GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 4, 1997, 355-372

ON THE GLOBAL SOLVABILITY OF THE CAUCHY

PROBLEM FOR SINGULAR FUNCTIONAL DIFFERENTIAL EQUATIONS

I. KIGURADZE AND Z. SOKHADZE

Abstract. Sufficient conditions are found for the global solvability of the weighted Cauchy problem

\frac{dx(t)}{dt} = f(x)(t), lim_{t→a} \frac{∥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 a function continuous and positive on ]a, b], and ∥ · ∥ is the norm in R^{n}.

Throughout the paper the following notation will be used: R is the set of real numbers, R_{+} = [0, +∞[ ; if u ∈ R, then [u]_{+} =

\frac{1}{2}(|u| + u);

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

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 · y is the scalar product of the vectors x and y ∈ R^{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};

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};

355

1991 Mathematics Subject Classification. 34K05.

Key words and phrases. Volterra operator, functional differential equation, weighted Cauchy problem, global solvability.

1072-947X/97/0700-0355$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

L_{loc}(]a, b]; R^{n}) is the space of vector functions x : ]a, b] → R^{n} which are summable on each segment of ]a, b] with the topology of convergence in the mean on each 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. An operator f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) is called a Volterra one if the equality f(x)(t) = f(y)(t) holds almost everywhere on ]a, t_{0}] for any t_{0} ∈]a, b] and any vector-functions x and y ∈ C([a, b]; R^{n}) satisfying the condition x(t) = y(t) when a < t ≤ t_{0}.

Definition 2. An operator f : C([a, b]; R^{n}) → L_{loc}(]a, b]; R^{n}) will be said to satisfy the local Carathéodory conditions if it is continuous and there exists a function γ : ]a, b] × R_{+} → R_{+} nondecreasing with respect to the second argument such that γ(·, ρ) ∈ L_{loc}(]a, b]; R) for ρ ∈ R_{+}, and the inequality

∥f(x)(t)∥ ≤ γ(t, ∥x∥_{C})

is fulfilled for any x ∈ C([a, b]; R^{n}) almost everywhere on ]a, b[.

If

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

then the operator f is called regular, and, otherwise, singular.

Here we will consider the vector functional differential equation

\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)

It is assumed everywhere that f : C([a, b]; R^{n})→L_{loc}(]a, b]; R^{n}) is a Volterra, generally speaking, singular operator satisfying the local Carathéodory conditions, c_{0} ∈ R^{n}, and h : [a, b] → [0, +∞[ is a continuous function nondecreasing and positive on ]a, b].

We will separately consider the case where h(a) > 0 so that condition (2) takes the form

x(a) = c_{0}.  (2_{1})

The vector differential equation with delay

\frac{dx(t)}{dt} = f_{0}(t, x(t), x(τ_{1}(t)), …, x(τ_{m}(t)))  (3)

is the important particular case of the functional differential equation (1).

Below, whenever equation (3) is discussed, it will be assumed that the vector function f_{0} : ]a, b[ ×R^{(m+1)n} → R^{n} satisfies the local Carathéodory 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_{m}) : ]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} there holds the inequality

∥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 with respect to 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).

Definition 3. If b_{0} ∈]a, b], then:

(i) For any 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) = \begin{cases}x(t) & for a ≤ t ≤ b_{0} \\ x(b_{0}) & for b_{0} < t ≤ b_{0} \end{cases} ;

(ii) a continuous vector function x : [a, b_{0}] → R^{n} is called a solution of equation (1) on the segment [a, b_{0}] if x is absolutely continuous on each segment contained in ]a, b_{0}] and satisfies equation (1) almost everywhere on ]a, b_{0}[;

(iii) a vector function x : [a, b_{0}] → R^{n} is called a solution of equation (1) on the semi-open segment [a, b_{0}[ if for each b_{1} ∈]a, b_{0}[ the restriction of x on [a, b_{1}] is a solution of this equation on the segment [a, b_{1}];

(iv) a solution x of equation (1) satisfying the initial condition (2) is called a solution of problem (1), (2).

Definition 4. Problem (1), (2) is said to be globally solvable if it has at least one solution on the segment [a, b].

Definition 5. A solution x of equation (1) defined on the segment [a, b_{0}] ⊂ [a, b[ (on the semi-open segment [a, b_{0}[⊂ [a, b[) is called continuable if for some b_{1} ∈]b_{0}, b] (b_{1} ∈ [b_{0}, b]) equation (1) has, on the segment [a, b_{1}], a solution y satisfying the condition x(t) = y(t) for a ≤ t ≤ b_{0}. Otherwise, the solution x is called noncontinuable.

Definition 6. An operator φ : C([a, b]; R_{+}) → L_{loc}(]a, b]; R) is called nondecreasing if the inequality φ(u)(t) ≤ φ(v)(t) is fulfilled almost everywhere on ]a, b[ for any u and v ∈ C([a, b]; R_{+}) satisfying the condition u(t) ≤ v(t) when a ≤ t ≤ b.

