Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 259730, 10 pages

http://dx.doi.org/10.1155/2013/259730

Research Article

Existence and Uniqueness of Positive Solutions to Nonlinear Second Order Impulsive Differential Equations with Concave or Convex Nonlinearities

Lingling Zhang¹ and Chengbo Zhai²

¹Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

²School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 29 January 2013; Accepted 15 May 2013

Academic Editor: Yanbin Sang

Copyright © 2013 Lingling Zhang and Chengbo Zhai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Using a new fixed point theorem of generalized concave operators, we present in this paper criteria which guarantee the existence and uniqueness of positive solutions to nonlinear two-point boundary value problems for second-order impulsive differential equations with concave or convex nonlinearities.

1. Introduction

In this paper, we study the existence and uniqueness of positive solutions to the following two-point boundary value problems for second-order impulsive differential equations:

where , , , , , , , , denote the right limit (left limit) of and at , respectively. , .

Impulsive differential equations have been studied extensively in recent years. Such equations arise in many applications such as spacecraft control, impact mechanics, chemical engineering, and inspection process in operations research. It is now recognized that the theory of impulsive differential equations is a natural framework for a mathematical modelling of many natural phenomena. There have appeared numerous papers on impulsive differential equations during the last ten years. Many of them are on boundary value problems, see [1–18], and it is interesting to note that some of them are about comparatively new applications like ecological competition, respiratory dynamics, and vaccination strategies, see [12, 19–25].

Second-order impulsive differential equations have been studied by many authors with much of the attention given to positive solutions. For a small sample of such work, we refer the reader to works by Feng and Xie [6], Hu et al. [8], Jankowski [10, 11], E. K. Lee and Y.-H. Lee [12], Lin and Jiang [13], Liu et al. [14], Agarwal and O’Regan [26], Wang et al. [27], Zhang [28], and Chu et al. [29]. The results of these papers are based on the Schauder fixed point theorem, Leggett-Williams theorem, fixed point index theorems in cones, Krasnoselski fixed point theorem, the method of upper-lower solutions, fixed point theorems in cones, and so on. But, in most of the existing works, in order to establish the existence of positive solutions, a key condition is the existence of upper-lower solutions. However, as we know, it is difficult to verify the existence of upper-lower solutions for concrete impulsive differential equations. In addition, few papers can be found in the literature on the existence and uniqueness of positive solutions for second-order impulsive differential equations. In this paper, we will study the problem (1) with concave or convex nonlinearities and not suppose the existence of upper-lower solutions and compactness condition. Different from the previously mentioned works, in this paper we will use a new fixed point theorem of generalized concave operators to show the existence and uniqueness of positive solutions for the problem (1).

For convenience, we list the following assumptions on the functions , and :

, , , and is decreasing in for each ,, , and is decreasing, and is increasing in , ,  for any , , and , there exist such that

 , , , and is increasing in for each and for ,  , for , and is increasing, and is decreasing in , ,  for any , , and , there exist such that

.

2. Preliminaries

In this section, we state some definitions, notations, and known results. For convenience of readers, we suggest that one refers to [30] and references therein for details.

Suppose that is a real Banach space which is partially ordered by a cone . That is, if and only if . By we denote the zero element of . Recall that a nonempty closed convex set is called a cone if it satisfies (i) , (ii) .

Moreover, is called normal if there exists a constant such that, for all implies . In this case is called the normality constant of . We say that an operator is increasing (decreasing) if implies .

For all , the notation means that there exist and such that . Clearly, is an equivalence relation. Given and ), we denote by the set . Clearly, is convex and for all .

We now present a fixed point theorem of generalized concave operators which will be used in the latter proof. See [30] for further information.

Theorem 1 (from [30, Lemma 2.1, and Theorem 2.1]). Let , and let be a normal cone. Assume that is increasing and ; for any and , there exists with respect to such that . Then (i) there are and such that ;  (ii) operator equation has a unique solution in .

Remark 2. An operator is said to be generalized concave if satisfies condition .

In what follows, for the sake of convenience, let , be continuous at and left continuous at , exists, , and let be continuous at and left continuous at , exists, . Evidently, is a Banach space with the norm , and is a Banach space with the norm . Let .

Definition 3. A function is called a solution of the problem (1) if it satisfies problem (1).

Lemma 4. is a solution of the problem (1) if and only if is the solution of the following integral equation:

where

Proof. First suppose that is a solution of the problem (1). It is easy to see by integration of (1) that

Integrate again, we can get

Letting in (6) and (7), we find

From the boundary conditions , and , we have

Then we obtain

Substituting (10) into (7), we have

Thus, the proof of sufficient is complete.

