The existence and uniqueness of periodic solutions for nonlinear neutral first order differential equation with functional delay

Abstract

We employ the fixed point theorem of Krasnoselskii, to show the existence and uniqueness of periodic solutions of the nonlinear neutral differential equation:

By modifying the given neutral differential equation into an equivalent integral equation using lemma (2.1). This is done by constructing appropriate operators, one is a contraction and the other is compact, which allow us to prove the existence of periodic solutions. In addition, we used the Banach fixed-point theorem to guarantee a unique periodic solution.