# 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.

## 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)
- (ii)

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.

Proof.

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:

## 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
.

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.

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.

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

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