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 may be singular at and/or and. Here, the solution of the BVP (1)-(3) (or the BVP (2)-(3)) is called positive if.

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 and is nonsingular at and, Guo et al. [1] [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, respectively.

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 be a real Banach space. A nonempty closed convex set is called a cone of if it satisfies the following two conditions:

1) implies;

2) implies.

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, , then the unique solution of the following equation

with boundary conditions (3) is given by

where

and

We need some properties of functions and in order to establish the existence and uniqueness of positive solutions.

Lemma 4 For all, we have

Proof The conclusion is obvious. The proof is completed.

Lemma 5 For all, we have

Proof For all, if, it follows from (7) that

and

If, then from (7) we have

The proof is completed.

Lemma 6 The Green’s function 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.

We shall consider the Banach space equipped with norm.

Theorem 7 Suppose that

(H1) for any;

(H2) There exist such that

(H3)

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

Constructing successively the sequence of functions

for any initial function, then must converge to uniformly on [0, 1] and the rate of convergence is

where, which depends on the initial function.

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 is increasing. Observe that the BVP (1)-(3) has a solution if and only if the operator T has a fixed point.

In what follows, we first prove In fact, for any there exist positive numbers such that

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, we let

and

Since the operator is increasing, (H1), (H2), (21) and (22) imply that

For, from (H1), (17) and (22), it can obtained by induction that

From (23) and (24) we know that

so that there exists a function such that

and

From (H1) and (22) we have

This together with (26) and uniqueness of the limit imply that u^* satisfy, thus is a solution of the BVP (1)-(3).

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

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

Therefore,

So that (15) holds. Since is arbitrary in D we know that is the unique solution of the BVP (1)-(3) in D.

Remark If is continuous on, then it is quite evident that the condition (H3) holds. Hence the unique solution is in.

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

Theorem 8 Suppose that

(H4) and

(H5) There exists positive number such that

where M is defined by (11).

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

Proof We consider the Banach space equipped with the norm

where.

For, define the operator S by

By Ascoli-Arzela Theorem, it is easy to known that the operator is a completely continuous operator. The BVP (2)-(3) has a solution if and only if is a fixed point of operator S defined by (32).

Let

then is a bounded closed convex set of E. We show that. For, by (31) we have

which implies that

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

and

Then (33) and (34) show that

i.e.,. Thus, by Leray-Schauder fixed point theorem, S has a fixed point, which implies that BVP (2)-(3) has at least one positive solution satisfying (30). This completes the proof.

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).