Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 684962, 10 pages

http://dx.doi.org/10.1155/2013/684962

Research Article

Positive Solutions for Fourth-Order Nonlinear Differential Equation with Integral Boundary Conditions

Qi Wang, Yanping Guo, and Yude Ji

Hebei University of Science and Technology, Shijiazhuang, Hebei 050018, China

Received 5 November 2012; Accepted 8 January 2013

Academic Editor: Cengiz Çinar

Copyright © 2013 Qi Wang et al. 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.

This paper investigates the existence and nonexistence of positive solutions for a class of fourth-order nonlinear differential equation with integral boundary conditions. The associated Green's function for the fourth-order boundary value problems is first given, and the arguments are based on Krasnoselskii's fixed point theorem for operators on a cone.

1. Introduction

In this paper, we consider the following fourth-order boundary value problems (BVPs) with integral boundary conditions

where may be singular at and(or) , is continuous, and are nonnegative.

The existence of solutions for nonlinear higher-order nonlocal BVPs has been studied by several authors, for example, see [1–9] and the references therein. However, there are a few papers dealing with the existence of positive solutions for the fourth-order boundary value problems with integral boundary conditions, see [7–15]. For the case of , BVP (1) reduces to the following two-point BVP

A great deal of research has been devoted to the existence of solutions for problem (2) by using the Leray-Schauder continuation method, topological degree, and the method of lower and upper solutions.

For the case of or , and , BVP (1) reduces to the following BVP with integral boundary conditions

or

By using the fixed point theorem of cone expansion and compression of norm type, Zhang and Ge [11] have obtained the existence and multiplicity of positive solutions for BVP (3) and (4).

The following assumptions will stand throughout this paper:

is nonnegative, and may have singularities at and(or) ;  ;   are nonnegative and

The paper is organized as follows. In Section 2, we provide some necessary background material such as the Krasnoselskii’s fixed point theorem in cones. The associated Green’s function for the fourth-order boundary value problem is first given, and we also look at some properties of the Green’s function associated with the problem (1). In Section 3, the main results of problem (1) will be stated and proved. In Section 4, we give an example to illustrate how the main results can be used in practice.

2. Preliminary Results

In our main results, we will make use of the following lemmas.

Lemma 1 (in [16]). Let and be two bounded open sets in a real Banach space , such that and . Let operator be continuous, where denotes the zero element of and is a cone in . Suppose that one of the two conditions

 , and , or  , and , is satisfied. Then, has at least one fixed point in .

Lemma 2. If are nonnegative, and , then for any , the BVP

has a unique solution which is given by

where

Proof. The general solution of can be written as

Now, we solve for by and , it follows that

that is,

Solving the above equations, we get

Therefore, (6) has a unique solution

The unique solution of (6) be expressed as

Therefore, the unique solution of (6) is .

Lemma 3 (in [11]). If are nonnegative, then,

, , for all , for all , Letting , for all , one has ,  Letting , we have , where is defined by (9).

Lemma 4. If are nonnegative, and , for function , we have the following properties:

,,,

where , .

Proof.

  From (8) and Lemma 3, we get (1).  For all , by (8) and Lemma 3, we have

For all , by (8) and Lemma 3, we have

Lemma 5. Assume that (A1)–(A3) holds. If is a solution of the following integral equation

then is a solution of BVP (1), where

Proof. By using Lemma 2, the conclusion is obvious.

Lemma 6. If (A3) holds, then one has the following three properties:

, for all ,,, for all ,

where

Proof. By using Lemma 3, the conclusions for (1) and the first part of (2) are obvious.

By using by (21) and Lemma 3, we have

Similarly, by (22), we get

So, by (20), (25), and (26), we concluded

By using Lemmas 3 and 4, we have

Let . Then is a Banach space with the norm , where .

For a fixed , define the cone by

It's easy to prove that is a closed convex cone, and

Lemma 7. If (A1)–(A3) hold, then , and is completely continuous.

Proof. By the properties of Green's function, if then and

For all , we have by (19) and (31)

On the other hand, by (19), (33), and Lemma 3, we obtain

Similarly, by (31), (34), and Lemma 3, we obtain

So , and then .

