Abstract and Applied Analysis

Volume 2013 (2013), Article ID 708281, 13 pages

http://dx.doi.org/10.1155/2013/708281

Research Article

Computation of Positive Solutions for Nonlinear Impulsive Integral Boundary Value Problems with -Laplacian on Infinite Intervals

Xingqiu Zhang

School of Mathematics, Liaocheng University, Liaocheng, Shandong 252059, China

Received 14 October 2012; Accepted 11 January 2013

Academic Editor: Patricia J. Y. Wong

Copyright © 2013 Xingqiu Zhang. 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.

This paper deals with the existence and iteration of positive solutions for nonlinear second-order impulsive integral boundary value problems with -Laplacian on infinite intervals. Our approach is based on the monotone iterative technique.

1. Introduction

The theory of impulsive differential equations has been emerging as an important area of investigation in recent years. It has been extensively applied to biology, biologic medicine, optimum control in economics, chemical technology, population dynamics, and so on. It is much richer because all the structure of its emergence has deep physical background and realistic mathematical model and coincides with many phenomena in nature. For an introduction of the basic theory of impulsive differential equations in , the reader is referred to see Lakshmikantham et al. [1, 2], Samoĭlenko and Perestyuk [3], and the references therein.

Boundary value problems on infinite intervals arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium; see [4–7], for example. In a recent paper [8], by means of a fixed-point theorem due to Avery and Peterson, Li and Nieto obtained some new results on the existence of multiple positive solutions for the following multipoint boundary value problem with a finite number of impulsive times on an infinite interval:

where , , , , , and .

Boundary value problems with integral boundary conditions for ordinary differential equations arise in different fields of applied mathematics and physics such as heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. Moreover, boundary value problems with Riemann-Stieltjes integral conditions constitute a very interesting and important class of problems. They include two-point, three-point, and multipoint boundary value problems as special cases; see [9–14]. For boundary value problems with other integral boundary conditions and comments on their importance, we refer the reader to the papers [11–20] and the references therein.

There are relatively few papers available for integral boundary value problems for impulsive differential equations on an infinite interval with an infinite number of impulsive times up to now. In [21], Zhang et al. investigated the existence of minimal nonnegative solution for the following second-order impulsive differential equation

where , , , , , , and with . denotes the jump of at , that is,

where and represent the right-hand limit and left-hand limit of at , respectively. has a similar meaning to .

In the past few years, the existence and the multiplicity of bounded or unbounded positive solutions to nonlinear differential equations on infinite intervals have been studied by several different techniques; we refer the reader to [5–8, 21–29] and the references therein. However, most of these papers only considered the existence of positive solutions under various boundary value conditions. Seeing such a fact, a natural question which arises is “how can we find the solutions when they are known to exist?” More recently, Ma et al. [30] and Sun et al. [31, 32] established iterative schemes for approximating the solutions for some boundary value problems defined on finite intervals by virtue of the iterative technique.

However, to the author's knowledge, the corresponding theory for impulsive integral boundary value problems with -Laplacian operator and infinite impulsive times on infinite intervals has not been considered till now. Motivated by previous papers, the purpose of this paper is to obtain the existence of positive solutions and establish a corresponding iterative scheme for the following impulsive integral boundary value problem of second-order differential equation with -Laplacian on an infinite interval

where , , , , , , , and with , , and .

It is clear that

Throughout this paper, we adopt the following assumptions.

() on any subinterval of , and when are bounded, is bounded on . () is a nonnegative measurable function defined in and does not identically vanish on any subinterval of , and

(), and there exist such that

with .

If , , , then BVP (4) reduces to the following two-point boundary value problem:

which has been studied in [23].

Compared with [8, 21], the main features of the present paper are as follows. Firstly, second-order differential operator is replaced by a more general -Laplacian operator. Secondly, in this paper, in boundary value conditions may not be zero which will bring about computational difficulties. Thirdly, by applying monotone iterative techniques, we construct successive iterative schemes starting off with simple known functions. It is worth pointing out that the first terms of our iterative schemes are simple functions. Therefore, the iterative schemes are significant and feasible.

The rest of this paper is organized as follows. In Section 2, we give some preliminaries and establish several lemmas. The main theorems are formulated and proved in Section 3. Then, in Section 4, an example is presented to illustrate the main results.

2. Preliminaries and Several Lemmas

Definition 1. Let be a real Banach space. A nonempty closed set is said to be a cone provided that

(1) for all and all ,(2) implies that .

Definition 2. A map is said to be concave on , if for all and .

Let is a map from into such that is continuous at , left continous at and exists for , exists and is continuous at , left continous at and exists for

Obviously, . It is clear that is a Banach space with the norm

and is also a Banach space with the norm

where . Let , . Define a cone by

Remark 3. If satisfies (4), then , and which implies that is nonincreasing on ; that is, is also nonincreasing on . Thus, is concave on . Moreover, if , then , , and so is monotone increasing on .

