GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 3, 1997, 259-278

ON SINGULAR FUNCTIONAL DIFFERENTIAL INEQUALITIES

I. KIGURADZE AND Z. SOKHADZE

Abstract. Classical theorems on differential inequalities [1, 2, 3] are generalized for initial value problems of the kind

\frac{dx(t)}{dt} ≤ f(x)(t), lim_{t→a} ∥[x(t) − c_{0}]_{+}∥/h(t) = 0

and

\frac{dx(t)}{dt} ≥ f(x)(t), lim_{t→a} ∥[x(t) − c_{0}]_{−}∥/h(t) = 0,

where f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) is a singular Volterra operator, c_{0} ∈ R^{n}, h : [a, b] → [0, +∞[ is continuous and positive on ]a, b], ∥ · ∥ is a norm in R^{n}, and [u]_{+} and [u]_{−} are respectively the positive and the negative part of the vector u ∈ R^{n}.

§ 1. Statement of the Basic Results

1.1. Main Notation and Definitions. Throughout this paper use will be made of the following notation:

R is the set of real numbers; R_{+} = [0, +∞[ ; R^{n} is the space of n-dimensional column vectors x = (x_{i})_{i=1}^{n} with ele-

ments x_{i} ∈ R (i = 1, …, n) and norm ∥x∥ = ∑_{i=1}^{n} |x_{i}|; R_{+}^{n}={x=(x_{i})_{i=1}^{n} ∈R^{n} : x_{i} ≥ 0 (i=1, …, n)}; R_{ρ}^{n}={x∈R^{n} : ∥x∥≤ ρ}; if x = (x_{i})_{i=1}^{n}, then

sgn(x) = (sgn x_{i})_{i=1}^{n}; [x]_{+} = (\frac{|x_{i}| + x_{i}}{2} )_{i=1}^{n}, [x]_{−} = (\frac{|x_{i}| − x_{i}}{2} )_{i=1}^{n};

if x and y ∈ R^{n}, then x ≤ y ⟺ y − x ∈ R_{+}^{n}; x · y is the scalar product of vectors x and y ∈ R^{n};

1991 Mathematics Subject Classification. 34K05. Key words and phrases. Singular Volterra operator, functional differential equa-

tion, functional differential inequality, weighted initial value problem, upper and lower solutions.

259

1072-947X/97/0500-0259$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

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};

C_{ρ}([a, b]; R^{n}) = {x ∈ C([a, b]; R^{n}) : ∥x∥_{C} ≤ ρ};
C([a, b]; R_{+}) = {x ∈ C([a, b]; R) : x(t) ≥ 0 for a ≤ t ≤ b};

if x ∈ C([a, b]; R^{n}) and a ≤ s ≤ t ≤ b, then

ν(x)(s, t) = max {∥x(ξ)∥ : s ≤ ξ ≤ t};

L_{loc}(]a, b]; R^{n}) is the space of vector functions x : ]a, b] → R^{n} summable on every segment contained in ]a, b] with the topology of mean convergence on every segment from ]a, b];

L_{loc}(]a, b]; R_{+}) = {x ∈ L_{loc}(]a, b]; R) : x(t) ≥ 0 for almost all t ∈ [a, b]}.

Definition 1.1. 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 every 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 1.2. We 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 function γ : ]a, b] × R_{+} → R_{+} nondecreasing in the second argument 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[.

If

∫_{a}^{b} γ(t, ρ)dt < +∞ for ρ ∈ R_{+},

then the operator f is called regular. Otherwise f is called singular.

Definition 1.3. An operator f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) is said to be nondecreasing if for every x and y ∈ C([a, b]; R^{n}) satisfying x(t) ≤ y(t) for a ≤ t ≤ b, the inequality f(x)(t) ≤ f(y)(t) is fulfilled a.e. on ]a, b[.

In the present paper we consider the following weighted initial value problems:

\frac{dx(t)}{dt} = f(x)(t),  (1.1)
lim_{t→a} \frac{∥x(t) − c_{0}∥}{h(t)} = 0;  (1.2)
\frac{dz(t)}{dt} ≤ f(z)(t),  (1.3)

lim_{t→a} \frac{∥[z(t) − c_{0}]_{+}∥}{h(t)} = 0;  (1.4)
\frac{dz(t)}{dt} ≥ f(z)(t),  (1.5)
lim_{t→a} \frac{∥[z(t) − c_{0}]_{−}∥}{h(t)} = 0,  (1.6)

where everywhere below f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) is assumed to be a Volterra, in general singular, operator satisfying the local Carathéodory conditions, c_{0} ∈ R^{n}, and h : [a, b] → [0, +∞[ is continuous, nondecreasing and positive on ]0, a].

We shall separately consider the case where h(a) > 0, so that the conditions (1.2), (1.4) and (1.6) have respectively the form

x(a) = c_{0},  (1.2_{1})
z(a) ≤ c_{0},  (1.4_{1})
z(a) ≥ c_{0}.  (1.6_{1})

The question of estimating solutions of functional differential inequalities (1.3) and (1.5) satisfying the initial conditions

z(a) < c_{0},  (1.4_{2})
z(a) > c_{0}  (1.6_{2})

will also be studied.

The vector differential equation and differential inequalities with delay:

\frac{dx(t)}{dt} = f_{0}(t, x(t), x(τ_{1}(t)), …, x(τ_{m}(t))),  (1.7)
\frac{dz(t)}{dt} ≤ f_{0}(t, z(t), z(τ_{1}(t)), …, z(τ_{m}(t))),  (1.8)
\frac{dz(t)}{dt} ≥ f_{0}(t, z(t), z(τ_{1}(t)), …, z(τ_{m}(t)))  (1.9)

are important particular cases of the vector functional differential equation (1.1) and of the vector functional differential inequalities (1.3) and (1.4).

In the sequel, when considering equation (1.7) or inequalities (1.8) and (1.9), it will be assumed that the vector function f_{0} : ]a, b[ ×R^{(m+1)n} → R^{n} satisfies the local Caratéodory conditions, i.e., f_{0}(t, ·, …, ·) : R^{(m+1)n} → R^{n} is continuous for almost all t ∈]a, b[, f_{0}(·, x_{0}, x_{1}, …, x_{n}) : ]a, b[ ×R^{n} is measurable for all x_{k} ∈ R^{n} (k = 0, 1, …, m), and on the set ]a, b[ ×R^{(m+1)n} the following inequality is fulfilled:

∥f_{0}(t, x_{0}, x_{1}, …, x_{m})∥ ≤ γ(t, ∑_{k=0}^{m} ∥x_{k}∥),

where γ : ]a, b] × R_{+} → R_{+} does not decrease in the second argument, and γ(·, ρ) ∈ L_{loc}(]a, b]; R_{+}) for ρ ∈ R_{+}. As for τ_{i} : [a, b] → [a, b] (i = 1, …, m), they are measurable, and

τ_{i}(t) ≤ t for a ≤ t ≤ b (i = 1, …, m).

