GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 3, 1998, 251-262

ON THE SOLVABILITY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL

EQUATIONS

I. KIGURADZE AND B. PŮŽA

Abstract. Sufficient conditions are established for the solvability of the boundary value problem

\frac{dx(t)}{dt} = p(x, x)(t) + q(x)(t),
l(x, x) = c(x) ,

where p : C(I; R^{n}) × C(I; R^{n}) → L(I; R^{n}), q : C(I; R^{n}) → L(I; R^{n}), l : C(I, R^{n}) × C(I; R^{n}) → R^{n}, and c_{n} : C(I, R^{n}) → R^{n} are continuous operators, and p(x, ·) and l(x, ·) are linear operators for any fixed x ∈ C(I; R^{n}).

1. Formulation of the Main Results

1.1. Formulation of the problem. Let n be a natural number, I = [a, b], −∞ < a < b + ∞ and p : C(I; R^{n}) × C(I; R^{n}) → L(I, R^{n}), q : C(I; R^{n}) → L(I; R^{n}), l : C(I; R^{n}) × C(I; R^{n}) → R^{n} and c : C(I; R^{n}) → R^{n} be continuous operators. We consider the vector functional differential equation

\frac{dx(t)}{dt} = p(x, x)(t) + q(x)(t)  (1.1)

with the boundary condition

l(x, x) = c(x) .  (1.2)

By a solution of (1.1) we mean an absolutely continuous vector function x : I → R^{n} which satisfies it almost everywhere in I, and by a solution of problem (1.1), (1.2) a solution of (1.1) satisfying condition (1.2).

251

1991 Mathematics Subject Classification. 34K10.

Key words and phrases. Functional differential equation, Volterra operator, boundary value problem, existence theorem.

1072-947X/98/0500-0251$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

In the present paper, we use the results proved in [1,2] to establish new sufficient conditions for the solvability of problem (1.1), (1.2). The results obtained are made more concrete for the boundary value problem

\frac{dx(t)}{dt} = ∑_{i=1}^{m} P_{i}(x)(t)x (τ_{i}(t)) + q(x)(t) ,  (1.3)

∑_{i=1}^{m_{0}} H_{i}(x)x (t_{i}) = c(x) ,  (1.4)

where P_{i} : C(I, R^{n}) → L(I; R^{n×n}) (i = 1, …, m) and H_{i} : C(I; R^{n}) → R^{n} (i = 1, …, m_{0}) are continuous operators, and t_{i} ∈ I (i = 1, …, m_{0}) and τ_{j} : I → I (i = 1, …, m) are measurable functions.

1.2. Basic notation and terms. Throughout this paper the following notation and terms are used.

R =] − ∞, +∞[, R_{+} = [0, +∞[ .

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^{n×n} is the space of n×n matrices X = (x_{ik})_{i,k=1}^{n} with elements 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}, 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}.

X^{−1} is the inverse matrix of X; E is the unit matrix; det(X) is the determinant of the matrix X; r(X) is the spectral radius of the matrix X.

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

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

∥x∥_{C} = max{∥x(t)∥ : t ∈ I} .

If x = (x_{i})_{i=1}^{n} ∈ C(I; R^{n}), then |x|_{C} = (∥x_{i}∥_{C})_{i=1}^{n}. C(I; R^{n×n}) is the space of continuous matrix functions X : I → R^{n×n}. L(I; R^{n}) is the space of summable vector functions x : I → R^{n} with the

norm

∥x∥_{L} = ∫_{a}^{b} ∥x(t)∥ dt .

L(I; R^{n×n}) is the space of summable matrix functions x : I → R^{n×n}. If Y ∈ C(I; R^{n×n}) and l_{0} : C(I; R^{n}) → R^{n} and p_{0} : C(I; R^{n}) → L(I; R^{n})

are linear operators, then l_{0}(Y) ∈ R^{n×n} and p_{0}(Y) ∈ L(I; R^{n×n}) are the matrix satisfying, for every u ∈ R^{n}, the equalities

l_{0}(Y u) = l_{0}(Y)u, p_{0}(Y u)(t) = p_{0}(Y)(t)u for t ∈ I .

