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</svg>-Laplacian Boundary Value Problem with Impulsive Effects and Two Parameters

Feng, Meiqiang

doi:https://doi.org/10.1155/2014/534787

Abstract and Applied Analysis

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Abstract Introduction Preliminaries Conclusion Acknowledgments References Copyright Related Articles

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Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential Equations 2014

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Research Article | Open Access

Volume 2014 | Article ID 534787 | https://doi.org/10.1155/2014/534787

Positive Solutions for a Second-Order -Laplacian Boundary Value Problem with Impulsive Effects and Two Parameters

Meiqiang Feng¹

Academic Editor: Patricia J. Y. Wong

Received01 Apr 2014

Accepted19 May 2014

Published18 Jun 2014

Abstract

The author considers an impulsive boundary value problem involving the one-dimensional p-Laplacian , where and are two parameters. Using fixed point theories, several new and more general existence and multiplicity results are derived in terms of different values of and . The exact upper and lower bounds for these positive solutions are also given. Moreover, the approach to deal with the impulsive term is different from earlier approaches. In this paper, our results cover equations without impulsive effects and are compared with some recent results by Ding and Wang.

1. Introduction

Impulsive differential equations, which provide a natural description of observed evolution processes, are regarded as important mathematical tools for the better understanding of several real-world problems in applied sciences, such as population dynamics, ecology, biological systems, biotechnology, industrial robotics, pharmacokinetics, and optimal control. Therefore, the study of this class of impulsive differential equations has gained prominence, and it is a rapidly growing field. For the general theory of impulsive differential equations, we refer the reader to [1–3], whereas the applications of impulsive differential equations can be found in [4–20]. In particular, we would like to mention some results of Lin and Jiang [8] and Feng and Xie [10]. In [8], Lin and Jiang investigated the following Dirichlet boundary value problem with impulse effects: and, by virtue of the fixed point index theory in cones, the authors obtained some sufficient conditions for the existence of multiple positive solutions.

Recently, using fixed point theorems in a cone, Feng and Xie [10] considered the existence of positive solutions for the following problem:

Moreover, differential equations with -Laplacian arise naturally in the study of flow through porous media , nonlinear elasticity , glaciology , and so forth. In recent years, many cases of the existence, multiplicity, and uniqueness of positive solution of differential equations with -Laplacian have attracted considerable attention [21–46]. Here, it is worth mentioning the studies by Dai and Ma [25] and Kajikiya et al. [26]. In [25], Dai and Ma considered the following one-dimensional -Laplacian problem: By using the global bifurcation theory, the authors showed the existence of nodal solutions.

In [26], Kajikiya et al. investigated the following one-dimensional -Laplacian problem: and, by virtue of the global bifurcation theory, they obtained the existence, nonexistence, uniqueness, and multiplicity of positive solutions as well as sign-changing solutions under suitable conditions imposed on the nonlinear term .

At the same time, we notice that there has been a considerable attention on impulsive differential equations with one-dimensional -Laplacian. For example, in [31], Ding and O’Regan studied the second-order -Laplacian boundary value problems involving impulsive effects: and, via Jensens inequality, the first eigenvalue of a relevant linear operator, and the Krasnoselskii-Zabreiko fixed point theorem, they obtained the existence and multiplicity of positive solutions under suitable conditions imposed on the nonlinear term and the impulsive terms .

In [33], employing the classical fixed point index theorem for compact maps, Zhang and Ge obtained some sufficient conditions for the existence of multiple positive solutions of the following problem:

However, to the best of our knowledge, no paper has considered the second-order impulsive differential equations with one-dimensional -Laplacian and two parameters till now; for example, see [4–20, 31, 32, 43–45] and the references therein. In this paper, we will use fixed point theorem to investigate the existence and multiplicity of positive solutions for a second-order impulsive differential equation involving one-dimensional -Laplacian and two parameters.

Consider the following second-order impulsive differential equation with one-dimensional -Laplacian: where , are two parameters, , , , , , may be singular at and/or , , where is fixed positive integer) are fixed points with , and denotes the jump of at ; that is, where and represent the right-hand limit and left-hand limit of at , respectively. In addition, , , , and satisfy the following: is a nonnegative measurable function on and on any open subinterval in ; with for all and ; with for all and ; is nonnegative and , where

Some special cases of (7) have been investigated. For example, Ding and Wang [14] considered problem (7) with , , , and , . By using Krasnoselskii’s fixed point theorem, they proved the existence results of positive solutions of problem (7). However, the authors only obtained that problem (7) has at least one positive solution.