Conversely, if is a solution of (4). Then we can easily get . Direct differentiation of (4) implies that, for ,

Further

So and it is easy to verify that , and the lemma is proved.

Define an operator by

Lemma 5. is a solution of problem (1) if and only if is a fixed point of the operator .

3. Existence and Uniqueness of Positive Solutions for Problem (1)

In this section, we apply Theorem 1 to study the problem (1), and we obtain a new result on the existence and uniqueness of positive solutions. The method used in this paper is new to the literature and so is the existence and uniqueness result to the second-order impulsive differential equations. This is also the main motivation for the study of (1) in the present work.

Set , the standard cone. It is clear that is a normal cone in and the normality constant is 1. Our main result is summarized in the following theorem.

Theorem 6. Assume that (H₁)–(H₄) hold. Then

 (i) there exist such that

 (ii) the nonlinear impulsive problem (1) has a unique positive solution in , where .

Remark 7. It is easy to see that .

Proof of Theorem 6. Firstly, we show that is increasing, generalized concave. For any ,

From , we know that , and . So we have for . By Lemma 4, . It follows from that is increasing. Now we prove that is generalized concave. Set . Then . For any and , from we have

That is, .

Secondly, we prove that . Set

Then from , we have . Further, from , and ,

Hence,

That is, . Finally, an application of Theorem 1 implies that (i) there are such that , (ii) operator equation has a unique solution in . That is,

and the problem (1) has a unique positive solution in . Moreover, from Lemmas 4 and 5 we know that . Evidently, is a positive solution of the problem (1).

Theorem 8. Assume that (H₁)′–(H₄)′ hold. Then

 (i) there exist such that

where

 (ii) the nonlinear impulsive problem (1) has a unique positive solution in , where , .

Proof. From the proof of Theorem 6, for any ,

From , we know that , . By Lemma 4, . It follows from that is decreasing. Set , . Then . For any and , from we have

That is, . Further, for and ,

So we obtain . Consequently, is increasing, and, for ,

Let . Then and . So the operator is generalized concave. Next we prove that . Set

Then from , we have . Further, from , and ,

Hence,

We can choose a sufficiently small number such that

Then we get . Further,

That is, . Finally, an application of Theorem 1 implies that (i) there are such that , (ii) operator equation has a unique solution in . Let . Then, . That is,

Next we show that is the unique fixed point of in . In view of , and by the uniqueness of solutions for the operator equation , we have . Suppose that is another fixed point of in . Then . Hence, by the uniqueness of solutions for the operator equation , we obtain . So the problem (1) has a unique positive solution in . Moreover, from Lemmas 4 and 5 we know that . Evidently, is a positive solution of the problem (1).

Remark 9. Here, we provide an alternative approach to study the same type of problems under different conditions. Our result can guarantee the existence of a unique positive solution without supposing the existence of upper-lower solutions. The method used in this paper is relatively new to the literature and so is the existence and uniqueness result to the impulsive differential equations.

In the following we consider two special cases of the problem (1):

where , , , .

From Theorems 6 and 8, we have the following conclusions.

Corollary 10. Assume that holds and

and is decreasing in with

for any and , there exist such that

Then

 (i) there exist such that

 (ii) the nonlinear impulsive problem (34) has a unique positive solution in , where , , and is given as in Lemma 4.

Corollary 11. Assume that hold and

  for and is increasing in with

for any and , there exist such that

Then

 (i) there exist such that

where

 (ii) the nonlinear impulsive problem (34) has a unique positive solution in , where and   and is given as in Lemma 4.

Corollary 12. Assume that holds and

and is increasing in with

for any and , there exist such that

Then

 (i) there exist such that

 (ii) the nonlinear impulsive problem (35) has a unique positive solution in , where , , , and is continuous at and left continuous at , exists, .

Corollary 13. Assume that hold and

  for and is decreasing in , with

  for any , , and , there exist such that

Then

 (i) there exist such that

where

 (ii) the nonlinear impulsive problem (35) has a unique positive solution in , where , , , and is continuous at and left continuous at exists, .

4. An Example

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

Example 1. Consider the following boundary value problem:

Conclusion. BVP (50) has a unique positive solution in , where , .

Proof. BVP (50) can be regarded as a BVP of the form (1), where , , , and . It is not difficult to see that the conditions , and hold. In addition, let , , and . Then, the condition of Theorem 6 holds. Hence, by Theorem 6, the conclusion follows, and the proof is complete.

Remark 14. Example 1 implies that there is a large number of functions that satisfy the conditions of Theorem 6. In addition, the conditions of Theorem 6 are also easy to check.

Acknowledgments

The research was supported by the Fund of National Natural Science of China (61250011, 11201272) and the Science Foundation of Shanxi Province (2012011004-4, 2010021002-1).

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