I_{t_{0},t} = [t_{0}, t] for t ≥ t_{0} and I_{t_{0},t} = [t, t_{0}] for t < t_{0} . A linear operator p is called a Volterra^{∗} operator with respect to t_{0} ∈ I,

if for arbitrary t ∈ I and x ∈ C(I; R^{n}) satisfying the condition x(s) = 0 for s ∈ I_{t_{0},t} we have p(x)(s) = 0 for almost all s ∈ I_{t_{0},t} .

1.3. Problem (1.1), (1.2). We consider the case where:

(i) the operators p : C(I; R^{n})×C(I; R^{n}) → L(I; R^{n}) and l : C(I; R^{n})× C(I; R^{n}) → R^{n} are continuous and for arbitrary x ∈ C(I; R^{n}) the operators p(x, ·) : C(I; R^{n}) → L(I; R^{n}) and l(x, ·) : C(I; R^{n}) → R^{n} are linear;

(ii) there exist a summable function α : [a, b] → R_{+} and a positive number α_{0} such that for arbitrary x and y ∈ C(I; R^{n}) the following inequalities are satisfied:

∥p(x, y)(t)∥ ≤ α(t)∥y∥_{C} for almost all t ∈ I, ∥l(x, y)∥ ≤ α_{0}∥y∥_{C} ;

(iii) the operators q : C(I; R^{n}) → L(I; R^{n}) and c : C(I; R^{n}) → R^{n} are continuous and

lim_{ϱ→∞} \frac{1}{ϱ} ∫_{a}^{b} η(t, ϱ) dt = 0, lim_{ϱ→∞} \frac{η_{0}(ϱ)}{ϱ} = 0 ,

where η(t, ϱ) = sup{∥q(x)(t)∥ : ∥x∥_{C} ≤ ϱ}, η_{0}(ϱ) = sup{∥c(x)∥ :

∥x∥_{C} ≤ ϱ}.

Following [2] we introduce

∗See [3–5].

Definition 1.1. Let p and l be the operators satisfying conditions (i) and p_{0} : C(I; R^{n}) → L(I; R^{n}) and l_{0} : C(I; R^{n}) → R^{n} be the linear operators. We say that the pair of operators (p_{0}, l_{0}) belongs to the set E_{p,l}^{n} if there is a sequence x_{k} ∈ C(I; R^{n}) (k = 1, 2, …) such that for any y ∈ C(I; R^{n}) we have

lim_{k→∞} ∫_{a}^{t} p(x_{k}, y)(s) ds = ∫_{a}^{t} p_{0}(y)(s) ds uniformly on I ,  (1.5)
lim_{k→∞} l(x_{k}, y) = l_{0}(y) .  (1.6)

Definition 1.2. We say that a pair of operators (p, l) belongs to the Opial class O_{0}^{n}, if p and l satisfy conditions (i) and (ii), and for any (p_{0}, l_{0}) ∈ E_{p,l}^{n} the problem

\frac{dy(t)}{dt} = p_{0}(y)(t) ,  (1.7)
l_{0}(y) = 0  (1.8)

has only the trivial solution.

Theorem 1.1. If

(p, l) ∈ O_{0}^{n}  (1.9)

and the operators q and c satisfy conditions (iii), then problem (1.1), (1.2) is solvable.

The proof of this theorem is contained in [2] (see [2], Corollary 1.1).

Let t_{0} be an arbitrary fixed point of I. We introduce a sequence of operators Y_{j} : C(I; R^{n}) → C(I; R^{n×n}) and z_{j} : C(I; R^{n}) × C(I; R^{n}) → C(I; R^{n}) (j = 1, 2, …) by

Y_{1}(x)(t) = E, Y_{j+1}(x)(t) = E + ∫_{t_{0}}^{t} p(x, Y_{j}(x))(s) (j = 1, 2, …) (1.10)

z_{1}(x, y)(t) = ∫_{t_{0}}^{t} p(x, y)(s) ds , z_{j+1}(x, y)(t) =

= ∫_{t_{0}}^{t} p(x, z_{j}(x, y))(s) ds (i = 1, 2, …) .  (1.11)