Motivated by the papers mentioned above, we will extend the results of [11, 14, 23, 31, 33, 47, 48] to problem (7). 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, [12, 18, 19, 45]. These results only dealt with the case that and . Many difficulties occur when we study problem (7); for example, it is difficult to construct the cone and the operator because its state is discontinuous. It is also difficult to deal with and because of with one-dimensional -Laplacian, and without one-dimensional -Laplacian in the same equation (21). In this paper, we try to solve this kind of problem. Moreover, we will use a different approach to deal with the impulsive term to obtain the existence and multiplicity of positive solutions for problem (7); for details, see the proof of Theorem 1.

The rest of the paper is organized as follows: in Section 2, we state the main results of problem (7). 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. The final section of the paper contains an example to illustrate the theoretical results.

2. Main Results

In this section, we state the main results, including existence and multiplicity results of positive solutions for problem (7).

For convenience, we introduce the following notations: Moreover, we choose four numbers , , , and satisfying where is defined in (23).

Theorem 1. Assume that ()–() hold and , , , and are positive constants. Then,(i)there exist and such that, for any and , problem (7) has a positive solution with (ii)there exist and such that, for any and , problem (7) has a positive solution with property (12).

Theorem 2. Assume that ()–() hold and , , , and are positive constants. Then,(i)there exist and such that, for any and , problem (7) has a positive solution with (ii)there exist and such that, for any and , problem (7) has a positive solution with property (13).

Remark 3. Some ideas of the proof of Theorems 1 and 2 are from Yan [47]. In [47], Yan studied a class of the periodic impulsive functional differential equations with two parameters and proved the following existence result by using a well-known fixed point index theorem due to Krasnoselskii.

Theorem 4 (see [47, Theorem 3.1]). Assume that ()–() hold and , , , and are positive constants. If then problem (7) has a positive -periodic solution.

It is not difficult to see that the conditions of Theorem 4 are not the optimal conditions which guarantee the existence of at least one positive -periodic solution for the related problem. In fact, if or we can prove that the problem studied in [47] has at least one positive -periodic solution, respectively.

Theorem 5. Assume that ()–() hold.(i)If and , , then there exist and such that, for any and , problem (7) has a positive solution with property (12).(ii)If and , , then there exist and such that, for any and , problem (7) has a positive solution with property (13).(iii)If , , then there exist and such that, for any and , problem (7) has at least two positive solutions and with

Theorem 6. Assume that ()–() hold. (i)If and , , then there exist and such that, for any and , problem (7) has a positive solution with property (12).(ii) If and , , then there exist and such that, for any and , problem (7) has a positive solution with property (13).(iii)If , , then there exist and such that, for any and , problem (7) has at least two positive solutions and with

Remark 7. Some ideas of the proof of Theorems 5 and 6 are from Graef et al. [48].

3. Preliminaries

Let and let be the Banach space: with . We denote for all in the sequel.

In our main results, we will make use of the following definitions and lemmas.

Definition 8 (see [49]). Let be a real Banach space over . A nonempty closed set is said to be a cone provided that(i) for all and all , ;(ii) implies .
Every cone induces an ordering in given by if and only if .

Definition 9. A function with is called a solution of (7) if it satisfies (7). If and on , then is called a positive solution of (7).

Lemma 10. Assume that ()–() hold. Then, with is a solution of problem (7) if and only if is a solution of the following impulsive integral equation: Moreover, if is a positive solution of problem (7), then where

Proof. First, suppose that is a solution of problem (7). It is easy to see by integration of (7) that By the boundary condition , we have If , integrating (25) from to we obtain If , integrating (25) from to we obtain and integrating (25) from to we obtain It follows that For , repeating the process we have Combining this with the boundary condition, we have Then, the proof of sufficient is complete.
Conversely, if is a solution of (21), then we have the following.
Direct differentiation of (21) implies Evidently,
Finally, we show that (22) holds. It is clear that , which implies that
As we assume that and , we see that any nontrivial solution of problem (7) is concave on ; that is, , and then we get that is nonincreasing on .
So, for every , we have that is, . Therefore, Noticing that is a positive solution of problem (7) and , we have . Thus, we obtain The lemma is proved.

