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# Positive solutions for classes of multi-parameter fourth-order impulsive differential equations with one-dimensional singular p-Laplacian

Xuemei Zhang1* and Meiqiang Feng2

Author Affiliations

1 Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, Republic of China

2 School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, Republic of China

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Boundary Value Problems 2014, 2014:112  doi:10.1186/1687-2770-2014-112

 Received: 29 January 2014 Accepted: 29 April 2014 Published: 13 May 2014

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 credited.

### Abstract

The authors consider the following impulsive differential equations involving the one-dimensional singular p-Laplacian: , , , , , , , , , where and are two parameters. Several new and more general existence and multiplicity results are derived in terms of different values of and . In this case, our results cover equations without impulsive effects and are compared with some recent results.

##### Keywords:
multi-parameter; impulsive differential equations; one-dimensional singular p-Laplacian; positive solution; cone and partial ordering

### 1 Introduction

The theory and applications of the fourth-order ordinary differential equation are emerging as an important area of investigation; it is often referred to as the beam equation. In [1], Sun and Wang pointed out that it is necessary and important to consider various fourth-order boundary value problems (BVPs for short) according to different forms of supporting. Owing to its importance in engineering, physics, and material mechanics, fourth-order BVPs have attracted much attention from many authors; see, for example [2-29] and the references therein.

Very recently, Zhang and Liu [30] studied the following fourth-order four-point boundary value problem without impulsive effect:

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

In this paper, we investigate the existence of positive solutions of fourth-order impulsive differential equations with two parameters

(1.1)

where and are two parameters, , , is a p-Laplace operator, i.e., , , , , ω is a nonnegative measurable function on , on any open subinterval in which may be singular at and/or , () (where m is fixed positive integer) are fixed points with , , where and represent the right-hand limit and left-hand limit of at , respectively. In addition, ω, f, , g, and h satisfy

(H1) ;

(H2) with for all t and ;

(H3) with () for all t and ;

(H4) are nonnegative and , , where

(1.2)

Some special cases of (1.1) have been investigated. For example, Bai and Wang [14] studied the existence of multiple solutions of problem (1.1) with , , and for . By using a fixed point theorem and degree theory, the authors proved the existence of one or two positive solutions of problem (1.1).

Feng [31] considered problem (1.1) with , , for and . By using a suitably constructed cone and fixed point theory for cones, the author proved the existence results of multiple positive solutions of problem (1.1).

Motivated by the papers mentioned above, we will extend the results of [14,30,31] to problem (1.1). We remark that on impulsive differential equations with a parameter only a few results have been obtained, not to mention impulsive differential equations with two parameters; see, for instance, [32-34]. However, these results only dealt with the case that and .

The rest of the paper is organized as follows: in Section 2, we state the main results of problem (1.1). In Section 3, we provide some preliminary results, and the proofs of the main results together with several technical lemmas are given in Section 4.

### 2 Main results

In this section, we state the main results, including existence and multiplicity of positive solutions for problem (1.1).

We begin by introducing the notation

We also choose four numbers r, , , and R satisfying

(2.1)

where δ is defined in (3.20).

Theorem 2.1Assume that (H1)-(H4) hold.

(i) Ifand, then there existandsuch that, for anyand, problem (1.1) has a positive solution, with

(2.2)

(ii) Ifand, then there existandsuch that, for anyand, problem (1.1) has a positive solutionwith

(2.3)

(iii) If, then there existandsuch that, for anyand, problem (1.1) has at least two positive solutionsandwith

(2.4)

Theorem 2.2Assume that (H1)-(H4) hold.

(i) Ifand, then there existandsuch that, for anyand, problem (1.1) has a positive solution, with property (2.2).

(ii) Ifand, then there existandsuch that, for anyand, problem (1.1) has a positive solution, with property (2.3).

(iii) If, then there existandsuch that, for anyand, problem (1.1) has at least two positive solutionsandwith

(2.5)

### 3 Preliminaries

Let , and

Then is a real Banach space with norm

(3.1)

where , .

A function with is called a solution of problem (1.1) if it satisfies (1.1).

We shall reduce problem (1.1) to an integral equation. To this goal, firstly by means of the transformation

(3.2)

we convert problem (1.1) into

(3.3)

and

(3.4)

Lemma 3.1If (H1), (H2), and (H4) hold, then problem (3.3) has a unique solutionxgiven by

(3.5)

where

(3.6)

(3.7)

Proof The proof of Lemma 3.1 is similar to that of Lemma 2.1 in [31]. □

Write . Then from (3.6) and (3.7), we can prove that and have the following properties.

Proposition 3.1If (H4) holds, then we have

(3.8)

(3.9)

(3.10)

(3.11)

where

(3.12)

Remark 3.1 From (3.6) and (3.11), we obtain

Lemma 3.2If (H1), (H3), and (H4) hold, then problem (3.4) has a unique solutionyandycan be expressed in the form

(3.13)

where

(3.14)

(3.15)

Proof The proof of Lemma 3.2 is similar to that of Lemma 2.2 in [31]. □

From (3.14) and (3.15), we can prove that and have the following properties.

