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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 157043, 8 pages
http://dx.doi.org/10.1155/2013/157043
Research Article

Positive Solutions for Discrete Boundary Value Problems to One-Dimensional -Laplacian with Delay

Faculty of Science, Jiangsu University, Zhenjiang 212013, China

Received 23 April 2013; Accepted 20 June 2013

Academic Editor: Junjie Wei

Copyright © 2013 Linjun Wang and Xumei Chen. 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

We study the existence of positive solutions for discrete boundary value problems to one-dimensional -Laplacian with delay. The proof is based on the Guo-Krasnoselskii fixed-point theorem in cones. Two numerical examples are also provided to illustrate the theoretical results.

1. Introduction

The -Laplacian differential equations have been vastly applied in many fields such as non-Newtonian mechanics, economics, ecology, neural networks, and nonlinear flow laws, [15].

One of the important examples was described in [6]. Let be the two Cartesian coordinates in the plane of the glacier occupying the Lipschitzian domain , we denote by the horizontal velocity component of the ice at the point . After a rescaling of the physical velocity of the ice, satisfies the following equation: where is a hydrostatic pressure force acting on the glacier and is a function resulting from a constitutive law for the ice. The typical case of (1) is the following equation: where with and are the open balls centred about the origin with radius , respectively. People are interested to consider the positive radial solutions of (2), and then (2) can be reduced to the following form [7]:

Let , , and . Then (3) is transformed to

With variable change and , (4) reads where . Equation (5) is a typical type of one-dimensional -Laplacian equation.

In the real world, some processes are more reasonably described as -Laplacian differential equations with delay [2, 8, 9]. The reason is that the differential of the unknown solutions depends not only on the values of the unknown solutions at the current time but also on the values prior to that. Such equations, to a certain extent, reflect much more exactly the physical reality than the equations without delay.

In reality, (5) is applied together with some boundary value conditions, see for example, [1, 4, 811]. By using the Guo-Krasnoselskii fixed-point theorem, Jin and Yin [9] proved the existence of one positive solutions for the following boundary value problem of one-dimensional -Laplacian with delay

Based on a fixed-point approach, Bai and Xu [8] obtained the existence of positive solutions for problem (6) with having some singularities.

The main motivation of our work is twofold. The one is to prove the existence of positive solutions for discrete boundary value problems with delay. It is of interest to note here that the existence of single and multiple positive solutions for discrete boundary value problems to one-dimensional -Laplacian have been studied in great detail in the literature [1218] and the references therein. However, there are few papers dealing with the existence of positive solutions for discrete boundary value problems to one-dimensional -Laplacian with delay.

The other motivation is coming from the numerical solutions of problem (6). In order to get the numerical solutions of problem (6), we can apply the standard Euler method to discretize problem (6) and approximate its solutions numerically. An immediate and natural question arises if the corresponding difference equation together with boundary conditions has positive solutions.

Motivated by above, our purpose in this paper is to show the existence of positive solutions for the discretization equations of problem (6). Namely, we will prove the existence of positive solutions of the following problem: where and is a positive integer.

This paper is organized as follows. In Section 2, we introduce some basic definitions and then we state the Guo-Krasnoselskii fixed-point theorem. In Section 3, we write a representation for a solution to problem (7) in terms of the fixed point of an appropriate operator. Then we prove that problem (7) has a positive solution with belonging to an open interval by employing the fixed-point theorem. In Section 4, two numerical examples are presented to illustrate the theoretical results.

2. Preliminaries

In this section, we introduce some basic definitions and recall the Guo-Krasnoselskii fixed-point theorem which plays a fundamental role in our subsequent analysis.

For convenience, we will let denote , where .

A sequence is said to be a positive solution of (7), if it satisfies (7) with , .

We collect some properties of the function .

Lemma 1. is increasing in and , with ; .

Definition 2. Let be a real Banach space. A nonempty closed convex set is called a cone if(i) and imply ,(ii) and imply .

