Mem. Differential Equations Math. Phys. 10(1997), 157–161

I. Kiguradze and B. Puža

ON THE SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS WITH

DEVIATING ARGUMENTS

(Reported on November 18–25, 1996)

In the present note, basing on the results of our previous work [1] we establish sufficient conditions for the existence and uniqueness of a solution of the boundary value problem

\frac{dx(t)}{dt} = g(t, x(τ_{1}(t)), …, x(τ_{m}(t))),  (1)
h(x) = 0,  (2)

where g : [a, b] × R^{nm} → R^{n} is a vector function satisfying the local Carathéodory conditions, τ_{i} : [a, b] → [a, b] (i = 1, …, m) are measurable functions and h : C([a, b]; R^{n}) → R^{n} is a continuous operator.

Under solution of the system (1) we understand an absolutely continuous vector function x : [a, b] → R^{n} which almost everywhere on [a, b] satisfies it, and under solution of the problem (1), (2) we mean a solution of the system (1) which satisfies the condition (2).

The use is made of the following notation: I = [a, b], R = ] − ∞, +∞[ , R_{+} = [0, +∞[ ; R^{n} – the space of n-dimensional column vectors x = (x_{i})_{i=1}^{n} with x_{i} ∈ R (i = 1, …, n)

and the norm

∥x∥ = ∑_{i=1}^{n} |x_{i}|;

R^{n×n} – the space of n × n matrices X = (x_{ik})_{i,k=1}^{n} with x_{ik} ∈ R (i, k = 1, …, n) and the norm

∥X∥ = ∑_{i,k=1}^{n} |x_{ik}|;

R_{+}^{n} = {(x_{i})_{i=1}^{n} ∈ R^{n} : x_{i} ≥ 0 (i = 1, …, n)},
R_{+}^{n×n} = {(x_{ik})_{i,k=1}^{n} ∈ R^{n×n} : x_{ik} ≥ 0 (i, k = 1, …, n)};

if x, y ∈ R^{n} and X, Y ∈ R^{n×n}, then

x ≤ y ⟺ y − x ∈ R_{+}^{n} and X ≤ Y ⟺ Y − X ∈ R_{+}^{n×n};

if x = (x_{i})_{i=1}^{n} ∈ R^{n} and X = (x_{ik})_{i,k=1}^{n} ∈ R^{n×n}, then

|x| = (|x_{i}|)_{i=1}^{n}, |X| = (|x_{ik}|)_{i,k=1}^{n};

1991 Mathematics Subject Classification. 34K10. Key words and phrases. System of nonlinear differential equations with deviating

arguments, boundary value problem, existence theorem, uniqueness theorem.

C(I; R^{n}) – the space of continuous vector functions^{∗} x : I → R^{n} with the norm

∥x∥_{C} = max{∥x(t)∥ : t ∈ I};

C(I; R_{+}^{n}) = {x ∈ C(I; R^{n}) : x(t) ≥ 0 for a ≤ t ≤ b};

L(I; R^{n}) – the space of summable vector functions x : I → R^{n} with the norm

∥x∥_{L} = ∫_{a}^{b} ∥x(t)∥ dt.

Definition 1. Let P : I × R^{n_{0}} → R^{n×n} be a matrix function satisfying the local Carathéodory conditions. We say that a summable matrix function P_{0} : I → R^{n×n} belongs to the set E_{P}^{n_{0}} if there exists a sequence u_{k} ∈ C(I; R^{n_{0}}) (k = 1, 2, …) such that

lim_{k→∞} ∫_{a}^{t} P(s, u_{k}(s)) ds = ∫_{a}^{t} P_{0}(s) ds uniformly on I.

Definition 2. Let l : C(I; R^{n_{0}}) × C(I; R^{n}) → R^{n} be a continuous operator. We say that a linear operator l_{0} : C(I; R^{n}) → R^{n} belongs to the set E_{l}^{n_{0}} if there exists a sequence u_{k} ∈ C(I; R^{n_{0}}) (k = 1, 2, …) such that

lim_{k→∞} l(u_{k}, v) = l_{0}(v) for v ∈ C(I; R^{n}).

Definition 3. An operator h_{0} : C(I; R_{+}^{n}) → R_{+}^{n} is said to be positively homogeneous if for any λ ∈ R_{+} and u ∈ C(I; R_{+}^{n}) we have h_{0}(λu) = λh_{0}(u). However, if for any u, v ∈ C(I; R_{+}^{n}) satisfying u(t) ≤ v(t) for t ∈ I the inequality h_{0}(u) ≤ h_{0}(v) is fulfilled, then h_{0} is said to be nondecreasing.

