- Research Article
- Open Access

# Iterative Solutions of Singular Boundary Value Problems of Third-Order Differential Equation

- Peiguo Zhang
^{1}Email author

**Received:**19 January 2011**Accepted:**6 March 2011**Published:**15 March 2011

## Abstract

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for singular third-order boundary value problems. The theorems obtained are very general and complement previous known results.

## Keywords

- Unique Solution
- Ordinary Differential Equation
- Functional Equation
- Natural Number
- Electromagnetic Wave

## 1. Introduction

where , .

Three-point boundary value problems (BVPs for short) have been also widely studied because of both practical and theoretical aspects. There have been many papers investigating the solutions of three-point BVPs, see [2–5, 10, 12] and references therein. Recently, the existence of solutions of third-order three-point BVP (1.1) has been studied in [2, 3]. Guo et al. [2] show the existence of positive solutions for BVP (1.1) when and is separable by using cone expansion-compression fixed point theorem. In [3], the singular third-order three-point BVP (1.1) is considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, where , is separable and is not necessary to be nonnegative, and the existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory. Motivated by the above works, we consider the singular third-order three-point BVP (1.1). Here, we give the unique solution of BVP (1.1) under the conditions that and is mixed nonmonotone in and does not need to be separable by using the cone theory and the Banach contraction mapping principle.

## 2. Preliminaries

It is shown in [2] that is the Green's function to , , and .

It is easy to see that .

is generating if and only if there exists a constant such that every element can be represented in the form , where and

## 3. Singular Third-Order Boundary Value Problem

This section discusses singular third-order boundary value problem (1.1).

Let . Obviously, is a normal solid cone of Banach space ; by [16, Lemma 2.1.2], we have that is a generating cone in .

Theorem 3.1.

converges to .

Remark 3.2.

Recently, in the study of BVP (1.1), almost all the papers have supposed that the Green's function is nonnegative. However, the scope of is not limited to in Theorem 3.1, so, we do not need to suppose that is nonnegative.

Remark 3.3.

The function in Theorem 3.1 is not monotone or convex; the conclusions and the proof used in this paper are different from the known papers in essence.

Proof.

It follows from (3.18) and (3.19) that the norms and are equivalent.

then , , , , and .

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 , that is, is the unique solution of (1.1). And, for any , let ; we have . By the equivalence of and again, we get . This completes the proof.

Example 3.4.

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

converges to .

Then (3.1) is satisfied for any , , , and .

Thus all conditions in Theorem 3.1 are satisfied.

## Declarations

### Acknowledgment

The author is grateful to the referees for valuable suggestions and comments.

## Authors’ Affiliations

## References

- Gregus M: Third order linear Differential equations. In
*Mathematics and Its Applications*. Reidel, Dordrecht, the Netherlands; 1987.Google Scholar - Guo L-J, Sun J-P, Zhao Y-H: Existence of positive solutions for nonlinear third-order three-point boundary value problems.
*Nonlinear Analysis. Theory, Methods & Applications*2008, 68(10):3151-3158. 10.1016/j.na.2007.03.008View ArticleMathSciNetGoogle Scholar - Wang F, Cui Y: On the existence of solutions for singular boundary value problem of third-order differential equations.
*Mathematica Slovaca*2010, 60(4):485-494. 10.2478/s12175-010-0027-5View ArticleMathSciNetGoogle Scholar - Sun Y: Positive solutions for third-order three-point nonhomogeneous boundary value problems.
*Applied Mathematics Letters*2009, 22(1):45-51. 10.1016/j.aml.2008.02.002View ArticleMathSciNetGoogle Scholar - Graef JR, Webb JRL: Third order boundary value problems with nonlocal boundary conditions.
*Nonlinear Analysis. Theory, Methods & Applications*2009, 71(5-6):1542-1551. 10.1016/j.na.2008.12.047View ArticleMathSciNetGoogle Scholar - Liu Z, Debnath L, Kang SM: Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem.
*Computers & Mathematics with Applications*2008, 55(3):356-367. 10.1016/j.camwa.2007.03.021View ArticleMathSciNetGoogle Scholar - Minhós FM: On some third order nonlinear boundary value problems: existence, location and multiplicity results.
*Journal of Mathematical Analysis and Applications*2008, 339(2):1342-1353. 10.1016/j.jmaa.2007.08.005View ArticleMathSciNetGoogle Scholar - Hopkins B, Kosmatov N: Third-order boundary value problems with sign-changing solutions.
*Nonlinear Analysis. Theory, Methods & Applications*2007, 67(1):126-137. 10.1016/j.na.2006.05.003View ArticleMathSciNetGoogle Scholar - Yao Q: Successive iteration of positive solution for a discontinuous third-order boundary value problem.
*Computers & Mathematics with Applications*2007, 53(5):741-749. 10.1016/j.camwa.2006.12.007View ArticleMathSciNetGoogle Scholar - Boucherif A, Al-Malki N: Nonlinear three-point third-order boundary value problems.
*Applied Mathematics and Computation*2007, 190(2):1168-1177. 10.1016/j.amc.2007.02.039View ArticleMathSciNetGoogle Scholar - Li S: Positive solutions of nonlinear singular third-order two-point boundary value problem.
*Journal of Mathematical Analysis and Applications*2006, 323(1):413-425. 10.1016/j.jmaa.2005.10.037View ArticleMathSciNetGoogle Scholar - Sun Y: Positive solutions of singular third-order three-point boundary value problem.
*Journal of Mathematical Analysis and Applications*2005, 306(2):589-603. 10.1016/j.jmaa.2004.10.029View ArticleMathSciNetGoogle Scholar - Du Z, Ge W, Lin X: Existence of solutions for a class of third-order nonlinear boundary value problems.
*Journal of Mathematical Analysis and Applications*2004, 294(1):104-112. 10.1016/j.jmaa.2004.02.001View ArticleMathSciNetGoogle Scholar - Guo D:
*Semi-Ordered Method in Nonlinear Analysis*. Shandong Scientific Technical Press, Jinan, China; 2000.Google Scholar - Guo DJ, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar - Guo D, Lakshmikantham V, Liu X:
*Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications*.*Volume 373*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:viii+341.View ArticleGoogle Scholar

## Copyright

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