Abstract and Applied Analysis

Volume 2013, Article ID 640183, 7 pages

http://dx.doi.org/10.1155/2013/640183

## Existence and Uniqueness of Solution to Nonlinear Boundary Value Problems with Sign-Changing Green’s Function

^{1}Department of Mathematics, Heze University, Heze, Shandong 274000, China^{2}School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China^{3}Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 20 July 2013; Accepted 23 August 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Peiguo Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.

#### 1. Introduction

Boundary value problems (BVPs for short) for nonlinear differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory. The study of multipoint BVPs for second-order differential equations was initiated by Bicadze and Samarskiĭ [1] and later continued by II'in and Moiseev [2, 3] and Gupta [4]. Since then, great efforts have been devoted to nonlinear multipoint BVPs due to their theoretical challenge and great application potential. Many results on the existence of solutions for multipoint BVPs have been obtained; the methods used therein mainly depend on the fixed point theorems, degree theory, upper and lower techniques, and monotone iteration. The existence results are available in the literature [5–25] and the references therein.

Recently, by applying the fixed point theorems on cones, the authors of papers [5–7] established the existence and multiplicity of positive solutions for the th-order three-point BVP: where and . The th-order -point BVP has been studied in [8–10], where , and with . The existence and multiplicity results of solutions were shown by using various fixed point theorems and fixed point index theory.

By using the cone theory and the Banach contraction mapping principle, the author [26] established the existence and uniqueness for singular third-order three-point boundary value problems.

The purpose of this paper is to investigate the existence and uniqueness of solution of the following higher-order differential equation boundary value problem: where , ,

Here, we give the unique solution of BVP (3) under the conditions that is mixed nonmonotone. The methods used in this paper are motivated by [26], and the arguments are based upon the cone theory and the Banach contraction mapping principle.

#### 2. The Preliminary Lemmas

Lemma 1. * For any , the BVP
**
has a unique solution , where
*

* Proof. *First, suppose that is a solution to problem (4) and (5). It is easy to see by integration of (4) that
Substituting (7) into (5), we obtain
and so
Substituting (9) into (7), we have
Conversely, suppose that ; then it is easy to verify that (4) and (5) are satisfied. The lemma is proved.

For any , let

Lemma 2. * If is a solution to problem (3), then is a fixed point of .** If is a fixed point of , then is a solution to problem (3). *

By Lemma 1, the proof follows by routine calculations.

Let

It is easy to see that .

Lemma 3 (see [27, 28]). * is a generating cone in Banach space if and only if there exists a constant such that every element can be represented in the form , where and .*

#### 3. Main Results

This section discusses the solution of nonlinear higher-order differential equation BVP (3).

Let . Obviously, is a normal solid cone of Banach space , by Lemma 2.1.2 in [29], and we have that is a generating cone in .

Theorem 4. *Suppose that , , and there exist positive constants with
**
such that for any , , with ,, one has
**
and there exist , such that
**
converges. Then, BVP (3) has a unique solution in , and moreover, for any , the iterative sequence
**
converges to in .*

*Remark 5. *Recently, in the study of BVP (3), almost all the papers have supposed that Green’s function is nonnegative. However, the scope of is not limited to in Theorem 4, so, we do not need to suppose that is nonnegative.

*Remark 6. *The function in Theorem 4 is not monotone or convex; the conclusions and the proof used in this paper are different from the known papers in essence.

* Proof of Theorem 4. *It is easy to see that, for any can be divided into finite partitioned monotone and bounded function on , and then, by (15), we have that
converges. Let ; then
converges.

For any , let and then . By (14), we have
Hence,
Following the former inequality, we can easily have that
converges, thus,
is converged.

Similarly, by ,
is converged, and we have that
converges.

Define the operator by
Let
By (14) and (25), for any , we have
So we can choose , which satisfies , and so there exists a positive integer such that

Since is a generating cone in , from Lemma 3, there exists such that every element can be represented in
this implies
Let
By (31), we know that is well defined for any . It is easy to verify that is a norm in . By (30)–(32), we get

On the other hand, for any which satisfies , we have ; thus, , where denotes the normal constant of . Since is arbitrary, we have
It follows from (33) and (34) that the norms and are equivalent. Now, for any and which satisfies , let
then .

It follows from (27) that
subtracting (37) from (36) + (38), we obtain
Let ; then we have

As and are both positive linear bounded operators, so is a positive linear bounded operator, and therefore, . Hence, by mathematical induction, it is easy to know that for natural number in (29), we have
since , we see that
which implies by virtue of the arbitrariness of that
By , we have . Thus, the Banach contraction mapping principle implies that has a unique fixed point in , and so has a unique fixed point in ; by the definition of has a unique fixed point in ; then, by Lemma 2, is the unique solution of (3). And, for any , let ; we have . By the equivalence of and again, we get . This completes the proof.

#### 4. Example

In this paper, the results apply to a very wide range of functions, and we are following only one example to illustrate.

Consider the following th-order three-point boundary value problem: where , .

Applying Theorem 4, we can find that (44) has a unique solution provided , and moreover, for any , the iterative sequence converges to uniformly for all in .

To see that, let then is Green’s function of (44). It is easy to verify that , and so .

Let where ; then it is easy to verify that all conditions in Theorem 4 are satisfied.

#### Acknowledgments

Peiguo Zhang and Lishan Liu were supported financially by the National Natural Science Foundation of China (11071141, 11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, and the Project of Shandong Province Higher Educational Science and Technology Program (J11LA06, J13LI02). Yonghong Wu was supported financially by the Australian Research Council through an ARC Discovery Project grant.

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