On the existence of Besicovitch Almost periodic solutions for a class of neutral delay differential equations

Abstract

We study the existence of a Besicovitch almost periodic solution for a class of second order neutral delay differential equations $$u''(t-r)+D_1F(u(t-r),u(t-2r),t-r)+ D_2F(u(t),u(t-r),t)=0,$$ in a Hilbert space, under some hyptoheses on the function $F(\cdot, \cdot, t)$. Here, $F: H\times H \times \mathbb{R} \rightarrow \mathbb{R}$ denotes a differentiable function, $D_j$, $j=1,2$, denotes the partial differential with respect to the $j$th vector variable and $r\in (0, \infty)$ is a fixed real number. The approach we use is based on a variational method applied on a Hilbert space of Besicovitch almost periodic functions.