Theorem 1. Let there exist a positive number ρ, summable functions p and q : [a, b] → R_{+}, and a continuous nondecreasing Volterra operator φ : C([a, b]; R_{+}) → L_{loc}(]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  (4)

and the inequalities

f(c_{0} + hy)(t) · sgn(y(t)) ≤ φ(∥y∥)(t),  (5)
φ(u)(t) ≤ p(t)ν(u)(a, t) + q(t)  (6)

are fulfilled for any y ∈ C([a, b]; R^{n}) and u ∈ C_{ρ}([a, b]; R_{+}) almost everywhere on ]a, b[. Let furthermore the problem

\frac{dv(t)}{dt} = φ(\frac{v}{h})(t); lim_{t→a} \frac{v(t)}{h(t)} = 0  (7)

have an upper solution on the segment [a, b]. Then problem (1), (2) is globally solvable and each of its noncontinuable solutions is defined on [a, b].

Proof. By virtue of Theorem 3.2 from [1] conditions (4)–(6) imply the existence of a noncontinuable solution of problem (1), (2). Let x be an arbitrary noncontinuable solution of problem (1), (2) defined on the segment I_{0}. Our aim is to prove that I_{0} = [a, b].

By (4) there exist b_{0} ∈]a, b] and α ∈]0, 1[ such that

∫_{a}^{t} p(s)ds ≤ αh(t), h_{0}(t) ≤ ρh(t) for a ≤ t ≤ b_{0},  (8)

where h_{0}(a) = 0, and

h_{0}(t) = \frac{h(t)}{1 − α} sup \{\frac{1}{h(s)} ∫_{a}^{s} q(ξ)dξ : a < s ≤ t\} for a < t ≤ b_{0}. (9)

By Lemma 2.3 and Corollary 3.1 from [1], conditions (4)–(6) and (8) imply that I_{0} ⊃ [a, b_{0}] and

∥x(t) − c_{0}∥ ≤ h_{0}(t) for a ≤ t ≤ b_{0}.  (10)

Let v^{∗} be an upper solution of problem (7). Then by conditions (4), (6),

(8) and Lemma 2.3 from [1]

v^{∗}(t) ≤ h_{0}(t) for a ≤ t ≤ b_{0}.  (11)

By (9) the function

ε(t) = \begin{cases}\frac{h_{0}(t)}{h(t)} + \frac{t − a}{b_{0} − t} & for a < t < b_{0} \\ 0 & for t = a \end{cases}  (12)

is continuous on [a, b_{0}[.

For any u ∈ C([a, b]; R) we set

χ(u)(t) = \begin{cases}ε(t) & for u(t) > ε(t), a ≤ t < b_{0} \\ [u(t)]_{+} & for u(t) ≤ ε(t), a ≤ t < b_{0} \\ [u(t)]_{+} & for b_{0} ≤ t ≤ b \end{cases}  (13)

and

\overline{φ}(u)(t) = φ(χ(\frac{u}{h}))(t).  (14)

Obviously, χ : C([a, b]; R)→C([a, b]; R_{+}) and \overline{φ} : C([a, b]; R)→L_{loc}(]a, b]; R_{+}) are continuous nondecreasing Volterra operators and the inequality

\overline{φ}(hu)(t) · sgn(u(t)) ≤ p(t)ν(u)(a, t) + q(t)  (15)

holds for any u ∈ C_{ρ}([a, b]; R) almost everywhere on ]a, b[ .

By Corollary 1.3 from [2] the problem

\frac{dv(t)}{dt} = \overline{φ}(v)(t); lim_{t→a} \frac{v(t)}{h(t)} = 0  (16)

has a noncontinuable upper solution \overline{v} defined on some interval I. On the other hand, by conditions (11)–(14),

φ(\frac{v^{∗}}{h} )(t) ≡ \overline{φ}(v^{∗})(t)

and therefore v^{∗} is a solution of problem (16). Thus

v^{∗}(t) ≤ \overline{v}(t) for t ∈ I.  (17)

By Lemma 2.3 and Corollary 3.1 from [1], conditions (4), (6), and (8) imply the estimate

0 ≤ \overline{v}(t) ≤ h_{0}(t) for a ≤ t ≤ b_{0}

and hence by (12)—(14) we obtain

\overline{φ}(\overline{v})(t) ≡ φ(\frac{\overline{v}}{h})(t).

Therefore \overline{v} is a solution of problem (7) so that

\overline{v}(t) ≤ v^{∗}(t) for t ∈ I.  (18)

By (17) and (18) it is obvious that I = [a, b] and \overline{v}(t) ≡ v^{∗}(t). Thus v^{∗} is an upper solution of problem (16).

Due to conditions (5), (10) and (12)–(14) we have

\frac{d}{dt}(∥x(t) − c_{0}∥) ≤ \overline{φ}(∥x − c_{0}∥)(t)  (19)

almost everywhere on I_{0}.