Theorem 3 (see [19]). Let be a Banach space and a cone. Assume , are bounded open subsets of with , , and let be a completely continuous operator such that one of the following holds:(i), , , ;(ii), , , .
Then has a fixed point in .

3. Existence of Positive Solutions

In this section, we study the existence of positive solutions for the following equation with delay: subject to the boundary condition

We assume that(A1) is a continuous function ( denotes the set of nonnegative reals);(A2) is a positive function defined on , .

Remark 4. Recall that a map is continuous if it is continuous as a map of the topological space into the topological space . Throughout this paper, the topology on will be the discrete topology.

Rewrite (8) as Summing up (10) with respect to from to leads to

Noticing Lemma 1, we have

Summing up (12) with respect to from 1 to and noticing the boundary condition (9) lead to where satisfies

Obviously, is continuous and strictly increasing. We have

Thus, there exists a unique solution to (14)

Let

Since is increasing and , we have . There exists that satisfied

Hence, where .

By (13) and (19), together with the definition of the positive solution, it is easy to get

Similarly, summing up (12) with respect to from to and noticing (9), we have Then by substituting (19) in (21), we obtain

Since (19) is the solution of (14), we have Namely, Thus,

Therefore, if is a positive solution of the boundary value problems (8) and (9), then it can be expressed by where .

Let denote the class of maps continuous on (discrete topology). We introduce a function space , namely, with norm given by . Then is a Banach space. Define a cone as follows: where  .

From the definition of , we have

Moreover, for any , we have , .

For any , define an operator on by where .

Lemma 5. Suppose that and hold. Then the operator is completely continuous.

Proof. For any , it is easily to check
According to Lemma  3.2 in [1], we have . Thus, . Moreover, and are both continuous, and is a finite space. This implies that the operator is a completely continuous. This completes the proof.

For convenience, we introduce some notations:

If is a positive solution of problems (8) and (9), then it is a fixed point of . So the existence of positive solution is transformed into the existence of fixed point. The following theorem is the main result of this paper.

Theorem 6. Suppose that and hold. Assume that the delay time is appropriately small, say is a positive integer not bigger than . If , , then there exists at least one positive solution to the boundary value problems (8) and (9) for , where while if , , then for any , where problems (8) and (9) admit at least one positive solution .

Proof. From Lemma 5, we see that is a completely continuous operator. Since , there exists an such that where
Let be fixed. By , there exists an such that for ,
Let . Then for any , we have
In view of Lemma 1 and the definition of operator , we further have Thus,
Next, by , it is easy to see that there exists an such that for all ,
Take . Let . For any , we have Thus, if , we have while if , then we have
Combining the two estimates above, we deduce , when . Therefore, by Theorem 3, has a fixed point . Namely, is a positive solution of problems (8) and (9).
The proof of the second half is parallel to above. There exists an such that where
Let be fixed. By , then it is not difficult to choose an such that ,
Let . Then for , it can be obtained that
If , we have Conversely, if , then we have
Next, by , there exists such that for
If is bounded, then there exists an such that for , . Take and . Then for any , we have
While if is unbounded, it is not difficult to see that there exists an , such that , . Take , then for , we get
Therefore, by Theorem 3, has a fixed point . Namely, is a positive solution of problema (8) and (9). This completes the proof.

4. Numerical Illustration

In this section, we present two numerical experiments to illustrate our results.

Example 7. Consider the following boundary value problem: where , . It is easy to see that , . By Theorem 6, problem (56) has a positive solution. In Figure 1, we show the numerical solution to problem (56).

fig1
Figure 1: The figures of the numerical solutions.

Example 8. Consider the following boundary value problem where , . We check that , . Then applying Theorem 6, problem (57) has a positive solution. Numerically, we obtain the solution of problem (57) which is shown in Figure 1.

Acknowledgments

This work is supported by Tian Yuan Fund of China (11226308), NNSF of China (11071102), Natural Science Fund for Colleges and Universities in Jiangsu Province (11KJD110001), the Research Fund (10JDG124) for Highlevel Group of Jiangsu University, and Postdoctoral Science Foundation (2011M500874, 1002030C).

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