Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 293734, 19 pages

doi:10.1155/2012/293734

Research Article

Positive Solutions for -Laplacian Fourth-Order Differential System with Integral Boundary Conditions

Jiqiang Jiang,¹ Lishan Liu,1,2 and Yonghong Wu²

¹School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

²Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 25 March 2012; Accepted 15 May 2012

Academic Editor: Cengiz Çinar

Copyright © 2012 Jiqiang Jiang 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 of positive solutions for a class of singular -Laplacian fourth-order differential equations with integral boundary conditions. By using the fixed point theory in cones, explicit range for and is derived such that for any and lie in their respective interval, the existence of at least one positive solution to the boundary value system is guaranteed.

1. Introduction

Boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics and so on. Fourth-order differential equations boundary value problems, including those with the -Laplacian operator, have their origin in beam theory [1, 2], ice formation [3, 4], fluids on lungs [5], brain warping [6, 7], designing special curves on surfaces [8], and so forth. In beam theory, more specifically, a beam with a small deformation, a beam of a material that satisfies a nonlinear power-like stress and strain law, and a beam with two-sided links that satisfies a nonlinear powerlike elasticity law can be described by fourth order differential equations along with their boundary value conditions. For more background and applications, we refer the reader to the work by Timoshenko [9] on elasticity, the monograph by Soedel [10] on deformation of structure, and the work by Dulcska [11] on the effects of soil settlement. Due to their wide applications, the existence and multiplicity of positive solutions for fourth-order (including -Laplacian operator) boundary value problems has also attracted increasing attention over the last decades; see [12–33] and references therein. In [28], Zhang and Liu studied the following singular fourth-order four-point boundary value problem

where , ,  , ,  , , , may be singular at and/or and . The authors gave sufficient conditions for the existence of one positive solution by using the upper and lower solution method, fixed point theorems, and the properties of the Green function.

In [32], Zhang et al. discussed the existence and nonexistence of symmetric positive solutions of the following fourth-order boundary value problem with integral boundary conditions:

where , , is nonnegative, symmetric on the interval is continuous, for all , and are nonnegative, symmetric on .

Motivated by the work of the above papers, in this paper, we study the existence of positive solutions of the following singular fourth-order boundary value system with integral boundary conditions:

where and are positive parameters, , , , , are nondecreasing functions of bounded variation, and the integrals in (1.3) are Riemann-Stieltjes integrals, and are two continuous functions, and may be singular at while may be singular at ;    , are continuous and may be singular at and/or , in which , .

Compared to previous results, our work presented in this paper has the following new features. Firstly, our study is on singular nonlinear differential systems, that is, and in (1.3) are allowed to be singular at and/or , meanwhile is allowed to be singular at while is allowed to be singular at , which bring about many difficulties. Secondly, the main tools used in this paper is a fixed-point theorem in cones, and the results obtained are the conditions for the existence of solutions to the more general system (1.3). Thirdly, the techniques used in this paper are approximation methods, and a special cone has been developed to overcome the difficulties due to the singularity and to apply the fixed-point theorem. Finally, we discuss the boundary value problem with integral boundary conditions, that is, system (1.3) including fourth-order three-point, multipoint and nonlocal boundary value problems as special cases. To our knowledge, very few authors studied the existence of positive solutions for -Laplacian fourth-order differential equation with boundary conditions involving Riemann-Stieltjes integrals. Hence we improve and generalize the results of previous papers to some degree, and so it is interesting and important to study the existence of positive solutions for system (1.3).

The rest of this paper is organized as follows. In Section 2, we present some lemmas that are used to prove our main results. In Section 3, the existence of positive solution for system (1.3) is established by using the fixed point theory in cone. Finally, in Section 4, one example is also included to illustrate the main results.

Definition 1.1. A vector is said to be a positive solution of system (1.3) if and only if satisfies (1.3) and , or , for any .

Let be a cone in a Banach space . For , let , , and . The proof of the main theorem of this paper is based on the fixed point theory in cone. We list one lemma [34, 35] which is needed in our following argument.

Lemma 1.2. Let be a positive cone in real Banach space and a completely continuous operator. If the following conditions hold

(i) for ;(ii)there exists such that for any and . Then has a fixed point in .

Remark 1.3. If (i) and (ii) are satisfied for and , respectively. Then Lemma 1.2 is still true.

2. Preliminaries and Lemmas

The basic space used in this paper is . Obviously, the space is a Banach space if it is endowed with the norm as follows:

for any . Denote . For convenience, we list the following assumptions:

are continuous and

where , . are nondecreasing functions of bounded variation, and , , where

are continuous and satisfy

where are continuous and nonincreasing in the second variable, and are continuous and for any constant ,

Similar to the proof of Lemmas 2.1 and  2.2 in [32], the following two lemmas are valid.

Lemma 2.1. If holds, then for any , the boundary value problem

has a unique solution

where

Lemma 2.2. If holds, then for any , the boundary value problem

has a unique solution

where

Remark 2.3. For , , we have

Remark 2.4. If holds, it is easy to testify defined by (2.8) that:

where

Remark 2.5. From (2.11), we can prove that the properties of are similar to those of .

Lemma 2.6. For , , we have

Proof. The proof of this lemma is easy, and we omit it.

Let

where

It is easy to see that is a cone of . For any , let .

Remark 2.7. By the definition of , , , , we have .

To overcome singularity, we consider the following approximate problem of (1.3):

where is a positive integer and

Clearly, .

