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Mathematical Problems in Engineering
Volume 2011, Article ID 705740, 10 pages
http://dx.doi.org/10.1155/2011/705740
Research Article

Existence and Iteration of Monotone Positive Solution of BVP for an Elastic Beam Equation

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 5 October 2010; Accepted 28 January 2011

Academic Editor: Jitao Sun

Copyright © 2011 Jian-Ping Sun and Xian-Qiang Wang. 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 is concerned with the existence of monotone positive solution of boundary value problem for an elastic beam equation. By applying iterative techniques, we not only obtain the existence of monotone positive solution but also establish iterative scheme for approximating the solution. It is worth mentioning that the iterative scheme starts off with zero function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results.

1. Introduction

It is well-known that beam is one of the basic structures in architecture. The deformations of an elastic beam in equilibrium state can be described by the following equation of deflection curve: where is Yang's modulus constant, is moment of inertia with respect to axes, and is loading at . If the loading of beam considered is in relation to deflection and rate of change of deflection, we need to study more general equation According to different forms of supporting, various boundary value problems (BVPs) should be considered.

Owing to its importance in engineering, physics, and material mechanics, BVPs for elastic beam equations have attracted much attention from many authors see, for example, [115] and the references 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 concerned with the computational methods of solutions or positive solutions. It is worth mentioning that Zhang [16] obtained the existence of positive solutions and established iterative schemes for approximating the solutions for an elastic beam equation with a corner. The main tools used were monotone iterative techniques. For monotone iterative methods, one can refer [1719] and the references therein.

Motivated greatly by the above-mentioned excellent works, in this paper we investigate the existence and iteration of monotone positive solution for the following elastic beam equation BVP In material mechanics, the equation in (1.3) describes the deflection or deformation of an elastic beam under a certain force. The boundary conditions in (1.3) mean that the elastic beam is simply fixed at the end and fastened with a sliding clamp at the end . By applying iterative techniques, we not only obtain the existence of monotone positive solution but also establish iterative scheme for approximating the solution. It is worth mentioning that the iterative scheme starts off with zero function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results.

Throughout this paper, we always assume that the following condition is satisfied:

.

2. Preliminary

In order to obtain the main results of this paper, we first present several fundamental lemmas in this section.

Lemma 2.1. Let . Then, the BVP has a unique solution where

Lemma 2.2. For any , we have

Proof. For any fixed , it is easy to know that which shows that and so, On the other hand, it follows from the expression of that

Let be equipped with the norm . Then, is a Banach space. Denote Then, it is easy to verify that is a cone in . Note that this induces an order relation in by defining if and only if . Now, we define an operator on by Obviously, fixed points of are monotone and nonnegative solutions of the BVP (1.3).

Lemma 2.3. is completely continuous.

Proof. First, we prove . Suppose that . In view of Lemma 2.2, on the one hand, which shows that On the other hand, which together with (2.12) implies that Again, by Lemma 2.2, we have So, it follows from (2.14), (2.15) and the definition of that .
Next, we show that is a compact operator. Let be a bounded set. Then, there exists such that for any . For any , there exist such that . Denote Then, for any positive integer , it follows from Lemma 2.2 that which indicates that is uniformly bounded. Similarly, we have This shows that is uniformly bounded, which implies that is equicontinuous. By Arzela-Ascoli theorem, we know that has a convergent subsequence in . Without loss of generality, we may assume that converges in .
On the other hand, for any , by the uniform continuity of , we know that there exists a such that for any with , So, for any positive integer , with , we get which shows that is equicontinuous. Again, it follows from Arzela-Ascoli theorem that has a convergent subsequence in . Therefore, has a convergent subsequence in .
Finally, we prove that is continuous. Suppose that and . Then, there exists such that for any positive integer , . Denote Then, for any positive integer and , by Lemma 2.2, we have So, it follows from Lebesgue dominated convergence theorem that which implies that is continuous. Therefore, is completely continuous.

3. Main Results

Theorem 3.1. Assume that for , and there exists a constant such that If we construct a iterative sequence , , where for , then converges to in , which is a monotone positive solution of the BVP (1.3) and satisfy

Proof. Let . We assert that . In fact, if , then which together with the condition (3.1) and Lemma 2.2 implies that Hence, we have shown that .
Now, we assert that converges to in , which is a monotone positive solution of the BVP (1.3) and satisfies
Indeed, in view of and , we have , . Since the set is bounded and is completely continuous, we know that the set is relatively compact.
In what follows, we prove that is monotone by induction. First, by Lemma 2.2, we have which implies that At the same time, it is obvious that It follows from (3.7) and (3.8) that , which shows that . Next, we assume that . Then, in view of Lemma 2.2 and (3.1), we have which implies that At the same time, by Lemma 2.2 and (3.1), we also have It follows from (3.10) and (3.11) that , which indicates that . Thus, we have shown that , .
Since is relatively compact and monotone, there exists a such that , which together with the continuity of and the fact that implies that . Moreover, in view of for , we know that the zero function is not a solution of the BVP (1.3). Thus, . So, it follows from that

4. An Example

Example 4.1. Consider the BVP
If we let for , then all the hypotheses of Theorem 3.1 are fulfilled with . It follows from Theorem 3.1 that the BVP (4.1) has a monotone positive solution satisfying Moreover, the iterative scheme is The first, second, third, and fourth terms of the scheme are as follows:

Acknowledgment

This work is supported by the National Natural Science Foundation of China (10801068).

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