Mem. Differential Equations Math. Phys. 5(1995), 127–130

I. Kiguradze and Z. Sokhadze

EXISTENCE AND CONTINUABILITY OF SOLUTIONS OF THE INITIAL VALUE PROBLEM FOR THE SYSTEM OF SINGULAR FUNCTIONAL DIFFERENTIAL EQUATIONS

(Reported on May 29 and June 5, 12, 1995)

In the present paper we consider the system of functional differential equations

\frac{dx(t)}{dt} = f(x)(t)  (1)

with the weighted initial condition

lim_{t→a} \frac{∥x(t) − c_{0}∥}{h(t)} = 0,  (2)

where f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) is a Volterra operator c_{0} ∈ R^{n}, and h : [a, b] → [0, +∞[ is a continuous function such that h(t) > 0 for a < t ≤ b.

A particular case of (1) is the differential system with the delay

\frac{dx(t)}{dt} = f_{0}(t, x(t), x(τ_{1}(t)), …, x(τ_{m}(t))),  (1′)

where f_{0} :]a, b] × R^{n} → R^{n} is a vector function satisfying the local Carathéodory conditions and τ_{k} : [a, b] → [a, b] (k = 1, …, m) are measurable functions satisfying the inequalities τ_{k}(t) ≤ t for a ≤ t ≤ b (k = 1, …, m).

The initial value problem for regular systems of the type (1) and (1′) has been studied fully enough (see, e.g., [1, 6]). We are interested in the singular systems, i.e., the systems where f(x)(·) and f_{0}(·, x_{0}, x_{1}, …, x_{m}) are not summable on [a, b] for some x ∈ C([a, b]; R^{n}) and x_{k} ∈ R^{n} (k = 0, …, m). So far little is known about the initial value problem for such systems. The exception is the system

\frac{dx(t)}{dt} = f_{0}(t, x(t))

(see [2–5]).

Below we shall give new results on the existence and continuability of solutions of the problems (1), (2) and (1′), (2). To formulate them we shall need the following notation and definitions.

R is the set of real numbers; R_{+} = [0, +∞[;

R^{n} is the space of n-dimensional vectors x = (x_{i})_{i=1}^{n} with elements x_{i} ∈ R (i = 1, …, n) and the norm ∥x∥ = ∑_{i=1}^{n} |x_{i}|; x · y is the scalar product of the vectors x and y ∈ R^{n};

if ρ ∈]0, +∞[, then R_{ρ}^{n} = {x ∈ R^{n} : ∥x∥ ≤ ρ};

if x = (x_{i})_{i=1}^{n}, then sgn(x) = (sgn x_{i})_{i=1}^{n};

C([a, b]; R^{n}) is the space of continuous vector functions x : [a, b] → R^{n} with the norm ∥x∥_{C} = max{∥x(t)∥ : a ≤ t ≤ b∥;

1991 Mathematics Subject Classification. 34K05.

Key words and phrases. System of singular functional differential equations, Volterra operator, initial value problem. if ρ ∈]0, +∞[, then C_{ρ}([a, b]; R^{n}) = {x ∈ C([a, b]; R^{n}) : ∥x∥_{C} ≤ ρ}; if a ≤ s ≤ t ≤ b and x ∈ C([a, b]; R^{n}), then ν(x)(s, t) = max{∥x(ξ)∥ : s ≤ ξ ≤ t}; L_{loc}(]a, b]; R^{n}) is the space of locally summable vector functions x :]a, b] → R^{n} with

the topology of uniform mean convergence on every segment contained in ]a, b].

Definition 1. An operator f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) is said to be Volterra if for every t_{0} ∈]a, b] and any vector functions x and y ∈ C([a, b]; R^{n}) satisfying x(t) = y(t) for a ≤ t ≤ t_{0}, the equality f(x)(t) = f(y)(t) is fulfilled a.e. on ]a, t_{0}.

