Research

# Positive solutions for a fourth-order p-Laplacian boundary value problem with impulsive effects

Keyu Zhang12*, Jiafa Xu1 and Wei Dong3

Author Affiliations

1 School of Mathematics, Shandong University, Jinan, Shandong, 250100, China

2 Department of Mathematics, Qilu Normal University, Jinan, Shandong, 250013, China

3 Department of Mathematics, Hebei University of Engineering, Handan, Hebei, 056038, China

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Boundary Value Problems 2013, 2013:120  doi:10.1186/1687-2770-2013-120

 Received: 31 January 2013 Accepted: 15 April 2013 Published: 10 May 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper is devoted to study the existence and multiplicity of positive solutions for the fourth-order p-Laplacian boundary value problem involving impulsive effects

where , , (). Based on a priori estimates achieved by utilizing the properties of concave functions and Jensen’s inequality, we adopt fixed point index theory to establish our main results.

MSC: 34B18, 47H07, 47H11, 45M20, 26D15.

##### Keywords:
p-Laplacian boundary value problem with impulsive effects; positive solution; fixed point index; concave function; Jensen inequality

### 1 Introduction

In this paper, we mainly investigate the existence and multiplicity of positive solutions for the fourth-order p-Laplacian boundary value problem with impulsive effects

(1.1)

Here , , . Let be fixed, , where and denote the right and left limit of at , respectively.

Fourth-order boundary value problems, including those with the p-Laplacian operator, have their origin in beam theory [1,2], ice formation [3,4], fluids on lungs [5], brain warping [6,7], designing special curves on surfaces [6,8], etc. In beam theory, more specifically, a beam with a small deformation, a beam of a material which satisfies a nonlinear power-like stress and strain law, and a beam with two-sided links which satisfies a nonlinear power-like elasticity law can be described by fourth-order differential equations along with their boundary value conditions. For the case of , , and , problem (1.1) reduces to the differential equation subject to boundary value conditions , which can be used to model the deflection of elastic beams simply supported at the endpoints [9-11]. This explains the reason that the last two decades have witnessed an overgrowing interest in the research of such problems, with many papers in this direction published. We refer the interested reader to [12-26] and references therein devoted to the existence of solutions for the equations with p-Laplacian operator.

In [17], Zhang et al. studied the existence and nonexistence of symmetric positive solutions of the following fourth-order boundary value problem with integral boundary conditions:

(1.2)

where is nonnegative, symmetric on the interval (i.e., for ), , for all , and are nonnegative, symmetric on . The arguments are based upon a specially constructed cone and the fixed point theory for cones. Moreover, they also studied the nonexistence of a positive solution.

In [16], Luo and Luo considered the existence, multiplicity, and nonexistence of symmetric positive solutions for (1.2) with a ϕ-Laplacian operator and the term f involving the first derivative.

Except that, many researchers considered and studied the existence of positive solutions for a lot of impulsive boundary value problems; see, for example, [21-29] and the references therein.

In [21], Feng considered the problem (1.2) with impulsive effects and he obtained the existence and multiplicity of positive solutions. The fundamental tool in this paper is Guo-Krasnosel’skii fixed point theorem on a cone. Moreover, the nonlinearity f can be allowed to grow both sublinear and superlinear. Therefore, he improved and generalized the results of [17] to some degree. However, we can easily find that these papers do only simple promotion based on their original papers, and no substantial changes.

Motivated by the works mentioned above, in this paper, we study the existence and multiplicity of positive solutions for (1.1). Nevertheless, our methodology and results in this paper are different from those in the papers cited above. The main features of this paper are as follows. Firstly, we convert the boundary value problem (1.1) into an equivalent integral equation. Next, we consider impulsive effect as a perturbation to the corresponding problem without the impulsive terms, so that we can construct an integral operator for an appropriate linear Dirichlet boundary value problem and obtain its first eigenvalue and eigenfunction. Our main results are formulated in terms of spectral radii of the linear integral operator, and our a priori estimates for positive solutions are derived by developing some properties of positive concave functions and using Jensen’s inequality. It is of interest to note that our nonlinearity f may grow superlinearly and sublinearly. The main tool used in the proofs is fixed point index theory, combined with the a priori estimates of positive solutions. Although our problem (1.1) merely involves Dirichlet boundary conditions, both our methodology and the results in this work improve and extend the corresponding ones from [21-29].

### 2 Preliminaries

Let , . Then is a real Banach space. Let and introduce the following space:

with the norm . Then is also a Banach space.

A function is called a solution of (1.1) if it satisfies the differential equation

and the function y satisfies the conditions , and the Dirichlet boundary conditions .

