- Research Article
- Open Access

# Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional -Laplacian

- Meiqiang Feng
^{1}Email author

**Received:**2 February 2010**Accepted:**5 June 2010**Published:**29 June 2010

## Abstract

By using the fixed point theory for completely continuous operator, this paper investigates the existence of positive solutions for a class of fourth-order impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian. Moreover, we offer some interesting discussion of the associated boundary value problems. Upper and lower bounds for these positive solutions also are given, so our work is new.

## Keywords

- Fixed Point Theorem
- Nonlocal Boundary
- Impulsive Differential Equation
- Nonlocal Boundary Condition
- Integral Boundary Condition

## 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 recent years and is much richer than the corresponding theory of differential equations (see, e.g., [3–18] and references cited therein).

Moreover, the theory of boundary-value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions. For boundary-value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo [19], Karakostas and Tsamatos [20], Lomtatidze and Malaguti [21], and the references therein. For more information about the general theory of integral equations and their relation with boundary-value problems, we refer to the book of Corduneanu [22] and Agarwal and O'Regan [23].

where is a positive parameter, , is the zero element of , and . The authors investigated the multiplicity of positive solutions to problem (1.1) by using the fixed point index theory in cone for strict set contraction operator.

where and may be singular at or ; are continuous and may be singular at , and ; , and , and , and are nonnegative, .

where denotes a linear functional on given by involving a Stieltjes integral, and is a function of bounded variation.

where , , and . By using upper and lower solution method, fixed-point theorems, and the properties of Green's function and , the authors give sufficient conditions for the existence of one positive solution.

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

For the case of , problem (1.6) reduces to the problem studied by Zhang et al. in [33]. By using the fixed point theorem in cone, the authors obtained some sufficient conditions for the existence and multiplicity of symmetric positive solutions for a class of -Laplacian fourth-order differential equations with integral boundary conditions.

For the case of , and , problem (1.6) is related to fourth-order two-points boundary value problem of ODE. Under this case, problem (1.6) has received considerable attention (see, e.g., [40–42] and references cited therein). Aftabizadeh [40] showed the existence of a solution to problem (1.6) under the restriction that is a bounded function. Bai and Wang [41] have applied the fixed point theorem and degree theory to establish existence, uniqueness, and multiplicity of positive solutions to problem (1.6). Ma and Wang [42] have proved that there exist at least two positive solutions by applying the existence of positive solutions under the fact that is either superlinear or sublinear on by employing the fixed point theorem of cone extension or compression.

Being directly inspired by [18, 34, 43], in the present paper, we consider some existence results for problem (1.6) in a specially constructed cone by using the fixed point theorem. The main features of this paper are as follows. Firstly, comparing with [39–43], we discuss the impulsive boundary value problem with integral boundary conditions, that is, problem (1.6) includes fourth-order two-, three-, multipoint, and nonlocal boundary value problems as special cases. Secondly, the conditions are weaker than those of [33, 34, 46], and we consider the case . Finally, comparing with [33, 34, 39–43, 46], upper and lower bounds for these positive solutions also are given. Hence, we improve and generalize the results of [33, 34, 39–43, 46] to some degree, and so, it is interesting and important to study the existence of positive solutions of problem (1.6).

The organization of this paper is as follows. We shall introduce some lemmas in the rest of this section. In Section 2, we provide some necessary background. In particular, we state some properties of the Green's function associated with problem (1.6). In Section 3, the main results will be stated and proved. Finally, in Section 4, we offer some interesting discussion of the associated problem (1.6).

To obtain positive solutions of problem (1.6), the following fixed point theorem in cones is fundamental, which can be found in [47, page 94].

Lemma 1.1.

Let and be two bounded open sets in Banach space , such that and . Let be a cone in and let operator be completely continuous. Suppose that one of the following two conditions is satisfied:

(a) , and ;

(b) , and .

Then, has at least one fixed point in .

## 2. Preliminaries

In order to define the solution of problem (1.6), we shall consider the following space.

where .

A function is called a solution of problem (1.6) if it satisfies (1.6).

To establish the existence of multiple positive solutions in of problem (1.6), let us list the following assumptions:

;

From , it is clear that .