Definition 4. Let Q_{k} : I → R_{+}^{n×n} (k = 1, …, m) be summable matrix functions, τ_{k} : [a, b] → [a, b] (k = 1, …, m) be measurable functions and h_{0} : C(I; R_{+}^{n}) → R_{+}^{n} be a positively homogeneous continuous nondecreasing operator. Then the writing

(P_{1}, …, P_{m}; l) ∈ O_{Q_{1},...,Q_{m};τ_{1},...,τ_{m};h_{0}}^{n_{1},n_{2}}

means that

(i) P_{k} : I × R^{n_{1}} → R^{n×n} (k = 1, …, n) are matrix functions satisfying the local Carathéodory conditions and l : C(I; R^{n_{2}}) × C(I; R^{n}) → R^{n} is a continuous operator; moreover, l(u, ·) : C(I; R^{n}) → R^{n} is linear for arbitrarily fixed u ∈ C(I; R^{n_{2}}).

(ii) there exist a summable function α : I → R_{+} and a positive number α_{0} such that the inequalities

∥P_{k}(t, x)∥ ≤ α(t) (k = 1, …, m) and ∥l(u, v)∥ ≤ α_{0}∥v∥_{C}

are fulfilled on I × R^{n_{1}} and C(I; R^{n_{2}}) × C(I; R^{n}), respectively; (iii) for any P_{0k} ∈ E_{P}^{n_{1}} (k = 1, …, m) and l_{0} ∈ E_{l}^{n_{2}}, the problem

|\frac{dv(t)}{dt} − ∑_{k=1}^{m} P_{0k}(t)v(τ_{k}(τ))| ≤ ∑_{k=1}^{m} Q_{k}(t)|v(τ_{k}(t)|, |l_{0}(v)| ≤ h_{0}(|v|)

has only the trivial solution.

∗A vector or matrix function is said to be continuous, absolutely continuous, summable, etc., if all its components have such a property.

Theorem 1. Let on I × R^{mn} the inequality

|g(t, x_{1}, …, x_{m}) − ∑_{k=1}^{m} P_{k}(t, x_{1}, …, x_{m})x_{k}| ≤ ∑_{k=1}^{m} Q_{k}(t)|x_{k}| + η(t)  (3)

and on C(I; R^{n}) the inequality

|h(x) − l(x, x)| ≤ h_{0}(|x|) + η_{0}  (4)

be fulfilled, where Q_{k} : I → R_{+}^{n×n} (k = 1, …, m) are summable matrix functions, η : I → R_{+}^{n} is a summable vector function, h_{0} : C(I; R_{+}^{n}) → R_{+}^{n} is a positively homogeneous continuous nondecreasing operator, η_{0} ∈ R_{+}^{n} and

(P_{1}, …, P_{m}; l) ∈ O_{Q_{1},...,Q_{m};τ_{1},...,τ_{m};h_{0}}^{mn,n}.  (5)

Then the problem (1), (2) has at least one solution.

Scheme of the proof. For any x, y ∈ C(I; R^{n}) and u ∈ C(I; R_{+}^{n}), put f(x)(t) = g(t, x(τ_{1}(t), …, x(τ_{m}(t))), p(x, y)(t) = ∑_{k=1}^{m} P_{k}(t, x_{1}(τ_{1}(t), …, x_{m}(τ_{m}(t)))y(τ_{k}(t)) and q_{0}(u)(t) = ∑_{k=1}^{m} Q_{k}(t)u(τ_{k}(t)). Then the system (1) and the condition (3) take respectively the form

\frac{dx(t)}{dt} = f(x)(t),  (1′)
|f(x)(t) − p(x, x)(t)| ≤ q_{0}(|x|)(t) + η(t).  (3′)

Owing to the restrictions imposed on g and τ_{k} (k = 1, …, n), the operator f : C(I; R^{n}) → L(I; R^{n}) is continuous. On the other hand, by Definition 4 and also by Definition 1.3 of [1], we can show that the condition (5) implies the condition

(p, l) ∈ O_{q_{0},h_{0}}^{n}.  (5′)

By Theorem 1.1 from [1], the conditions (3′), (4) and (5′) ensure the solvability of the problem (1′), (2). ■

According to Theorem 1, we can easily prove

Theorem 2. Let on I × R^{mn} the inequality

|g(t, x_{1}, …, x_{m}) − g(t, y_{1}, …, y_{m}) − ∑_{k=1}^{m} P_{k}(t, x_{1}, …, x_{m}, y_{1}, …, y_{m})(x_{k} − y_{k})| ≤

≤ ∑_{k=1}^{m} Q_{k}(t)|x_{k} − y_{k}|,

and on C(I; R^{n}) the inequality

|h(x) − h(y) − l(x, y, x − y)| ≤ h_{0}(|x − y|)

be fulfilled, where Q_{k} : I → R_{+}^{n×n} (k = 1, …, m) are summable matrix functions, h_{0} : C(I; R_{+}^{n}) → R_{+}^{n} is a positively homogeneous continuous nondecreasing operator and

(P_{1}, …, P_{m}; l) ∈ O_{Q_{1},...,Q_{m};τ_{1},...,τ_{m};h_{0}}^{2mn,2n}.

