- Research Article
- Open Access

# Positive Solutions of th-Order Nonlinear Impulsive Differential Equation with Nonlocal Boundary Conditions

- Meiqiang Feng
^{1}Email author, - Xuemei Zhang
^{2}and - Xiaozhong Yang
^{2}

**2011**:456426

https://doi.org/10.1155/2011/456426

© Meiqiang Feng et al. 2011

**Received:**25 March 2010**Accepted:**9 May 2010**Published:**10 June 2010

## Abstract

This paper is devoted to study the existence, nonexistence, and multiplicity of positive solutions for the th-order nonlocal boundary value problem with impulse effects. The arguments are based upon fixed point theorems in a cone. An example is worked out to demonstrate the main results.

## Keywords

- Banach Space
- Fixed Point Theorem
- Point Theory
- Direct Differentiation
- Real Banach Space

## 1. Introduction

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. [1]; for an overview of existing results and of recent research areas of impulsive differential equations, see Benchohra et al. [2]. The theory of impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations; see, for instance, [3–14] and their references.

where . Furthermore, they obtained the existence of positive solutions by means of fixed point index theory.

Recently, Yang and Wei [23] and the author of [24] improved and generalized the results of Pang et al. [22] by using different methods, respectively.

On the other hand, it is well known that fixed point theorem of cone expansion and compression of norm type has been applied to various boundary value problems to show the existence of positive solutions; for example, see [7, 8, 11, 19, 23, 24]. However, there are few papers investigating the existence of positive solutions of th impulsive differential equations by using the fixed point theorem of cone expansion and compression. The objective of the present paper is to fill this gap. Being directly inspired by [19, 22], using of the fixed point theorem of cone expansion and compression, this paper is devoted to study a class of nonlocal BVPs for th-order impulsive differential equations with fixed moments.

Here (where is fixed positive integer) are fixed points with where and represent the right-hand limit and left-hand limit of at , respectively, is nonnegative.

For the case of , problem (1.3) reduces to the problem studied by Samoĭlenko and Perestyuk in [4]. By using the fixed point index theory in cones, the authors obtained some sufficient conditions for the existence of at least one or two positive solutions to the two-point BVPs.

Motivated by the work above, in this paper we will extend the results of [4, 19, 22–24] to problem (1.3). On the other hand, it is also interesting and important to discuss the existence of positive solutions for problem (1.3) when , and . Many difficulties occur when we deal with them; for example, the construction of cone and operator. So we need to introduce some new tools and methods to investigate the existence of positive solutions for problem (1.3). Our argument is based on fixed point theory in cones [45].

To obtain positive solutions of (1.3), the following fixed point theorem in cones is fundamental which can be found in [45, page 93].

Lemma 1.1.

- (i)
;

- (ii)
.

Then, has at least one fixed point in .

## 2. Preliminaries

In order to define the solution of problem (1.3), we will consider the following space.

where

A function is called a solution of problem (1.3) if it satisfies (1.3).

To establish the existence of multiple positive solutions in of problem (1.3), let us list the following assumptions:

;

, where .

Lemma 2.1.

Proof.

Then, the proof of sufficient is complete.

So and , and it is easy to verify that , and the lemma is proved.

Similar to the proof of that from [22], we can prove that , and have the following properties.

Proposition 2.2.

The function defined by (2.5) satisfyong is continuous for all .

Proposition 2.3.

where is defined in (2.20).

Proposition 2.4.

If , then one has

(i) is continuous for all ;

(ii) .

Proof.

From the properties of and the definition of , we can prove that the results of Proposition 2.4 hold.

Proposition 2.5.

If , the function defined by (2.4) satisfies

(i) is continuous for all ;

(ii) for each , and

is defined in Proposition 2.3.

- (i)
From Propositions 2.2 and 2.4, we obtain that is continuous for all , and .

- (ii)
From (ii) of Proposition 2.2 and (ii) of Proposition 2.4, we have for each .

Now, we show that (2.19) holds.

Then the proof of Proposition 2.5 is completed.

Remark 2.6.

From the definition of , it is clear that .

Lemma 2.7.

Assume that and hold. Then, the solution of problem (1.3) satisfies

Proof.

It is an immediate subsequence of the facts that on .

Remark 2.8.

Lemma 2.9.

Assume that and hold. Then, , and is completely continuous.

Proof.

Thus, .

Next, by similar arguments to those in [8] one can prove that is completely continuous. So it is omitted, and the lemma is proved.

## 3. Main Results

where denotes or

In this section, we apply Lemma 1.1 to establish the existence of positive solutions for BVP (1.3).

Theorem 3.1.

Assume that and hold. In addition, letting and satisfy the following conditions:

and ;

or ,

BVP (1.3) has at least one positive solution.

Proof.

Therefore, , which is a contraction. Hence, (3.2) holds.

In fact, if , then , for . Since . Hence, , which contracts So, (3.15) holds. Therefore, , this is also a contraction. Hence, (3.10) holds.

We prove that (3.10) holds.

Similar to the proof in case ( ), we can show that . Then, from (3.23), we have , which is a contraction. Hence, (3.10) holds.

Applying (i) of Lemma 1.1 to (3.2) and (3.10) yields that has a fixed point . Thus, it follows that BVP (1.3) has at least one positive solution, and the theorem is proved.

Theorem 3.2.

Assume that and hold. In addition, letting and satisfy the following conditions:

and ;

or ,

BVP (1.3) has at least one positive solution.

Proof.

Considering , there exists such that , for , where satisfy .

Applying (i) of Lemma 1.1 to (3.24) and (3.25) yields that has a fixed point . Thus, it follows that BVP (1.3) has one positive solution, and the theorem is proved.

Theorem 3.3.

Assume that and hold. In addition, letting and satisfy the following condition:

where satisfy , BVP (1.3) has at least two positive solutions and with .

Proof.

that is, , which is a contraction. Hence, (3.29) holds.

Applying Lemma 1.1 to (3.27), (3.28), and (3.29) yields that has two fixed points with . Thus it follows that BVP (1.3) has two positive solutions with . The proof is complete.

Theorem 3.4.

Assume , and , then problem (1.3) has no positive solution.

Proof.

which is a contradiction, and this completes the proof.

To illustrate how our main results can be used in practice we present an example.

Example 3.5.

Conclusion.

BVP (3.34) has at least one positive solution.

Proof.

Then, conditions and of Theorem 3.1 hold. Hence, by Theorem 3.1, the conclusion follows, and the proof is complete.

## Declarations

### Acknowledgment

This work is supported by the National Natural Science Foundation of China (10771065), the Natural Sciences Foundation of Heibei Province (A2007001027), the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education(KM201010772018) and Beijing Municipal Education Commission(71D0911003). The authors thank the referee for his/her careful reading of the paper and useful suggestions.

## Authors’ Affiliations

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