By Lemmas 2.1 and 2.2, for each , , , let us define operators , , and by

and , respectively.

Lemma 2.8. Assume that – hold, then for each , , , is a completely continuous operator.

Proof. Let , , and be fixed. For any , by (2.21), we have

which implies that is nonnegative and concave on . Similarly, by (2.22) we can obtain that is nonnegative and concave on . For any and , it follows from (2.13) that

Thus

On the other hand, by (2.13) and (2.18), we have

This implies that

Similar to (2.27), we also have

Therefore, .

Next, we prove that is completely continuous. Suppose and with . We notice that . Using the Lebesgue dominated convergence theorem, we have

Therefore,

Similarly, we also have

So and are continuous. Therefore, is also continuous.

Let be any bounded set, then for any , we have , , , and then , for any . By , we have

It is easy to show that is uniformly bounded. In order to show that is a compact operator, we only need to show that is equicontinuous. By the uniformly continuity of on , for all , there is such that for any , , and , we have

This together with (2.15) and (2.32) implies

This means that is equicontinuous. By the Arzela-Ascoli theorem, is a relatively compact set and that is a completely continuous operator.

In the same way, we can show that is also completely continuous, and so is completely continuous. Now since , , and are given arbitrarily, the conclusion of this lemma is valid.

3. Main Results

For notational convenience, we denote by

where denotes or . The main results of this paper are the following.

Theorem 3.1. Assume that – hold. Then we have:

If , , and , then for each , , the system (1.3) has at least one positive solution. If , , and , then for each , , the system (1.3) has at least one positive solution. If , , , then for each , , the system (1.3) has at least one positive solution. If , , , then for each , , the system (1.3) has at least one positive solution. If , , then for each , , the system (1.3) has at least one positive solution. If , or , , then for each , , the system (1.3) has at least one positive solution. If , , and , , then for each , or , , the system (1.3) has at least one positive solution.

Proof. We only prove the condition in which holds. The other cases can be proved similarly.

Let , , choose such that and

It follows from of that there exists such that for any ,

Let . For any , , by (2.13), (3.3), we have

Similarly, we also have

Therefore, we have

On the other hand, by , there exists such that

Let , . Next, we take , and for any , , , we will show

Otherwise, there exist and such that

From , we know that or . Without loss of generality, we may suppose that , then for any . So, by (2.13), (3.8), we have

This implies that , which is a contradiction. This yields that (3.9) holds. By (3.7), (3.9), and Lemma 1.2, for any and , , we obtain that has a fixed point in satisfying ,.

Let be the sequence of solutions of boundary value problems (2.19), where is a fixed integer. It is easy to see that they are uniformly bounded. Next we show that are equicontinuous on . From , we know that , , . For any , by the continuous of in , there exists such that for any , , and , we have

This combining with (2.15), (2.32) implies that for any , and , we have

Similarly, are also equicontinuous on . By the Ascoli-Arzela theorem, the sequence has a subsequence being uniformly convergent on . From Lemma 2.2, we know that

Since the properties of are similar to those of , so have the similar properties of , that is, also has a subsequence being uniformly convergent on . Without loss of generality, we still assume that itself uniformly converges to on and itself uniformly converges to on , respectively. Since , so we have , . By (2.19), we have

From (3.15) and (3.16), we know that , , , , , are bounded sets. Without loss of generality, we may assume as . Then by (3.15), (3.16), and the Lebesgue dominated convergence theorem, we have

By (3.17) and (3.18), direct computation shows that

On the other hand, satisfies the boundary condition of (1.3). In fact, , ,  ,  , and so the conclusion holds by letting .

Theorem 3.2. Assume that – hold. Then we have:

If , , and , then for each , , the system (1.3) has at least one positive solution. If , , and , then for each , , the system (1.3) has at least one positive solution. If , , , then for each , , the system (1.3) has at least one positive solution. If , , , then for each , , the system (1.3) has at least one positive solution. If , , then for each , , the system (1.3) has at least one positive solution. If , or , , then for each , , the system (1.3) has at least one positive solution. If , , and , , then for each , or , , the system (1.3) has at least one positive solution.

Proof. We may suppose that condition holds. Similarly, we can prove the other cases.

Let , . We can choose such that , and

It follows from and (2.16) that there exists such that for any

Let , . For any , , by (2.13), (3.21), we have

Similarly, by (3.22) we have . Therefore,

On the other hand, choose such that . By the condition of and (2.16), there exists such that

Let ,  . Next, we take , , and for any , , we will show

Otherwise, there exist and such that

From , we know that or . Without loss of generality, we may suppose that , then for any . So, we have

This implies that , which is a contradiction. This yields that (3.26) holds. By (3.24), (3.26), and Lemma 1.2, for any and , , we obtain that has a fixed point in and ,  . The rest of proof is similar to Theorem 3.1.

4. An Example

Example 4.1. We consider system (1.3) with ,   ,  ,  ,

Obviously, ,   are singular at and , is singular at and is singular at . Choose , , , and . Let

By direct calculation, we have ,  ,  ,  , . It is easy to check that , , and the conditions – and are satisfied. By Theorem 3.2, system (1.3) has at least one positive solution provided , .

Remark 4.2. Example 4.1 not only implies that , can be singular at and , respectively, but also indicates that there is a large number of functions that satisfy the conditions of Theorem 3.2. In addition, the condition is also easy to check.

Acknowledgments

The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11126231) and the Natural Science Foundation of Shandong Province of China (ZR2009AL014, ZR2011AQ008). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.

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