By f^{∗}(·, c_{0}, ρ) and f_{0}^{∗}(·, c_{0}, ρ) are meant the functions given by the equalities

f^{∗}(t, c_{0}, ρ) = sup {f(t, c_{0} + y)(t) · sgn(y(t)) : y ∈ C_{ρ}([a, b]; R^{n})},
f_{0}^{∗}(t; c_{0}, ρ) = sup {f_{0}(t, c_{0} + y_{0}, c_{0} + y_{1}, …, c_{0} + y_{m}) · sgn(y_{0}) :
y_{0} ∈ R_{ρ}^{m}, …, y_{m} ∈ R_{ρ}^{n}}.

Definition 1.4. If b_{0} ∈]a, b], then

(i) for every x ∈ C([a, b_{0}]; R^{n}) by f(x) is understood the vector function given by the equality 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}(t) = x(b_{0}) for b_{0} < t ≤ b.

(ii) a continuous vector function x : [a, b_{0}] → R^{n} is said to be a solution of equation (1.1) (of inequality (1.3) or (1.5)) on the segment [a, b_{0}] if x is absolutely continuous on every segment contained in ]a, b_{0}] and satisfies equation (1.1) (inequality (1.3) or (1.5)) a. e. on ]a, b_{0}[;

(iii) a vector function x : [a.b_{0}[→ R^{n} is said to be a solution of equation (1.1) (of inequality (1.3) or (1.5)) on a half-open interval [a, b_{0}[ if for every b_{1} ∈]a, b_{0}[ the restriction of x on [a, b_{1}] is a solution of this equation (inequality) on the segment [a, b_{1}];

(iv) a solution x of equation (1.1) (of inequality (1.3) or (1.5)) satisfying the initial condition (1.2) (the initial condition (1.4) or (1.6)) is said to be a solution of problem (1.1), (1.2) (of problem (1.1), (1.4) or (1.5), (1.6)).

Definition 1.5. A solution x of equation (1.1) 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]) equation (1.1) has a solution y on the segment [a, b_{1}] satisfying the condition x(t) = y(t) for a ≤ t ≤ b_{0}. Otherwise x is said to be noncontinuable.

Definition 1.6. A solution x^{∗} (solution x_{∗}) of problem (1.1), (1.2) defined in the interval I_{0} ⊂ [a, b] is said to be upper (lower) if an arbitrary solution x of this problem defined in some interval I ⊂ [a, b] satisfies

x(t) ≤ x^{∗}(t) (x(t) ≥ x_{∗}(t)) for t ∈ I_{0} ∩ I.

1.2. Existence of a Non-Continuable Solution of Problem (1.1), (1.2). In [4] the following propositions are proved.

Theorem 1.1. Let there exist a positive number ρ and summable functions p and q : [a, b] → R_{+} such that

lim sup_{t→a} (\frac{1}{h(t)} ∫_{a}^{t} p(s)ds) < 1, lim_{t→a} (\frac{1}{h(t)} ∫_{a}^{t} q(s)ds) = 0 (1.10)

and for every 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)  (1.11)

is fulfilled a.e. on [a, b[. Then problem (1.1), (1.2) has at least one noncontinuable solution and every continuable solution of this problem is a restriction of some noncontinuable solution.

Corollary 1.1. Let for some ρ > 0, on the set ]a, b[ ×R_{ρ}^{(m+1)n} the following inequality be fulfilled:

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),

where p_{k} : [a, b] → R_{+} (k = 0, …, m) and q : [a, b] → R_{+} are summable functions satisfying conditions (1.10), where p(t) = ∑_{k=0}^{m} p_{k}(t). Then problem (1.7), (1.2) has at least one noncontinuable solution, and every continuable solution of this problem is a restriction of some noncontinuable solution.

1.3. Theorems on the Existence of Upper and Lower Solution of Problem (1.1), (1.2).

Theorem 1.2. Let f be a nondecreasing operator satisfying the conditions of Theorem 1.1. Let, moreover, b^{∗} ∈]a, b] be such that the interval of definition of an arbitrary noncontinuable solution of problem (1.1), (1.2) contains the segment [a, b^{∗}]. Then problem (1.1), (1.2) has on the segment [a, b^{∗}] the upper and the lower solution.

Corollary 1.2. Let f be a nondecreasing operator and for some ρ > 0 and b^{∗} ∈]a, b] let the inequality

∫_{a}^{b^{∗}}f^{∗}(t; c_{0}, ρ)dt < ρ  (1.12)

be fulfilled. Then the interval of definition of an arbitrary noncontinuable solution of problem (1.1), (1.2_{1}) contains the segment [a, b^{∗}] and this problem has on [a, b^{∗}] the upper and the lower solution.

Let n = 1 and f(x)(t) = g(t, x(t)), where f_{0} : [a, b] × R → R is a function from the Carathéodory class. Then we can exclude from Corollary 1.2 the requirement for f to be a non-decreasing operator (see [1], Ch. II, §1, Theorem 1.2). However, if f is an arbitrary operator, then this requirement is essential and it cannot be neglected. To verify the above-said, let us consider

Example 1.1. J. Heidel [5]^{1} proved the existence of a continuous function p : [a, b] →] − ∞, 0[ such that the equation has on the segment [a, b] a non-trivial solution u′′(t) = p(t)u^{1/3}(t) which in any neighborhood of the point b changes sign, and u(b) = u′(b) = 0. The function x(t) = u(b + a − t) is obviously a non-zero solution of the problem

\frac{dx(t)}{dt} = ∫_{a}^{t} p(a + b − s)x^{1/3}(s)ds,  (1.13)
x(a) = 0.  (1.14)

On the other hand, if for ρ = 1 we choose a number b^{∗} such that

(b^{∗} − a) ∫_{a}^{b^{∗}}|p(a + b − s)|ds < 1,

then the operator f(x)(t) = ∫_{a}^{t} p(a + b − s)x^{1/3}(s)ds will satisfy condition (1.12). However, since p is negative, the operator f is not nondecreasing. Suppose that problem (1.13), (1.14) has on the interval [a, b^{∗}] the upper solution x^{∗}. Since this problem has as well the zero solution, x^{∗}(t) ≥ 0 for a ≤ t ≤ b^{∗}, whence, owing to the fact that p is negative and because of (1.14), we obtain \frac{dx^{∗}(t)}{dt} ≤ 0 for a ≤ t ≤ b^{∗} and x^{∗}(t) = 0 for a ≤ t ≤ b^{∗}. But this contradicts the inequality x^{∗}(t) ≥ x(t) for a ≤ t ≤ b^{∗}, since the solution x takes positive values in any right-hand neighborhood of the point a. The above-obtained contradiction proves that problem (1.13), (1.14) has no upper solution. The fact that this problem has no lower solution either is proved analogously.