By Theorem 1.3 from [2], (19) and (2) imply the estimate

∥x(t) − c_{0}∥ ≤ v^{∗}(t) for t ∈ I_{0}.

Hence by Corollary 3.1 from [1], we conclude that I_{0} = [a, b].

For our further discussion we need

Definition 7. Let D ⊂ C([a, b]; R^{k}) and M ⊂ L_{loc}(]a, b]; R^{l}). An operator g : D → M is called a strictly Volterra operator if there exists a continuous nondecreasing function τ : [a, b] → [a, b] such that

τ(t) < t for a < t ≤ b

and the equality

g(x)(t) = g(y)(t)

holds almost everywhere on [a, t_{0}] for any t_{0} ∈]a, b] and any vector functions x and y ∈ D satisfying the condition

x(t) = y(t) for a ≤ t ≤ τ(t_{0}).

Corollary 1. Let, for any y ∈ C([a, b]; R^{n}), the inequality

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

be fulfilled almost everywhere on ]a, b[ , where p and q : C([a, b]; R_{+}) → L_{loc}([a, b]; R_{+}) are continuous, nondecreasing, strictly Volterra operators satisfying, for some ρ > 0, the conditions

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

and φ_{0} : R_{+} → R_{+} is a continuous, nondecreasing function such that

lim sup_{s→0} \frac{φ_{0}(s)}{s} < 1,  (20)

φ_{0}(s) > 0, ∫_{s}^{+∞}\frac{dξ}{φ_{0}(ξ)} = +∞ for s > 0.  (21)

Then the conclusion of Theorem 1 is valid.

Proof. By (20) the number ρ > 0 can be assumed to be so small that

\frac{φ_{0}(s)}{s} < ρ for 0 < s ≤ ρ.

We set

φ(u)(t) = p(u)(t)φ_{0}(ν(u)(a, t)) + q(u)(t).  (22)

Then inequalities (5) and (6), where

p(t) ≡ p(ρ)(t), q(t) ≡ q(ρ)(t),

are fulfilled almost everywhere on ]a, b[ for any y ∈ C([a, b]; R^{n}) and u ∈ C_{ρ}([a, b]; R_{+}).

By virtue of Theorem 1, to prove the corollary it is sufficient to show that problem (7) has an upper solution on the segment [a, b].

Choose b_{0} ∈]a, b] and α ∈]0, 1[ such that inequalities (8) are fulfilled. Let h_{0}, ε, χ and \overline{φ} be the functions and operators given by equalities (9)

and (12)–(14). As shown while proving Theorem 1, problem (7) is equivalent to problem (16).

Following Corollary 1.3 from [2], problem (16) has an upper solution v^{∗} defined on some interval I. Since problems (7) and (16) are equivalent it remains for us only to show that I = [a, b].

By virtue of Corollary 3.1 and Lemma 2.3 from [1], I ⊃ [a, b_{0}] and inequality (11) is fulfilled.

Now assume that I ≠ [a, b]. Then, by Corollary 3.1 from [1] and the nonnegativeness of the operator \overline{φ}, there exists b_{1} ∈]b_{0}, b] such that I = [a, b_{1}[ and

lim_{t→b_{1}} v^{∗}(t) = +∞.  (23)

By (8) and (11)–(13)

\frac{v^{∗}(t)}{h(t)} < ρ for 0 ≤ t ≤ b_{0}

and

ν(χ(\frac{v^{∗}}{h} ))(a, t) ≤ ρ + \frac{v^{∗}(t)}{h(b_{0})} for b_{0} ≤ t < b_{1},

due to which (14) and (22) imply

\overline{φ}(v^{∗})(t) ≤ p_{0}(t)φ_{0}(ρ + \frac{v^{∗}(t)}{h(b_{0})}) for b_{0} ≤ t < b_{1},

where

p_{0}(t) = p(χ(\frac{v^{∗}}{h} ))(t) + \frac{q(χ(\frac{v^{∗}}{h}))(t)}{φ_{0}(ρ)}.

On the other hand, since the operators p and q are the strictly Volterra ones, the function p_{0} : [b_{0}, b_{1}] → R_{+} is summable.

From the above discussion it follows that the inequality

\frac{dv^{∗}(t)}{dt}/φ_{0}(ρ + \frac{v^{∗}(t)}{h(b_{0})}) ≤ p_{0}(t)

is fulfilled almost everywhere on ]b_{0}, b_{1}[.

If we integrate both parts of this inequality from b_{0} to b_{1}, by (23) we will obtain

h(b_{0}) ∫_{s_{0}}^{+∞}\frac{ds}{φ_{0}(s)} ≤ ∫_{b_{0}}^{b_{1}}p_{0}(s)ds < +∞,

which contradicts condition (21). The obtained contradiction proves the corollary.

The next corollary is proved similarly.

