Abstract and Applied Analysis

Volume 2013 (2013), Article ID 621315, 8 pages

http://dx.doi.org/10.1155/2013/621315

Research Article

Successive Iteration of Positive Solutions for Fourth-Order Two-Point Boundary Value Problems

Yongping Sun, Xiaoping Zhang, and Min Zhao

College of Electron and Information, Zhejiang University of Media and Communications, Hangzhou, Zhejiang 310018, China

Received 1 March 2013; Revised 12 June 2013; Accepted 14 June 2013

Academic Editor: Chuanzhi Bai

Copyright © 2013 Yongping Sun 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.

We are concerned with a fourth-order two-point boundary value problem. We prove the existence of positive solutions and establish iterative schemes for approximating the solutions. The interesting point of our method is that the nonlinear term is involved with all lower-order derivatives of unknown function, and the iterative scheme starts off with a known cubic function or the zero function. Finally we give two examples to verify the effectiveness of the main results.

1. Introduction

The bending of an elastic beam can be described with some fourth-order boundary value problems. Owing to their importance in engineering, physics, and material mechanics, boundary value problems for elastic beam equations have attracted much attention of many authors. Many methods, such as upper and lower solutions method, shooting method, coincidence degree theory, or fixed point theorem and so on, have been developed to derive existence of solutions for fourth-order differential equations with two-point boundary conditions; see [1–20] and many references cited therein. However, almost all of the papers we mentioned focused their attention on the existence of solutions or positive solutions. In the existing literature, there are few papers that concern with the computational methods of solutions or positive solutions; see [21–24]. In this paper, by the successively iterative technique, we study the existence and iteration of monotone positive solutions for the following fourth-order differential equation:

together with two-point boundary conditions

Throughout this paper, the following conditions are satisfied:

  ;    is nonnegative and .

In material mechanics, (1) describes the deflection of a beam under a certain force. The boundary conditions of the forms (2) mean that the elastic beam is simply fixed at the end and fastened with a sliding clamp at the end .

Fourth-order differential equations with boundary conditions (2) have been studied. For example, Graef and Yang in [7] studied the nonlinear fourth-order differential equation:

together with boundary conditions (2). The authors established some sufficient conditions for the existence and nonexistence of positive solutions. The main tool is the Kranosel’skii fixed point theorem. Using the Leggett-Williams fixed point theorem, Yang [18] established an existence criterion for triple positive solutions of the nonlinear fourth-order differential equation:

with two-point boundary conditions (2). By applying iterative techniques, Sun and Wang [21] investigated the existence and iteration of monotone positive solution for the following elastic beam equation

with boundary conditions (2). By using Leray-Schauder nonlinear alternate, Leray-Schauder fixed point theorem and a fixed point theorem due to Avery and Peterson, Sun [15] presented some results on the existence and multiplicity of positive solutions to problem (1)-(2). Bai [3] proposed a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the following differential equations

or

with boundary conditions (2), if there exists an upper solution and a lower solution with . By placing some restrictions on the nonlinear term , Bai [4] obtained the existence results for the fourth-order two-point boundary value problem (1)-(2) via the lower and upper solution method. In particular, a new truncating technique and an appropriate Nagumo-type condition are introduced and employed. In addition, Yao studied (1) with nonhomogeneous boundary condition

In [20] the author obtained local existence theorem via Leray-Schauder fixed point theorem. In [22] Yao constructed an iterative sequence by the monotonic technique and proved that the sequence approximates successively to the solution of the equation under suitable assumptions.

Motivated by the above-mentioned works, this paper investigates the existence and iteration of positive solutions for problem (1)-(2). It is noted that the successively iterative scheme starts off with the zero function or a simple cubic function, which is feasible for the computational purpose. At the end of the paper, two examples are included to illustrate the main results.

2. Preliminaries

In this section, we introduce some necessary definitions and properties of the Green function.

