Mathematical Problems in Engineering

Volume 2011, Article ID 609189, 9 pages

http://dx.doi.org/10.1155/2011/609189

## Monotone Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions

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

Received 19 April 2011; Accepted 2 August 2011

Academic Editor: Angelo Luongo

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 solutions for an elastic beam equation with nonlinear boundary conditions. By applying monotone iteration method, we not only obtain the existence of monotone positive solutions but also establish iterative schemes for approximating the solutions. It is worth mentioning that these iterative schemes start off with zero function or quadratic function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results obtained.

#### 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 a more general equation: According to different forms of supporting, various boundary value problems (BVPs for short) should be considered.

Owing to its importance in engineering, physics, and material mechanics, fourth-order BVPs have attracted much attention from many authors; see, for example [1–15] and the references therein. However, almost all of the papers we mentioned focused their attention on the null boundary conditions. When the boundary conditions are nonzero or nonlinear, fourth-order equations can model beams resting on elastic bearings located in their extremities. Up to now, a little work has been done for fourth-order BVPs with nonlinear boundary conditions. It is worth mentioning that, in 2009, Alves et al. [16] studied some fourth-order BVPs with nonlinear boundary conditions, which models an elastic beam whose left end is fixed and right end is attached to a bearing device or sliding clamped. Their main tool was monotone iterative method. For more on monotone iterative techniques, one can refer to [17–20] and the references therein.

Motivated greatly by the above-mentioned excellent works, in this paper we consider the existence and iteration of monotone positive solutions for the following fourth-order BVP with nonlinear boundary conditions: which models an elastic beam whose left end is simply supported and right end is sliding clamped, given by the function . By applying monotone iterative method, we not only obtain the existence of monotone positive solutions for the BVP (1.3) but also establish iterative schemes for approximating the solutions. These iterative schemes start off with zero function or quadratic function, which is very useful and feasible for computational purpose. Our main tool is the following theorem [21].

Theorem 1.1. *Let be a normal cone of a Banach space and . Suppose that ** is completely continuous,** is monotone increasing on ,** is a lower solution of , that is, ,** is an upper solution of , that is, . **Then the iterative sequences
**
satisfy
**
and converge to, respectively, and , which are fixed points of .*

Throughout this paper, we always assume that the following conditions are 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 be a constant and . Then the BVP
**
has a unique solution
**
where
*

*Proof. *Let be a solution of the BVP (2.1). Then we may suppose that
By the boundary conditions in (2.1), we have
Therefore, the BVP (2.1) has a unique solution

Lemma 2.2. *For any , one has
*

*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

Lemma 2.3. *For any , one has
*

*Proof. *It is obvious.

#### 3. Main Results

Theorem 3.1. *Assume that for and there exists a constant such that the following conditions are satisfied:**;
** .**Then the BVP (1.3) has monotone positive solutions.*

*Proof. *Let be equipped with the norm
Then is a normal cone in Banach space . Note that this induces an order relation ≤ in by defining if and only if . If we define an operator by
then
which together with , , and Lemmas 2.2 and 2.3 implies that . Obviously, fixed points of are monotone nonnegative solutions of the BVP (1.3).

Let and , . First, it is easy to verify that is completely continuous by an application of Arzela-Ascoli theorem. Now, we divide our proof into the following steps.*Step 1. *We assert that is monotone increasing on .

Suppose that and . Then and . By and , we have
which shows that .*Step 2. *We prove that is a lower solution of .

For any , we know that
which implies that .*Step 3. *We show that is an upper solution of .

It follows from Lemmas 2.2 and 2.3, , and that
which indicates that .*Step 4. *We claim that the BVP (1.3) has monotone positive solutions.

In fact, if we construct sequences and as follows:
then it follows from Theorem 1.1 that
and and converge to, respectively, and , which are monotone solutions of the BVP (1.3). Moreover, for any , by Lemmas 2.2 and 2.3, we know that
and so
which shows that and are positive solutions of the BVP (1.3).

#### 4. An Example

*Example 4.1. *Consider the BVP

If we let forandfor, then all the hypotheses of Theorem 3.1 are fulfilled with . It follows from Theorem 3.1 that the BVP (4.1) has monotone positive solutions and . Moreover, the two iterative schemes are
The first, second, and third terms of the two schemes are as follows:

#### Acknowledgment

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

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