Corollary 2. Let, for any y ∈ C([a, b]; R^{n}), the inequality

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

be fulfilled almost everywhere on ]a, b[ , where p : C([a, b]; R_{+})→L([a, b]; R_{+}) is a continuous, nondecreasing, strictly Volterra operator satisfying, for some ρ > 0, the condition

lim_{t→a} (\frac{1}{h(t)} ∫_{a}^{t} p(ρ)(s)ds) = 0,

and φ_{0} : R_{+} →]0, +∞[ is a continuous, nondecreasing function such that

∫_{0}^{+∞}\frac{ds}{φ_{0}(s)} = +∞.  (24)

Then the conclusion of Theorem 1 is valid.

Corollary 3. Let, for any y ∈ C([a, b]; R^{n}), the inequality

f(c_{0} + y)(t) · sgn(y(t)) ≤ \frac{α}{b − a}(\frac{b − t}{b − a})^{β} exp [ν(y)(a, τ(t))] (25)

be fulfilled almost everywhere on ]a, b[ , where

τ(t) = b − (b − a)(1 + ln \frac{b − a}{b − t})^{−1},  (26)

α > 0 and β are constants such that

β > [α − 1]_{+} − 1.  (27)

Then problem (1), (2) is globally solvable and each of its noncontinuable solutions is defined on the entire [a, b].

Proof. By Theorem 1 it is sufficient to show that the problem

\frac{du(t)}{dt} = \frac{α}{b − a}(\frac{b − t}{b − a})^{β} exp [ν(u)(a, τ(t))], u(a) = 0

has an upper solution on the segment [a, b]. But this problem is equivalent to the problem

\frac{du(t)}{dt} = \frac{α}{b − a}(\frac{b − t}{b − a})^{β} exp [u(τ(t))], u(a) = 0.  (28)

By Theorem 3.2 from [1] problem (28) has a noncontinuable solution. It is not difficult to verify that this solution is unique. We denote it by u. Let I be the definition interval of u. Our aim is to show that I = [a, b].

Consider the function

v(t) = α(\frac{b − a}{b − t} − 1) for a ≤ t < b.

By virtue of (26) and (27)

a ≤ τ(t) < t for a < t < b

and

\frac{dv(t)}{dt} = \frac{α}{b − a}(\frac{b − t}{b − a})^{α−2} exp [v(τ(t))] ≥

≥ \frac{α}{b − a}(\frac{b − t}{b − a})^{β} exp [v(τ(t))] for a ≤ t < b.

Hence, by Corollary 1.8 from [2], we have

v(t) ≥ u(t) for t ∈ I.

However, by Corollary 3.1 from [2] this estimate implies that I ⊃ [a, b[ and

u(t) ≤ α(\frac{b − a}{b − t} − 1) for a ≤ t < b.  (29)

On account of (27) there is ε ∈]0, 1[ such that β > [α − 1]_{+} − ε. If, along with this inequality, we use equality (26) and estimate (29), then (28) will yield

u(t) ≤ \frac{α}{b − a} ∫_{a}^{t} (\frac{b − s}{b − a})^{[α−1]_{+}−ε} exp [u(τ(s))]ds for a ≤ t < b, (30)

u(t) ≤ \frac{α}{b − a} ∫_{a}^{t} (\frac{b − s}{b − a})^{α−1−ε}(\frac{b − s}{b − a})^{−α}ds =

= α(b − a)^{ε} ∫_{a}^{t}(b − s)^{−1−ε}ds < \frac{α}{ε}(\frac{b − a}{b − t})^{ε} for a ≤ t < b,

and hence

u(t) ≤ α_{1} + \frac{1 − ε}{2}(\frac{b − a}{b − t}) for a ≤ t < b,  (31)

where

α_{1} = \frac{1}{ε}(\frac{2α}{1 − ε})^{\frac{1}{1−ε}} .

By (26) and (31), from (30) we obtain

u(t) ≤ \frac{α}{b − a} exp(α_{1}) ∫_{a}^{t} (\frac{b − s}{b − a})^{[α−1]_{+}−ε}(\frac{b − s}{b − a})^{\frac{ε−1}{2}} ds ≤

≤ \frac{α}{b − a} exp(α_{1}) ∫_{a}^{t} (\frac{b − s}{b − a})^{− \frac{1+ε}{2}} ds ≤ α_{2} for a ≤ t < b,

where

α_{2} = \frac{2α}{1 − ε}.

Hence by Corollary 3.1 from [1] it follows that I = [a, b].

Corollaries 1–3 imply the following propositions for equation (3).

