Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 783509, 9 pages

http://dx.doi.org/10.1155/2013/783509

Research Article

Upper and Lower Solutions for -Point Impulsive BVP with One-Dimensional -Laplacian

Junfang Zhao,¹ Jianfeng Zhao,² and Weigao Ge³

¹School of Science, China University of Geosciences, Beijing 100083, China

²College of Science, Hebei University of Engineering, Handan 056038, China

³Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received 30 July 2013; Revised 13 October 2013; Accepted 15 October 2013

Academic Editor: Gabriele Bonanno

Copyright © 2013 Junfang Zhao et al. 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.

Upper and lower solutions theories are established for a kind of -point impulsive boundary value problems with -Laplacian. By using such techniques and the Schauder fixed point theorem, the existence of solutions and positive solutions is obtained. Nagumo conditions play an important role in the nonlinear term involved with the first-order derivatives.

1. Introduction

In this paper, we study the following -point impulsive boundary value problem with one-dimensional -Laplacian:

where , . , (, where is a fixed positive integer) are fixed points with . . , . , , where and represent the right-hand limit and left-hand limit of at , respectively.

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations in , see Lakshmikantham et al. [1], Baĭnov and Simeonov [2], Samoilenko and Perestyuk [3], and the references therein. The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see for instance [4–9] and the references therein).

At the same time, it is well known that the method of lower and upper solutions is a powerful tool for proving the existence results for a large class of boundary value problems. For a few of such works, we refer the readers to [10–14].

In [15], Cabada and Pouso considered

by using the upper and lower method, the authors get the existence of solution to the above BVP.

In [16], Lü et al. studied

by giving conditions on involving pairs of lower and upper solutions, they get the existence of at least three solutions to the above BVP.

In [12], Shen and Wang studied

they prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.

Motivated by the works mentioned above, in this paper, we considered BVP (1), the main tool is upper and lower method, and the Schäuder fixed point theorem. We not only get the existence of solutions, but also the existence of positive solutions.

The main structure of this paper is as follows. In Section 2, we give the preliminary and present some lemmas in order to prove our main results. Section 4 presents the main theorems of this paper, and at the end of Section 4, we give an example to illustrate our main results.

2. Preliminary

Define is continuous at , , exists, left continuous}, and , is continuous at , and exists, , . Then, is a real Banach space with the norm , where .

By a solution of (1), we mean a function which satisfies (1), where .

Definition 1. A function , with , will be called a lower solution of (1) if

A function , with , satisfying the reversed inequalities is an upper solution of problem (1).

Next, we define the Nagumo condition we are going to use. Note that the condition does not depend on the boundary data of the problem.

Definition 2. We say that satisfies a Nagumo condition relative to the pair and , with , in , if there exists a function such that

where ; , and also that

here, , .

From Definition 2, we can find a real number such that

In addition, we assume that the following three conditions hold:

) with on and ;(); (), . is nondecreasing in for all .

Firstly, we define

One can find the next result, with its proof, in [17].

Lemma 3. For each , the next two properties hold:

(i) exists for a.e. ,(ii)if , and in , then

We consider the following modified problem:

with

where is defined by

Thus is a continuous function on and satisfies

for some constant . Moreover, we may choose so that .

Lemma 4. Suppose that , denote , then the following boundary value problem:

has a unique solution as follows:

Proof.

For , integrating (17) from to , we get

For , integrating (17) from to , we get

Combining (18), (19), and the boundary condition, for , we have

For , integrating (17) from to , and combining (20), we have

By induction, for , there holds

Integrating (22) from to , we have

For , integrating (22) from to , we have

By the boundary condition , (23), and (24), for , we have

Repeating in the above manner, for , we have

By (26), we have

By the boundary condition , we obtain

Substituting (26) into (28), we have (16) which holds. The proof is complete.

Lemma 5. If is a solution of BVP (11), and are lower and upper solutions of (1), respectively, , and

then,

Proof. Denote , we will only see that for every . An analogous reasoning shows that for all .

Otherwise, if , does not hold, then . Since , there are three cases.

Case  1. Denote , . There exists , such that . Then, . Let , . If for all , let , and if for all , let . Thus,

As a consequence, for all and since is increasing and, in particular, one to one, we have

Thus, , which is a contradiction.

Case  2. According to Case  1, if , then or , . Since , cannot be .

Next, we claim that for , if , then . Otherwise, . Suppose that and .

From the boundary condition and (29), we have

Hence,

Suppose that and is nonincreasing on some interval , where is sufficiently small such that on . For ,

Note that as . Hence, when is sufficiently small, , , which contradicts the assumption of monotonicity of . Thus, we obtain

We use the preceding procedure and deduce by induction that

which contradicts that . Essentially, by the same analysis, we can get that , cannot hold.

Case  3. . Easily, it holds that . While by the boundary condition and , we have

which is also a contradiction.

Consequently, holds for all . The proof is complete.

Lemma 6. If is a solution of (11), then for all .

Proof. Let be a solution of (11). From Lemma 5, we have , and so

By the mean value theorem, there exists with , , and as a result,

Let . Suppose that there exists a point in the interval for which or . From the continuity of , we can choose , such that one of the following situations holds:

(i), and for all ;(ii), and for all ;(iii), and for all ;(iv), and for all .

Without loss of generality, suppose that for all . Then,

and so

As a result,

Note also that for , so we have and thus , which leads to the following:

a contradiction. The proof is complete.

Theorem 7 (Schäuder fixed point theorem). Let be a convex subset of a normed Linear space . Each continuous, compact map has a fixed point.

