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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 945467, 15 pages
http://dx.doi.org/10.1155/2012/945467
Research Article

Positive Periodic Solutions for Second-Order Ordinary Differential Equations with Derivative Terms and Singularity in Nonlinearities

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 17 July 2012; Accepted 26 August 2012

Academic Editor: Gabriel N. Gatica

Copyright © 2012 Yongxiang Li and Xiaoyu Jiang. 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

The existence results of positive -periodic solutions are obtained for the second-order ordinary differential equation where, is a continuous function, which is -periodic in and may be singular at . The discussion is based on the fixed point index theory in cones.

1. Introduction

In this paper, we discuss the existence of positive -periodic solutions of the second-order ordinary differential equation with first-order derivative term in the nonlinearity where the nonlinearity is a continuous function, which is -periodic in t and may be singular at .

The existence problems of periodic solutions for nonlinear second-order ordinary differential equations have attracted many authors’ attention and concern, and most works are on the special equation that does not contain explicitly first-order derivative term in nonlinearity. Many theorems and methods of nonlinear functional analysis have been applied to the periodic problems of (1.2). These theorems and methods are mainly the upper and lower solutions method and monotone iterative technique [14], the continuation method of topological degree [57], variational method and critical point theory, [810] and so forth.

In recent years, the fixed point theorems of cone mapping, especially the fixed point theorem of Krasnoselskii’s cone expansion or compression type, have been extensively applied to two-point boundary value problems of second-order ordinary differential equations, and some results of existence and multiplicity of positive solutions have been obtained, see [1115]. Lately, the authors of [1618] have also applied the Krasnoselskii’s fixed point theorem to periodic problems of second-order nonlinear ordinary differential equations, and obtained existence results of positive periodic solutions. In these works, the new discovered positivity of Green function of the corresponding linear second-order periodic boundary value problems plays an important role. The positivity guarantees that the integral operators of the second-order periodic problems are cone-preserving in the cone in the Banach space , where is a constant. Hence the fixed point theorems of cone mapping can be applied to the second-order periodic problems. For more precise results using the theory of the fixed point index in cones to discuss the existence of positive periodic solutions of second-order ordinary differential equation, see [1922]. However, all of these works are on the special second-order equation (1.2), and few people consider the existence of the positive periodic solutions for the general second-order equation (1.1) that explicitly contains the first order derivative term.

The purpose of this paper is to extend the results of [1622] to the general second-order equation (1.1). We will use the theory of the fixed point index in cones to discuss the existence of positive periodic solutions of (1.1). For the periodic problem of (1.1), since the corresponding integral operator has no definition on the cone in , the argument methods used in [1622] are not applicable. We will use a completely different method to treat (1.1). Our main results will be given in Section 3. Some preliminaries to discuss (1.1) are presented in Section 2.

2. Preliminaries

Let denote the Banach space of all continuous -periodic function with norm . Let be the Banach space of all continuous differentiable -periodic function with the norm Generally, denotes the th-order continuous differentiable -periodic function space for . Let be the cone of all nonnegative functions in .

Let be a constant. For , we consider the linear second-order differential equation The -periodic solutions of (2.2) are closely related with the linear second-order boundary value problem see [19]. It is easy to see that Problem (2.3) has a unique solution, which is explicitly given by where . We have the following Lemma.

Lemma 2.1. Let . Then for every , the linear equation (2.2) has a unique -periodic solution , which is given by Moreover, is a completely continuous linear operator.

Proof. Taking the derivative in (2.5) and using the boundary condition of , we obtain that Therefore, satisfies (2.2). Let ; it follows from (2.5) that Hence, is an -periodic solution of (2.2). From the maximum principle for second-order periodic boundary value problems [4], it is easy to see that is the unique -periodic solution of (2.2).
From (2.5) and (2.6), we easily see that is a linear bounded operator. By the compactness of the embedding , is a completely continuous operator.

Since for every , by (2.5), if and , then the -periodic solution of (2.2) for every , and we term it the positive -periodic solution. Let Define the cone in by We have the following Lemma.

Lemma 2.2. Let . Then for every , the positive -periodic solution of (2.2) . Namely, .

