On some boundary value problems for systems of linear functional-differential equations

1991 Mathematics Subject Classification. 34K10 Key words and phrases. System of linear functional differential equations, linear differential system with deviating arguments, boundary value problem

Here ik : C(I; R) → L(I; R) are linear bounded operators, p ik and q i ∈ L(I; R), c i ∈ R (i, k = 1, . . ., n), ϕ i : I → R (i = 1, . . ., n) are the functions with bounded variations, and τ ik : I → I (i, k = 1, . . ., n) are measurable functions.The simple but important particular case of the conditions (2) are the two-point boundary conditions the periodic boundary conditions and the initial conditions where t 0 ∈ I and λ i ∈ R (i = 1, . . ., n).By a solution of the system (1) (of the system (1 )) we understand an absolutely continuous vector function (x i ) n i=1 : I → R which satisfies the system (1) (the system (1 )) almost everywhere on I.A solution of the system (1) (of the system (1 )) which satisfies the condition (j), where j ∈ {2, 3, 4, 5}, is said to be a solution of the problem (1), (j).
1.2.Basic Notation.Throughout this paper the following notation and terms are used: R n is the space of n-dimensional column vectors x = (x i ) n i=1 with the components . ., n}; the inequalities between vectors x and y ∈ R n , and between matrices X and Y ∈ R n×n are considered componentwise, i.e., r(X) is the spectral radius of the matrix X ∈ R n×n ; X −1 is the inverse matrix to X ∈ R n×n ; E is the unit matrix; C(I; R n ) is the space of continuous 1 vector functions x : I → R n with the norm holds almost everywhere on I; for any u ∈ L(I; R) 1.3.Criterion on the Unique Solvability of the Problem (1), (2).The results in general theory of boundary value problems (see [12], Theorems 1.1 and 1.4) yield the following Theorem 1.If ik ∈ L I (i, k = 1, . . ., n), then the boundary value problem (1), (2) with arbitrary c i ∈ R and q i ∈ L(I; R) (i = 1, . . ., n) is uniquely solvable if and only if the corresponding homogeneous problem has only the trivial solution.If the latter condition is fulfilled, then the solution of the problem (1), (2) admits the representation where y ik ∈ C(I; R) (i, k = 1, . . ., n), and g i : linear continuous operators such that the vector function is the solution of the problem (1 0 ), (2), and the vector function (g i (q 1 , . . ., q n )) n i=1 is the solution of the problem (1), (2 0 ).

Remark 1. The operator (g
0 ).According to Danford-Pettis Theorem (see [1], Ch.XI, §1, Theorem 6), there exists the unique matrix function G = (g ik ) n i,k=1 : I × I → R n×n with the essentially bounded components g ik : Consequently, the formula ( 6) can be rewritten as follows: This formula is called the Green's formula for the problem (1), (2), and the matrix G is called the Green's matrix of the problem (1 0 ), (2 0 ).
The aim of the following is to find effective criteria for the unique solvability of the above formulated problems.With a view to achieve this goal, we will need one lemma which is proved in Section 2.

Lemma on Boundary Value Problem for the System of Functional Differential Equations
Consider the system of differential inequalities with the boundary conditions b a where Along with (7), (8) for every i ∈ {1, . . ., n} consider the homogeneous problem Let for every i ∈ {1, . . ., n} the homogeneous problem (9 i ) have only the trivial solution and there exist a matrix A = (a ik where g i is the Green's function of the problem (9 i ).Then the problem (7), ( 8) has only the trivial solution.