Corollary 4. Let there exist m_{0} ∈ {1, …, m} and a continuous nondecreasing function τ : [a, b] → [a, b] such that

τ_{k}(t) ≤ τ(t) < t for a < t ≤ b (k = m_{0}, …, m).  (32)

Let furthermore on the set ]a, b[ ×R^{(m+1)n} there hold 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}) ≤

≤ p(t, ∑_{k=m_{0}}^{m} ∥y_{k}∥)φ_{0}( ∑_{k=0}^{m_{0}−1} η_{k}∥y_{k}∥) + q(t, ∑_{k=m_{0}}^{m} ∥y_{k}∥),

where p and q : [a, b] × R_{+} → R_{+} are functions summable with respect to the first argument and continuously nondecreasing with respect to the second argument, and satisfying, for some ρ > 0, the conditions

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

φ_{0} : R_{+} → R_{+} is a continuous nondecreasing function satisfying conditions (20) and (21), η_{k} (k = 0, …, m_{0}) are non-negative constants such that

∑_{k=0}^{m_{0}} η_{k} = 1.

Then problem (3), (2) is globally solvable and each of its noncontinuable solutions is defined on [a, b].

Corollary 5. Let the function h be nondecreasing and inequalities (32) be fulfilled, where m_{0} ∈ {1, …, m} and τ : [a, b] → [a, b] is a continuous nondecreasing function. Let, furthermore, on the set ]a, b[ ×R^{(m+1)n} there hold 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}) ≤

≤ p(t, ∑_{k=m_{0}}^{m} ∥y_{k}∥)φ_{0}( ∑_{k=0}^{m_{0}−1} ∥y_{k}∥),

where p : [a, b] × R_{+} → R_{+} is a function summable with respect to the first argument and continuously nondecreasing with respect to the second argument, and satisfying, for some ρ > 0, the condition

lim_{t→a} (\frac{1}{h(t)} ∫_{a}^{t} p(s, ρ)ds) = 0,

φ_{0} : R_{+} →]0, +∞[ is a continuous nondecreasing function satisfying condition (21). Then the conclusion of Corollary 4 is valid.

Corollary 6. Let on the set ]a, b[ ×R^{(m+1)n} and the segment ]a, b[ there hold respectively the inequalities

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

≤ \frac{α}{b − a}(\frac{b − t}{b − a})^{β} exp ( ∑_{k=1}^{m} η_{k}∥y_{k}∥)

and

τ_{k}(t) ≤ b − (b − a)(1 + ln \frac{b − a}{b − t})^{−1} (k = 1, …, m),

where α > 0, β and η_{k} ≥ 0 (k = 1, …, m) are constants satisfying condition (27) and

∑_{k=1}^{m} η_{k} = 1.

Then problem (3),(2_{1}) is globally solvable and each of its noncontinuable solutions is defined on the entire [a, b].

Example 1. Consider problem (28) where α > 1, β = α − 2, τ is the function given by equality (26). All conditions of Corollary 6 except (27) are fulfilled. Nevertheless it has the noncontinuable solution

u(t) = α(\frac{b − a}{b − t} − 1),

which is defined not on the segment [a, b] but on [a, b[ . This example shows that in Corollaries 3 and 6 the strict inequality (27) cannot be replaced by the nonstrict inequality β ≥ [α − 1]_{+} − 1.

Theorem 2. Let there exist δ : ]a, b] →]0, ∞[ and c : ]a, b] → R^{n} such that δ(s) < s − a for a < s ≤ b, and for any numbers s ∈]a, b], ρ > 0 and any vector function y ∈ C([a, s]; R^{n}) satisfying the condition

∥y(t)∥ ≤ ρ for a ≤ t ≤ s − δ(s),  (33)

let the inequality

f(c(s) + y)(t) · sgn(y(t)) ≤ φ_{s,ρ}(∥y∥)(t)  (34)

hold almost everywhere on ]s − δ(s), s[ , where φ_{s,ρ} : C([s − δ(s), s]; R_{+}) → L([s − δ(s), s]; R_{+}) is a continuous nondecreasing Volterra operator. Let furthermore the problem

\frac{du(t)}{dt} = φ_{s,ρ}(u)(t), u(s − δ(s)) = ρ  (35)

have an upper solution on the interval [s − δ(s), s] for any s ∈]a, b] and ρ > 0. Then each noncontinuable solution of equation (1) is defined on [a, b].

Proof. Assume that the theorem is not valid. Then by virtue of Corollary 3.1 from [1] there exist s ∈]a, b] and a noncontinuable solution x : [a, s[→ R^{n} of equation (1) such that

lim sup_{t→s} ∥x(t)∥ = +∞.  (36)

We set y(t) = x(t) − c(s) and choose ρ > 0 such that inequality (33) is fulfilled. Then by (34) the inequality

\frac{d∥y(t)∥}{dt} ≤ φ_{s,ρ}(∥y∥)(t)

is fulfilled almost everywhere on ]s − δ(s), s[.

Moreover,

∥y(s − δ(s))∥ ≤ ρ.

By Corollary 1.7 from [2] the latter two inequalities yield the estimate

∥y(t)∥ ≤ u(t) for s − δ(s) ≤ t < s.