Corollary 1.3. Let f be a nondecreasing operator satisfying the conditions of Theorem 1.1. Then problem (1.1), (1.2) has the noncontinuable upper and the noncontinuable lower solution.

1See also [6], Theorem 17.7.

Corollary 1.4. Let the vector function f_{0} be nondecreasing in the last (m + 1)n variables and let the conditions of Corollary 1.1 be fulfilled. Moreover, let b^{∗} ∈]a, b] be such that the interval of definition of an arbitrary noncontinuable solution of problem (1.7), (1.2) contains the segment [a, b^{∗}]. Then problem (1.7), (1.2) has on the segment [a, b^{∗}] the upper and the lower solution.

Corollary 1.5. Let the vector function f_{0} be nondecreasing in the last (m + 1)n variables and let for some ρ > 0 and b^{∗} ∈]a, b] the inequality

∫_{a}^{b^{∗}}f_{0}^{∗}(t; c_{0}, ρ)dt < ρ

be fulfilled. Then problem (1.7), (1.2_{1}) has on the segment [a, b^{∗}] the upper and the lower solution.

Corollary 1.6. Let the vector function f_{0} be nondecreasing in the last (m + 1)n variables and let the conditions of Corollary 1.2 be fulfilled. Then problem (1.7), (1.2) has the noncontinuable upper and the noncontinuable lower solution.

1.4. Theorems on Functional Differential Inequalities.

Theorem 1.3. Let f be a nondecreasing operator satisfying the conditions of Theorem 1.1. Moreover, let x^{∗} be the upper solution (x_{∗} be the lower solution) of problem (1.1), (1.2) on the interval I_{0} ⊂ [a, b], and let z be a solution of problem (1.3), (1.4) (of problem (1.5), (1.6)) in the interval I ⊂ [a, b]. Then

z(t) ≤ x^{∗}(t) (z(t) ≥ x_{∗}(t)) for t ∈ I ∩ I_{0}.  (1.15)

Corollary 1.7. Let f be a nondecreasing operator satisfying f^{∗}(·; c_{0}, ρ) ∈ L([a, b]; R) for some ρ > 0. Moreover, let x^{∗} be the upper solution (x_{∗} be the lower solution) of problem (1.1), (1.2_{1}) on the interval I ⊂ [a, b], and let z be a solution of problem (1.3), (1.4_{1}) (of problem (1.5), (1.6_{1})) on the interval I ⊂ [a, b]. Then estimate (1.15) is valid.

Corollary 1.8. Let the vector function f_{0} be nondecreasing in the last (m + 1)n variables and let the conditions of Corollary 1.1 be fulfilled. Moreover, let x^{∗} be the upper solution (x_{∗} be the lower solution) of problem (1.7), (1.2) on the interval I_{0} ⊂ [a, b], and let z be a solution of problem (1.8), (1.4) (of problem (1.9), (1.6)) on the interval I ⊂ [a, b]. Then the estimate (1.15) is valid.

Corollary 1.9. Let the vector function f_{0} be nondecreasing in the last (m + 1)n variables and let f_{0}^{∗}(·; c_{0}, ρ) ∈ L([a, b]; R) for some ρ > 0. Moreover, let x^{∗} be the upper solution (x_{∗} be the lower solution) of problem (1.7), (1.2_{1}), and let z be a solution of problem (1.8), (1.4_{1}) (of problem (1.9), (1.6_{1})) on the interval I ⊂ [a, b]. Then estimate (1.15) is valid.

Theorem 1.4. Let f be a nondecreasing operator satisfying the conditions of Theorem 1.1. Moreover, let x_{∗} be the upper solution (x_{∗} be the lower solution) of problem (1.1), (1.2) on the interval I_{0}, and let z be a solution of problem (1.3), (1.4_{2}) (of problem (1.5), (1.6_{2})) on the interval I. Then

z(t) < x_{∗}(t) (z(t) > x^{∗}(t)) for t ∈ I ∩ I_{0}.  (1.16)

Corollary 1.10. Let the vector function f_{0} be nondecreasing in the last (m + 1)n variables and let the conditions of Corollary 1.1 be fulfilled. Moreover, let x_{∗} be the upper solution (x^{∗} be the lower solution) of problem (1.7), (1.2), and let z be a solution of problem (1.8), (1.4_{2}) (of problem (1.9), (1.6_{2})). Then the estimate (1.16) is valid.

§ 2. Auxiliary Propositions

2.1. Lemma on an a priori estimate. In [4] the following Lemma is proved.

Lemma 2.1. Let p and q : [a, b] → R_{+} be summable functions satisfying conditions (1.10) and let numbers b_{0} ∈]a, b] and α ∈]0, 1[ be such that

∫_{a}^{t} p(s)ds ≤ αh(t) for a ≤ t ≤ b_{0}.

Then any solution x : [a, b_{0}] → R^{n} of the differential inequality

x′′(t) · sgn(x(t) − c_{0}) ≤ p(t)ν(\frac{1}{h}(x − c_{0}))(a, t) + q(t)  (2.1)

satisfying the initial condition (1.2) admits the estimate

∥x(t) − c_{0}∥ ≤ h(t)ε(t) for a ≤ t ≤ b_{0},  (2.2)

where ε(a) = 0 and

ε(t) = \frac{1}{1 − α} sup \{\frac{1}{h(s)} ∫_{a}^{s} q(ξ)dξ : a < s ≤ t\} for a < t ≤ b_{0}. (2.3)

2.2. Lemma on the Boundedness of a Set of Solutions of Problem (1.1), (1.2). Denote by I^{∗}(f; c_{0}, h) the set of those b^{∗} ∈]a, b] for which the interval of definition of every noncontinuable solution of problem (1.1), (1.2) contains the segment [a, b^{∗}].

Lemma 2.2. If the conditions of Theorem 1.1 are fulfilled, then I^{∗}(f; c_{0}, h) ≠ ∅. Moreover, for every b^{∗} ∈ I^{∗}(f; c_{0}, h) the set X of restrictions of all noncontinuable solutions of problem (1.1), (1.2) on [a, b^{∗}] is a compact set of the space C([a, b^{∗}]; R^{n}). Moreover, there exists a bounded function h^{∗} : [a, b^{∗}] → R_{+} such that

lim_{t→a} \frac{h^{∗}(t)}{h(t)} = 0  (2.4)

and an arbitrary solution x ∈ X admitting the estimate

∥x(t) − c_{0}∥ ≤ h^{∗}(t) for a ≤ t ≤ b^{∗}.  (2.5)

Proof. Let ρ, b_{0}, α and ε be the numbers and the function appearing in Theorem 1.1 and Lemma 2.1. By (1.10) and (2.3), without loss of generality, we may assume that

ε(t) < ρ for a ≤ t ≤ b_{0}.  (2.6)

Let x be an arbitrary noncontinuable solution of problem (1.1), (1.2) which is defined on the interval I_{0}. Let us show that [a, b_{0}] ⊂ I_{0} and