Definition 1 (see [25, 26]). Let be a real Banach space. A nonempty closed convex set is called a of if it satisfies the following two conditions:

(1) imply ;(2) imply .

Definition 2 (see [25, 26]). An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

The Green’s function for problem (1)-(2) is given by

so solving the boundary value problem (1)-(2) is equivalent to finding a solution to the integral equation

Some properties of functions are summarized as follows.

Lemma 3. For all , the following inequalities hold true:

Proof. For any fixed , from (9) we obtain

then,

For all , it follows from (9) that

For all , if , it yields

While , we have

The proof is completed.

In what follows we will consider the Banach space equipped with norm

Define the cone by

For , we give an operator by

Obviously, the fixed points of are monotone and nonnegative solutions of the problem (1)-(2).

Lemma 4. Consider that is completely continuous.

Proof. Suppose that . Using , , (19), and Lemma 3, we obtain , , , , for all . Then . Using arguments similar to those of [21, 22], one can prove that is a completely continuous operator.

3. Main Results

In this section, we give the main results of this paper.

Theorem 5. Assume that there exists such that

   for any , , , , ;  , where ;  .

Then problem (1)-(2) has two monotone positive solutions and satisfying , and , , where , , , ,  and , for .

Proof. Let ; it follows that . In fact, if , then ; thus, for any ,

which together with the conditions and implies that

Hence, we have shown that , that is, .

Let ; then and . Denote . Since , we have . Since is completely continuous, we assert that is a sequentially compact set.

Since , by Lemma 3, , and , we obtain

Thus,

Furthermore, it follows from that

By the induction, we obtain

Hence, there exists such that . Applying the continuity of and using , we obtain , which implies that is a nonnegative solution of problem (1)-(2). Furthermore, by we know that the zero function is not a solution of problem (1)-(2). Thus, . From the definition of the cone , it yields ; that is, is a monotone positive solution of problem (1)-(2).

Let ; then . Let . With respect to , we obtain . Since is completely continuous, we assert that is a sequentially compact set.

Since , we obtain

So by , we have

By the induction, we further obtain

Hence, there exists such that , and is a monotone positive solution of problem (1)-(2). The proof is completed.

Corollary 6. Suppose that and hold. Assume

 .

Then problem (1)-(2) has at least two monotone positive solutions and satisfying the conclusion of Theorem 5.

Remark 7. and may coincide with each other and then the problem (1)-(2) has only one monotone positive solution in .

Remark 8. By the proof of Theorem 5, and converge in to the monotone positive solution and of the problem (1)-(2), respectively. In addition, the sequences , and are increasing, while the sequences , and are decreasing.

Remark 9. The iterative schemes in Theorem 5 start off with a known cubic function or the zero function, which is feasible for the computational purpose.

4. Examples

In this section we provide two examples to illustrate the application of our results.

Example 1. Consider the problem

For this problem, and . By the calculations, the conditions , and of Theorem 5 hold for . From Theorem 5, we have that problem (29) has two monotone positive solutions and satisfying and .

Moreover, the two iterative schemes are

Using Matlab 7.5, the second and third terms of the two schemes are as follows:

Example 2 (see [22]). Consider the problem

In this problem, and . Obviously, and satisfy conditions and . In addition, is increasing with regard to and and is decreasing with and , and . Moreover,

Therefore, assumptions of Theorem 5 are satisfied with . From Theorem 5, problem (32) has two monotone positive solutions and satisfying and , .

Moreover, the two iterative schemes are

Using Mathlab 7.5, the second terms of the two schemes are given as follows:

Remark 10. This example was carried out firstly in [22], in which the author showed that the solution is nontrivial, , and the iterative scheme starts off with

In the present paper, we proved that the solution is monotone positive, , and the iterative scheme starts off with or , which is much more simple than that of [22].

Acknowledgments

The authors would like to thank the reviewer for his/her careful reading, suggestions, and comments which improved the presentation of this paper. This work is supported by the Natural Science Foundation of Zhejiang Province of China (LY12A01012).

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