Positive Periodic Solutions for a Kind of Second-Order Neutral Differential Equations with Variable Coefficient and Delay

Abstract

In this article, we discuss a type of second-order neutral differential equations with variable coefficient and delay: $$\begin{aligned} (x(t)-c(t)x(t-\tau (t)))''+a(t)x(t)=f(t,x(t-\delta (t))), \end{aligned}$$ where $$c(t)\in C({\mathbb {R}},{\mathbb {R}})$$ and $$|c(t)|\ne 1$$ . By employing Krasnoselskii’s fixed-point theorem and properties of the neutral operator $$(Ax)(t):=x(t)-c(t)x(t-\tau (t))$$ , some sufficient conditions for the existence of periodic solutions are established.