Proof. Let , . For every , from (2.5) it follows that and therefore, Using (2.5), we obtain that For every , since we have Hence, .

Now we consider the nonlinear equation (1.1). Hereafter, we assume that the nonlinearity satisfies the following condition. (F0) There exists such that Let , then for ,  , and (1.1) is rewritten to For , if , then and by the definition of , for every . Hence is well defined, and we can define the integral operator by By the definition of operator , the positive -periodic solution of (1.1) is equivalent to the nontrivial fixed point of . From Assumption (F0), Lemmas 2.1 and 2.2, we easily see the following Lemma.

Lemma 2.3. , and is completely continuous.

We will find the nonzero fixed point of by using the fixed point index theory in cones. Since the singularity of at implies that has no definition at , the fixed point index theory in the cone cannot be directly applied to . We need to make some Preliminaries.

We recall some concepts and conclusions on the fixed point index in [23, 24]. Let be a Banach space and a closed convex cone in . Assume is a bounded open subset of with boundary , and . Let be a completely continuous mapping. If for any , then the fixed point index has a definition. One important fact is that if , then has a fixed point in . The following two lemmas are needed in our argument.

Lemma 2.4 (see [24]). Let be a bounded open subset of with and a completely continuous mapping. If for every and , then .

Lemma 2.5 (see [24]). Let be a bounded open subset of and a completely continuous mapping. If there exists an such that for every and , then .

We use Lemmas 2.4 and 2.5 to show the following fixed-point theorem in cones which is applicable to the operator defined by (2.18).

Theorem 2.6. Let be a Banach space and a closed convex cone. Assume and are bounded open subsets of with , . Let be a completely continuous mapping. If satisfies the following conditions:(1) for , ; (2) there exists such that for , , or the following conditions:(3) there exists such that for , ; (4)  for , , then has a fixed-point in .

Proof. By Dugundji’s extension theorem, the operator can be extended into a completely continuous operator from to , says .
If satisfies conditions (1) and (2) of Theorem 2.6, then also satisfies them. By Lemmas 2.4 and 2.5, respectively, we have By the additivity of the fixed point index, we have Hence has a fixed-point in . Since is an extension of , it follows that has a fixed-point in .
If satisfies conditions (3) and (4) of Theorem 2.6, with a similar count, we obtain that This means that has a fixed-point in . Hence, has a fixed-point in .

Theorem 2.6 is an improvement of the fixed point theorem of Krasnoselskii’s cone expansion or compression. We will use it to discuss the existence of positive -periodic solutions of (1.1) in the next section.

3. Main Results

We consider the the existence of positive -periodic solutions of (1.1). Let satisfy Assumption (F0) and be -periodic in . Let be the constant defined by (2.8) and . To be convenient, we introduce the notations Our main results are as follows.

Theorem 3.1. Let be continuous and be -periodic in t. If satisfies Assumption (F0) and the condition (F1), then (1.1) has at least one positive -periodic solution.

Theorem 3.2. Let be continuous and be -periodic in t. If satisfies Assumption (F0) and the conditions (F2), then (1.1) has at least one positive -periodic solution.

Noting that is an eigenvalue of the associated linear eigenvalue problems of (1.1) with periodic boundary condition, if one inequality concerning comparison with in (F1) or (F2) of Theorem 3.1 or Theorem 3.2 is not true, the existence of periodic solution to (1.1) cannot be guaranteed. Hence, the is the optimal value in condition (F1) and (F2).

In Theorem 3.1, the condition (F1) allows to have superlinear growth on and . For example, satisfies (F0) with and (F1) with and .

In Theorem 3.2, the condition (F2) allows that has singularity at . For example, satisfies (F0) with , and (F2) with and . The existence of periodic solutions for singular ordinary differential equations has been studied by several authors, see [20, 25, 26]. But the equations considered by these authors do not contain derivative term .

