Abstract and Applied Analysis

Volume 2013 (2013), Article ID 310469, 7 pages

http://dx.doi.org/10.1155/2013/310469

Research Article

Positive Periodic Solution for Second-Order Singular Semipositone Differential Equations

Xiumei Xing

School of Mathematics and Statistics, Yili Normal University, Yining City 835000, China

Received 7 November 2012; Accepted 20 December 2012

Academic Editor: Jifeng Chu

Copyright © 2013 Xiumei Xing. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

In this paper, we study the existence of positive -periodic solutions for the following singular semipositone differential equation:

where and the nonlinearity satisfies for some . In particular, the nonlinearity may have a repulsive singularity at , which means that

Electrostatic or gravitational forces are the most important examples of singular interactions.

During the last two decades, the study of the existence of periodic solutions for singular differential equations has attracted the attention of many researchers [1–4]. Some strong force conditions introduced by Gordon [5] are standard in the related earlier works [6, 7]. Compared with the case of a strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity is more recent [2, 8, 9], but has also attracted many researchers. Some classical tools have been used to study singular differential equations in the literature, including the method of upper and lower solutions [10], degree theory [11], some fixed point theorem in cones for completely continuous operators [12], Schauder’s fixed point theorem [8, 9, 13], and a nonlinear Leray-Schauder alternative principle [2, 3, 14, 15].

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 [14, 16, 17].

The aim of this paper is to further show that the nonlinear Leray-Schauder alternative principle can be applied to (1) in the semipositone cases, that is, for some .

The remainder of the paper is organized as follows. In Section 2, we state some known results. In Section 3, the main results of this paper are stated and proved. To illustrate our result, we select the following system:

where ,  ,  ,   is a positive parameter, is a -periodic function.

In this paper, let us fix some notations to be used in the following: given , we write if for almost everywhere and it is positive in a set of positive measure. The usual -norm is denoted by . and the essential supremum and infinum of a given function , if they exist.

2. Preliminaries

We say that

associated to the periodic boundary conditions

is nonresonant when its unique solutions is the trival one. When (4)-(5) is nonresonant, as a consequence of Fredholm’s alternative, the nonhomogeneous equation

admits a unique -periodic solution, which can be written as

where is the Green’s function of problem (4)-(5). Throughout this paper, we assume that the following standing hypothesis is satisfied.

(A) The Green function , associated with (4)-(5), is positive for all .

In other words, the strict antimaximum principle holds for (4)-(5).

Definition 1. We say that (4) admits the antimaximum principle if (6) has a unique -periodic solution for any and the unique -periodic solution for all if .

Under hypothesis (A), we denote

Thus and . We also use to denote the unique periodic solution of (6) with under condition (5), that is, . In particular, .

With the help of [18, 19], the authors give a sufficient condition to ensure that (4) admits the antimaximum principle in [14]. In order to state this result, let us define the functions

Lemma 2 (see [14, Corollary 2.6]). Assume that and the following two inequalities are satisfied:

where . Then the Green’s function , associated with (5), is positive for all .

Next, recall a well-known nonlinear alternative principle of Leray-Schauder, which can be found in [20] and has been used by Meehan and O’Regan in [4].

Lemma 3. Assume is an open subset of a convex set in a normed linear space and . Let be a compact and continuous map. Then one of the following two conclusions holds:

(I) has at least one fixed point in .(II)There exists and such that .

In applications below, we take with the norm and define .

3. Main Results

In this section, we prove a new existence result of (1).

Theorem 4. Suppose that (4) satisfies (A) and

Furthermore, assume that there exist three constants such that:

(H₁) for all .(H₂) for , where the nonincreasing continuous function satisfies and .(H₃),  for  all  , where is nonincreasing in and ,   are nondecreasing in . (H₄)

 where

 Then (1) has at least one positive periodic solution with .

Proof. For convinence, let us write , , where . Let

First we show that

has a solution satisfying (5), and for . If this is true, it is easy to see that will be a positive solution of (1)–(5) with .

Choose such that , and then let .

Consider the family of equations

where ,  ,   and .

A -periodic solution of (17) is just a fixed of the operator equation

where and is a completely continuous operator defined by

where we have used the fact

We claim that for any -periodic solution of (17) satisfies

Note that the solution of (17) is also satisfies the following equivalent equation

Integrating (22) from to , we obtain

By the periodic boundary conditions, we have for some . Therefore,

where we have used the assumption (11) and . Therefore,

which implies that (21) holds. In particular, let in (17), we have

Choose such that , and then let . The following lemma holds.

Lemma 5. There exists an integer large enough such that, for all ,

Proof. The lower bound in (27) is established by using the strong force condition of . By condition (H₂), there exists and a continuous function such that

for all , where satisfies also the strong force condition like in (H₂).

For , let ,  .

If , due to , (27) holds.

If , we claim that, for all ,

Otherwise, suppose that for some . Then it is easy to verify

In fact, if , we obtain from (28)

and, if , we have

Integrating (22) (with ) from to , we deduce that

where estimation (30) and the fact are used. This is a contradiction. Hence (29) holds.

Due to , that is, for some . By (29), there exists (without loss of generality, we assume .) such that and for .

It can be checked that

where is defined by (15).

In fact, if is such that , we have

and, if is such that , we have

So (34) holds.

Using (17) (with ) for and the estimation (34), we have, for

As , for all , so is strictly increasing on . We use to denote the inverse function of restricted to .

Suppose that (27) does not hold, that is, for some , . Then there would exist such that and

Multiplying (17) (with ) by and integrating from to , we obtain

By the facts ,  ,   and the definition of , we can obtain , together with , implies that the second term and the third term are bounded. The first term is

which is also bounded. As a consequence, there exists a such that

On the other hand, by (H₂), we can choose large enough such that

for all . So (27) holds.

Furthermore, we can prove has a uniform positive lower bound .

Lemma 6. There exist a constant such that, for all ,

Proof. Multiplying (17) (with ) by and integrating from to , we obtain

In the same way as in the proof of (41), one way readily prove that the right-hand side of the above equality is bounded. On the other hand, if , by (H₂),

if . Thus we know that there exists a constant such that . Hence (43) holds.

Next, we will prove (17) has periodic solution .

For , we can choose such that , which together with (H₄) imply

Let . For , consider (17).

Next we claim that any fixed point of (18) for any must satisfy . So, by using the Leray-Schauder alternative principle, (17) (with ) has a periodic solution . Otherwise, assume that is a fixed point of (18) for some such that . Note that

For , we have

By (27) and assumption (H₃), for all and , we have

Therefore,

This is a contradiction to the choice of and the claim is proved.

The fact and show that is a bounded and equicontinuous family on . Now Arzela-Ascoli Theorem guarantees that has a subsequence , converging uniformly on to a function . From the fact and , satisfies for all . Moreover, satisfies the integral equation

Letting , we arrive at

where the uniform continuity of on is used. Therefore, is a positive periodic solution of (16) and . Thus we complete the prove of Theorem 4.

Corollary 7. Let the nonlinearity in (1) be

where ,  ,  ,   is a positive parameter, is a -periodic function.

(i)If , then (1) has at least one positive periodic solution for each .(ii)If , then (1) has at least one positive periodic solution for each , where is some positive constant.

Proof. We will apply Theorem 4 with and ,  ,  . Then condition (H₁)–(H₃) are satisfied and existence condition (H₄) becomes

So (1) has at least one positive periodic solution for

Note that if and if . We have the desired results.

Acknowledgments

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.

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