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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.

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 [

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.

In this section, we introduce definitions and preliminary facts which are used throughout this paper.

Definition 1 Let

1)

2)

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 [

with boundary conditions (3) is given by

where

and

We need some properties of functions

Lemma 4 For all

Proof The conclusion is obvious. The proof is completed.

Lemma 5 For all

Proof For all

and

If

The proof is completed.

Lemma 6 The Green’s function

Proof After direct computations, we easily get

From (11) and (12) we can get (9) and (10) respectively. The proof is completed.

We shall consider the Banach space

Theorem 7 Suppose that

(H1)

(H2) There exist

(H3)

Then the BVP (1)-(3) has an unique positive, nondecreasing solution

Constructing successively the sequence of functions

for any initial function

where

Proof Obviously, from (H1) we obtain

Let

In view of Lemma 3, we define an operator T as

By (H1) it is easy to see that the operator

In what follows, we first prove

It follows from (H2) and (16) that

Using (17), (18), (8) and the condition (H1), we obtain

and

Equations (19), (20) and (H5) imply that

For any

and

Since the operator

For

From (23) and (24) we know that

so that there exists a function

and

From (H1) and (22) we have

This together with (26) and uniqueness of the limit imply that u^{*} satisfy

Form (22), (23) and (H1), we obtain

It follows from (26), (27) and (28) that

Therefore,

So that (15) holds. Since

Remark If

Now we are ready to discuss the existence of positive solutions for the BVP (2)-(3).

Theorem 8 Suppose that

(H4)

(H5) There exists positive number

where M is defined by (11).

Then the BVP (2)-(3) has at least one positive solution

Proof We consider the Banach space

where

For

By Ascoli-Arzela Theorem, it is easy to known that the operator

Let

then

which implies that

Therefore, by (9), (10), (29) and (32) we get

and

Then (33) and (34) show that

i.e.,

The authors thank the referee for her/his careful reading of the paper and useful suggestions. This work is supported by Hangzhou Polytechnic (KZYZ-2009-2) and the Natural Science Foundation of Zhejiang Province of China (LY12A01012).