∥x(t) − c_{0}∥ < ρh(t) for a < t ≤ b_{0}.  (2.7)

Assume the contrary. Then by Corollary 3.1 from [4], there exists t_{0} ∈ ]a, b_{0}] ∩ I_{0} such that

∥x(t) − c_{0}∥ < ρh(t) for a < t < t_{0}, ∥x(t_{0}) − c_{0}∥ = ρh(t_{0}). (2.8)

Due to conditions (1.11) and (2.8), the vector function x is a solution of problem (2.1), (1.2) on the interval [a, t_{0}]. By Lemma 2.1, it admits the estimate ∥x(t) − c_{0}∥ ≤ ε(t)h(t) for a ≤ t ≤ t_{0}. From this, owing to (2.6), we find that ∥x(t_{0}) − c_{0}∥ < ρh(t_{0}), which contradicts condition (2.8). The obtained contradiction proves that b_{0} ∈ I^{∗}(f; c_{0}, h), and an arbitrary noncontinuable solution x of problem (1.1), (1.2) admits estimate (2.7). Conditions (1.10), (2.7) and Lemma 2.1 result in estimate (2.2).

Let b^{∗} ∈ I^{∗}(f, c_{0}, h), and let X be the set of restrictions of all noncontinuable solutions of problem (1.1), (1.2) on [a, b^{∗}]. If b^{∗} = b_{0}, then according to the above arguments, X is the bounded set. Consider the case where b_{0} < b^{∗}, and prove that X is bounded. For any t ∈]a, b^{∗}] assume

r(t) = sup{∥x(t)∥ : x ∈ X}, r^{∗}(t) = sup{r(s) : a ≤ s ≤ t}.

Because of (2.2), r^{∗}(b_{0}) < +∞. Denote by t^{∗} the exact upper bound of the set of t ∈ [a, b^{∗}] for which r^{∗}(t) < +∞. Obviously, t^{∗} ∈ [b_{0}, b^{∗}]. Show that r^{∗}(t^{∗}) < +∞. Assume the contrary. Then t^{∗} > b_{0}, and there exist a sequence t_{k} ∈]b_{0}, t^{∗}[ (k = 1, 2, …) and a sequence (x_{k})_{k=1}^{∞} of noncontinuable solutions of problem (1.1), (1.2) such that

lim_{k→∞} t_{k} = t^{∗}, lim_{k→+∞} ∥x_{k}(t_{k})∥ = +∞.  (2.9)

By virtue of (1.10) and (2.2), for every β ∈ [b_{0}, t^{∗}[ and natural k we have ∥x_{k}(t) − c_{0}∥ ≤ h_{β}(t) for a ≤ t ≤ β, where

h_{β}(t) = \begin{cases}ε(t)h(t) & for a ≤ t ≤ b_{0} \\ ∥c_{0}∥ + r^{∗}(β) & for b_{0} < t ≤ β \end{cases} , lim_{t→a} \frac{h_{β}(t)}{h(t)} = 0.

Applying now Lemma 2.1 from [4], it becomes clear that (x_{k})_{k=1}^{∞} contains a subsequence (x_{k_{j}})_{j=1}^{∞} which converges uniformly on every segment contained in [a, t^{∗}[ , and

x_{0}(t) = lim_{j→+∞} x_{k_{j}}(t) for a ≤ t < t^{∗}  (2.10)

is a solution of problem (1.1), (1.2) on [a, t^{∗}[ . According to one of the conditions of the lemma, x_{0} is a continuable solution, and thus

r_{0} = sup {∥x_{0}(t)∥ : a ≤ t < t^{∗}} < +∞.  (2.11)

Let

γ(t; δ) = sup {∥f(y)(t)∥; y ∈ C_{δ}([a, b]; R^{n})}.  (2.12)

Then γ(·; δ) ∈ L_{loc}(]a, b]; R_{+}) for δ ∈ R_{+}, and for every y ∈ C([a, b]; R^{n}) the inequality

f(y)(t) · sgn(y(t)) ≤ γ(t; ν(y)(a, t))  (2.13)

is fulfilled a.e. on ]a, b[ .

By (2.10), (2.11), and (2.13), there exist t_{∗} ∈]b_{0}, t^{∗}[ and a natural number k_{0} such that

∫_{t_{∗}}^{t^{∗}} γ(t, r_{0} + 2)dt < 1, ∥x_{k}(t)∥ < r_{0} + 1 for a ≤ t ≤ t_{∗} (k = k_{0}, k_{0} + 1, …),

x′_{k}(t) · sgn(x_{k}(t)) ≤ γ(t, ν(x_{k})(a, t)) for a < t < t^{∗} (k = k_{0}, k_{0} + 1, …).

This, owing to Lemma 3.1 from [4], implies that

∥x_{k}(t)∥ < r_{0} + 2 for a < t < t^{∗} (k = k_{0}, k_{0} + 1, …).

But this contradicts condition (2.9). The obtained contradiction proves that r^{∗}(t^{∗}) < +∞.

If now we again apply Lemma 3.1 from [4], then from the condition r^{∗}(t^{∗}) < +∞ and the definition of t^{∗} we conclude that t^{∗} = b^{∗}. Thus the boundedness of X is proved. Moreover, we have proved that an arbitrary solution x ∈ X admits estimate (2.5), where h^{∗}(t) = ε(t)h(t) for a ≤ t ≤ b_{0}, h^{∗}(t) = r^{∗}(b^{∗}) for b_{0} < t ≤ b^{∗}, the function h^{∗} being bounded and satisfying condition (2.4). This, due to Lemma 2.1 from [4], implies that X is a compactum from C([a, b^{∗}]; R^{n}).

2.3. Lemmas on Differential Inequalities.

Lemma 2.3. Let f be a nondecreasing operator satisfying the conditions of Theorem 1.1, and b^{∗} ∈ I^{∗}(f; c_{0}, h). Then for an arbitrary solution z of problem (1.3), (1.4) (of problem (1.5), (1.6)) defined on some segment [a, \overline{b}] ⊂ [a, b^{∗}] there exists a solution \overline{x} (a solution \underline{x}) of problem (1.1), (1.2) such that

z(t) ≤ \overline{x}(t) (z(t) ≥ \underline{x}(t)) for a ≤ t ≤ \overline{b}.  (2.14)

Proof. Let ρ be the number appearing in Theorem 1.1 and let z : [a, \overline{b}] → R^{n} be a solution of problem (1.3), (1.4).

