obtained. Our results extend some known results from the nonsingular case to the singular case, and we also improve and extend some results of the singular cases.

The existence of solutions for a nonlinear fractional order differential inclusion is investigated. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.

In this paper, we study a nonlinear boundary value system with $p$-Laplacian operator

$$\left\{\begin{array}{lll}

(\phi_{p_1}(u'))'+a_1(t)f(u,v)=0, 0<t<1,\cr

(\phi_{p_2}(v'))'+a_2(t)g(u,v)=0, 0<t<1,\cr

\alpha_1\phi_{p_1}(u(0))-\beta_1\phi_{p_1}(u'(0))=\gamma_1\phi_{p_1}(u(1))+\delta_1\phi_{p_1}(u'(1))=0,

\alpha_2\phi_{p_2}(v(0))-\beta_2\phi_{p_2}(v'(0))=\gamma_2\phi_{p_2}(v(1))+\delta_2\phi_{p_2}(v'(1))=0,

\end{array}\right.$$

where $\phi_{p_i}(s)=|s|^{p_i-2}s, p_i>1, i=1,2$. We obtain some sufficient conditions for the existence of two positive solutions or infinitely many positive solutions by using a fixed-point theorem in cones. Especially, the nonlinear terms $f,g $ are allowed to change sign. The conclusions essentially extend and improve the known results.

$$\left\{\begin{array}{lll}

(\phi_{p_1}(u'))'+a_1(t)f(u,v)=0, 0<t<1,\cr

(\phi_{p_2}(v'))'+a_2(t)g(u,v)=0, 0<t<1,\cr

\alpha_1\phi_{p_1}(u(0))-\beta_1\phi_{p_1}(u'(0))=\gamma_1\phi_{p_1}(u(1))+\delta_1\phi_{p_1}(u'(1))=0,

\alpha_2\phi_{p_2}(v(0))-\beta_2\phi_{p_2}(v'(0))=\gamma_2\phi_{p_2}(v(1))+\delta_2\phi_{p_2}(v'(1))=0,

\end{array}\right.$$

where $\phi_{p_i}(s)=|s|^{p_i-2}s, p_i>1, i=1,2$. We obtain some sufficient conditions for the existence of two positive solutions or infinitely many positive solutions by using a fixed-point theorem in cones. Especially, the nonlinear terms $f,g $ are allowed to change sign. The conclusions essentially extend and improve the known results.

In this paper, we study a kind of higher-order nonlinear fractional differential equation with integral boundary