Existence and Uniqueness Theorems
Throughout the following we will assume that ik ∈ L I (i, k = 1, . . ., n) and for any u ∈ C(I; R) the inequalities hold almost everywhere on I, where ik ∈ P I (i, k = 1, . . ., n). (i) any solution of the system (1 0 ) is also a solution of the system (ii) for any y ∈ C(I; R), the inequalities holds almost everywhere on I; (iii) for every i ∈ {1, . . ., n} the problem (9 i ) has only the trivial solution and the inequalities (11) are fulfilled, where g i is the Green's function of the problem (9 i ).
Proof.Let (y i ) n i=1 be a solution of the problem (1 0 ), (2 0 ).Then by ( 17) and ( 18) it is also a solution of the problem ( 7), (8).Now it is obvious that all the assumptions of Lemma 1 are fulfilled.Therefore y i (t) ≡ 0 (i = 1, . . ., n).Thus the homogeneous problem (1 0 ), (2 0 ) has only the trivial solution and consequently, by Theorem 1, the problem (1), (2) has a unique solution.2 Corollary 1.Let there exist operators i ∈ L ) such that: (i) for any y ∈ C(I; R), the inequalities . ., n) holds almost everywhere on I; (ii) for every i ∈ {1, . . ., n} the problem (9 i ) has only the trivial solution and the inequalities (11) are fulfilled, where g i is the Green's function of the problem (9 i ).
Then the assumptions of Theorem 2 are fulfilled.Consequently, the problem (1), (2) has a unique solution.2 EJQTDE, 1999 No. 10, p. 7 and there exist a matrix A = (a ik ) n i,k=1 ∈ R n×n + satisfying (10) such that (11) is fulfilled, where g i is the Green's function of the problem and Then the problem (1), ( 2) has a unique solution.
Proof.The condition ( 19) is necessary and sufficient for the problem (20 i ) to have only the trivial solution for every i ∈ {1, . . ., n}.
On the other hand, every solution (x i ) n i=1 of the system (1 0 ) satisfies Then any solution of the system (1 0 ) is also a solution of the system (17).Now it is obvious that all the assumptions of Theorem 2 are fulfilled.Consequently, the problem (1), ( 2) has a unique solution.2 then the system (1) has the form (1 ).In that case and there exist a matrix A = (a ik ) n i,k=1 ∈ R n×n + satisfying (10) such that (11) is fulfilled, where g i is the Green's function of the problem Then the problem (1 ), ( 2) has a unique solution.
and there exist a matrix A = (a ik ) n i,k=1 ∈ R n×n + satisfying (10) such that (11) is fulfilled, where g i is the Green's function of the problem and h ik is defined by (21).Then the problem (1), ( 4) has a unique solution.
Proof.The condition (24) is necessary and sufficient for the problem (25 i ) to have only the trivial solution for every i ∈ {1, . . ., n} and its Green's function is of the form Now it is obvious that all the assumptions of Corollary 2 are fulfilled.Consequently, the problem (1), (4) has a unique solution.2 and there exist a matrix A = (a ik ) n i,k=1 ∈ R n×n + satisfying (10) such that (11) is fulfilled, where g i is the Green's function of the problem and h ik is defined by (21 ).Then the problem (1 ), (4) has a unique solution.
Proof.At first note that according to Theorems 1.13 and 1.18 in [10] the negativeness of real parts of the eigenvalues of the matrix A yields the inequality where .
On the other hand, from (27) it follows that for every i ∈ {1, . . ., n} the problem (25 i ) has only the trivial solution and its Green's function g i is given by (26 i ).Put Then for every i ∈ {1, . . ., n} from (26 i ) and ( 27) we obtain Define the functions h ik by (21).Then from (28) and (30) we get Taking into account (29) we conclude that all the assumptions of Corollary 2 are fulfilled.Consequently, the problem (1), (4) has a unique solution.5) has a unique solution.
At the end of this subsection we give the examples verifying the optimality of the above formulated conditions in the existence and uniqueness theorems.
, +∞[.On the segment I = [0, 1] consider the system (1 ) with the boundary conditions (3), where and c i ∈ R (i = 1, 2).Then all the assumptions of Corollary 3 with are fulfilled except the condition (10) instead of which we have On the other hand, the homogeneous problem x has the nontrivial solution This example shows that the strict inequality (10) in Corollaries 3 and 3 cannot be replaced by the nonstrict one.On the other hand, the vector (γ i ) n i=1 is a nontrivial solution of the homogeneous problem x i (b) = x i (a).
The last example shows that in Corollaries 6 and 6 we cannot choose α = 1.

1 . Introduction 1 . 1 .
Statement of the Problem.On the segment I = [a, b] consider the system of linear functional differential equations the space of absolutely continuous vector functions x : I → R n with the norm x e C = x C + x L ; P I is the set of linear operators : C(I; R) → L(I; R) mappings C(I; R + ) into L(I; R + ); L I is the set of linear continuous operators : C(I; R) → L(I; R), for each of them there exists an operator ∈ P I such that for any u ∈ C(I; R) the inequalities