Therefore

lim sup_{t→s} ∥x(t)∥ ≤ ∥c(s)∥ + lim sup_{t→s} ∥y(t)∥ ≤ ∥c(s)∥ + u(s) < +∞,

which contradicts equality (36). The obtained contradiction proves the theorem.

Remark. It is obvious that if the conditions of Theorem 2 are fulfilled, then the local solvability of problem (1), (2) guarantees its global solvability. Therefore if the conditions of Theorem 2, as well as of Theorem 2.1 from [1], are fulfilled, then problem (1), (2) is globally solvable and each of its noncontinuable solutions is defined on [a, b].

Corollary 7. Let there exist functions δ : ]a, b] →]0, 1[, c : ]a, b] → R^{n}, α : ]a, b]×R_{+} → R_{+}, β : ]a, b]×R_{+} →]−1, 0], and λ_{k} : ]a, b]×R_{+} →[1, +∞[ (k = 1, …, m) such that δ(s) < s − a for a < s ≤ b, and for any numbers s ∈]a, b], ρ > 0 and any vector function y ∈ C([a, b]; R^{n}) satisfying condition (33), there hold, almost everywhere on ]s − δ(s), s[, the inequality

f(c(s) + y)(t) · sgn(y(t)) ≤

≤ α(s, ρ)(s − t)^{β(s,ρ)}(1 + ∑_{k=1}^{m} [ν(y)(s − δ(s), τ_{ksρ}(t))]^{λ_{k}(s,ρ)}) ×

× ln (2 + ∑_{k=1}^{m} ν(y)(s − δ(s), τ_{ksρ}(t))),  (37)

where

τ_{ksρ}(t) = \begin{cases}s − δ(s) & for s − δ(s) ≤ t ≤ s − [δ(s)]^{λ_{k}(s,ρ)} \\ s − (s − t)^{\frac{1}{λ_{k}(s,ρ)}} & for s − [δ(s)]^{λ_{k}(s,ρ)} < t ≤ s \end{cases} . (38)

Then each noncontinuable solution of equation (1) is defined on [a, b].

Proof. Let s ∈]a, b] and ρ > 0 be arbitrarily fixed. We introduce ε ∈]0, \frac{1}{2}] such that

β(s, ρ) ≥ 2ε − 1.

By virtue of Theorem 2 and inequality (37), to prove the corollary it is sufficient to establish that problem (35), where

φ_{sρ}(u)(t)= α(s, ρ)(s − t)^{2ε−1}(1 + ∑_{k=1}^{m} [ν([u]_{+})(s − δ(s), τ_{ksρ}(t))]^{λ_{k}(s,ρ)}) ×

×ln (2 + ∑_{k=1}^{m} ν([u]_{+})(s − δ(s), τ_{ksρ}(t))),  (39)

has an upper solution on the interval [s − δ(s), s].

By Corollary 1.3 from [2] and equalities (38) and (39), problem (35) has a noncontinuable upper solution u^{∗} on some interval I ⊂ [s − δ(s), s]. On the other hand, from Corollary 2 it immediately follows that I ⊃ [s − δ(s), s[ . It remains for us to show that s ∈ I.

Choose numbers δ_{0} ∈]0, δ(s)[ and ρ_{0} ∈]ρ, +∞[ such that

\frac{1}{ε} < ln \frac{1}{δ_{0}} < [α(s, ρ)(m + 1)(1 + ln(2 + m))]^{−1}δ_{0}^{−ε},  (40)
u^{∗}(s − δ_{0}) < ρ_{0},  (41)

and for any k ∈ {1, …, m} and u ∈ C([s − δ_{0}, s]; R) put

τ_{k}^{∗}(t) = \begin{cases}s − δ_{0} & for s − δ_{0} ≤ t ≤ s − δ_{0}^{λ_{k}(s,ρ)} \\ s − (s − t)^{\frac{1}{λ_{k}(s,ρ)}} & for s − δ_{0}^{λ_{k}(s,ρ)} < t ≤ s \end{cases} ,  (42)

ν_{k}^{∗}(u)(t) = \begin{cases}ρ_{0} & for s − δ_{0} ≤ t ≤ s − δ_{0}^{λ_{k}(s,ρ)} \\ [u(τ_{k}^{∗}(t)]_{+} & for s − δ_{0}^{λ_{k}(s,ρ)} < t ≤ s \end{cases} ,  (43)

φ^{∗}(u)(t) = α(s, ρ)(s − t)^{2ε−1}(1 + ∑_{k=1}^{m} [ν_{k}^{∗}(u)(t)]^{λ_{k}(s,ρ)}) ×

× ln (2 + ∑_{k=1}^{m} ν_{k}^{∗}(u)(t)).  (44)

Then by virtue of (38) and (39) the inequality φ_{s,ρ}(u^{∗})(t) ≤ φ^{∗}(u^{∗})(t) holds almost everywhere on ]s − δ_{0}, s[ and therefore