By (1.4), there exists an absolutely continuous vector function z_{0} : [a, \overline{b}] → R^{n} such that

z(t) < z_{0}(t) for a < t ≤ \overline{b}  (2.15)

and

ε_{0}(t) = sup \{\frac{∥z_{0}(s) − c_{0}∥}{h(s)} : a < s ≤ t\} → 0 for t → a. (2.16)

On the other hand, according to (1.10), we can choose the numbers b_{0} ∈]a, \overline{b}[ and α ∈]0, 1[ such that

∫_{a}^{t} p(s)ds ≤ αh(t), ε(t) = t − a + \frac{1 + α}{1 − α}ε_{0}(t) +

+\frac{1}{1 − α} sup \{\frac{1}{h(s)} ∫_{a}^{s} q(ξ)dξ : a < s ≤ t\} < ρ for a < t ≤ b_{0}. (2.17)

It is also clear that

lim_{t→a} ε(t) = 0.  (2.18)

For any natural k suppose

t_{k} = a + \frac{b − a}{2k}, f_{k}(x)(t) = \begin{cases}z′_{0}(t) & for a < t ≤ t_{k} \\ f(x)(t) & for t_{k} < t < b \end{cases}  (2.19)

and consider the differential equation

\frac{dx(t)}{dt} = f_{k}(x)(t).  (2.20)

By virtue of Theorem 1.1, for every natural k problem (2.20), (1.2_{1}) has a noncontinuable solution x_{k} defined on some interval I_{k}.

By (2.16) and (2.19), it is clear that I_{k} ⊃ [a, t_{k}],

x_{k}(t) = z_{0}(t) for a ≤ t ≤ t_{k}  (2.21)

and on ]a, t_{k}] the following inequality is fulfilled:

∥x_{k}(t) − c_{0}∥ < ε(t)h(t).  (2.22)

Let us show that I_{k} ⊃ [a, b_{0}] and inequality (2.22) is fulfilled on the whole ]a, b_{0}]. Assume the contrary. Then by Corollary 3.1 from [4], there exists b_{k} ∈ I_{k}∩]t_{k}, b_{0}[ such that

∥x_{k}(t) − c_{0}∥ < ε(t)h(t) for a < t < b_{k},  (2.23)
∥x_{k}(b_{k}) − c_{0}∥ = ε(b_{k})h(b_{k}).  (2.24)

Owing to (1.11), (2.17), (2.19) and (2.23), the inequality

∥x_{k}(t) − c_{0}∥′ = f(x_{k})(t) · sgn(x_{k}(t) − c_{0}) ≤

≤ p(t)ν(\frac{1}{h}(x_{k} − c_{0}))(a, t) + q(t)

is fulfilled a.e. on ]t_{k}, b_{k}[ . On the other hand, ∥x_{k}(t_{k})−c_{0}∥ = ∥z_{0}(t_{k})−c_{0}∥ ≤ ε_{0}(t_{k})h(t_{k}) ≤ ε_{0}(t)h(t) for t_{k} ≤ t ≤ b_{k} and

ν(\frac{1}{h}(x_{k} − c_{0}))(a, t) ≤ ν(\frac{1}{h}(x_{k} − c_{0}))(a, t_{k}) + ν(\frac{1}{h}(x_{k} − c_{0}))(t_{k}, t) ≤

≤ ν(\frac{1}{h}(z_{0} − c_{0}))(a, t_{k}) + ν(\frac{1}{h}(x_{k} − c_{0}))(t_{k}, t) ≤

≤ ε_{0}(t) + ν(\frac{1}{h}(x_{k} − c_{0}))(t_{k}, t) for t_{k} ≤ t ≤ b_{k}.

Therefore

∥x_{k}(t) − c_{0}∥ ≤ ∥x_{k}(t_{k}) − c_{0}∥ + ∫_{t_{k}}^{t} p(s)ν(\frac{1}{h}(x_{k} − c_{0}))(a, s)ds + ∫_{t_{k}}^{t} q(s)ds ≤

≤ ε_{0}(t)[h(t) + ∫_{a}^{t} p(s)ds] + ν(\frac{1}{h}(x_{k} − c_{0}))(t_{k}, t) ∫_{a}^{t} p(s)ds + ∫_{a}^{t} q(s)ds,

\frac{∥x_{k}(t) − c_{0}∥}{h(t)} ≤ (1 + α)ε_{0}(t) + αν(\frac{1}{h}(x_{k} − c_{0}))(t_{k}, t) +

+\frac{1}{h(t)} ∫_{a}^{t} q(s)ds for a < t ≤ b_{k}

which by (2.17), implies

ν(\frac{1}{h}(x_{k} − c_{0}))(t_{k}, t) ≤ (1 − α)[ε(t) − t + a] +

+αν(\frac{1}{h}(x_{k} − c_{0}))(t_{k}, t) for a < t ≤ b_{k}.

Therefore ν(\frac{1}{h}(x_{k} − c_{0}))(t_{k}, b_{k}) ≤ ε(b_{k}) − t_{k} + a < ε(b_{k}), which contradicts equality (2.24). Thus we have shown that I_{k} ⊃ [a, b_{0}] and inequality (2.22) is fulfilled on ]a, b_{0}].

By (2.15) and (2.21), if t̃∈]t_{k}, b_{0}] and t̃− t_{k} is sufficiently small, then on ]a, t̃] we have

z(t) < x_{k}(t).  (2.25)

Denote by t^{∗} the exact upper bound of the set of t̃ ∈]t_{k}, b_{0}] for which inequality (2.25) is fulfilled on the interval [a, t̃]. Then because of inequality (1.3) and the fact that the operator f is nondecreasing, we have

x_{k}(t^{∗}) = z_{0}(t_{k}) + ∫_{t_{k}}^{t^{∗}} f(x_{k})(s)ds ≥ z_{0}(t_{k}) + ∫_{t_{k}}^{t^{∗}} f(z)(s)ds ≥

≥ z_{0}(t_{k}) + ∫_{t_{k}}^{t^{∗}} z′(s)ds = z_{0}(t_{k}) − z(t_{k}) + z(t^{∗}) > z(t^{∗}).

This, by the definition of t^{∗}, implies that t^{∗} = b_{0}, and hence inequality (2.25) is fulfilled on ]a, b_{0}].

By Lemma 2.1 from [4], conditions (2.18), (2.19), and (2.22) ensure the existence of a subsequence (x_{k_{j}})_{j=1}^{∞} of the sequence (x_{k})_{k=1}^{∞} such that (x_{k_{j}})_{j=1}^{∞} converges uniformly on [a, b_{0}], and x_{0}(t) = lim_{j→∞} x_{k_{j}}(t) is a solution of problem (1.1), (1.2) in [a, b_{0}]. On the other hand, from (2.25) it is clear that

z(t) ≤ x_{0}(t) for a ≤ t ≤ b_{0}.  (2.26)

For any x ∈ C([b_{0}, \overline{b}]; R^{n}) assume

χ(x)(t) = \begin{cases}x_{0}(t) + [x(b_{0}) − z(b_{0})]_{−} & for a ≤ t ≤ b_{0} \\ z(t) + [x(t) − z(t)]_{+} & for b_{0} < t ≤ \overline{b} \end{cases} ,  (2.27)
f̃(x)(t) = f(χ(x))(t) for b_{0} ≤ t ≤ \overline{b}.  (2.28)

Obviously, f̃: C([b_{0}, \overline{b}]; R^{n}) → L([b_{0}, \overline{b}]; R^{n}) is a Volterra operator satisfying the local Carathéodory conditions. Moreover, for any x ∈ C([b_{0}, \overline{b}]; R^{n}) the inequality ∥f̃(x)(t)∥ ≤ γ̃(t, ∥x∥_{C}) is fulfilled a.e. on [b_{0}, b], where γ̃ : [b_{0}, \overline{b}] × R_{+} → R_{+} does not decrease in the second argument, and γ̃(·; s) ∈ L([b_{0}, \overline{b}]; R) for s ≥ 0.

