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

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 offwith 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.


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 E is Yang's modulus constant, I z is moment of inertia with respect to z axes, and q t is loading at t.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, fourthorder 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 g.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 K be a normal cone of a Banach space E and v 0 ≤ w 0 .Suppose that Then the iterative sequences and converge to, respectively, v and w ∈ v 0 , w 0 , which are fixed points of T .

Mathematical Problems in Engineering 3
Throughout this paper, we always assume that the following conditions are satisfied:

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

2.3
Proof.Let u 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 t, s ∈ 0, 1 × 0, 1 , one has Proof.For any fixed s ∈ 0, 1 , it is easy to know that ∂G t, s ∂t and so On the other hand, it follows from the expression of G t, s that

Main Results
Theorem 3.1.Assume that f t, 0, 0 / ≡ 0 for t ∈ 0, 1 and there exists a constant a > 0 such that the following conditions are satisfied: Then the BVP 1.3 has monotone positive solutions.
Proof.Let E C 1 0, 1 be equipped with the norm Let v 0 t ≡ 0 and w 0 t a 2t − t 2 /2, t ∈ 0, 1 .First, it is easy to verify that T : v 0 , w 0 → K is completely continuous by an application of Arzela-Ascoli theorem.Now, we divide our proof into the following steps.
Step 1.We assert that T is monotone increasing on v 0 , w 0 .
Suppose that u, v ∈ v 0 , w 0 and u ≤ v. Then 0 ≤ u t ≤ v t ≤ a and 0 ≤ u t ≤ v t ≤ a for t ∈ 0, 1 .By H 1 and H 2 , we have which shows that Tu ≤ Tv.
Step 2. We prove that v 0 is a lower solution of T .For any t ∈ 0, 1 , we know that which implies that v 0 ≤ Tv 0 .
Step 3. We show that w 0 is an upper solution of T .It follows from Lemmas 2.2 and 2.3, H 1 , and H 2 that which indicates that Tw 0 ≤ w 0 .
Step 4. We claim that the BVP 1.3 has monotone positive solutions.
In fact, if we construct sequences {v n } ∞ n 1 and {w n } ∞ n 1 as follows: then it follows from Theorem 1.1 that and {v n } ∞ n 0 and {w n } ∞ n 0 converge to, respectively, v and w ∈ v 0 , w 0 , which are monotone solutions of the BVP 1.3 .Moreover, for any t ∈ 0, 1 , by Lemmas 2.2 and 2.3, we know that 3.9 and so 0 < Tv 0 t ≤ Tv t v t ≤ w t , t ∈ 0, 1 , 3.10 which shows that v and w are positive solutions of the BVP 1.3 .

An Example
Example 4.1.Consider the BVP

4.1
If we let f t, u, v 1/4 u 1/16 1 t v 2 1 for t, u, v ∈ 0, 1 × 0, ∞ × 0, ∞ and g u − 1/4 u 2 for u ∈ 0, ∞ , then all the hypotheses of Theorem 3.1 are fulfilled with a 2. It follows from Theorem 3.1 that the BVP 4.1 has monotone positive solutions v and w.Moreover, the two iterative schemes are

1
Then K is a normal cone in Banach space E. Note that this induces an order relation ≤ in E by defining u ≤ v if and only if v − u ∈ K.If we define an operator T : K → E by 1 , A 2 , and Lemmas 2.2 and 2.3 implies that T : K → K. Obviously, fixed points of T are monotone nonnegative solutions of the BVP 1.3 .