If for some x ∈ C(I; R^{n}) and natural j the matrix l(x, Y_{j}(x)) is nonsingular, we set

z_{j,k}(x, y)(t) = z_{k}(x, y)(t) −
−Y_{k}(x)(t)[l(x, Y_{j}(x))]^{−1}l(x, z_{j}(x, y)) (k = 1, 2, …) .  (1.12)

Theorem 1.2. Suppose that conditions (i)–(iii) are satisfied, and there exist t_{0} ∈ I, δ > 0, A ∈ R_{+}^{n×n} and natural numbers j_{0} and k_{0} such that

r(A) < 1  (1.13)

and for arbitrary x and y ∈ C(I; R^{n}) the inequalities

| det(l(x, Y_{j_{0}}(x))| > δ  (1.14)

and

|z_{j_{0},k_{0}}(x, y)|_{C} ≤ A|y|_{C}  (1.15)

hold, where Y_{j} and z_{j,k} are the operators given by (1.10)–(1.12). Then problem (1.1), (1.2) is solvable.

If p satisfies conditions (i) and (ii) and for some t_{0} ∈ I and arbitrary fixed x ∈ C(I; R^{n}) the operator p(x, ·) : C(I; R^{n}) → L(I; R^{n}) is a Volterra operator with respect to t_{0}, then for any x ∈ C(I; R^{n}) the differential equation

\frac{dy(t)}{dt} = p(x, y)(t)  (1.16)

has a unique fundamental matrix^{∗} Y (x) : I → R^{n×n} satisfying the initial condition

Y (x)(t_{0}) = E  (1.17)

(see [1], Lemma 1.2). In this case, Theorem 1.2 takes the following form.

Theorem 1.3. Suppose that conditions (i)–(iii) are satisfied, and there is t_{0} ∈ I such that for any x ∈ C(I; R^{n}) the operator p(x, ·) is a Volterra operator with respect to t_{0}. Furthermore, let

inf {| det (l(x, Y (x)))| : x ∈ C(I; R^{n})} > 0  (1.18)

where Y (x) is the fundamental matrix of system (1.16), satisfying the initial condition (1.17). Then problem (1.1), (1.2) is solvable.

∗That is, the matrix whole columns form a basis of the solution space of equation (1.16).

Remark 1.1. Theorem 1.3 is a generalization of a theorem of R. Conti ([6], Theorem 2) and Corollary 1.4′ from [2] to differential systems of form (1.1).

1.4. Problem (1.3), (1.4). Problem (1.3), (1.4) is obtained from (1.1), (1.2) for

p(x, y)(t) = ∑_{i=1}^{m} P_{i}(x)(t)y(τ_{i}(t)), l(x, y) = ∑_{i=1}^{m_{0}} H_{i}(x)y(t_{i}) . (1.19)

In order to guarantee that conditions (i) and (ii) be satisfied, we consider the case where:

(iv) the operators P_{i} : C(I; R^{n}) → L(I; R^{n×n}) (i = 1, …, m) and H_{i} : C(I; R^{n}) → R^{n×n} (i = 1, …, m_{0}) are continuous and there are a summable function α : I → R_{+} and a positive number α_{0} such that for any x ∈ C(I; R^{n}) and t ∈ I the following inequalities hold:

∑_{i=1}^{m} ∥P_{i}(x)(t)∥ ≤ α(t) , ∑_{i=1}^{m_{0}} ∥H_{i}(x)∥ ≤ α_{0} .

From (1.19) and (1.10) we obtain

Y_{1}(x)(t) = E, Y_{j+1}(x)(t) =
= E + ∑_{i=1}^{m} ∫_{t_{0}}^{t} P_{i}(x)(s)Y_{j}(τ_{i}(s)) ds (j = 1, 2, …) .  (1.20)

Together with this we introduce the sequence of operators

\begin{matrix}Z_{1}(x)(t) = ∑_{i=1}^{m} | ∫_{t_{0}}^{t} |P_{i}(x)(s)| ds|, \\ Z_{j+1}(x)(t) = ∑_{i=1}^{m} | ∫_{t_{0}}^{t} |P_{i}(x)(s)|Z_{j}(x)(τ_{i}(s)) ds| (j = 1, 2, …) . \end{matrix}(1.21)

