Mediterr. J. Math.
Springer International Publishing AG,
part of Springer Nature 2018
Positive Periodic Solutions for a Kind
of Second-Order Neutral Diﬀerential
Equations with Variable Coeﬃcient
Zhibo Cheng and Feifan Li
Abstract. In this article, we discuss a type of second-order neutral dif-
ferential equations with variable coeﬃcient and delay:
(x(t) − c(t)x(t − τ(t)))
+ a(t)x(t)=f(t, x(t − δ(t))),
where c(t) ∈ C(R, R)and|c(t)| = 1. By employing Krasnoselskii’s ﬁxed-
point theorem and properties of the neutral operator (Ax)(t):=x(t) −
c(t)x(t − τ (t)), some suﬃcient conditions for the existence of periodic
solutions are established.
Mathematics Subject Classiﬁcation. 34C25, 34K13, 34K40.
Keywords. Neutral operator, second-order equation, positive periodic
solution, two operators, Krasnoselskii’s ﬁxed-point theorem.
Neutral functional diﬀerential equations are widely used in many ﬁelds, such
as Biology, Economics, and population models [3,7]. The existence of periodic
solutions to neutral functional diﬀerential equations plays a very important
role in solving these practical problems. In recent years, a great deal of work
has been performed on the existence of periodic solutions for ﬁrst-order and
second-order neutral diﬀerential equations (see [1,4–6,9–11,13–15]). In 2007,
Wu and Wang  discussed the second-order neutral diﬀerential equation
(x(t) − cx(t − τ))
+ a(t)x(t)=λb(t)f(x(t − δ(t))), (1.1)
Research is supported by the National Natural Science Foundation of China (no. 11501170),
China Postdoctoral Science Foundation funded project (no. 2016M590886), Fundamental
Research Funds for the Universities of Henan Province (NSFRF170302), Education Depart-
ment of Henan Province Project (no. 14A110002), Henan Polytechnic University Outstand-
ing Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055).