UNIQUE SOLVABILITY OF SECOND ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NON-LOCAL BOUNDARY CONDITIONS

Some general conditions sufficient for unique solvability of the boundary-value problem for a system of linear functional differential equations of the second order are established. The class of equations considered covers, in particular, linear equations with transformed argument, integro-differential equations and neutral equations. An example is presented to illustrate the general theory. 1. Problem formulation The purpose of this paper, which has been motivated in part by the recent works [11–16,18], is to establish new general conditions sufficient for the unique solvability of the non-local boundary-value problem for systems of linear functional differential equations on the assumptions that the linear operator l = (lk) n k=1, appearing in (1.1) can be estimated by certain other linear operators generating problems with conditions (1.2), (1.3) for which the statement on the integration of differential inequality holds. The precise formulation of the property mentioned is given by Definition 1.1. The proof of the main result obtained here is based on the application of [10, Theorem 49.4], which ensures the unique solvability of an abstract equation with an operator satisfying Lipschitz-type conditions with respect to a suitable cone. We consider the linear boundary-value problem for a second order functional differential equation u(t) = (lu)(t) + q(t), t ∈ [a, b], (1.1) u(a) = r1(u), (1.2) u(a) = r0(u), (1.3) where l : W ([a, b], R) → L1([a, b], R ) is linear operator, ri : W ([a, b], R) → R, i = 0, 1, are linear functionals. By a solution of problem (1.1)–(1.3), as usual (see, e. g., [1]), we mean a vector function u = (uk) n k=1 : [a, b] → R n whose components are absolutely continuous, satisfy system (1.1) almost everywhere on the interval [a, b], and possess properties (1.2), (1.3). Definition 1.1. A linear operator l = (lk) n k=1 : W ([a, b], R) → L1([a, b], R ) is said to belong to the set Sr0,r1 if the boundary value problem (1.1), (1.2), (1.3) has a unique solution u = (uk) n k=1 for any q ∈ L1([a, b], R ) and, moreover, the solution 2000 Mathematics Subject Classification. 34K10.


Problem formulation
The purpose of this paper, which has been motivated in part by the recent works [11][12][13][14][15][16]18], is to establish new general conditions sufficient for the unique solvability of the non-local boundary-value problem for systems of linear functional differential equations on the assumptions that the linear operator l = (l k ) n k=1 , appearing in (1.1) can be estimated by certain other linear operators generating problems with conditions (1.2), (1.3) for which the statement on the integration of differential inequality holds.The precise formulation of the property mentioned is given by Definition 1.1.
The proof of the main result obtained here is based on the application of [10,Theorem 49.4], which ensures the unique solvability of an abstract equation with an operator satisfying Lipschitz-type conditions with respect to a suitable cone.

Notation
Throughout the paper, we fix a bounded interval [a, b] and a natural number n.We use the following notation.
(1) 1) absolutely continuous on [a, b] and the norm given by the formula R n ) such that the components of u (m) are non-negative a.e. on [a, b] and u ) for m = 0, 2. The symbols defined above will usually appear in the text in a shortened form, e. g., the sets R n ) will be referred to simply as W 2 and W 2 (m;r0,r1) , etc. Since a, b, and n are fixed, no confusion will arise.

Auxiliary statements
To prove our main results, we use the following statement on the unique solvability of an equation with a Lipschitz type non-linearity established in [9] (see also [10]).
Let us consider the abstract operator equation where and, furthermore, let the order relation be satisfied for any pair (x, y) ∈ E 2 1 such that x ≧ K1 y.Then equation (3.1) has a unique solution x ∈ E 1 for an arbitrary element z ∈ E 2 .
with respect to the partial ordering ≦ E2 generated by K 2 is also bounded with respect to the norm.
A cone K 1 is said to be generating in E 1 if an arbitrary element x ∈ E 1 can be represented in the form x = u − v, where {u, v} ⊂ K 1 .
To prove that the cone W 2 (2;0,0) is generating in the space W 2 (0,0) , it is sufficient to show that every element x of W 2 (0,0) admits a majorant in W 2 (2;0,0) .Indeed, let x ∈ W 2 (0,0) be arbitrary.Then x has the form EJQTDE, 2012 No. 14, p. 3 where X ∈ L 1 .Equality (3.5) implies that, componentwise, where It is obvious from (3.6) that u(a) = 0, u ′ (a) = 0, and u ′′ is non-negative and, therefore, u is an element of W 2 (2;0,0) .This, due to the arbitrariness of x, proves that W 2 (2;0,0) is generating.Let us define a linear operator V l,r0,r1 : W 2 (r0,r1) → W 2 (0,0) by putting for all u ∈ W 2 (r0,r1) .Then the following assertion is straightforward.Lemma 3.2.A function u from W 2 is a solution of the equation where q ∈ L 1 , if and only if it is a solution of the non-local boundary value problem The lemma below sets the relation between the property described by Definition 1.1 and the positive invertibility of operator (3.7).
then the operator V l,r0,r1 is invertible and, moreover, its inverse V −1 l,r0,r1 satisfies the inclusion Proof.Let the mapping l belong to the set S r0,r1 .Given an arbitrary function Since y ∈ W 2 (0,0) , we have that, in particular, In view of assumption (3.9), there exists a unique function u such that u ′ is absolutely continuous, the equation holds, and By Lemma 3.2, it follows that u is, in fact, the unique solution of equation (3.11).