Proof of Theorem 3.1. Choose the working space . Let be the closed convex cone in defined by (2.9) and the operator defined by (2.18). Then the positive -periodic solution of (1.1) is equivalent to the nontrivial fixed point of . Let and set We show that the operator has a fixed point in by Theorem 2.6 when is small enough and large enough.
By and the definition of , there exist and , such that Let . We now prove that satisfies the Condition (1) of Theorem 2.6, namely, for every and . In fact, if there exist and such that , then by definition of and Lemma 2.1, satisfies the differential equation Since , by the definitions of and , we have Hence from (3.5) it follows that By this, (3.6), and the definition of we have Integrating both sides of this inequality from to and using the periodicity of , we obtain that Since , it follows that , which is a contradiction. Hence the Condition (1) of Theorem 2.6 holds.
On the other hand, since , by the definition of , there exist and such that Define a function by Then is continuous. By (3.5) and Assumption (F0), This implies that Hence for every , , and , we have Combining this with (3.11), it follows that Choose . Clearly,. We show that satisfies the Condition (2) of Theorem 2.6 if is large enough, namely, for every and . In fact, if there exist and such that , since , by definition of and Lemma 2.1, satisfies the differential equation From (3.17) and (3.16), it follows that Integrating this inequality on and using the periodicity of , we get that Since , by the definition of , we have By the first inequality of (3.20), we have From this and (3.19), it follows that By this and the second inequality of (3.20), we have Therefore, choose , then satisfies the Condition (2) of Theorem 2.6.
Now by the first part of Theorem 2.6, has a fixed point in , which is a positive -periodic solution of (1.1).

Proof of Theorem 3.2. Let be defined by (3.4). We use Theorem 2.6 to prove that the operator has a fixed point in if is small enough and large enough.
By and the definition of , there exist and , such that Let and . We prove that satisfies the Condition (3) of Theorem 2.6, namely, for every and . In fact, if there exist and such that , since , by definition of and Lemma 2.1, satisfies the differential equation Since , by the definitions of and , satisfies (3.7). From (3.7), and (3.24) it follows that By this, (3.25), and the definition of , we have Integrating this inequality on and using the periodicity of , we obtain that Since , from this inequality it follows that , which is a contradiction. Hence satisfies the Condition (3) of Theorem 2.6.
Since , by the definition of , there exist and such that Choosing , we show that satisfies the Condition (4) of Theorem 2.6, namely, for every and . In fact, if there exist and such that , then by the definition of and Lemma 2.1, satisfies the differential equation Since , by the definition of , satisfies (3.20). By the second inequality of (3.20), we have Consequently, By (3.32) and the first inequality of (3.20), we have From this, the second inequality of (3.20) and (3.29), it follows that By this and (3.30), we have Integrating this inequality on and using the periodicity of , we obtain that Since , from this inequality it follows that , which is a contradiction. This means that satisfies the Condition (4) of Theorem 2.6.
By the second part of Theorem 2.6, has a fixed point in , which is a positive -periodic solution of (1.1).

Example 3.3. Consider the second-order differential equation where . If and for , then satisfies the conditions (F0) and (F1). By Theorem 3.1, (3.37) has at least one positive -periodic solution.

Example 3.4. Consider the singular differential equation: where . If and for , then satisfies the conditions (F0) and (F2). By Theorem 3.2, the (3.38) has a positive -periodic solution.

4. Remarks

Our discussion on the existence of the positive -periodic solutions to (1.1) is applicable to the following ordinary differential equation: where the nonlinearity is continuous and is -periodic in . For (4.1), we need the following assumption. There exists such that Similarly to Lemma 2.1, we have the following conclusion.

Lemma 4.1. Let be a constant. Then for every , the linear second order differential equation has a unique -periodic solution , which is given by where is the unique solution of the linear second-order boundary value problem which is explicitly given by with .

Since we renew to define and by

Now, using the similar arguments to Theorems 3.1 and 3.2, we can obtain the following results.

Theorem 4.2. Let be continuous and be -periodic in t. If satisfies Assumption (F0)* and the condition (F1), then (4.1) has at least one positive -periodic solution.

Theorem 4.3. Let be continuous and be -periodic in t. If satisfies Assumption (F0)* and the conditions (F2), then (4.1) has at least one positive -periodic solution.

Theorems 4.2 and 4.3 improve and extend some results in References [18, 19, 22].

Acknowledgment

This paper is supported by NNSFs of China (11261053 and 11061031).

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