We study the existence of a positive periodic solution for second-order singular semipositone differential equation by a nonlinear alternative principle of Leray-Schauder. Truncation plays an important role in the analysis of the uniform positive lower bound for all the solutions of the equation. Recent results in the literature (Chu et al., 2010) are generalized.

In this paper, we study the existence of positive

During the last two decades, the study of the existence of periodic solutions for singular differential equations has attracted the attention of many researchers [

However the singular differential equations, in which there is the damping term, that is, the nonlinearity is dependent on the derivative, has not attracted much attention in the literature. Several existence results can be found in [

The aim of this paper is to further show that the nonlinear Leray-Schauder alternative principle can be applied to (

The remainder of the paper is organized as follows. In Section

In this paper, let us fix some notations to be used in the following: given

We say that

The Green function

In other words, the strict antimaximum principle holds for (

We say that (

Under hypothesis (A), we denote

With the help of [

Assume that

Next, recall a well-known nonlinear alternative principle of Leray-Schauder, which can be found in [

Assume

There exists

In applications below, we take

In this section, we prove a new existence result of (

Suppose that (

where

Then (

For convinence, let us write

Choose

Consider the family of equations

A

We claim that for any

By the periodic boundary conditions, we have

Choose

There exists an integer

The lower bound in (_{2}), there exists _{2}).

For

If

If

Due to

It can be checked that

In fact, if

Using (

Suppose that (

By the facts _{2}), we can choose

Furthermore, we can prove

There exist a constant

Multiplying (

In the same way as in the proof of (_{2}),

Next, we will prove (

For _{4}) imply

Next we claim that any fixed point _{3}), for all

The fact

Let the nonlinearity in (

If

If

We will apply Theorem _{1})–(H_{3}) are satisfied and existence condition (H_{4}) becomes

The research of X. Xing is supported by the Fund of the Key Disciplines in the General Colleges and Universities of Xin Jiang Uygur Autonomous Region (Grant no. 2012ZDKK13). It is a pleasure for the author to thank Professor J. Chu for his encouragement and helpful suggestions.