A general theorem on the solvability
The theorems presented below allow one to deduce conditions under which problem (1.3), (3.18) always has a unique solution.
Applying Theorem 3.1, we establish the unique solvability of the boundary value problem (3.18), (1.3) for arbitrary q ∈ L 1 .Taking Remark 3.1 into account, we complete the proof of Theorem 4.1.

Corollaries
The following statements are true.
Corollary 5.1.Assume that there exist certain linear operators f i : hold.Moreover, let the inclusions Proof.This statement is proved similarly to [6, Theorem 2].Indeed, it is obvious that, for any u from W 2 (0;r0,r1) , condition (5.1) is equivalent to the relation

Let us put
for any k = 1, 2, . . ., n.We see that, under conditions (5.1) and (5.2), the operators ) and are such that the inequalities Proof.It follows from assumption (5.5) the positivity of the operator g 1 that the relations are true for any u from W 2 (0;r0,r1) .This means that l = (l k ) n k=1 admits estimate (5.1) with the operators f 1 and f 2 defined by the equalities Moreover, assumption (5.4) guarantees that inclusions (5.2) hold for f 1 and f 2 of form (5.6).Thus, we can apply Corollary 5.1, which leads us to the required assertion.
Corollary 5.3.Assume that there exist positive linear operators and such that the inequalities 3) has a unique solution for any q ∈ L 1 .
Proof.It is sufficient to put g 0 := p 1 + p 2 , g 1 := p 1 , notice that g 0 and g 1 are positive, and apply Corollary 5.2.
It should be noted that conditions of the statements presented above are optimal in a certain sense and cannot be improved.For example, assumption (5.4) cannot be replaced by any of the weaker conditions where ε > 0, because after such a replacement the assertion of Corollary 5.2 is not true any more.The optimality of the conditions is proved by analogy to [3,16].
6.The case of l defined on W 1 In the general case, l from equation (1.1) is given on W 2 only and, thus, the right-hand side term of equation (1.1) may contain u ′′ , which corresponds to an equation of neutral type.
If the operator l in equation (1.1) is defined not only on W 2 but also on the entire space W 1 , then a statement equivalent to Theorem 4.1 can be obtained with the help of results established in [6,16].
Given an operator p : W 1 → L 1 , we put for any u from W 1 , so that I p is a map from W 1 to itself.We need the following definition [6].
Definition 6.1.Let r : W 1 → R n be a continuous linear vector functional.A linear operator p : W 1 → L 1 is said to belong to the set S r if the boundary value problem u(a) = r(u) (6.3) has a unique solution u = (u k ) n k=1 for any v = (v k ) n k=1 ∈ L 1 and, moreover, the solution of (6.2), (6.3) has non-negative components provided that the functions v k , k = 1, 2, . . ., n, are non-negative almost everywhere on [a, b].
In the case where the operator l, which determines the right-hand side of equation (1.1), is well defined on the entire space W 1 , results of the preceding sections admit an alternative formulation.In particular, the following statements hold.Theorem 6.1.Suppose that there exist certain linear operators p i = (p ik ) n k=1 : W 1 → L 1 , i = 1, 2, satisfying the inclusions and such that inequalities (4.2) hold for an arbitrary u from W 1 (0;r1,r1) .Then the non-local boundary value problem (1.1)-( 1.3) has a unique solution for any q ∈ L 1 .Theorem 6.2.Let there exist certain positive linear operators g i = (g ik ) n k=1 : W 1 → L 1 , i = 0, 1, which satisfy inequalities (5.5) for arbitrary u from W 1 (0;r1,r1) , and, moreover, are such that the inclusions hold.
Then the non-local boundary value problem (1.1)-( 1.3) has a unique solution for an arbitrary q ∈ L 1 .
EJQTDE, 2012 No. 14, p. 9 The proof of Theorems 6.1 and 6.2 is based on the following Lemma 6.1.If l : W 1 → L 1 is a bounded linear operator, then the inclusion implies that l ∈ S r0,r1 .

An example of a second order equation with argument deviations
Let us consider the two-point boundary value problem for the nonlinear scalar differential equation with argument deviations u ′ (a) = 0, (7.2) where N ≥ 1, µ ∈ R, {q, α 1 , α 2 , . . ., α N } ⊂ L 1 and ω 1 , ω 2 , . . ., ω N are Lebesgue measurable functions mapping the interval [a, b] into itself and such that The following statement is true.To prove Corollary 7.1, we use the following propositions concerning the scalar linear functional differential equation where p is a map from We shall say that p is positive if it maps the non-negative functions from C to almost everywhere non-negative elements of L 1 .Then the boundary value problem (7.7), (7.3) is uniquely solvable for an arbitrary integrable q and, moreover, the non-negativity of q implies the non-negativity of the solution.(7.9) Then the boundary value problem (7.7), (7.3) is uniquely solvable for every integrable q.Moreover, if q is non-negative, then so does the solution of problem (7.7), (7.3).
Similarly, it follows from (6.1), (7.12), and (7.By assumption (7.4), G 1 is a Volterra operator, and it is obvious from (7.12) that −G 1 is positive.In view of (7.14), assumption (7.5) guarantees that (7.9) is satisfied.Consequently, by Proposition 7.2, we have G 1 ∈ S r0 .Thus, we have shown that all the conditions of Theorem 6.2 are satisfied.Applying that theorem, we complete the proof.

)
be satisfied.Then the non-local boundary value problem (1.1)-(1.3)has a unique solution for an arbitrary q ∈ L 1 .