0 < \frac{du^{∗}(t)}{dt} ≤ φ^{∗}(u^{∗})(t).  (45)

Let l be a natural number so large that

δ_{0}^{− \frac{(l+ \frac{1}{2} )ε}{λ_{k}(s,ρ)}} > ρ_{0} (k = 1, …, m).  (46)

ON THE GLOBAL SOLVABILITY OF THE CAUCHY PROBLEM 369

Setting v(t) = (s − t)^{−(l+ \frac{1}{2} )ε}, we obtain

v(s − δ_{0}) > ρ_{0}.  (47)

Moreover, with (42), (43), and (46) taken into account we find

ν_{k}^{∗}(v)(t) = \begin{cases}ρ_{0} & for s − δ_{0} ≤ t ≤ s − δ_{0}^{λ_{k}(s,ρ)} \\ (s − t)^{− \frac{(l+ \frac{1}{2} )ε}{λ_{k}(s,ρ)}} & for s − δ_{0}^{λ_{k}(s,ρ)} < t ≤ s \end{cases} ,

and

ν_{k}^{∗}(v)(t) ≤ (s − t)^{− \frac{(l+ \frac{1}{2} )ε}{λ_{k}(s,ρ)}} for s_{0} − δ_{0} ≤ t < s (k = 1, …, m). (48)

By (44) and (48) the inequality

φ^{∗}(v)(t) ≤ α(s, ρ)(s − t)^{2ε−1}(1 + m(s − t)^{−(l+ \frac{1}{2} )ε}) ×

× ln (2 + m(s − t)^{−(l+ \frac{1}{2} )ε}) ≤ α(s, ρ)(m + 1) × × (s − t)^{2ε−1−(l+ \frac{1}{2} )ε} ln [(2 + m)(s − t)^{−(l+ \frac{1}{2} )ε}]

holds almost everywhere on ]s − δ_{0}, s[.

On the other hand, using (40) we have

(s − t)^{ε} ln \frac{1}{s − t} ≤ δ_{0}^{ε} ln \frac{1}{δ_{0}} ≤

≤ [α(s, ρ)(m + 1)(1 + ln(2 + m))]^{−1} for s − δ_{0} ≤ t < s and

ln [(2 + m)(s − t)^{−(l+ \frac{1}{2} )ε}] = ln(2 + m) + (l + \frac{1}{2})ε ln \frac{1}{s − t} ≤

≤ (1 + ln(2 + m))(l + \frac{1}{2})ε ln \frac{1}{s − t} ≤

≤ \frac{(l + \frac{1}{2})ε}{α(s, ρ)(m + 1)}(s − t)^{−ε} for s − δ_{0} ≤ t < s.

Therefore

φ^{∗}(v)(t) ≤ (l + \frac{1}{2})ε(s − t)^{ε−1−(l+ \frac{1}{2})ε}  (49)

and

\frac{dv(t)}{dt} > φ^{∗}(v)(t).  (50)

By Theorem 1.4 from [2] inequalities (41), (45), (50) imply the estimate 0 < u^{∗}(t) < (s − t)^{−(l+ \frac{1}{2} )ε} for s − δ_{0} ≤ t < s, by means of which we find from (45) and (49) that

0 ≤ \frac{du^{∗}(t)}{dt} ≤ ρ_{1}(s − t)^{−(l−1+ \frac{1}{2} )ε−1},

where ρ_{1} = (l + \frac{1}{2})ε. Assume now that the inequality

0 ≤ \frac{du^{∗}(t)}{dt} ≤ ρ_{k}(s − t)^{−(l−k+ \frac{1}{2} )ε−1},  (51)

where ρ_{k} is a positive constant, holds almost everywhere on ]s − δ_{0}, s[ for some k ∈ {1, …, l}. Then

0 < u^{∗}(t) ≤ ρ_{0} + (l − k + \frac{1}{2})^{−1}ε^{−1}(s − t)^{−(l−k+ \frac{1}{2} )ε} ≤

≤ ρ_{1k}(s − t)^{−(l−k+ \frac{1}{2} )ε} for s − δ_{0} ≤ t < s,

where ρ_{1k} = ρ_{0}+(l−k+ \frac{1}{2})^{−1}ε^{−1}ρ_{k}. On account of this, estimates (42)–(46) yield

ν_{k}^{∗}(u^{∗})(t) ≤ ρ_{0} + ρ_{1k}(s − t)^{− \frac{(l−k+ \frac{1}{2} )ε}{λ_{k}(s,ρ)}} ≤
≤ (ρ_{0} + ρ_{1k})(s − t)^{− \frac{(l−k+ \frac{1}{2} )ε}{λ_{k}(s,ρ)}} ,

1 + ∑_{k=1}^{m} [ν_{k}^{∗}(u^{∗})(t)]^{λ_{k}(s,ρ)} ≤ 1 + ∑_{k=1}^{m}(ρ_{0} + ρ_{1k})^{λ_{k}(s,ρ)}(s − t)^{−(l−k+ \frac{1}{2} )ε} ≤

