Monotone Iterative Technique and Symmetric Positive Solutions to Fourth-Order Boundary Value Problem with Integral Boundary Conditions

Abstract

The purpose of this paper is to investigate the existence of symmetric positive solutions for a class of fourth-order boundary value problem: , , , , where , . By using a monotone iterative technique, we prove that the above boundary value problem has symmetric positive solutions under certain conditions. In particular, these solutions are obtained via the iteration procedures.

1. Introduction

The deformation of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by a fourth-order ordinary differential equation BVP (short for boundary value problem). At present, two-point situation of fourth-order BVP has been studied by many authors, generally using the nonlinear alternatives of Leray-Schauder, the fixed point index theory, and the method of upper and lower solutions, monotone iteration; see [1–6].

Recently, problems with integral boundary value conditions arise naturally in thermal conduction problems [7], semiconductor problems [8], and hydrodynamic problems [9]. Hence, the existence results of positive solutions to this kind of problems have received a great deal of attentions. We refer the readers to [10–15].

In [13], Ma studied the following problem: where and and are continuous. The existence of at least one symmetric positive solution is obtained by the application of the fixed point index in cones.

In [14], authors study the existence and nonexistence of symmetric positive solutions of the following fourth-order BVP: The argument was based on the fixed point theory in cones.

For fourth-order differential equation subject to boundary value conditions (2), author in [15] established the existence of positive solutions by the use of the Krasnoseliis fixed point theorem in cone.

The existing literature indicates that researches of fourth-order two point BVPs are excellent and methods are developed to be various. However, as to fourth-order BVPs with integral boundary value conditions, methods applied are relatively limited. Most of results are obtained by the use of fixed point theory in the cone or the fix point index theorem.

In this paper, we will apply the monotone iterative technique to the following fourth-order BVP with integral boundary conditions: We do not assume that the upper and lower solutions to the boundary value problem should exist but construct the specific form of the symmetric upper and lower solutions. And we will construct successive iterative schemes for approximating solutions. In addition, it is worth stating that the first term of our iterative scheme is a simple function or a constant function. Therefore, the iterative scheme is feasible. Under the appropriate assumptions on nonlinear term, a new and general result to the existence of symmetric positive solution of BVP (5) and (6) is obtained.

We assume that the following conditions hold throughout the paper:(S₁); (S₂); (S₃),  ,  ,  .

2. Preliminaries

Given , let and ,  . Denoted by ,  , the Green’s function of the following problem: Then, careful calculation yields

Denote

Lemma 1 (see [15]). Suppose that hold. Then, for any , solves the problem if and only if , where
During the process of getting the above solution, we can also know

Lemma 2. If is satisfied, the following results are true:(1), for ,  ;(2), for ,  .

Denote As , it is easy to check that and , for . Hence, from the symmetry and concavity of , we have In addition, for , the following results hold: Further, and therefore

We consider Banach space equipped with the norm , where . In this paper, a symmetric positive solution of (5) means a function which is symmetric and positive on and satisfies (5) as well as the boundary conditions (6).

In this paper, we always suppose that the following assumptions hold:(H₁) for ,  ,  ;(H₂), for ,  ;(H₃), for .

Denote It is easy to see that is a cone in .

We define the operator as follows: By the above argument, we know that, for any , and

Lemma 3. If are satisfied, is completely continuous; that is, is continuous and compact.

Proof. For any , from (21) and (22), combining Lemma 2 and (), we know that and for . We now prove that is symmetric about .
For , So, . The continuity of is obvious. We now prove that is compact. Let be a bounded set. Then, there exists such that For any , we have Therefore, from (17) and (18), we have and from (19), we have So, is uniformly bounded. Next we prove that is equicontinuous.
For , we have where and According to the Lagrange mean value theorem, we obtain that Similarly, we have Hence, there exists a positive constant such that And the similar results can be obtained for and .
The Arzelà-Ascoli theorem guarantees that is relatively compact which means that is compact.

3. Existence and Iterative of Solutions for BVP (5) and (6)

Theorem 4. Assume that hold. If there exists two positive numbers such that where and satisfy Then, problem (5) and (6) has concave symmetric positive solution with where where

Proof. We denote . In what follows, we first prove that .
Let ; then , .
By assumption and (33), for , we have
For any , by Lemma 3, we know that . According to (17), (18), and (33), we get and from (19) and (33), we get
Hence, . Thus, we get . Let , for ; then and . Let ; then . We denote
From the definition of , (16), (18), and (38), it follows that On the other hand, from (15), (18), and (38), we have From , it follows that By induction,
Since , we have , . From Lemma 3, is completely continuous. We assert that has a convergent subsequence and there exists such that .
Let ,  ; then . Let ; then ; we denote
Similarly to , we assert that has a convergent subsequence and there exists , such that .
Since , we have Hence, By induction, , , , (). Hence, we assert that , .
If ,  , then the zero function is not the solution of BVP (5) and (6). Thus, ; we have
It is well known that the fixed point of operator is the solution of BVP (5) and (6). Therefore, and are two positive, concave, and symmetric solutions of BVP (5) and (6).

Example 5. Consider the following fourth-order boundary value problem with integral boundary conditions: where
The calculation yields It is easy to check that assumptions hold. Set , . Then we can verify that conditions and and (33) are satisfied. Then applying Theorem 4, BVP (50) has two concave symmetric positive solutions with where where

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Beijing Higher Education Young Elite Teacher Project (Project no. YETP0322) and Chinese Universities Scientific Fund (Project no. 2013QJ004).