Báo Cáo Báo cáo khoa học On a periodic boundary value problem for second-order linear functional differentia

ON A PERIODIC BOUNDARY VALUE PROBLEM FOR SECOND-ORDER LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS S. MUKHIGULASHVILI Received 26 October 2004 and in revised form 7 March 2005 Unimprovable efficient sufficient conditions are established for the unique solvability of the periodic problem u (t) = (u)(t) + q(t) for 0 ≤ t ≤ ω, u(i) (0) = u(i) (ω) (i = 0,1), where ω 0, : C([0,ω]) → L([0,ω]) is a linear bounded operator, and q ∈ L([0,ω]). 1. Introduction Consider the equation u (t) = (u)(t) + q(t) for 0 ≤ t ≤ ω with the periodic boundary conditions u(i) (0) = u(i) (ω) (i = 0,1), (1.2) (1.1) where ω 0, : C([0,ω]) → L([0,ω]) is a linear bounded operator and q ∈ L([0,ω]). By a solution of the problem (1.1), (1.2) we understand a function u ∈ C ([0,ω]), which satisfies (1.1) almost everywhere on [0, ω] and satisfies the conditions (1.2). The periodic boundary value problem for functional differential equations has been studied by many authors