Lemma 2.1 (see [21])

Ifyis a solution of the integral equation

(2.1)

thenyis a solution of (1.1), where, . Note that if, , thenis a completely continuous operator, and the existence of positive solutions for (1.1) is equivalent to that of positive fixed points ofA.

Remark 2.1 By (2.1), we easily find y is concave on . Indeed,

implies y is concave on . Furthermore, () leads to , .

Let P be a cone in which is defined as

In what follows, we prove that .

Lemma 2.2.

Proof We easily see that , . Consequently, on the one hand, we find

On the other hand,

Therefore, , for any , as required. This completes the proof. □

We denote for in the sequel.

Lemma 2.3 (see [30])

Supposeis a completely continuous operator and has no fixed points on.

1. Iffor all, then, whereiis fixed point index onP.

2. Iffor all, then.

Lemma 2.4 (see [30])

Ifis a completely continuous operator. If there existssuch that, , , then.

Lemma 2.5 (see [30])

Ifandis a completely continuous operator. If, , , then.

Lemma 2.6Let. Then

(2.2)

Lemma 2.7 (Jensen’s inequalities)

Let, , (), and. Then

### 3 Main results

Let , , , , , , , . We now list our hypotheses.

(H1) There is a such that and imply

where satisfy

(H2) There exist and , satisfying

such that

(3.1)

where .

(H3) There exist and , satisfying

such that

(3.2)

(H4) There is a such that and imply

where satisfy

(H5) There exist and , satisfying

such that

(3.3)

(H6) There exist and , satisfying

such that

(3.4)

Theorem 3.1Suppose that (H1)-(H3) are satisfied. Then (1.1) has at least two positive solutions.

Proof If , it follows from (H1) that

Now Lemma 2.3 yields

(3.5)

Let . Then for , we find

(3.6)

where . Let , where . Next, from (H2), we prove . Indeed, implies . Lemma 2.6, together with this, leads to

(3.7)

Multiply both sides of the above by and integrate over and use (2.2) to obtain

(3.8)

Combining this and (3.1), we get

(3.9)

In what follows, we will distinguish three cases.

Case 1. . By (H2), we know . (3.9) implies

Therefore, (), and then , by Remark 2.1, which contradicts .

Case 2. . Equation (3.9) implies

and thus , , which also contradicts .

Case 3. . Since , we have by (3.6) and (3.9),

Therefore,

which contradicts (H2). So, we have for all and . Now, by virtue of Lemma 2.4, we obtain

(3.10)

On the other hand, by (H3), we prove is bounded in P. By (3.2) together with (3.8), we obtain

(3.11)

where . Now we distinguish the following two cases.

Case 1. . (H3) implies

Combining this and (3.11), we have

Therefore,

Case 2. . (3.11) implies

and thus

Therefore, we obtain the boundedness of , as claimed. Taking , we have for all and . Now, by virtue of Lemma 2.4, we obtain

(3.12)

Combining (3.5), (3.10), and (3.12), we arrive at

Now A has at least two fixed points, one on and the other on . Hence (1.1) has at least two positive solutions. The proof is completed. □

Theorem 3.2Suppose that (H4)-(H6) are satisfied. Then (1.1) has at least two positive solutions.

Proof If , then we find

(3.13)

By (H4),

so that

Now Lemma 2.3 yields

(3.14)

Let . Then for , we find

(3.15)

Let . Next, from (H5), we prove . Indeed, if , we have

Multiply both sides of the above by and integrate over and use (2.2) to obtain

(3.16)

Combining this and (3.3), we have

Consequently,

which contradicts (H5). This implies , and thus for all and . Now Lemma 2.5 yields

(3.17)

On the other hand, by (H6), we prove is bounded in P. By (3.4) together with (3.16), we obtain

where . Therefore,

namely,

This proves the boundedness of , as required. Choosing and , we have for all and . Now Lemma 2.5 yields

(3.18)

Combining (3.14), (3.17), and (3.18), we obtain

Hence A has at least two fixed points, one on and the other on , and thus (1.1) has at least two positive solutions. The proof is completed. □

### 4 An example

Let us consider the problem

(4.1)

Taking in (H1), , and is chosen such that . Set , , . Therefore, , , and

As a result, (H1) holds. On the other hand, by simple computation, we have

Therefore,

(i) There exist and , such that (H2) holds.

(ii) There exist and , such that (H3) holds.

Consequently, the problem (4.1) has at least two positive solutions by Theorem 3.1.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

KZ and JX obtained the results in a joint research. All the authors read and approved the final manuscript.

### Acknowledgements

Research supported by the NNSF-China (10971046), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007, A2012402036), GIIFSDU (yzc12063), IIFSDU (2012TS020) and the Project of Shandong Province Higher Educational Science and Technology Program (J09LA55).

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