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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 826451, 12 pages
http://dx.doi.org/10.1155/2011/826451
Research Article

Existence of Positive Solutions for a Fourth-Order Periodic Boundary Value Problem

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

Received 10 April 2011; Revised 7 July 2011; Accepted 14 July 2011

Academic Editor: Ferhan M. Atici

Copyright © 2011 Yongxiang Li. 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 solutions are obtained for the fourth-order periodic boundary value problem , , ,   , where is continuous, and satisfy , . The discussion is based on the fixed point index theory in cones.

1. Introduction

This paper concerns the existence of positive solutions for the fourth-order periodic boundary value problem (PBVP) where and is continuous, . PBVP (1.1) describes the deformations of an elastic beam in equilibrium state with periodic boundary condition. In the equation, the denotes the bending moment term which represents bending effect. Owing to its importance in physics, the existence of solutions to this problem has been studied by some authors, see [16]. In practice, only its positive solutions are significant. In this paper, we discuss the existence of positive solutions of PBVP (1.1).

In [1, 2], Cabada and Lois obtained the maximum principles for fourth-order operator in periodic boundary condition and then they proved the existence of solutions and the validity of the monotone method in the presence of lower and upper solutions for the periodic boundary problem In [3], the present author established a strongly maximum principle for operator in periodic boundary condition, and showed that if satisfy the assumption then is strongly inverse positive in space As an application of this strongly maximum principle, the author considered the existence of positive solutions for the special fourth-order periodic boundary problem and obtained the following result.

Theorem A. Let be continuous and the assumption (1.3) hold. If satisfies one of the following conditions
(G1) ;
(G2) ,
where then PBVP (1.5) has at least one positive solution.

Based upon this strongly maximum principle, the authors of [4, 5] further consider the existence and multiplicity of positive solutions of PBVP (1.5). In [6], Bereanu obtained existence results for PBVP (1.5) by using the method of topological degree. However, all of these works are on the special equation (1.5), and few people consider the existence of positive solutions of PBVP (1.1) that nonlinearity contains the bending moment term . The purpose of this paper is to discuss the existence of positive solutions of PBVP (1.1).

The strongly maximum principle implies that the fourth-order linear boundary value problem (LBVP) has a unique positive solution , see [3, Lemma 3]. This function has been introduced in [2, Lemma 2.1 and Remark 2.1]. Let , and set Let be continuous. To be convenient, we introduce the notations Our main result is as follows.

Theorem 1.1. Let be continuous, and let the assumption (1.3) hold. If satisfies one of the following conditions:
(F1) ,
(F2) ,
then PBVP (1.1) has at least one positive solution.

Clearly, Theorem 1.1 is an extension of Theorem A. Since that is an eigenvalue of linear eigenvalue problem with periodic boundary condition, if one inequality in (F1) or (F2) of Theorem 1.1 is not true, the existence of solution to PBVP (1.1) cannot be guaranteed. Hence, (F1) and (F2) are the optimal conditions for the existence of the positive of PBVP (1.1).

In Theorem 1.1, the condition (F1) allows that may be superlinear growth on and , for example, , and the condition (F2) allows that may be sublinear growth on and , for example, .

The proof of Theorem 1.1 is based on the theory of the fixed point index in cones. Since the nonlinearity of PBVP (1.1) contains , the argument of Theorem A in [3] is not applicable to Theorem 1.1. We will prove Theorem 1.1 by choosing a proper cone of in Section 3. Some preliminaries to discuss PBVP (1.1) are presented in Section 2.

2. Preliminaries

Let be the Banach space of all continuous functions on the unit interval with the norm . Let denote the cone of all nonnegative functions in . Generaly, for , we use to denote the Banach space of the th-order continuous differentiable functions on with the norm . In , we define a new norm by Then is equivalent to . In fact, for every , it is clear that . On the other hand, by the Lagrange mean-value theorem, there exists such that . For , we have Hence, . By this, we have

Therefore, the norms and are equivalent.

Let satisfy the assumption (1.3). For , we consider the fourth-order linear periodic boundary value problem (LPBVP) Let be the unique positive solution of LBVP(1.7), and set By [3, Lemma 1], we have the following result.

Lemma 2.1. Let satisfy the assumption (1.3). Then for every , LPBVP (2.4) has a unique solution which is given by Moreover, is a linear bounded operator.

Let and be the positive constants given by (1.8). Choose a cone in by We have the following.

Lemma 2.2. Let satisfy the assumption (1.3). Then for every , the solution of LPBVP (2.4) . Namely, .

Proof. Let , . For every , from (2.6) it follows that which implies that By this and (2.6), we have For , by the definition of and , we have Making derivation to both sides of this equality, we have from which it follows that Therefore, . This means that .

For every , since is continuous, we see that . By Lemma 2.2, . Define an operator by We have the following.