Corollary 1.1. Suppose that conditions (iii), (iv) are satisfied, and there exist t_{0} ∈ I, δ > 0, A ∈ R_{+}^{n×n}, and natural numbers j_{0} and k_{0}, such that r(A) < 1 and for arbitrary x ∈ C(I; R^{n}) the inequalities

| det (∑_{i=1}^{m_{0}} H_{i}(x)Y_{j_{0}}(x)(t)) | > δ  (1.22)

and

|Y_{k_{0}}(x)(t) (∑_{i=1}^{m_{0}} H_{i}(x)Y_{j_{0}}(x)(t_{i}))^{−1} | ∑_{i=1}^{m_{0}} |H_{i}(x)|Z_{j_{0}}(x)(t_{i}) +

+Z_{k_{0}}(x)(t) ≤ A for t ∈ I  (1.23)

hold, where Y_{j} and Z_{k} are the operators given by (1.20) and (1.21). Then problem (1.3), (1.4) is solvable.

Corollary 1.2. Suppose that conditions (iii), (iv) are satisfied, and there is t_{0} ∈ I such that

(t − τ_{i}(t))(t − t_{0}) ≥ 0 for t ∈ I (i = 1, …, m) .  (1.24)

Furthermore, let

inf {| det (∑_{i=1}^{m_{0}} H_{i}(x)Y (x)(t_{i})) | : x ∈ C(I; R^{n})} > 0 , (1.25)

where Y (x) id the fundamental matrix of the differential system

\frac{dy(t)}{dt} = ∑_{i=1}^{m} P_{i}(x)(t)y(τ_{i}(t)) ,

satisfying the initial condition

Y (x)(t_{0}) = E .

Then problem (1.3), (1.4) is solvable.

2. Auxiliary Statements

Lemma 2.1. Let p_{0} : C(I; R^{n}) → L(I; R^{n}) and l_{0} : C(I; R^{n}) → R^{n×n} be linear operators, and let there exist a summable function α : I → R_{+} such that for any y ∈ C(I; R^{n}) the inequality

∥p_{0}(y)(t)∥ ≤ α(t)∥y∥_{C} for almost all t ∈ I  (2.1)

holds. Furthermore, let p_{0} be a Volterra operator with respect to some t_{0} ∈ I. Then equation (1.7) has a unique fundamental matrix Y_{0} satisfying the initial condition

Y (t_{0}) = E  (2.2)

and the condition

det (l_{0}(Y_{0})) ≠ 0  (2.3)

is necessary and sufficient for problem (1.7), (1.8) to have only the trivial solution.

Proof. According to Lemma 1.2 in [1], for any c_{0} ∈ R, equation (1.7) has a unique solution satisfying the initial condition

y(t_{0}) = c_{0} .

It follows that this equation has a unique fundamental matrix satisfying the initial condition (2.2) and each of its solutions admits a representation

y(t) = Y_{0}(t)c_{0} .

It follows that problem (1.7), (1.8) has only the trivial solution if and only if the system of algebraic equations

l_{0}(Y_{0})c_{0} = 0

has only the trivial solution, i.e., condition (2.3) is satisfied.

Lemma 2.2. Let p and l be the operators satisfying conditions (i), (ii), and C(I; R^{n}) → L(I; R^{n}) and l_{0} : C(I; R^{n}) → R^{n} be linear operators. Furthermore, let there exist t_{0} ∈ I and x_{k} ∈ C(I; R^{n}) (k = 1, 2, …) such that for any natural k the operator p(x_{k}, ·) is a Volterra operator with respect to t_{0} and for any y ∈ C(I; R^{n}) conditions (1.5) and (1.6) hold. Then p_{0} is a Volterra operator with respect to t_{0}, satisfying condition (2.1), and

lim_{k→∞} l(x_{k}, Y (x_{k})) = l_{0}(Y_{0}) ,  (2.4)

where Y (x_{k}) is the fundamental matrix of the equation

\frac{dy(t)}{dt} = p(x_{k}, y)(t) ,  (2.5)

satisfying the condition

Y (x_{k})(t_{0}) = E ,

and Y_{0} is the fundamental matrix of (1.7) satisfying condition (2.2).

