Results Math 72 (2017), 71–86
2017 Springer International Publishing
published online January 7, 2017
Results in Mathematics
Positive Periodic Solutions for Singular
Higher Order Delay Diﬀerential Equations
Fanchao Kong, Shiping Lu, and Zhiguo Luo
Abstract. In this article, we consider the singular higher order delay
diﬀerential equations. Under certain assumptions, some new criteria for
guaranteeing the existence of positive periodic solutions are presented.
Moreover, two examples are provided to demonstrate the eﬀectiveness
and applications of the main results.
Mathematics Subject Classiﬁcation. 34B16, 34K13.
Keywords. Positive periodic solution, high order, singularity, delay.
The purpose of this article is to study higher-order delay diﬀerential equations
of the form
(t)+g(t, x(t − τ)) = e(t), (1.1)
where m ≥ 1andn ≥ 1 are integers, 0 ≤ τ<Tis a constant, f : R → R
is continuous, g :[0,T] × (0, +∞) → R is continuous, T -periodic in the ﬁrst
argument and can be singular at x =0,r(t) is a positive continuous function,
e :[0,T] → R is a continuous periodic function with
The existence problem of periodic solutions is an important topic in qual-
itative analysis of ordinary diﬀerential equations. In recent years, the existence
of positive periodic solutions of some ordinary diﬀerential equations with spe-
cial forms has been researched by several authors. Lui et al.  considered the
periodic problem of the second-order equation