Define a cone in by where is defined in (23). It is easy to see that is a closed convex cone of .

Define by From (40) and Lemma 10, it is easy to obtain the following result.

Lemma 11. Assume that ()–() hold. Problem (7) is equivalent to the fixed point problem of in .

Lemma 12 (see [47, Lemmas and 2.2]). Assume that ()–() hold. Then, is completely continuous.

The following well-known result of the fixed point is crucial in our arguments.

Lemma 13 (see [49]). Let be a cone in a real Banach space . Assume that , are bounded open sets in with , . If is completely continuous such that either (a), , and , , or(b), , and , ,then has at least one fixed point in .

4. Proofs of the Main Results

For convenience, we introduce the following notation:

Proof of Theorem 1. Part (i). Noticing that , for all and , we can define where , , .
Let Then, for and , , we have which implies that
If , , then there exist , and and such that where satisfies satisfies
Let . Thus, when , we have and then we get This yields
Applying (b) of Lemma 13 to (46) and (52) yields that has a fixed point with . Hence, since for we have for , it follows that (12) holds. This gives the proof of Part (i).
Part (ii). Noticing that , for all and , we can define where , , .
Let Then, for and , , we have This implies that
If , , then there exist , and and such that where satisfies satisfies
Let . Then, for , we obtain and it follows from (40) that which implies that
Applying (a) of Lemma 13 to (56) and (62) yields that has a fixed point with . Hence, since for we have for , it follows that (13) holds. This finishes the proof of Part (ii).

Proof of Theorem 2. Part (i). Noticing that , for all and , we can define where , , .
Let Then, for and , , we have which implies that
If , , then there exist , and such that where and satisfy (48) and (49), respectively.
Therefore, for , we have This yields
Applying (a) of Lemma 13 to (66) and (69) yields that has a fixed point with . Hence, since for we have for , it follows that (13) holds. This gives the proof of Part (i).
Part (ii). Noticing that , for all and , we can define where , , .
Let Then, for and , , we have This implies that
If , , then there exist , and such that where and satisfy (58) and (59), respectively.
Therefore, for , we obtain which implies that
Applying (a) of Lemma 13 to (73) and (76) yields that has a fixed point with . Hence, since for we have for , it follows that (13) holds. This finishes the proof of Part (ii).

Proof of Theorem 5. Similar to the proof of Theorems 1(i) and 2(i), respectively, one can show that Theorems 5(i) and (ii) hold.
Considering Part (iii), choose two numbers and satisfying (11). By Theorems 1(i) and 2(i), there exist and such that
Since , from the proof of Theorem 1(i) and Theorem 2(i) and from (11), it follows that
Applying Lemma 12 to (77)–(79) yields that has two fixed points and such that and . These are the desired distinct positive solutions of problem (7) for and satisfying (17). Then, the result of Part (iii) follows.

Proof of Theorem 6. Similar to the proof of Theorems 1(ii) and 2(ii), respectively, one can show that Theorems 6(i) and (ii) hold.
Now, considering Part (iii), choose two numbers and satisfying (11). By Theorems 1(ii) and 2(ii), there exist and such that
Since , from the proof of Theorems 1(ii) and 2(ii) and from (11), it follows that
Applying Lemma 13 to (80)–(82) yields that has two fixed points and such that and . These are the desired distinct positive solutions of problem (7) for and satisfying (18). Then, proof of Part (iii) is complete.

5. An Example

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

Example 1. For , consider the following boundary value problem: Evidently, is the trivial solution of problem (83).

6. Conclusion

Problem (83) has at least one positive solution for any and .

Proof. Problem (83) can be regarded as a problem of the form (7), where , , and
It follows from the definition of , , and that (H₁)–(H₄) hold, and is singular at and . By calculating, we have where is a constant.
Hence, by Theorem 1(i), the conclusion follows, and the proof is complete.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is sponsored by the Project NSFC (11301178 and 11171032) and the Improving Project of Graduate Education of Beijing Information Science and Technology University (YJT201416).

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