By Theorem 1.1, the problem

\frac{dx(t)}{dt} = f̃(x)(t),  (2.29)
x(b_{0}) = x_{0}(b_{0})  (2.30)

has a noncontinuable solution x̃ defined on some interval Ĩ ⊂ [b_{0}, \overline{b}] whose left end is equal to b_{0}.

Because of conditions (1.3), (2.26)–(2.28) and the fact that the operator f is nondecreasing, we have

χ(x̃)(t) = \begin{cases}x_{0}(t) & for a ≤ t ≤ b_{0} \\ z(t) + [x̃(t) − z(t)]_{+} & for t ∈ Ĩ \end{cases} ,  (2.31)
χ(x̃)(t) ≥ z(t) for t ∈ [a, b_{0}] ∪ Ĩ

and x′(t) = f(χ(x̃))(t) ≥ f(z)(t) ≥ z′(t) for almost all t ∈ Ĩ. Taking into account (2.26), we arrive at

x̃(t) ≥ x_{0}(b_{0}) + z(t) − z(b_{0}) ≥ z(t) for t ∈ Ĩ.  (2.32)

Assume \overline{I} = [a, b_{0}] ∪ Ĩ and \overline{x}(t) = x_{0}(t) for t ∈ [a, b_{0}], \overline{x}(t) = x̃(t) for t ∈ Ĩ.

By virtue of (2.31) and (2.32), \overline{x}(t) = χ(x̃)(t) for t ∈ \overline{I}. This implies that \overline{x} is a solution of problem (1.1), (1.2) on the interval \overline{I}. If now we take into consideration that x̃ is a noncontinuable solution of problem (2.29), (2.30) and the interval of definition of an arbitrary noncontinuable solution of problem (1.1), (1.2) contains the segment [a, \overline{b}], then it becomes clear that Ĩ = [b_{0}, \overline{b}] and \overline{I} = [a, \overline{b}]. Consequently, \overline{x} is a solution of problem (1.1), (1.2) on the interval [a, \overline{b}] and, as follows from (2.26), (2.31), it satisfies the inequality z(t) ≤ \overline{x}(t) for a ≤ t ≤ \overline{b}. Analogously, we can prove that if z : [a, \overline{b}] → R^{n} is a solution of problem (1.5), (1.6), then there exists a solution \underline{x} of problem (1.1), (1.2) satisfying the inequality z(t) ≥ \underline{x}(t) for a ≤ t ≤ \overline{b}.

Lemma 2.4. Let f be a nondecreasing operator, b_{0} ∈]a, b[ and let x be a solution of equation (1.1) on the interval [a, b_{0}]. Let, moreover,

ρ_{0} = max {∥x(t)∥ : a ≤ t ≤ b_{0}},  (2.33)

and b_{1} ∈]b_{0}, b] be such that

∫_{b_{0}}^{b_{1}}γ(t, ρ_{0} + 2)dt < 1,  (2.34)

where γ is a function given by equality (2.12). Then equation (1.1) has on the interval [a, b_{1}] a solution \overline{x} (a solution \underline{x}) such that \overline{x}(t) = x(t) (x(t) = \underline{x}(t)) for a ≤ t ≤ b_{0}, and for any \overline{b} ∈]b_{0}, b_{1}] an arbitrary solution z : [a, \overline{b}] → R^{n} of differential inequality (1.3) (of differential inequality (1.5)) satisfying on z(t) ≤ x(t) (z(t) ≥ x(t)) for a ≤ t ≤ b_{0} admits the estimate z(t) ≤ \overline{x}(t) (z(t) ≥ \underline{x}(t)) for a ≤ t ≤ \overline{b}.

Proof. We will prove only the existence of the solution \overline{x}, since the existence of the solution \underline{x} can be proved analogously.

For any y ∈ C([b_{0}, b]; R^{n}) we assume

χ(y)(t) = \begin{cases}x(t) + y(b_{0}) − x(b_{0}) & for a ≤ t ≤ b_{0} \\ y(t) & for b_{0} < t ≤ b \end{cases} ,  (2.35)
f_{1}(y)(t) = f(χ(y))(t).  (2.36)

Let c_{k} ∈ R^{n} (k = 1, 2, …) be an arbitrary sequence satisfying the inequalities

∥x(b_{0}) − c_{k}∥ < \frac{1}{k} (k = 1, 2, …),  (2.37)
c_{k} > x(b_{0}) (k = 1, 2, …).  (2.38)

For every natural k consider the initial value problem

\frac{dy}{dt} = f_{1}(y)(t),  (2.39)
y(b_{0}) = c_{k}.  (2.40)

Because of (2.12) and (2.33)–(2.35),

f_{1}^{∗}(t; c_{k}, 1) = sup {∥f_{1}(c_{k} + y)(t) · sgn(y(t))∥ : y ∈ C_{1}([b_{0}, b]; R^{n})} ≤

≤ γ(t, ρ_{0} + 2) and ∫_{b_{0}}^{b_{1}}f_{1}^{∗}(t; c_{k}, 1)dt ≤ 1.

This, according to Corollary 2.1 from [4], implies that problem (2.39), (2.40) has on the interval [b_{0}, \overline{b}] a solution y_{k} such that ∥y_{k}(t)∥ ≤ ∥c_{k}∥ + ∥y_{k}(t) − c_{k}∥ ≤ ρ_{0} + 2, ∥y′_{k}(t)∥ ≤ γ(t; ρ_{0} + 2) for b_{0} ≤ t ≤ b_{1}.

Hence, the sequence (y_{k})_{k=1}^{∞} is uniformly bounded and equicontinuous. By Arzela–Ascoli’s lemma, it can be assumed to be uniformly convergent. Let

\overline{y}(t) = lim_{k→∞} y_{k}(t) for b_{0} ≤ t ≤ b_{1}  (2.41)

and

\overline{x}(t) = x(t) for a ≤ t ≤ b_{0}, \overline{x}(t) = \overline{y}(t) for b_{0} < t ≤ b_{1}. (2.42)

Since the operator f_{1} is continuous, by condition (2.37) \overline{y} is a solution of equation (1.1) under the initial condition \overline{y}(b_{0}) = x(b_{0}). Taking this fact into account, it follows from (2.35), (2.36) that χ(\overline{y})(t) ≡ \overline{x}(t), f_{1}(y)(t) ≡ f(\overline{x})(t), and hence \overline{x} is a solution of equation (1.1) on the interval [a, b_{1}].

