On a Periodic Boundary Value Problem for Second-order Linear Functional Differential Equations

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 ∈ C̃′([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 (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 (1.1), (1.2) has a unique solution if the inequality ∫ ω