Existence and Uniqueness of Positive Solution for Third-Order Three-Point Boundary Value Problems

This paper is devoted to the study of the existence and uniqueness of the positive solution for a type of the nonlinear third-order three-point boundary value problem. Our results are based on an iterative method and the Leray-Schauder fixed point theorem.


Introduction
In this paper, we consider the uniqueness and existence of the positive solution for the following third-order differential equation or with the following three-point boundary conditions ( ) ( ) ( ) ( ) Throughout this paper, we assume that In the past few years, because of the extensive applications in mechanics and engineering, the existence of solutions or positive solutions for nonlinear singular or nonsingular three-point boundary value problems for third-order ordinary differential equations has been studied extensively in the literature (see [1]- [13] and references therein).For example, in the case of ( ) [2] established some existence results of at least one and at least three positive solutions for the BVP (1)-(3) by using the well-known Krasnosel'skii fixed point theorem and the Leggett-Williams fixed point theorem, respectively.By using the upper and lower solutions and the maximum principle, Yao and Feng in [14] and Feng and Liu in [15] studied the existence of solutions for the BVP (1)-(3) and BVP ( 2)-( 3) with 0 a = , respec- tively.
Motivated mainly by the papers mentioned above, in this paper we will consider the uniqueness of the positive solution, the iteration and the rate of the convergence by the iteration for the nonlinear singular third-order three-point BVP (1)-( 3).We study the existence of the positive solution for the nonlinear third-order three-point BVP ( 2)-( 3) by using the Leray-Schauder fixed point theorem.
The rest of this paper is organized as follows.After this section, we present some notations and lemmas that will be used to prove our main results in Section 2. We discuss the uniqueness in Section 3. Finally, we discuss the existence in Section 4.

Preliminaries
In this section, we introduce definitions and preliminary facts which are used throughout this paper.
Definition 1 Let E be a real Banach space.A nonempty closed convex set K E ⊂ is called a cone of E if it satisfies the following two conditions: Definition 2 An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
The following lemma plays a pivotal role in the forthcoming analysis.Lemma 3 [9] Suppose that 1 aη ≠ , [ ] , then the unique solution of the following equation with boundary conditions (3) is given by where We need some properties of functions ( ) , G t s in order to establish the existence and uniqueness of positive solutions.
Proof The conclusion is obvious.The proof is completed.
The proof is completed.Lemma 6 The Green's function ( ) , G t s has the following properties: Proof After direct computations, we easily get ( ) ( ) ( ) ( ) From ( 11) and ( 12) we can get ( 9) and ( 10) respectively.The proof is completed.
Constructing successively the sequence of functions for any initial function ( ) uniformly on [0, 1] and the rate of convergence is where 0 1 θ < < , which depends on the initial function ( ) , 0,1 0, .
From ( 23) and ( 24) we know that so that there exists a function ( ) .

Existence
Now we are ready to discuss the existence of positive solutions for the BVP (2)-(3).Theorem 8 Suppose that (H4) where M is defined by (11).
Then the BVP ( 2)-( 3) has at least one positive solution ( ) Proof We consider the Banach space where ( ) Ω is a bounded closed convex set of E. We show that ( ) Observe that the BVP (1)-(3) has a solution if and only if the operator T has a fixed point.