Lemma 4. Let conditions hold. Then, with is a solution of BVP (4) if and only if is a fixed point of the following operator equation:

Proof. Suppose that with is a solution of BVP (4). For , integrating (4) from to , we have

That is

which implies that

If , integrating (17) from 0 to , we get that

Integrating (17) from to , we obtain

Adding (18) and (19) together, we have

Repeating previous process, we get that

It follows that

Substituting (22) into (21), we get that

For , there exists such that . Set , and we have by and that

By (6), (24), we have

It follows from (24) and (25) that the right term in (23) is well defined. Thus, we have proved that is a fixed point of the operator defined by (14).

Conversely, suppose that is a fixed point of the operator equation (14). Evidently,

Direct differentiation of (14) implies that, for ,

which means that . It is easy to verify that ,  . The proof of Lemma 4 is complete.

To obtain the complete continuity of , the following lemma is still needed.

Lemma 5 (see [33, 34]). Let be a bounded subset of . Then, is relatively compact in if and are both equicontinuous on any finite subinterval for any , and for any , there exists such that

uniformly with respect to as , where , , .

This lemma is a simple improvement of the Corduneanu theorem in [33, 34].

Lemma 6. Let hold. Then is completely continuous.

Proof. For any , by (14), we have

It follows from (14), (29), and that , , , that is, . Now, we prove that is continuous and compact respectively. Let , as , then there exists such that . Let . By and , we have

It follows from (30) and dominated convergence theorem that

which implies that

By (30)–(32), and dominated convergence theorem, we get that

It follows from (33) that is continuous.

Let be any bounded subset. Then, there exists such that for any . Obviously,

From (34), , and , we know that is bounded.

For any , with , by the absolute continuity of the integral, we have

Thus, we have proved that is equicontinuous on any .

Next, we prove that for any , there exits sufficiently large such that

For any , we have

It follows from (37) that

On the other hand, we arrive at

Thus, (36) can be easily derived from (38) and (39). So, by Lemma 5, we know that is relatively compact. Thus, we have proved that is completely continuous.

3. Main Results

For notational convenience, we denote that

Theorem 7. Assume that hold, and there exists

such that

   for any ,    

  , for any .

Then, the boundary value problem (4) admits positive, nondecreasing on and concave solutions and such that , and , where

and , , where .

Proof. We only prove the case that . Another case can be proved in a similar way. By Lemma 6, we know that is completely continuous. From the definition of , , and , we can easily get that for any with , . Denote that

In what follows, we first prove that . If , then . By (6), (40), (42), (44), , , and , we get that

Thus, we get that . Hence, we have proved that .

Let , , then . Let , , then by Lemma 6, we have that and . Denote that

Since , we have that

It follows from the complete continuity of that is a sequentially compact set. We assert that has a convergent subsequence , and there exists such that .

By (51), , we get that

So, by (53) we have

By induction, we get that

Hence, we claim that as . Applying the continuity of and , we get that .

Let , , then . Let . By Lemma 6, we have that and . Denote

Since , we have that . It follows from the complete continuity of that is a sequentially compact set. And, we assert that has a convergent subsequence and there exists such that .

Since , we have

By , we have

By induction, we get that

Hence, we claim that as . Applying the continuity of and , we get that .

Since , then the zero function is not the solution of BVP (4). Thus, is a positive solution of BVP (4). By Lemma 4 we know that and are positive, nondecreasing on and concave solutions of the BVP (4).

We can easily get that Theorem 7 holds for in a similar manner.

Remark 8. The iterative schemes in Theorem 7 are , , and , , . They start off with a known simple linear function and the zero function respectively. This is convenient in application.

Theorem 9. Assume that hold, and there exist

such that

for any , , .  

, for any .

Then, the boundary value problem (4) admits positive nondecreasing on and concave solutions and , such that , and , where

and , , where , .

Remark 10. It is easy to see that and in Theorem 7 may coincide, and then the boundary value problem (4) has only one solution in . Similarly, positive solutions and may also coincide.

4. An Example

Example 11. Consider the following impulsive integral boundary value problem:

where

It is clear that conditions , , and hold for , ,  . By direct computation, we obtain that

which implies that holds.

Obviously, . Using a simple inequality

we get that

Thus, holds for , . Considering that

we can obtain that

Take . In this case, we have

On the other hand, nonlinear term satisfies

which means that holds. Thus, we have checked that all the conditions of Theorem 7 are satisfied. Therefore, the conclusion of Theorem 7 holds.

Acknowledgments

This paper is supported financially by the Foundation for Outstanding Middle-Aged and Young Scientists of Shandong Province (BS2010SF004), the Project of Shandong Province Higher Educational Science and Technology Program (no. J10LA53), and the National Natural Science Foundation of China (10971179).

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