Let \overline{b} ∈]b_{0}, b_{1}] and let z : [a, \overline{b}] → R^{n} be a solution of differential inequality (1.3) satisfying the condition

z(t) ≤ x(t) for a ≤ t ≤ b_{0}.  (2.43)

Our aim is to prove that

z(t) ≤ \overline{x}(t) for a ≤ t ≤ \overline{b}.  (2.44)

Due to (2.38) and (2.43), for every natural k the inequality

z(t) < y_{k}(t)  (2.45)

is fulfilled in a right-hand neighborhood of the point b_{0}. Let us show that this inequality is fulfilled on the whole [b_{0}, \overline{b}]. Assume the contrary. Then there exists \underline{b} ∈]b_{0}, \overline{b}] such that on [b_{0}, \underline{b}[ inequality (2.45) is fulfilled but for t = \underline{b} it does not hold. By (1.3), (2.35), (2.36) and the fact that the operator f is nondecreasing, χ(y_{k})(t) > z(t) for a ≤ t < \underline{b} and the inequality y′_{k}(t) ≥ f(z)(t) ≥ z′(t) is fulfilled a.e. on ]a, b_{1}[. Therefore y_{k}(\underline{b}) − z(\underline{b}) ≥ y_{k}(b_{0}) − z(b_{0}) ≥ c_{k} − x(b_{0}) > 0, which contradicts our assumption that inequality (2.45) does not hold for t = \underline{b}. Thus we have proved that inequality (2.45) is fulfilled on the whole [a, \overline{b}].

By (2.41) and (2.42), from (2.43) and (2.45) we obtain estimate (2.44).

§ 3. Proof of the Main Results

Proof of Theorem 1.2. We will prove only the existence of the upper solution, since that of the lower solution is proved analogously.

Let X be the set of restrictions of all noncontinuable solutions of problem (1.1), (1.2) on the segment [a, b^{∗}]. By Lemma 2.2, X is a compactum in C([a, b^{∗}]; R^{n}). Moreover, there exists a bounded function h^{∗} : [a, b^{∗}] → R_{+} satisfying condition (2.4) such that every solution x ∈ X admits estimate (2.5). On the other hand, there exists γ^{∗} ∈ L_{loc}(]a, b^{∗}]; R_{+}) such that

∥x(t) − x(s)∥ ≤ ∫_{s}^{t} γ^{∗}(ξ)dξ for x ∈ X, a < s < t ≤ b^{∗}. (3.1)

Assume x_{i}^{∗}(t) = sup{x_{i}(t) : x = (x_{j})_{j=1}^{n} ∈ X} and x^{∗}(t) = (x_{i}^{∗}(t))_{i=1}^{n}. Then from (2.5) and (3.1) we find

∥x^{∗}(t) − c_{0}∥ ≤ nh^{∗}(t) for a ≤ t ≤ b^{∗},  (3.2)
∥x^{∗}(t) − x^{∗}(s)∥ ≤ n ∫_{s}^{t} γ^{∗}(ξ)dξ for a < s < t ≤ b^{∗}.

Consequently, x^{∗} is continuous on [a, b^{∗}] and absolutely continuous on every segment contained in ]a, b^{∗}].

If we take into account that the operator f is nondecreasing, then from the equality

x(t) = x(s) + ∫_{s}^{t} f(x)(ξ)dξ for x ∈ X, a < s < t ≤ b^{∗}

we have

x^{∗}(t) < x^{∗}(s) + ∫_{s}^{t} f(x^{∗})(ξ)dξ for a < s < t ≤ b^{∗},

that is,

\frac{1}{t − s}(x^{∗}(t) − x^{∗}(s)) ≤ \frac{1}{t − s} ∫_{s}^{t} f(x^{∗})(ξ)dξ for a < s < t ≤ b^{∗}.

From the above it is clear that the inequality \frac{dx^{∗}(t)}{dt} ≤ f(x^{∗})(t) is fulfilled a.e. on ]a, b^{∗}[. On the other hand, by (2.4) and (3.2),

lim_{t→a} \frac{∥x^{∗}(t) − c_{0}∥}{h_{0}(t)} = 0.

Hence z(t) = x^{∗}(t) is a solution of problem (1.3), (1.4) in [a, b^{∗}]. Therefore, as follows from Lemma 2.3, problem (1.1), (1.2) has on the interval [a, b^{∗}] a solution \overline{x} such that x^{∗}(t) ≤ \overline{x}(t) for a ≤ t ≤ b^{∗}. However, by the definition of x^{∗}, \overline{x}(t) ≤ x^{∗}(t) for a ≤ t ≤ b^{∗}. Clearly, x^{∗}(t) ≡ \overline{x}(t), and x^{∗} is the upper solution of problem (1.1), (1.2).

Proof of Corollary 1.2. By Theorem 1.2, it suffices to establish that the interval of definition of every noncontinuable solution of problem (1.1), (1.2_{1}) contains the segment [a, b^{∗}].

Let x be an arbitrary noncontinuable solution of problem (1.1), (1.2_{1}) defined on the interval I. Suppose y(t) = x(t) − c_{0}. Then the inequality y′(t) sgn(y(t)) ≤ f^{∗}(t; c_{0}, ν(y)(a, t)) is fulfilled a.e. on I, and ν(y)(a, a) = 0. This, by Lemma 3.1 from [4] and inequality (1.12), implies that ∥y(t)∥ < ρ for t ∈ [a, b^{∗}] ∩ I. From this estimate, by virtue of Corollary 3.1 from [4], we have I ⊃ [a, b^{∗}].

Proof of Corollary 1.3. By Lemma 1.2 and Theorem 1.2, the set I^{∗}(f; c_{0}, h) is non-empty and for arbitrary fixed b_{0} ∈ I^{∗}(f; c_{0}, h) problem (1.1), (1.2) has in the interval [a, b_{0}] the upper solution x_{0}^{∗}. Assume ρ_{0} = max{∥x_{0}^{∗}(t)∥ : a ≤ t ≤ b_{0}}.

Let γ be the function given by equality (2.12) and let (t_{k})_{k=1}^{∞} be a sequence satisfying b_{0} < t_{k} < t_{k+1} < b (k = 1, 2, …), lim_{k→∞} t_{k} = b.

If ∫_{b_{0}}^{t_{1}} γ(t, ρ_{0} +2)dt ≤ 1, then we assume b_{1} = t_{1}. If, however, ∫_{b_{0}}^{t_{1}} γ(t, ρ_{0} +

2)dt > 1, then we choose b_{1} ∈]b_{0}, t_{1}[ such that ∫_{b_{0}}^{b_{1}} γ(t, ρ_{0} + 2)dt = 1. By virtue of Lemma 2.4, equation (1.1) has in the interval [a, b_{1}] a solution x_{1}^{∗} such that x_{1}^{∗}(t) = x_{0}^{∗}(t) for a ≤ t ≤ b_{0} and for any \overline{b} ∈]b_{0}, b_{1}] an arbitrary solution x : [a, \overline{b}] → R^{n} of equation (1.1) satisfying the condition x(t) ≤ x_{0}^{∗}(t) admits for a ≤ t ≤ b_{0} the estimate x(t) ≤ x_{1}^{∗}(t) for a ≤ t ≤ \overline{b}.