Clearly, the operator is continuous; thus by the Arzela-Ascoli theorem, it follows that is completely continuous.

3. Main Results

For convenience, we introduce the following notations:

Theorem 8. Assume that (A1)–(A3) hold. In addition, one supposes that one of the following conditions is satisfied

, for all ,, for all .

Then, the BVP (1) has no positive solution.

Proof. Case 1. The condition () holds. Assume towards a contraction that is a positive solution of BVP (1), from the proof of Lemma 7 and (29), then for . By Lemma 6 and (29), we have

So, , which is a contradiction.

Case 2. The condition () holds. Assume towards a contraction that is a positive solution of BVP (1), from the proof of Lemma 7 and (29), then , , for . By Lemmas 4 and 6 and (30), we have

So, , which is a contradiction, and this completes the proof.

Theorem 9. Assume that (A1)–(A3) hold. In addition, one supposes that one of the following conditions is satisfied

(),().

Then, the BVP (1) has at least one positive solution.

Proof. Case 1. The condition () holds.

Considering , then there exists , such that , for , where , satisfies .

Then, for , by (19), (26), and Lemma 6, we have

which means that

Next, turning to , then there exists , such that , for , where , satisfies .

Let , for all . By (30) and Lemma 6, we have

which means that

Case 2. The condition () holds.

Considering , then there exists , such that , for , where , satisfies .

Then, for , by (30) and Lemma 6, we have

which means that

Next, turning to , then there exists , such that , for , where , satisfies .

Let

for .

Let , for . By (26) and Lemma 6, we have

which means that

Applying Lemma 1 to (41) and (43), or (45), and (48) yields that has a fixed point with , . Thus, it follows that BVP (1) has at least one positive solution, and the theorem is proved.

Theorem 10. Assume that (A1)–(A3) hold, as do the following two conditions:

   and , there exists , such that

Then, the BVP (1) has at least two positive solutions , which satisfy

Proof. We choose with .

If , then by the proof of (45), we have

If , then by the proof of (43), we have

On the other hand, by , for , we have

where

By (52), we have

Applying Lemma 1 to (50), (51), and (53) yields that has a fixed point , and a fixed point . Thus it follows that BVP (1) has at least two positive solutions and . Noticing (53), we have and . Therefore, (49) holds, and the proof is complete.

Theorem 11. Assume that (A1)–(A3) hold, as do the following two conditions:

   and , there exist and such that for all and .

Then, the BVP (1) has at least two positive solutions , which satisfy

Proof. We choose with .

If , then by the proof of (41), we have

If , then by the proof of (48), we have

On the other hand, by , for , we have

Applying Lemma 1 to (55), (56), and (57) yields that has a fixed point , and a fixed point . Thus it follows that BVP (1) has at least two positive solutions and . Noticing (57), we have and . Therefore, (54) holds, and the proof is complete.

4. Example

Example 12. Consider the following fourth-order BVP

By calculation, we obtain

   About the nonexistence of positive solution, we consider BVP (58) with

where and are two positive real numbers, then

  If or , by Theorem 8, BVP (58) has no positive solution.  About the existence of positive solution, we consider BVP (58) with

where and are two positive real numbers, then

 If and , then we have . By Theorem 9, BVP (58) has at least one positive solution.  About the multiplicity of positive solution, we consider BVP (58) with

where and are two positive real numbers, then

If and , then we have and .

Furthermore, letting , then we have . By Theorem 10, BVP (58) has at least two positive solutions.

Remark 13. In [4, 7–10], the existence of solutions for fourth-order ordinary differential equations BVP has been treated but did not discuss problems with singularities. Although, [11] seems to have considered the existence of positive solutions for the fourth-order BVP with integral boundary conditions for the singularity allowed to appear at and/or . However, in [11], only the boundary conditions , or have been considered. It is clear that the boundary conditions of Example 12 are , and . Hence, we generalize the fourth-order ordinary differential equations BVP.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of PR China (10971045), the Natural Science Foundation of Hebei Province (A2013208147) and (A2011208012), and the Foundation of Hebei University of Science and Technology (XL201246). The authors thank the referee for his/her careful reading of the paper and useful suggestions that have greatly improved this paper.

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