Definition 2. If f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) is a Volterra operator, then:

(i) for every x ∈ C([a, b]; R^{n}) under f(x) is understood the vector function given by

f(x)(t) = f(\overline{x})(t) for a ≤ t ≤ b_{0},

where \overline{x}(t) = x(t) for a ≤ t ≤ b_{0} and \overline{x} = x(b_{0}) for b_{0} < t ≤ b;

(ii) a continuous vector function x : [a, b_{0}] → R^{n} is said to be the solution of the system (1) on [a, b_{0}] if it is absolutely continuous on every segment contained in ]a, b_{0}] and satisfies (1) a.e. on ]a, b_{0}];

(iii) x : [a, b_{0}[→ R^{n} is said to be the solution of the system (1) in a half-open interval [a, b_{0}[ if for every b_{1} ∈]a, b_{0}[ the restriction of x on [a, b_{1}] is the solution of the same system on [a, b_{1}].

Definition 3. We shall say that an operator f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) satisfies the local Carathéodory conditions if it is continuous and there exists a nondecreasing in the second argument function γ :]a, b] × R_{+}) → R_{+} such that γ(·, ρ) ∈ L_{loc}(]a, b]; R) for ρ ∈ R_{+}, and for any x ∈ C([a, b]; R^{n}) the inequality ∥f(x)(t)∥ ≤ γ(t, ∥x∥_{C}) is fulfilled a.e. on ]a, b[.

In the sequel we shall assume that f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) is a Volterra operator satisfying the local Carathéodory conditions.

Definition 4. A solution x of the system (1) defined on a segment [a, b_{0}] ⊂ [a, b[ (on a half-open interval [a, b_{0}[⊂ [a, b[) is said to be continuable if for some b_{1} ∈]b_{0}, b] (b_{1} ∈ [b_{0}, b]) the system (1) has on the segment [a, b_{1}] a solution y satisfying x(t) = y(t) for a ≤ t ≤ b_{0}. Otherwise x is said to be noncontinuable.

Definition 5. The problem (1), (2) is said to be locally solvable if the system (1) has on a segment [a, b_{0}] a solution x satisfying the initial condition (2).

Theorem 1. Let there exist a positive number ρ and summable functions p and q : [a, b] → R_{+} such that

lim_{t→a} sup (\frac{1}{h(t)} ∫_{a}^{t} p(s)ds) < 1,

lim_{t→a} sup (\frac{1}{h(t)} ∫_{a}^{t} q(s)ds) = 0

(3)

and let for any y ∈ C_{ρ}([a, b]; R^{n}) the inequality

f(c_{0} + hy)(t) · sgn(y(t)) ≤ p(t)ν(y)(a, t) + q(t).

be fulfilled a.e. on ]a, b[. Then the problem (1), (2) is locally solvable.

Corollary 1. Let there exist summable functions p_{k} : [a, b] → R_{+} (k = 0, …, m)

and q : [a, b] → R_{+} such that the conditions (3) are fulfilled, where p(t) = ∑_{k=0}^{m} p_{k}(t), and let for some ρ > 0 the inequality

f_{0}(t, c_{0} + h(t)y_{0}, c_{0} + h(τ_{1}(t))y_{1}, …, c_{0} + h(τ_{m}(t))y_{m}) · sgn(y_{0}) ≤

≤ ∑_{k=0}^{m} p_{k}(t)∥y_{k}∥ + q(t).

be fulfilled on ]a, b[×R_{ρ}^{(m+1)n}. Then the problem (1′), (2) is locally solvable.

Example 1. Let α_{k} and β_{k} ∈ R, μ ∈]0, +∞[, λ_{k} ∈]1, +∞[ (k = 1, …, m), a = 0, b = 1, n = 1 and ∑_{k=1}^{m} |α_{k}| < μ. Then because of Corollary 1, the problem

\frac{dx(t)}{dt} = − ∑_{k=1}^{l} exp (\frac{k}{t} )x^{2k−1}(t) +

+ ∑_{k=1}^{m} [α_{k} \frac{x(t^{k})}{t^{(k−1)μ+1}} + β_{k} \frac{|x(t^{k})|^{λ_{k}}}{t^{kμλ_{k}−μ+1}} ] + \frac{t^{μ−1}}{1 + | ln t|} ,

lim_{t→0} \frac{x(t)}{t^{μ}} = 0

is solvable. Consequently, under the conditions of Corollary 1, the right-hand side of the system (1′) with respect to the first argument may have nonintegrable singularities of arbitrary order.