Since x_{0}^{∗} is the upper solution of problem (1.1), (1.2) on the interval [a, b_{0}], it is clear that x_{1}^{∗} is the upper solution of this problem on the interval [a, b_{1}].

Proceeding from the above-said, we prove by induction the existence of an increasing sequence b_{k} ∈]b_{0}, b[ (k = 1, 2, …) and of a sequence of vector functions x_{k}^{∗} : [a, b_{k}] → R^{n} (k = 1, 2, …) such that for every natural k:

(i) x_{k}^{∗} is the upper solution of problem (1.1), (1.2) on the interval [a, b_{k}]; (ii) either b_{k}=t_{k} or ∫_{b_{k−1}}^{b_{k}}γ(t, ρ_{k−1} + 2)dt=1, where ρ_{k−1}=max{∥x_{k−1}^{∗}(t)∥ :

a ≤ t ≤ b_{k−1}}.

Assume b^{∗} = lim_{k→∞} b_{k}, x^{∗}(t) = x_{0}^{∗}(t) for a ≤ t < b_{0}, x^{∗}(t) = x_{k}^{∗}(t) for b_{k−1} < t < b_{k} (k = 1, 2, …). Clearly, x^{∗} is the upper solution of problem (1.1), (1.2) on [a, b^{∗}[ . If b^{∗} < b, then owing to (ii), x^{∗} is a noncontinuable solution.^{1} The solution x^{∗} is noncontinuable as well when b^{∗} = b and sup{∥x^{∗}(t)∥ : a ≤ t < b} = +∞. If, however, sup{∥x^{∗}(t)∥ : a ≤ t < b} < +∞, then x^{∗} has at the point b the left-hand limit x^{∗}(b−). Assume \overline{x}(t) = x^{∗}(t) for a ≤ t < b, \overline{x}(b) = x^{∗}(b−) for t = b. Then \overline{x} is the noncontinuable upper solution of problem (1.1), (1.2).

1See the proof of Theorem 3.2 in [4].

When

f(x)(t) ≡ f_{0}(t, x(t), x(τ_{1}(t)), …, x(τ_{m}(t))),  (3.3)

from Theorem 1.1 and Corollary 1.3 follow respectively Corollaries 1.4 and 1.6, while from Corollary 1.2 follows Corollary 1.5.

Proof of Theorem 1.3. Let z be a solution of problem (1.3), (1.4) on the interval I. Prove that in I_{0} ∩ I the inequality

z(t) ≤ x^{∗}(t)  (3.4)

is fulfilled. Assume the contrary. Then there exists \overline{b} ∈ (I_{0} ∩ I)∩]a, b] such that for t = \overline{b} inequality (3.4) does not hold.

By Lemmas 2.2, 2.3 and Theorem 1.2, the set I^{∗}(f; c_{0}, h) is non-empty and for any t_{0} ∈]a, \overline{b}[ ∩I^{∗}(f; c_{0}, h) inequality (3.4) is fulfilled on [a, t_{0}]. Denote by b_{0} the exact upper bound of the set of t_{0} ∈]a, \overline{b}[ for which inequality (3.4) is fulfilled on the segment [a, t_{0}]. Then according to our assumption, b_{0} < \overline{b}.

By Lemma 2.4, there exist b_{1} ∈]b_{0}, \overline{b}] and a solution \overline{x} : [a, b_{1}] → R^{n} of equation (1.1) such that \overline{x}(t) = x^{∗}(t) for a ≤ t ≤ b_{0} and z(t) ≤ \overline{x}(t) for a ≤ t ≤ b_{1}. Obviously, \overline{x} is a solution of problem (1.1), (1.2). Therefore, \overline{x}(t) ≤ x^{∗}(t) for a ≤ t ≤ b_{1}. Consequently, inequality (3.4) is also fulfilled on the segment [a, b_{1}], which contradicts the definition of b_{0}. The obtained contradiction proves that inequality (3.4) is fulfilled on I_{0} ∩ I.

It can be proved analogously that if z is a solution of problem (1.5), (1.6) on the interval I, then inequality z(t) ≥ x_{∗}(t) is fulfilled on I_{0} ∩ I.

When h(t) ≡ 1, from the above proved theorem we obtain Corollary 1.7, while when identity (3.3) is fulfilled, from Theorem 1.3 and Corollary 1.7 follow Corollaries 1.8 and 1.9.

Proof of Theorem 1.4. Let z be a solution of problem (1.3), (1.4_{2}) on the interval I. Then in a right-hand neighborhood of the point a the inequality

z(t) < x_{∗}(t)  (3.5)

is fulfilled.

Let us show that it is fulfilled on the whole interval I ∩ I_{0}. Assume the contrary. Then there exists b_{0} ∈ I ∩ I_{0}∩]a, b] such that inequality (3.5) is fulfilled on ]a, b_{0}[, but for t = b_{0} it does not hold. On the other hand, since the operator f is nondecreasing, we have (x_{∗}(t) − z(t))′ ≥ f(x_{∗})(t) − f(z)(t) ≥ 0 a.e. on ]a, b_{0}[ . Therefore x_{∗}(b_{0}) − z(b_{0}) ≥ x_{∗}(a) − z(a) > 0, which contradicts our assumption that (3.5) does not hold for t = b_{0}. Thus we have established that (3.5) is fulfilled on the whole I ∩ I_{0}. It can be proved analogously that an arbitrary solution z : I → R^{n} of problem (1.5), (1.6_{2}) admits the estimate z(t) > x^{∗}(t) for t ∈ I ∩ I_{0}.

If identity (3.3) is fulfilled, then from Theorem 1.9 we obtain Corollary 1.10.

References

1. E. A. Coddington and N. Levinson, Theory of ordinary differential equations. Mc Graw-Hill Book Company, Inc., New York–Toronto–London, 1995.

2. P. Hartman, Ordinary differential equations. John Wiley & Sons, New York–London–Sydney, 1964.

3. W. Walter, Differential and integral inequalities. Springer-Verlag, Berlin–Heidelberg–New York, 1970.

4. I. Kiguradze and Z. Sokhadze, On the Cauchy problem for evolutionary singular functional differential equations. (Russian) Differentsial’nye Uravneniya (to appear).

5. J. W. Heidel, Uniqueness, continuation and nonoscillation for a second order differential equation. Pacific J. Math. 32(1970), No. 3, 715–721.

6. I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. Kluwer Academic Publishers, Dordrecht–Boston–London, 1992.

(Received 21.05.1995)

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