3. The Main Results

Theorem 8. Suppose that conditions ()–() hold. Then, BVP (1) has at least one solution such that

Proof. Solving (11) is equivalent to finding a which satisfies

where , , .

Now, define the following operator by

It is obvious that is completely continuous.

By the Schäuder fixed point Theorem 7, we can easily obtain that has a fixed point , which is a solution of BVP (11). And by Lemmas 5 and 6, we know that , , then BVP (11) becomes BVP (1), therefore is a solution of BVP (1). The proof is complete.

Theorem 9. Suppose that conditions ()–() hold. Assume that there exist two lower solutions and and two upper solutions and for problem (1), satisfying the following:

(i); (ii); (iii), which means that there exists such that ;(iv)if is a solution of (1) with , then on ;(v)if is a solution of (1) with , then on .

If satisfies the Nagumo condition with respect to , , then problem (1) has at least three solutions , , and satisfying

Proof. We consider the following modified problem:

Now, define the following operator by

where , , and .

It is standard that is completely continuous.

Let , where , , , .

Let .

It is immediate from the argument above that . Thus,

Let

Since , , (i.e., choose such that , . It follows that , , and .

By assumptions (iv) and (v), there are no solutions in . Thus,

We show that , then

and there are solutions in , , as required.

We now show . The proof that is similar and hence omitted. We define , the extension to of the restriction of to as follows. Let

where (in replace by ) and are previously defined. Thus, is a continuous function on and satisfies

for some constants . Moreover, we may choose so that , .

Consider the following problem:

Now, define the following operator:

where , , and . Again, it is easy to check (from a previous argument and (v)) that is a solution of (57) if and (note that is compact).

Thus, . Moreover, it is easy to see that . By assumptions (iv) and (v), there are no solutions in . Thus,

Thus there are three solutions, as required. The proof is complete.

A slight modification of the argument in Theorem 9 yields the next result.

Theorem 10. Suppose that conditions ()–() hold. Assume that there exist two lower solutions and and two upper solutions and for problem (1), satisfying

(i); (ii); (iii); (iv)there exists such that all , the function , and are, respectively, lower and upper solution of (1);(v).

If satisfies the Nagumo condition with respect to , , then problem (1) has at least three solutions , , and satisfying

Proof. In the proof of Theorem 9, define

where is defined in Theorem 9.

Theorem 11. Let and satisfies (). Furthermore, the following conditions hold.

(i)BVP (1) has a pair of positive upper and lower solutions , satisfying

(ii) satisfies a Nagumo condition relative to the pair and .

Then, BVP (1) has at least one positive solution such that

Proof. It can be proved easily, we omit it here.

4. Examples

Example 1. Consider the following BVP:

where , . Let ; then, , which means that .

It is clear that

are lower and upper solutions of BVP (64), respectively. The figures of and are as shown in Figure 1.

Figure 1

Clearly, . After substituting , into , , denote and , we can get Figure 2. From Figure 2, we can see that when , and . Other inequalities can be verified easily. So, according to Definition 1, we can get that and are upper and lower solutions of BVP (64), respectively.

Figure 2

Let , and it is clear that satisfies Nagumo condition relative to and . Obviously, all the conditions of Theorem 11 hold. Hence, (64) has at least one positive solution .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation (no. 11226133), the Fundamental Research Funds for the Central Universities (no. 2652012141), and the Young Talents Programme of Beijing.

References

V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989.

D. D. Baĭnov and P. S. Simeonov, Systems with Impulse Effect, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1989.

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science: Series A: Monographs and Treatises, World Scientific, Singapore, 1995.

Z. L. Wei, “Periodic boundary value problems for second order impulsive integrodifferential equations of mixed type in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 195, no. 1, pp. 214–229, 1995.

S. G. Hristova and D. D. Bainov, “Monotone-iterative techniques of V. Lakshmikantham for a boundary value problem for systems of impulsive differential-difference equations,” Journal of Mathematical Analysis and Applications, vol. 197, no. 1, pp. 1–13, 1996.

X. Liu and D. Guo, “Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 284–302, 1997.

R. P. Agarwal and D. O'Regan, “Multiple nonnegative solutions for second order impulsive differential equations,” Applied Mathematics and Computation, vol. 114, no. 1, pp. 51–59, 2000.

W. Ding and M. Han, “Periodic boundary value problem for the second order impulsive functional differential equations,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 709–726, 2004.

E. K. Lee and Y. H. Lee, “Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations,” Applied Mathematics and Computation, vol. 158, no. 3, pp. 745–759, 2004.

C. D. Coster and P. Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Elsevier, New York, NY, USA, 2006.

D. Jiang, “Upper and lower solutions method and a singular superlinear boundary value problem for the one-dimensional -Laplacian,” Computers & Mathematics with Applications, vol. 42, no. 6-7, pp. 927–940, 2001.

J. Shen and W. Wang, “Impulsive boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 4055–4062, 2008.

I. Rachůnková and J. Tomeček, “Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 525–539, 2004.

D. Jiang, “Upper and lower solutions method and a superlinear singular boundary value problem,” Computers & Mathematics with Applications, vol. 44, no. 3-4, pp. 323–337, 2002.

A. Cabada and R. L. Pouso, “Existence results for the problem with nonlinear boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 35, no. 2, pp. 221–231, 1999.

H. Lü, D. O'Regan, and R. P. Agarwal, “Triple solutions for the one-dimensional -Laplacian,” Glasnik Matematički, vol. 38, no. 2, pp. 273–284, 2003.

M. X. Wang, A. Cabada, and J. J. Nieto, “Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions,” Annales Polonici Mathematici, vol. 58, no. 3, pp. 221–235, 1993.