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.
Positive Solution Uniqueness and Existence Third-Order Three-Point BVPs1. 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 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.
2. Preliminaries
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.
3. Uniqueness
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.
4. Existence
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.
Acknowledgements
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).
ReferencesGUO L., SUN J. , ZHAO Y. ,et al. (2007)GUO, L., SUN, J. AND ZHAO, Y. MULTIPLE POSITIVE SOLUTIONS FOR NONLINEAR THIRD-ORDER THREE-POINT BOUNDARY-VALUE PROBLEMS 112, 1-7.GUO, L., SUN, J. AND ZHAO, Y. (2008) EXISTENCE OF POSITIVE SOLUTION FOR NONLINEAR THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM. NONLINEAR ANALYSIS, THEORY, METHODS AND APPLICATIONS, 68, 3151-3158.ERSON, D. ,et al. (1998)MULTIPLE POSITIVE SOLUTIONS FOR A THREE-POINT BOUNDARY VALUE PROBLEM 27, 49-57.HTTP://DX.DOI.ORG/10.1016/S0895-7177(98)00028-4GRAEF, J.R. AND YANG, B. (2005) MULTIPLE POSITIVE SILUTIONS TO A THREE POINT THIRD ORDER BOUNDARY VALUE PROBLEM. DISCRETE AND CONTINUOUS DYNMICAL SYSTEMS, 1-8.PALAMIDES P.K. , PALAMIDES A.P. ,et al. (2008)PALAMIDES, P.K. AND PALAMIDES, A.P. A THIRD-ORDER 3-POINT BVP. APPLYING KRASNOSEL’SKII’S THEOREM ON THE PLANE WITHOUT A GREEN’S FUNCTION 14, 1-15.PALAMIDES A.P. , STAVRAKAKIS N.M. ,et al. (2010)PALAMIDES, A.P. AND STAVRAKAKIS, N.M. EXISTENCE AND UNIQUENESS OF A POSITIVE SOLUTION FOR A THIRD-ORDER THREE-POINT BOUNDARY-VALUE PROBLEM 155, 1-12.SUN J., REN Q. , ZHAO Y. ,et al. (2010)SUN, J., REN, Q. AND ZHAO, Y. THE UPPER AND LOWER SOLUTION METHOD FOR NONLINEAR THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM 26, 1-8.SUN Y. ,et al. (2005)POSITIVE SOLUTIONS OF SINGULAR THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEMS 306, 589-603.HTTP://DX.DOI.ORG/10.1016/J.JMAA.2004.10.029SUN Y. ,et al. (2009)POSITIVE SOLUTIONS FOR THIRD-ORDER THREE-POINT NONHOMOGENEOUS BOUNDARY VALUE PROBLEMS 22, 45-51.HTTP://DX.DOI.ORG/10.1016/J.AML.2008.02.002SUN Y. ,et al. (2008)EXISTENCE OF TRIPLE POSITIVE SOLUTIONS FOR A THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM 221, 194-201.HTTP://DX.DOI.ORG/10.1016/J.CAM.2007.10.064TORRES F.J. ,et al. (2013)POSITIVE SOLUTIONS FOR A THIRD-ORDER THREE-POINT BOUNDARY-VALUE PROBLEM 147, 1-11.YAO Q. ,et al. (2009)POSITIVE SOLUTIONS OF SINGULAR THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEMS 354, 207-212.HTTP://DX.DOI.ORG/10.1016/J.JMAA.2008.12.057ZHANG X. , LIU L. ,et al. (2008)NONTRIVIAL SOLUTION OF THIRD-ORDER NONLINEAR EIGENVALUE PROBLEMS (II) 204, 508-512.HTTP://DX.DOI.ORG/10.1016/J.AMC.2008.06.048YAO Q. , FENG Y. ,et al. (2002)THE EXISTENCE OF SOLUTIONS FOR A THIRD ORDER TWO-POINT BOUNDARY VALUE PROBLEM 15, 227-232.HTTP://DX.DOI.ORG/10.1016/S0893-9659(01)00122-7FENG Y. , LIU S. ,et al. (2005)SOLVABILITY OF A THIRD-ORDER TWO-POINT BOUNDARY VALUE PROBLEM 18, 1034-1040.HTTP://DX.DOI.ORG/10.1016/J.AML.2004.04.016