Example 2. Let n = 1, a = 0, b = 1, α_{k} ∈ R_{+} (k = 1, …, m) and

q(t) = \frac{1}{ln(2 + | ln t|)} + \frac{1}{(2 + | ln t|) ln^{2}(2 + | ln t|)} ,

p_{k}(t) = α_{k} ln(2 + | ln t^{k}|)q(t), p(t) = ∑_{k=1}^{m} p_{k}(t).

Then

lim_{t→0} (\frac{1}{t} ∫_{0}^{t} p(s)ds) = ∑_{k=1}^{m} α_{k}.

Therefore, owing to Corollary 1, the inequality ∑_{k=1}^{m} α_{k} < 1 guarantees solvability of the problem

\frac{dx(t)}{dt} = ∑_{k=1}^{m} p_{k}(t) \frac{|x(t^{k})|}{t^{k}} + q(t), lim_{t→0} \frac{x(t)}{t} = 0.  (4)

On the other hand, it is not difficult to show that if ∑_{k=1}^{m} α_{k} ≥ 1, then the problem (4) has no solution. Hence, the condition

lim_{t→a} sup (\frac{1}{t} ∫_{0}^{t} p(s)ds) < 1

in Theorem 1 and Corollary 1 is optimal and it cannot be replaced by the condition

lim_{t→a} sup (\frac{1}{h(t)} ∫_{0}^{t} p(s)ds) ≤ 1.

Theorem 2. Let there exist c ∈ R^{n}, a nondecreasing function τ : [a, b] → [a, b] and a decreasing in the second argument function φ : [a, b] × R_{+} → R_{+} such that τ(t) ≤ t for a ≤ t ≤ b, φ(·, ρ) ∈ L_{loc}(]a, b]; R) for ρ ∈ R_{+} and let for any y ∈ C([a, b]; R^{n}) the inequality

f(c + y)(t) · sgn(y(t)) ≤ φ(t, ν(y)(τ(t), t))

be fulfilled a.e. on [a, b]. Moreover, let x be the solution of the system (1) on an interval [a, b_{0}[⊂ [a, b[. Then for the x to be noncontinuable, it is necessary and sufficient that

lim_{t→b_{0}} ν(x)(τ(t), t) = +∞.

Corollary 2. If x is a solution of the system (1′) on an interval [a, b_{0}[⊂ [a, b[, then for its noncontinuability it is necessary and sufficient that

τ(t) = ess min{τ_{i}(s) : t ≤ s ≤ b_{0}; i = 1, …, m}.

Theorem 3. If the conditions of Theorem 1 are fulfilled, then the problem (1), (2) has at least one noncontinuable solution.

Corollary 3. If the conditions of Corollary 1 are fulfilled, then the problem (1′), (2) has at least one noncontinuable solution.

References

1. N. V. Azbelev, V. P. Maksimov and L. F. Rakhmatullina, Introduction to the theory of functional differential equations. (Russian) Nauka, Moscow, 1990.

2. B. A. Chechik, Investigation of systems of ordinary differential equations with singularity. (Russian) Trudy Moskov. Mat. Obshch. 8(1959), 155–197.

3. I. T. Kiguradze, On the Cauchy problem for singular systems of ordinary differential equations. (Russian) Differentsial’nye Uravneniya 1(1965), No. 10, 1271–1291.

4. I. T. Kiguradze, Some singular boundary value problems for ordinary differential equations. (Russian) Tbilisi University Press, Tbilisi, 1975.

5. I. T. Kiguradze, On the singular Cauchy problem for systems of linear ordinary differential equations. (Russian) Differentsial’nye Uravneniya (to appear).

6. A. D. Myshkis, General theory of differential equations with deviating argument. (Russian) Uspekhi Mat. Nauk 4(1949), No. 5, 99–141.

Authors’ addresses: I. Kiguradze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia Z. Sokhadze A. Tsereteli Kutaisi State University 55, Queen Tamar St., Kutaisi 384000 Georgia