- Research Article
- Open Access

# Positive Solutions for Impulsive Equations of Third Order in Banach Space

- Jingjing Cai
^{1}Email author

**2010**:185701

https://doi.org/10.1155/2010/185701

© Jingjing Cai. 2010

**Received:**4 September 2010**Accepted:**30 November 2010**Published:**15 December 2010

## Abstract

Using the fixed-point theorem, this paper is devoted to study the multiple and single positive solutions of third-order boundary value problems for impulsive differential equations in ordered Banach spaces. The arguments are based on a specially constructed cone. At last, an example is given to illustrate the main results.

## Keywords

- Banach Space
- Partial Order
- Nonlinear Term
- Differential System
- Normal Constant

## 1. Introduction

where , , , . , , . is the zero element of .

Recently, third-order boundary value problems (cf. [1–9]) have attracted many authors attention due to their wide range of applications in applied mathematics, physics, and engineering, especially in the bridge issue. To our knowledge, most papers in literature concern mainly about the existence of positive solutions for the cases in which the spaces are real and the equations have no parameters. And many authors consider nonlinear term have same linearity. In this paper, we consider the existence of solutions when the nonlinear terms have different properties, the space is abstract and the equations have two different parameters.

where , . The authors obtained at least one positive solutions of BVP (1.2) by using fixed-point theorem when is sublinear or suplinear.

Inspired by the above work, the aim of this paper is to establish some simple criteria for the existence of nontrivial solutions for BVP (1.1) under some weaker conditions. The new features of this paper mainly include the following aspects. Firstly, we consider the system (1.1) in abstract space while [3, 8] talk about equations in real space ( ). Secondly, we obtained the positive solutions when the two parameters have different ranges. Thirdly, and in system (1.1) may have different properties. Fourthly, in system (1.1) not only contains but also , which is much more complicated. Finally, the main technique used here is the fixed-point theory and a special cone is constructed to study the existence of nontrivial solutions.

We recall some basic facts about ordered Banach spaces . The cone in induces a partial order on , that is, if and only if , is said to be normal if there exists a positive constant such that implies , without loss of generality, suppose, in present paper, the normal constant . denotes the measure of noncompactness (cf. [10]).

Some preliminaries and a number of lemmas to the derivation of the main results are given in Section 2, then the proofs of the theorems are given in Section 3, followed by an example, in Section 4, to demonstrate the validity of our main results.

## 2. Preliminaries and Lemmas

In this paper we will consider the Banach space , denote and , is continuous at and is left continuous at , the right limit exists, . For any we define and for .

For convenience, let us list the following assumption.

(A) , , , . For any and , is relatively compact in , where .

Lemma 2.1.

Proof.

The proof is similar to Lemma 2.2 in [3], we omit it.

Lemma 2.2 (see [3]).

Assume that and . Then for any , where , .

Lemma 2.3 (see [3]).

Lemma 2.4 (see [10]).

- (i)
for any and for any or

- (ii)
for any and for any .

Then has a fixed-point in .

Lemma 2.5.

As we know, BVP (1.1) has a positive solution if and only if is the fixed-point of .

Lemma 2.6.

is completely continuous.

Proof.

So .

So is continuous. Similarly, is continuous. It follows that is continuous.

which implies that . So, , it follows that is compact. The lemma is proved.

where or , and . is a dual cone of .

We list the assumptions:

(H_{1})
,
, where
;

(H_{2})
,
,
, where
and
;

(H_{3})
,
,
,
, where
and
.

## 3. Main Results

Theorem 3.1.

Assume that (A), (H_{1}) and the following condition (H)^{'} hold, then BVP (1.1) has at least two positive solution while
and
.

(H)^{'}:
;
;
, where
,
.

Proof.

_{1}) and the definition of , we obtain

By (3.5), (3.7), (3.9) and Lemma 2.4 we get that BVP (1.1) has at least two positive solutions with .

Corollary 3.2.

Theorem 3.3.

Assume that (A) and (H_{2}) hold, then BVP (1.1) has at least one positive solution when
and
.

Proof.

_{2}), there exists , , such that for ,

By (3.16), (3.21) and Lemma 2.4, it is easily seen that has a fixed-point .

Corollary 3.4.

Theorem 3.5.

Let (A) and (H_{3}) hold, then BVP (1.1) has at least one positive solution while
and
.

Proof.

By (3.25), (3.31), and Lemma 2.4, has a fixed-point .

Corollary 3.6.

## 4. An Example

Conclusion.

BVP(4.1) has at least one positive solution.

Proof.

and .

Above all, the conditions of Theorem 3.3 are satisfied. Then for any and , BVP (4.1) has at least one positive solution.

## Declarations

### Acknowledgments

The author thanks Professor Liu and Professor Lou for many useful discussions and helpful suggestions. The work was partially supported by NSFC (10971155) and Innovation program of Shanghai Municipal Education Commission (09ZZ33).

## Authors’ Affiliations

## References

- 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.2004.02.001MATHMathSciNetView ArticleGoogle Scholar - Feng Y, Liu S:
**Solvability of a third-order two-point boundary value problem.***Applied Mathematics Letters*2005,**18**(9):1034-1040. 10.1016/j.aml.2004.04.016MATHMathSciNetView ArticleGoogle 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.2007.03.008MATHMathSciNetView ArticleGoogle Scholar - Gupta CP:
**Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation.***Journal of Mathematical Analysis and Applications*1992,**168**(2):540-551. 10.1016/0022-247X(92)90179-HMATHMathSciNetView ArticleGoogle Scholar - Ma R:
**Multiplicity results for a third order boundary value problem at resonance.***Nonlinear Analysis: Theory, Methods & Applications*1998,**32**(4):493-499. 10.1016/S0362-546X(97)00494-XMATHMathSciNetView ArticleGoogle 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.2004.10.029MATHMathSciNetView ArticleGoogle Scholar - Webb JRL:
**Positive solutions of some three point boundary value problems via fixed point index theory.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(7):4319-4332. 10.1016/S0362-546X(01)00547-8MATHMathSciNetView ArticleGoogle Scholar - Yao Q, Feng Y:
**The existence of solution for a third-order two-point boundary value problem.***Applied Mathematics Letters*2002,**15**(2):227-232. 10.1016/S0893-9659(01)00122-7MATHMathSciNetView ArticleGoogle Scholar - Yao Q:
**The existence and multiplicity of positive solutions for a third-order three-point boundary value problem.***Acta Mathematicae Applicatae Sinica*2003,**19**(1):117-122. 10.1007/s10255-003-0087-1MATHMathSciNetView ArticleGoogle Scholar - Guo D:
*Nonlinear Functional Analysis*. Shangdong Science and Technology Press, Jinan, China; 1985.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.