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.