Proof. By (1.5), for any s and t ∈ I we have

∫_{s}^{t} p_{0}(y)(ξ) dξ = lim_{k→∞} ∫_{s}^{t} p(x_{k}, y)(ξ) dξ .

Since p(x_{k}, ·) is a Volterra operator with respect to t_{0} it clearly follows that p_{0} is also Volterra operator with respect to t_{0}. On the other hand, using condition (ii), we get from the last equality

∥ ∫_{s}^{t} p_{0}(y)(ξ) dξ∥ ≤ | ∫_{s}^{t} α(ξ) dξ| ∥y∥_{C} .

Dividing both sides by t − s and passing to the limit for s → t we get (2.1).

By Corollary 1.6 in [1], it follows from (i), (ii), (1.5), (2.1) and from the unique solvability of the Cauchy problem for (1.7) with the initial condition at t_{0} that

lim_{k→∞} ∥Y (t) − Y_{0}∥_{C} = 0 .  (2.6)

On the other hand, by (1.6)

lim_{k→∞} ∥l(x_{k}, Y_{0}) − l_{0}(Y_{0})∥ = 0 .  (2.7)

Taking into account (ii), (2.6) and (2.7), we find

∥l(x_{k}, Y (x_{k})) − l_{0}(Y_{0})∥ ≤
≤ ∥l(x_{k}, Y (x_{k}) − Y_{0})∥ + ∥l(x_{k}, Y_{0}) − l_{0}(Y)∥ ≤
≤ α_{0}∥Y_{k} − Y_{0}∥_{C} + ∥l(x_{k}, Y_{0}) − l_{0}(Y_{0})∥ → 0 for k → ∞ .

Consequently, equality (2.4) holds.

Lemma 2.3. Let conditions (i), (ii) hold and let there exist t_{0} ∈ I such that for any x ∈ C(I; R^{n}) the operator p(x, ·) is a Volterra operator with respect to t_{0}. Then inequality (1.18) is necessary and sufficient for condition (1.9) to hold.

Proof. First we prove the necessity. Assume the contrary, that (1.9) holds, but (1.18) is violated. Then there is a sequence x_{k} ∈ C(I; R^{n}) (k = 1, 2, …) such that

lim_{k→∞} det (l(x_{k}, Y (x_{k}))) = 0 .  (2.8)

It follows from (i), (ii) and Lemma 2.1 in [1] that, without loss of generality, we can assume that conditions (1.5) and (1.6) hold for any y ∈ C(I; R^{n}), where (p_{0}, l_{0}) ∈ E_{p,c}^{n}. Then, by Lemma 2.2 and equality (2.8), the operator p_{0} is a Volterra operator with respect to t_{0}, inequality (2.1) holds, and

det (l_{0}(Y_{0})) = 0 ,

where y_{0} is the fundamental matrix of (1.7), satisfying condition (2.2). Therefore, it follows from Lemma 2.1 that problem (1.7), (1.8) has a nontrivial solution. This is a contradiction to (1.9), which proves that (1.18) holds.

Now we turn to proving the sufficiency. Let (p_{0}, l_{0}) ∈ E_{p,l}^{n}. Then by Definition 1.1 there is a sequence x_{k} ∈ C(I; R^{n}) (k = 1, 2 …) such that for any y ∈ C(I; R^{n}) conditions (1.5) and (1.6) hold. If we now apply Lemma 2.2, it becomes clear that the operator p_{0} is a Volterra operator with respect to t_{0}, and conditions (2.1), (2.4) hold, where Y_{0} is the fundamental matrix of equation (1.7), satisfying condition (2.2). On the other hand, by (1.18) condition (2.4) implies (2.3). By Lemma 2.1, this inequality implies that problem (1.7), (1.8) has only the trivial solution. Now since (p_{0}, l_{0}) ∈ E_{p,l}^{n} was arbitrary, condition (1.9) is obvious.

3. Proofs of Main Results

Proof of Theorem 1.2. According to Theorem 1.1 it is enough to show that for any (p_{0}, l_{0}) ∈ E_{p,l}^{n} problem (1.7), (1.8) has only the trivial solution.