≤ [1 + ∑_{k=1}^{m}(ρ_{0} + ρ_{1k})^{λ_{k}(s,ρ)}](s − t)^{−(l−k+ \frac{1}{2} )ε},

ln (2 + ∑_{k=1}^{m} ν_{k}^{∗}(u)(t)) ≤ ln [(2 + ∑_{k=1}^{m}(ρ_{0} + ρ_{1k}))(s − t)^{−l}] ≤

≤ ln (2 + ∑_{k=1}^{m}(ρ_{0} + ρ_{1k})) + l ln (\frac{1}{s − t}) ≤

≤ [ ln (2 + ∑_{k=1}^{m}(ρ_{0} + ρ_{1k})) + l] ln \frac{1}{s − t}

and

0 < \frac{du^{∗}(t)}{dt} ≤ ρ_{k+1}(s − t)^{−(l−k−1+ \frac{1}{2} )ε}(s − t)^{ε} ln \frac{1}{s − t} ≤

≤ ρ_{k+1}(s − t)^{−(l−k−1+ \frac{1}{2} )ε}δ_{0}^{ε} ln \frac{1}{δ_{0}} < ρ_{k+1}(s − t)^{−(l−k−1+ \frac{1}{2} )ε},

where

ρ_{k+1} = α(s, ρ)[1 + ∑_{k=1}^{m}(ρ_{0} + ρ_{1k})^{λ_{k}(s,ρ)}][ ln (2 + ∑_{k=1}^{m}(ρ_{0} + ρ_{1k})) + l].

We have thus shown by induction that inequality (51) holds almost everywhere on ]s − δ_{0}, s[ for each k ∈ {1, …, l + 1}. Therefore

0 < \frac{du^{∗}(t)}{dt} ≤ ρ_{l+1}(s − t)^{\frac{ε}{2} −1}

and 0 < u^{∗}(t) < ρ^{∗} for s − δ_{0} ≤ t < s, where ρ^{∗} = ρ_{0} + \frac{2}{ε}δ0^{\frac{ε}{2}} . By Corollary 3.3, the latter estimate implies s ∈ I.

The proved proposition immediately implies

Corollary 8. Let there exist functions δ : ]a, b] →]0, 1[, c : ]a, b] → R^{n}, α : ]a, b]×R_{+} → R_{+}, β : ]a, b] →]−1, 0], λ_{k} : ]a, b] → [1, +∞[ (k = 1, …, m) and a number m_{0} ∈ {1, …, m} such that the inequalities

τ_{k}(t) ≤ s − (s − t)^{\frac{1}{λ_{k}(s)}} (k = 1, …, m_{0}),
τ_{k}(t) ≤ s − δ(s) (k = m_{0} + 1, …, m),

and

f_{0}(t, c(s) + y_{0}, …, c(s) + y_{m}) · sgn(y_{0}) ≤

≤ α(s, ∑_{k=m_{0}+1}^{m} ∥y_{k}∥)(s − t)^{β(s)}(1 + ∑_{k=1}^{m_{0}} ∥y_{k}∥^{λ_{k}(s)}) ×

× ln (2 + ∑_{k=1}^{m_{0}} ∥y_{k}∥)  (52)

hold respectively on ]s − δ(s), s[ and ]s − δ(s), s[ ×R^{(m+1)n}. Then each noncontinuable solution of equation (3) is defined on [a, b].

Example 2. ^{1} Let b − a ≤ 1, λ ≥ 1 and ε > 0. Then the differential equation

\frac{dx(t)}{dt} = \frac{λ}{ε}|x(b − (b − t)^{\frac{1}{λ}})|^{λ+ε}

has the noncontinuable solution x(t) = (b − t)^{− \frac{λ}{2}} defined on the interval [a, b[ .

1See [3].

This example shows that the index λ_{k}(s) on the right-hand side of (52) cannot be replaced by λ_{k}(s)+ε for any k ∈ {1, …, m} no matter how small ε > 0 is.

To conclude, note that Corollaries 1, 2, 4, and 5 are analogues of the well-known theorem of A. Wintner ([4], Ch. III, §3.5) for problems (1), (2) and (3), (2), and Corollaries 7 and 8 are generalizations of the theorem of A. Myshkis and Z. Tsalyuk [3] (see also [5]).

References

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

2. I. Kiguradze and Z. Sokhadze, On singular functional differential inequalities. Georgian Math. J. 4(1997), No. 4, 259–278.

3. A. D. Myshkis and Z. B. Tsalyuk. On nonlocal continuability of solutions of differential systems with deviating argument. (Russian) Differential’nye Uravneniya 5(1996), No. 6, 1128–1130.

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

5. Z. Sokhadze, On a theorem of Myshkis–Tsalyuk. Mem. Differential Equations Math. Phys. 5(1995), 131–132.

(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