Proposition 3.2If (H4) holds, then we have

(3.16)

(3.17)

(3.18)

where

Suppose that y is a solution of problem (1.1). Then from Lemma 3.1 and Lemma 3.2, we have

Define a cone in by

(3.19)

where

(3.20)

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

Define an operator by

(3.21)

From (3.21), we know that is a solution of problem (1.1) if and only if y is a fixed point of operator .

Lemma 3.3Suppose that (H1)-(H4) hold. Thenandis completely continuous.

Proof The proof of Lemma 3.3 is similar to that of Lemma 2.4 in [31]. □

To obtain positive solutions of problem (1.1), the following fixed point theorem in cones is fundamental, which can be found in [[35], p.94].

Lemma 3.4LetPbe a cone in a real Banach spaceE. Assume, are bounded open sets inEwith, . If

is completely continuous such that either

(a) , and, , or

(b) , and, ,

thenAhas at least one fixed point in.

Remark 3.2 To make the reader clear what , , , and mean, we give typical examples of and , e.g.,

with , where .

### 4 Proofs of the main results

For convenience we introduce the following notation:

and

where is a constant.

Proof of Theorem 2.1 Part (i). Noticing that , () for all t and , we can define

where , and

Let

Then, for and , , we have

which implies that

(4.1)

If , , then there exist , , and such that

where satisfies

(4.2)

satisfies

(4.3)

Let . Thus, when we have

and then we get

(4.4)

(4.5)

where

and

It follows from (4.4) and (4.5) that

(4.6)

Applying (b) of Lemma 3.4 to (4.1) and (4.6) shows that has a fixed point with . Hence, since for we have , , it follows that (2.2) holds. This gives the proof of part (i).

Part (ii). Noticing that , () for all t and , we can define

where , and

Let

Then, for and , , we have

which implies that

(4.7)

If , , then there exist , , and such that

where and satisfy (4.2) and (4.3), respectively.

Similar to the proof of (4.6), we can prove that

(4.8)

Applying (a) of Lemma 3.4 to (4.7) and (4.8) shows that has a fixed point with . Hence, since for we have for , it follows that (2.3) holds. This gives the proof of part (ii).

Consider part (iii). Choose two numbers and satisfying (2.1). By part (i) and part (ii), there exist and such that

(4.9)

Since , from the proof of part (i) and part (ii), it follows that

(4.10)

and

(4.11)

Applying Lemma 3.4 to (4.9)-(4.11) shows that has two fixed points and such that and . These are the desired distinct positive solutions of problem (1.1) for and satisfying (2.4). Then the result of part (iii) follows. □

Proof of Theorem 2.2 Part (i). Noticing that , () for all t and , we can define

where , and

Let

Then, for and , , we have

(4.12)

Similar to the proof of (4.5), we can prove

(4.13)

It follows from (4.12) and (4.13) that

(4.14)

If , , then there exist , , and such that

where satisfies

(4.15)

satisfies

(4.16)

Let . Thus, when we have

and then we get

This yields

(4.17)

Applying (b) of Lemma 3.4 to (4.14) and (4.17) shows that has a fixed point with . Hence, since for we have , , it follows that (2.2) holds. This gives the proof of part (i).

Part (ii). Noticing that , () for all t and , we can define

where , and

Let

Then, for and , , we have

(4.18)

Similar to the proof of (4.5), we can prove

(4.19)

It follows from (4.18) and (4.19) that

(4.20)

If , , then there exist , , and such that

where and satisfy (4.15) and (4.16), respectively.

Therefore, for , we obtain

This yields

(4.21)

Applying (a) of Lemma 3.4 to (4.20) and (4.21) shows that has a fixed point with . Hence, since for we have , , it follows that (2.3) holds. This gives the proof of part (ii).

Consider part (iii). Choose two numbers and satisfying (2.1). By part (i) and part (ii), there exist and such that

(4.22)

Since , from the proof of part (i) and part (ii), it follows that

(4.23)

and

(4.24)

Applying Lemma 3.4 to (4.22)-(4.24) shows that has two fixed points and such that and . These are the desired distinct positive solutions of problem (1.1) for and satisfying (2.5). Then the proof of part (iii) is complete. □

Remark 4.1 Comparing with Feng [31], the main features of this paper are as follows.

(i) Two parameters and are considered.

(ii) , not only for .

(iii) It follows from the proof of Theorem 2.1 that the conditions of Corollary 3.2 in [31] are not the optimal conditions, which guarantee the existence of at least one positive solution for problem (1.1). In fact, if , or , , we can prove that problem (1.1) has at least one positive solution, respectively.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

XZ completed the main study and carried out the results of this article. MF checked the proofs and verified the calculation. All the authors read and approved the final manuscript.

### Acknowledgements

This work is sponsored by the project NSFC (11301178, 11171032) and the Fundamental Research Funds for the Central Universities (2014MS58). The authors are grateful to anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.

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