Discrete Dynamics in Nature and Society

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.