Then the problem (1), (2) has a unique solution.

In the case where the matrix functions P_{k} (k = 1, …, m) depend only on t and l : C(I; R^{n}) → R^{n} is a linear operator, Theorems 1 and 2 will respectively take the following form.

Corollary 1. Let on I × R^{mn} the inequality

|g(t, x_{1}, …, x_{m}) − ∑_{k=1}^{m} P_{k}(t)x_{k}| ≤ ∑_{k=1}^{m} Q_{k}(t)|x_{k}| + η(t)

and on C(I; R^{n}) the inequality

|h(x) − l(x)| ≤ h_{0}(|x|) + η_{0}

be fulfilled, where P_{k} : I → R^{n×n}, Q_{k} : I → R_{+}^{n×m} (k = 1, …, m) are summable matrix functions, η : I → R_{+}^{n} is a summable vector function, l : C(I; R^{n}) → R^{n} is a linear bounded operator and h_{0} : C(I; R_{+}^{n}) → R_{+}^{n} is a positively homogeneous continuous nondecreasing operator. Let, moreover, the problem

|\frac{dv(t)}{dt} − ∑_{k=1}^{m} P_{k}(t)v(τ_{k}(t))| ≤ ∑_{k=1}^{m} Q_{k}(t)|v(τ_{k}(t))|, |l(v)| ≤ h_{0}(|v|)  (6)

have only the trivial solution. Then the problem (1), (2) has at least one solution.

Corollary 2. Let on I × R^{mn} the inequality

|g(t, x_{1}, …, x_{m}) − g(t, y_{1}, …, y_{m}) − ∑_{k=1}^{m} P_{k}(t)(x_{k} − y_{k})| ≤ ∑_{k=1}^{m} Q_{k}(t)|x_{k} − y_{k}|

and on C(I; R^{n}) the inequality

|h(x) − h(y) − l(x − y)| ≤ h_{0}(|x − y|)

be fulfilled, where P_{k} : I → R^{n×n}, Q_{k} : I → R_{+}^{n×n} (k = 1, …, m) are summable matrix functions, l : C(I; R^{n}) → R^{n} is a linear bounded operator and h_{0} : C(I; R_{+}^{n}) → R_{+}^{n} is a positively homogeneous continuous nondecreasing operator such that the problem (6) has only the trivial solution. Then the problem (1), (2) has a unique solution.

Consider now the case where the boundary conditions (2) have the form

φ(x(t_{1}), …, x(t_{m_{0}})) = 0,  (7)

where φ : R^{m_{0}n} → R^{n} is a continuous vector function and t_{k} ∈ I (i = 1, …, m_{0}). For the problem (1), (7), we have from Theorem 2 the following

Corollary 3. Let: (i) for almost all t ∈ I there exist \frac{∂g(t,x_{1},...,x_{m})}{∂x_{k}} (k = 1, …, m) which are continuous with respect to x_{1}, …, x_{m} in R^{mn} and satisfy

P_{1k}(t) ≤ \frac{∂g(t, x_{1}, …, x_{m})}{∂x_{k}} ≤ P_{2k}(t) (k = 1, …, m),

where P_{1k} and P_{2k} : I → R^{n×n} (k = 1, …, m) are summable matrix functions; (ii) the vector function φ have the first order continuous partial derivatives and

A_{1k} ≤ \frac{∂φ(x_{1}, …, x_{m_{0}})}{∂x_{k}} ≤ A_{2k} (k = 1, …, m_{0})

on R^{m_{0}n}, where A_{1k} and A_{2k} ∈ R^{n×n} (k = 1, …, m_{0}); (iii) for any summable matrix functions P_{k} : I → R^{n×n} (k = 1, …, m) and matrices A_{k} ∈ R^{n×n} (k = 1, …, m_{0}) satisfying

P_{1k}(t) ≤ P_{k}(t) ≤ P_{2k}(t) for t ∈ I (k = 1, …, m),
A_{1k} ≤ A_{k} ≤ A_{2k} (k = 1, …, m_{0}),

the boundary value problem

\frac{dv(t)}{dt} = ∑_{k=1}^{m} P_{k}(t)v(τ_{k}(t)), ∑_{k=1}^{m_{0}} A_{k}v(t_{k}) = 0

have only the trivial solution. Then the problem (1), (7) has a unique solution.

Remark. This work was supported by Grant 201/96/0410 of the Grant Agency of the Czech Republic (Prague) and by Grant 619/1996 of the Development Fund of Czech Universities.

References

1. I. T. Kiguradze and B. Puža, Conti-Opial type theorems for nonlinear functional differential equations. (Russian) Differentsial’nye Uravneniya 33(1997), No. 2 (to appear).

Authors’ addresses:

I. Kiguradze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia

B. Puža Department of mathematics Masaryk University Janáčkovo nám. 2a 66 295 Brno Czech Republic