- Open Access
Positive solutions for a second-order p-Laplacian impulsive boundary value problem
Advances in Difference Equations volume 2012, Article number: 159 (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.
In , Lian and Ge investigated the Sturm-Liouville-like boundary value problem
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 .
In , Xu et al. studied the existence of multiple positive solutions for the following boundary value problem with the p-Laplacian operator and impulsive effects
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.
Motivated by the above works, in this paper, we investigate the existence and multiplicity of positive solutions for the second-order p-Laplacian boundary value problems involving impulsive effects
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 )
Let f and be as in (1.3). Then the problem (1.3) is equivalent to
It is clear that , and , which implies that is increasing on . Furthermore, for given with , we have . Hence, is nonincreasing on , and thus
i.e., . Therefore,
We denote P by
Then P is a cone on .
Define an operator
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).
In what follows, we consider the following eigenvalue problem:
where λ is a parameter. We easily know that (2.4) has a nontrivial solution if . Furthermore, we have
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:
Lemma 2.2 If , then
where , .
Lemma 2.3 (see )
Let E be a real Banach space and W a cone of E. Suppose is a completely continuous operator with . If either
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 )
Let , , (), and . Then
3 Main results
Let , , , , , , , . We now list our hypotheses.
(H1) There exist and , satisfying
(H2) There exist and , satisfying
(H3) There exist and , satisfying
(H4) There exist and , satisfying
(H5) There exists such that , and implies
where and .
(H6) There exists such that , and implies
where , .
Theorem 3.1 Suppose that (H 1)-(H 2) are satisfied. Then (1.3) has at least one positive solution.
Proof If (H1) is satisfied, then we obtain for all . Indeed, if the claim is false, there is a such that , i.e.,
Now apply Lemma 2.4 to obtain
Multiply both sides of (3.5) by and then integrate over and use (2.5) to obtain
The above and (H1) imply that
By (3.7), we have . If , then , , and in view of the concavity and the nondecreasing nature of u, we find , , contradicting . So, .
Since , . Therefore,
Combining (3.7) and (2.2), we obtain
Therefore, , which contradicts (H1). Thus we have
On the other hand, by (H2), we shall prove that there exists a sufficiently large number such that , . Suppose there exists such that . This, together with Lemma 2.4, yields
Multiply both sides of the above by and integrate over and use (2.5) to obtain
Combining this and (H2), we get
where . Consequently, by (2.2) we have
Choosing and , we have
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.
Proof If (H3) is satisfied, we will prove that there exists a sufficiently large number such that , . Suppose there exists such that , and then
In view of , from (3.3), we know . Accordingly, . Similarly, . These and (3.6) imply that
Now, we consider two cases.
Case 1. If , by (2.2) we obtain
Case 2. If , by (2.2) we have
In view of , we obtain
Choosing (r is determined by (H4)), we get
On the other hand, if (H4) is satisfied, then , . If not, there exists such that . It follows from (3.10) and (H4) that
which contradicts (H4). Thus
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.
Proof If , it follows from (H5) that
from which we obtain
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.
Proof For any , for all . It follows from (H6) that
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. □
Example 4.1 Consider the impulsive boundary value problem
Case 1. Let , , , . Then
From (4.2) we see that (H1) is satisfied. In fact, since and , , . Taking , , we get
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.
Case 3. Let , , . Thus, we get
From (4.4) we see that (H1) is satisfied. Note
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.
Glowinski R, Rappaz J: Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology. Math. Model. Numer. Anal. 2003, 37: 175–186. 10.1051/m2an:2003012
Diaz J, de Thélin F: On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 1994, 25: 1085–1111. 10.1137/S0036141091217731
Lian H, Ge WG: Positive solutions for a four-point boundary value problem with the p -Laplacian. Nonlinear Anal. 2008, 68: 3493–3503. 10.1016/j.na.2007.03.042
Zhang X, Ge W: Impulsive boundary value problems involving the one-dimensional p -Laplacian. Nonlinear Anal. 2009, 70: 1692–1701. 10.1016/j.na.2008.02.052
Cabada A, Tomeček J: Extremal solutions for nonlinear functional p -Laplacian impulsive equations. Nonlinear Anal. 2007, 67: 827–841. 10.1016/j.na.2006.06.043
Xu J, Kang P, Wei Z: Singular multipoint impulsive boundary value problem with p -Laplacian operator. J. Appl. Math. Comput. 2009, 30: 105–120. 10.1007/s12190-008-0160-2
Feng M: Multiple positive solutions of four-order impulsive differential equations with integral boundary conditions and one-dimensional p -Laplacian. Bound. Value Probl. 2011. doi:10.1155/2011/654871
Li P, Chen H, Wu Y: Multiple positive solutions of n -point boundary value problems for p -Laplacian impulsive dynamic equations on time scales. Comput. Math. Appl. 2010, 60: 2572–2582. 10.1016/j.camwa.2010.08.067
Zhang X: Existence and successive iteration of positive solutions for some impulsive multi-point boundary value problem with p -Laplacian on infinite interval. J. Syst. Sci. Math. Sci. 2010, 30: 619–632.
Yang Z: Positive solutions for a system of p -Laplacian boundary value problems. Comput. Math. Appl. 2011, 62: 4429–4438. 10.1016/j.camwa.2011.10.019
Bai L, Dai B: Three solutions for a p -Laplacian boundary value problem with impulsive effects. Appl. Math. Comput. 2011, 217: 9895–9904. 10.1016/j.amc.2011.03.097
Yang Z, O’Regan D: Positive solutions for one-dimensional p -Laplacian equations. Math. Comput. Model. 2012, 55: 1942–1950. 10.1016/j.mcm.2011.11.052
Xu J, Yang Z: Positive solutions for a fourth order p -Laplacian boundary value problem. Nonlinear Anal. 2011, 74: 2612–2623. 10.1016/j.na.2010.12.016
Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, Orlando; 1988.
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).
The authors declare that they have no competing interests.
The first author conceived of the study and carried out the proof. Authors read and approved the final manuscript.
About this article
Cite this article
Ding, Y., O’Regan, D. Positive solutions for a second-order p-Laplacian impulsive boundary value problem. Adv Differ Equ 2012, 159 (2012). https://doi.org/10.1186/1687-1847-2012-159
- impulsive boundary value problem
- fixed point theorem
- positive solution
- Jensen inequality