- Open Access
Positive solutions for a second-order p-Laplacian impulsive boundary value problem
© Ding and O’Regan; licensee Springer 2012
- Received: 18 April 2012
- Accepted: 28 August 2012
- Published: 11 September 2012
In this work, we study the existence and multiplicity of positive solutions for a second-order p-Laplacian boundary value problem involving impulsive effects. We establish our main results via Jensen’s inequality, the first eigenvalue of a relevant linear operator and the Krasnoselskii-Zabreiko fixed point theorem. Some examples are presented to illustrate the main results.
MSC:34B15, 34B18, 34B37, 45G15, 45M20.
Second-order differential equations with the p-Laplacian operator arise in modeling some physical and natural phenomena and can occur, for example, in non-Newtonian mechanics, nonlinear elasticity, glaciology, population biology, combustion theory, and nonlinear flow laws, see [1, 2]. Recently, many cases of the existence and multiplicity of positive solutions for boundary value problems of differential equations with the p-Laplacian operator have appeared in the literature. For details, see, for example, [3–13] and the references therein.
and by virtue of Krasonsel’skii’s fixed point theorem, they obtained the existence of positive solutions and multiple positive solutions under suitable conditions imposed on the nonlinear term .
where the nonlinear term may be singular on . The main tools are fixed point index theorems for compact maps in Banach spaces. They stated the proofs by considering an approximating completely continuous operator.
In , Feng studied an integral boundary value problem of fourth-order p-Laplacian differential equations involving the impulsive effect , . Using a suitably constructed cone and fixed point theory for cones, the existence of multiple positive solutions was established. Furthermore, upper and lower bounds for these positive solutions were given.
where , , (, where m is a fixed positive integer) are fixed points with ; is the p-Laplacian operator, i.e., , , , ; denotes the jump of at , i.e., , where and represent the right-hand limit and left-hand limit of at , respectively. In addition, we suppose that , .
The main features of this paper are as follows. Firstly, we convert the boundary value problem (1.3) into an equivalent integral equation so that we can construct a special cone. 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 Robin boundary value problem and obtain its first eigenvalue and eigenfunction, which are used in the proofs of main theorems by Jensen’s inequalities. Finally, employing the Krasnoselskii-Zabreiko fixed point theorem, we establish the existence and multiplicity of positive solutions of (1.3). Although our problem (1.3) merely involves Robin boundary conditions, our methods are different from those in [3, 6, 7], and our main results are optimal.
This paper is organized as follows. Section 2 contains some preliminary results. Section 3 is devoted to the existence and multiplicity of positive solutions for (1.3). Section 4 contains some illustrative examples.
Let . Then is a Banach space with norm . We denote for in the sequel.
A function is called a solution of (1.3) if it satisfies the boundary value problem (1.3).
Lemma 2.1 (see )
Then P is a cone on .
Clearly, the operator A is a completely continuous operator, and the existence of positive solutions for (1.3) is equivalent to that of positive fixed points of A. Moreover, it is easy to see by (2.2).
where , are constants and . implies that , and thus , . and are called the first eigenvalue and the corresponding eigenfunction associated with , respectively. Consequently, it is easy to have the following result:
where , .
Lemma 2.3 (see )
for each and for each or
for each and for each ,
then A has at least one fixed point in .
Lemma 2.4 (Jensen’s inequalities, see )
3 Main results
Let , , , , , , , . We now list our hypotheses.
where and .
where , .
Theorem 3.1 Suppose that (H 1)-(H 2) are satisfied. Then (1.3) has at least one positive solution.
By (3.7), we have . If , then , , and in view of the concavity and the nondecreasing nature of u, we find , , contradicting . So, .
Therefore, (3.8) and (3.12), together with Lemma 2.3, guarantee that (1.3) has at least one positive solution in . □
Theorem 3.2 Suppose that (H 3)-(H 4) are satisfied. Then (1.3) has at least one positive solution.
Now, we consider two cases.
By Lemma 2.3, (3.16) and (3.18) imply that (1.3) has at least one positive solution in . □
Theorem 3.3 Suppose that (H 1), (H 3) and (H 5) are satisfied. Then (1.3) has at least two positive solutions.
On the other hand, by (H1) and (H3), we may take and such that , and , (see the proofs of Theorems 3.1 and 3.2). Now Lemma 2.3 guarantees that the operator A has at least two fixed points, one in and the other in . The proof is completed. □
Theorem 3.4 Suppose that (H 2), (H 4) and (H 6) are satisfied. Then (1.3) has at least two positive solutions.
On the other hand, by (H2) and (H4), we may take and such that , and , (see the proofs of Theorems 3.1 and 3.2). Thus Lemma 2.3 indicates that the operator A has at least two fixed points, one in and the other in . The proof is completed. □
From (4.3) we see that (H2) is satisfied, for example taking , . All the assumptions in Theorem 3.1 are satisfied, and the problem (4.1) has at least one positive solution by Theorem 3.1.
Case 2. Take , , , . One can easily verify conditions (H3) and (H4). Thus the problem (4.1) has at least one positive solution by Theorem 3.2.
From (4.5) we see that (H3) is satisfied. Take , , in (H5), and note for and that , . As a result, (H5) holds. From Theorem 3.3, the problem (4.1) has at least two positive solutions.
The authors are grateful to the anonymous referee for his/her valuable suggestions. The first author thanks Prof. Zhongli Wei for his valuable help. This work is supported financially by the Shandong Provincial Natural Science Foundation (ZR2012AQ007) and Graduate Independent Innovation Foundation of Shandong University (yzc12063).
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