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). | 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 u u i 0 u i u i 0 1 where u 0 Ể C 0 L 0 u is a linear bounded operator and q G L 0 u . 1. Introduction Consider the equation u t u t q t for0 t u with the periodic boundary conditions u i 0 u i u i 0 1 where u 0 f C 0 u L 0 u is a linear bounded operator and q G L 0 u . By a solution of the problem we understand a function u G C 0 u which satisfies almost everywhere on 0 u and satisfies the conditions . The periodic boundary value problem for functional differential equations has been studied by many authors see for instance 1 2 3 4 5 6 8 9 and the references therein . Results obtained in this paper on the one hand generalise the well-known results of Lasota and Opial see 7 Theorem 6 page 88 for linear ordinary differential equations and on the other hand describe some properties which belong only to functional differential equations. In the paper 8 it was proved that the problem has a unique solution if the inequality c . d f 1 s ds -f 0 u with d 16 is fulfilled. Moreover there was also shown that the condition is non-improvable. This paper attempts to find a specific subset of the set of linear monotone operators in which the condition guarantees the unique solvability of the problem Copyright 2006 Hindawi Publishing Corporation BoundaryValue Problems 2005 3 2005 247-261 DOI 248 On a periodic BVP for second-order linear FDE even for d 16 see Corollary . It turned out that if A satisfies some conditions dependent only on the constants d and w then K 0 W A see Definition is such a subset of the set of linear monotone operators. The following notation is used throughout. N is the set of .