Lemma 2.1.

Proof.

The proof follows by routine calculations.

Write . Then from (2.9) and (2.10), we can prove that have the following properties.

Proposition 2.2.

Proposition 2.3.

Proposition 2.4.

Proof.

The proof of Proposition 2.4 is complete.

Remark 2.5.

Lemma 2.6.

and is defined in (2.10).

Proof.

First suppose that is a solution of problem (2.7).

and the proof of sufficient is complete.

Conversely, if is a solution of (2.18).

The Lemma is proved.

Remark 2.7.

From (2.19), we can prove that the properties of are similar to that of .

It is easy to see that is a closed convex cone of .

From (2.35), we know that is a solution of problem (1.6) if and only if is a fixed point of operator .

Definition 2.8 (see [1]).

We present the following result about relatively compact sets in which is a consequence of the Arzela-Ascoli Theorem. The reader can find its proof partially in [1].

Lemma 2.9.

is relatively compact if and only if is bounded and quasi-equicontinuous on .

where .

Lemma 2.10.

Suppose that and hold. Then and is completely continuous.

Proof.

From (2.35) and Remark 2.5, we obtain the following cases.

Case 1.

Case 2.

Therefore, , that is, . Also, we have since . Hence we have .

Next, we prove that is completely continuous.

It is obvious that is continuous. Now we prove is relatively compact.

Therefore is uniformly bounded.

and then is quasi-equicontinuous. It follows that is relatively compact on by Lemma 2.9. So is completely continuous.

## 3. Main Results

In this section, we apply Lemma 1.1 to establish the existence of positive solutions of problem (1.6). We begin by introducing the following conditions on and .

There exist numbers such that

where denotes or

Theorem 3.1.

Proof.

Applying (b) of Lemma 1.1 to (3.7) and (3.12) yields that has a fixed point with . Hence, since for we have , it follows that (3.4) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.1, we can prove the following results.

Corollary 3.2.

Assume that and hold. If , and , then, for being sufficiently small and being sufficiently large, BVP (1.6) has at least one positive solution with property (3.4).

Proof.

The proof is similar to that of Theorem of [6].

In Theorem 3.3, we assume the following condition on and .

There exist numbers such that

Theorem 3.3.

Proof.

Applying (a) of Lemma 1.1 to (3.19) and (3.24) yields that has a fixed point with . Hence, since for we have , it follows that (3.17) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.3, we can prove the following results.

Corollary 3.4.

Assume that and hold. If and ; then, for being sufficiently small and being sufficiently large, BVP (1.6) has at least one positive solution with property (3.17).

Proof.

The proof is similar to that of Theorem of [6].

Theorem 3.5.

Assume that (3.1) of and (3.14) and (3.15) of hold. In addition, letting and satisfy the following condition:

Proof.

which implies that (3.30) holds.

Applying Lemma 1.1 to (3.28), (3.29), and (3.30) yields that has two fixed point with , and . Hence, since for we have , it follows that (3.26) holds. This and Lemma 2.9 complete the proof.

Remark 3.6.

Similar to the proof of that of [5], we can prove that problem (1.6) can be generalized to obtain many positive solutions.

## 4. Discussion

In this section, we offer some interesting discussions associated with problem (1.6).

Discussion.

Generally, it is difficult to obtain the upper and lower bounds of positive solutions for nonlinear higher-order boundary value problems (see, e.g., [33, 34, 39–43, 46, 48, 49] and their references).

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

Lemma 4.1.

is defined in (2.10).

Lemma 4.2.

It is not difficult to prove that and have the similar properties to that of and . But for , and have no property (2.13). In fact, if , then we can prove that and have the following properties.

Proposition 4.3.

Proof.

Similarly, we can prove that (4.10) holds, too.

Remark 4.4.

Since , .

which implies that we cannot obtain the lower bounds for the positive solutions of problem (4.1).

## 5. Example

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

Example 5.1.

Conclusion.

Proof.

By Theorem 3.1, (5.1) has a positive solution with .

## Declarations

### Acknowledgments

The authors thank the referee for his/her careful reading of the manuscript and useful suggestions. This work is sponsored by 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).

## Authors’ Affiliations

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