Lemma 2.3. is a completely continuous operator.

Proof. Let be a bounded set in . By the continuity of , is a bounded set in . By the boundedness of the operator , is a bounded set in . By the compactness of the embedding , is a precompact set in . So is completely continuous.

By the definition of and , the positive solution of PBVP (1.1) is equivalent to the nontrivial fixed point of . We will find the nonzero fixed point of by using the fixed point index theory in cones.

We recall some concepts and conclusions on the fixed point index in [7, 8]. Let be a Banach space, and let be 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 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 [8]). Let be a bounded open subset of with , and let be a completely continuous mapping. If for every and , then .

Lemma 2.5 (see [8]). Let be a bounded open subset of , and let be a completely continuous mapping. If there exists an such that for every and , then .

3. Proof of the Main Result

Proof of Theorem 1.1. Choose the working space with the norm . Let be the closed convex cone in defined by (2.7), and let be the operator defined by (2.14). By Lemma 2.3 and the definition of , the nonzero fixed of the operator is the positive solution of PBVP (1.1). Let , and set We show that, if is small enough and large enough, the operator has a fixed point in in either case that (F1) holds or (F2) holds.Case 1. Assume that (F1) holds.
Since , by the definition of , we may choose and , such that Let . We prove that satisfies the condition of Lemma 2.4 in ; 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 and the periodic boundary condition Since , by the definitions of and , we have From this and (3.2), it follows that By this inequality and (3.3), we have Integrating this inequality from 0 to 1 and using the periodic boundary condition (3.4), we obtain that Since , form this inequality it follows that , which is a contradiction. Hence, satisfies the condition of Lemma 2.4 in . By Lemma 2.4 we have
On the other hand, since , by the definition of , there exist and such that Choose , and let . Clearly, . We show that satisfies the condition of Lemma 2.5 in ; namely, , for every and . In fact, if there exist and such that , since , by definition of and Lemma 2.1, satisfies the differential equation and the periodic boundary condition (3.4). Since , by the definition of , we have By the second inequality of (3.12), we have Consequently, By (3.14) and the first inequality of (3.12), we have From this, the second inequality of (3.12), and (3.10), it follows that By this and (3.11), we have Integrating this inequality on and using the periodic boundary condition (3.4), we get that Since , from this inequality it follows that , which is a contradiction. This means that satisfies the condition of Lemma 2.5 in . By Lemma 2.4,
Now, by the additivity of fixed point index, (3.9), and (3.19), we have Hence, has a fixed point in , which is the positive solution of PBVP (1.1).Case 2. Assume that (F2) holds.
By the assumption of and the definition of , there exist and , such that Let , and let . We prove that satisfies the hypothesis of Lemma 2.5 in ; namely, for every and . In fact, if there exist and such that , since , by the definition of and Lemma 2.1, satisfies the differential equation and the periodic boundary condition (3.4). Since , by the definitions of and , satisfies (3.5). From (3.5) and (3.22), it follows that By this inequality and (3.22), we have Integrating this inequality on and using the periodic boundary condition (3.4), we have Since , from this inequality, it follows that , which is a contradiction. Hence satisfies the hypothesis of Lemma 2.5 in . By Lemma 2.5,
Since , by the definition of , there exist and such that Choosing , we show that satisfies the condition of Lemma 2.4 in ; 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 and the periodic boundary condition (3.4). Since , by the definition of , satisfies (3.12). From (3.12), we can show that satisfies (3.14). By (3.14) and the first inequality of (3.12), we have From this, the second inequality of (3.12), and (3.27), it follows that By this inequality and (3.28), we have Integrating this inequality on and using the periodic boundary condition (3.4), we obtain that Since , form this inequality it follows that , which is a contradiction. This means that satisfies the condition of Lemma 2.4 in . By Lemma 2.4,
From (3.26) and (3.33), it follows that Hence, has a fixed point in , which is the positive solution of PBVP (1.1).
The proof of Theorem 1.1 is completed.

Example 3.1. Consider the superlinear fourth-order periodic boundary problem where and for . It is easy to verify that and satisfy the assumption (P). satisfies the condition (F1), in which and . Hence, by Theorem 1.1, (3.35) has at least one positive solution.

Example 3.2. Consider the sublinear fourth-order periodic boundary problem where and for . For PBVP (3.36), it is easy to verify that and satisfy the assumption (1.3), and satisfies the condition (F2) with and . By Theorem 1.1, (3.36) has a positive solution.
Since (3.35) and (3.36) have nonlinear terms of , which are not in the range considered by [16], the existence results in Example 3.1, and Example 3.2 cannot be obtained from [16].

Acknowledgment

The author thanks to the referee for the helpful comments and suggestions. Research supported by NNSFs of China (10871160, 11061031) and Project of NWNU-KJCXGC-3-47.

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