By Definition 1.1 there is a sequence x_{k} ∈ C(I; R^{n}) (k = 1, 2, …) such that for any y ∈ C(I; R^{n}) conditions (1.5) and (1.6) hold. Using (ii), (1.5), and (1.6), we get from (1.10)

lim_{k→∞} Y_{j}(x_{k})(t) = Y_{j}^{0}(t) uniformly on I (j = 1, 2, …), (3.1)
lim_{k→∞} z_{j}(x_{k}, y)(t) = z_{j}^{0}(y)(t) uniformly on I (j = 1, 2, …)  (3.2)

for any y ∈ C(I; R^{n}) and

lim_{k→∞} l(x_{k}, Y_{j}(x_{k})) = l_{0}(Y_{j}^{0}) (j = 1, 2, …),  (3.3)

where

Y_{1}^{0}(t) = E, Y_{j+1}^{0}(t) = E + ∫_{t_{0}}^{t} p_{0}(Y_{j}^{0})(s) ds (j = 1, 2, …) ,

z_{1}^{0}(y)(t) = ∫_{t_{0}}^{t} p_{0}(y)(s) ds, z_{j+1}^{0}(y)(t) = ∫_{t_{0}}^{t} p_{0}(z_{j}^{0}(y))(s) ds (j = 1, 2, …) .

By (3.1)–(3.3) we get from (1.12), (1.14) and (1.15)

det(l_{0}(Y_{j_{0}}^{0})) ≠ 0  (3.4)

and

|z_{j_{0},k_{0}}^{0}(y)|_{C} ≤ A|y|_{C} ,  (3.5)

where

z_{j_{0},k_{0}}(y)(t) = z_{k_{0}}^{0}(y)(t) − Y_{k_{0}}^{0}(t)[l_{0}(Y_{j_{0}}^{0})]^{−1}l_{0}(z_{j_{0}}^{0}) .

According to Theorem 1.2 in [1], it follows from (1.13), (3.4), and (3.5) that problem (1.7), (1.8) has only the trivial solution.

Theorem 1.1 and Lemma 2.3 immediately imply Theorem 1.3.

Remark 3.1. In the case where p(x, ·) is a Volterra operator with respect to t_{0} for any x ∈ C(I; R^{n}), then by Lemma 2.3, Theorem 1.1 and 1.3 are equivalent.

Proof of Corollary 1.1. As we already noted, problem (1.3), (1.4) is obtained from problem (1.1), (1.2) when the operators p and l are given by (1.19). In this case, obviously, conditions (1.14) and (1.22) are equivalent. On the other hand, using (1.19) and (1.21), it follows from (1.11) that

|z_{j}(x, y)(t)| ≤ Z_{j}(x)(t)|y|_{C} for t ∈ I (j = 1, 2, …)

and

|l(x, z_{j}(x, y))| ≤ ∑_{i=1}^{m_{0}} |H_{i}(x)|Z_{j}(x)(t_{i})|y|_{C} (j = 1, 2, …) .

Using these inequalities, we get from (1.12) and (1.23)

|z_{j_{0},k_{0}}(x, y)(t)| ≤ Z_{k_{0}}(x)(t)|y|_{C} + |Y_{k_{0}}(x)(t)|( ∑_{i=1}^{m_{0}} H_{i}(x)Y_{j_{0}}(x)(t_{i}))^{−1}| ×

× ∑_{i=1}^{m_{0}} |H_{i}(x)|Z_{j_{0}}(x)(t_{i})|y|_{C} ≤ A|y|_{C} for t ∈ I .

Consequently, inequality (1.15) holds. This argument proves that all conditions of Theorem 1.2 are satisfied, which guarantees the unique solvability of the problem under consideration.

Corollary 1.2 is obtained from Theorem 1.3. It is enough to take into account that condition (1.24) and the equality

p(x, y)(t) = ∑_{i=1}^{m} P_{i}(x)(t)y(τ_{i}(t))

guarantee that p(x, ·) is a Volterra operator with respect to t_{0} for any x ∈ C(I; R^{n}).

Acknowledgement

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.

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(Received 21.10.1996)

Authors’ addresses:

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

Bedřich Půža Department of Mathematical Analysis Faculty of Science, Masaryk University Janáčkovo nám. 2a, 662 95 Brno, Czech Republic