On a non-local boundary value problem for linear functional differential equations

Abstract

where ` : C([a, b];R) → L([a, b];R) and h : C([a, b];R) → R are linear bounded operators, q ∈ L([a, b];R), and c ∈ R. By a solution to the problem (1.1), (1.2) we understand an absolutely continuous function u : [a, b] → R satisfying the equation (1.1) almost everywhere on the interval [a, b] and verifying also the boundary condition (1.2). The question on the solvability of various types of boundary value problems for functional differential equations and their systems is a classical topic in the theory of differential equations (see, e.g., [1,3–5,7–9,11–14] and references therein). Many particular cases of the boundary condition (1.2) are studied in detail (namely, periodic, anti-periodic and multi-point conditions), but only a few efficient conditions is known in the case, where a general non-local boundary condition is considered. In the present paper, new efficient conditions are found sufficient for the unique solvability of the problem (1.1), (1.2). It is clear that the ordinary differential equation u′ = p(t)u + q(t), (1.3)