Advances in Difference Equations

, 2009:312058 | Cite as

Existence of Positive Solutions for Multipoint Boundary Value Problem with Open image in new window -Laplacian on Time Scales

Open Access
Research Article
Part of the following topical collections:
  1. Boundary Value Problems on Time Scales

Abstract

We consider the existence of positive solutions for a class of second-order multi-point boundary value problem with Open image in new window -Laplacian on time scales. By using the well-known Krasnosel'ski's fixed-point theorem, some new existence criteria for positive solutions of the boundary value problem are presented. As an application, an example is given to illustrate the main results.

Keywords

Banach Space Eating Disorder West Nile Virus Fixed Point Theorem Epidemic Model 

1. Introduction

The theory of time scales has become a new important mathematical branch since it was introduced by Hilger [1]. Theoretically, the time scales approach not only unifies calculus of differential and difference equations, but also solves other problems that are a mix of stop start and continuous behavior. Practically, the time scales calculus has a tremendous potential for application, for example, Thomas believes that time scales calculus is the best way to understand Thomas models populations of mosquitoes that carry West Nile virus [2]. In addition, Spedding have used this theory to model how students suffering from the eating disorder bulimia are influenced by their college friends; with the theory on time scales, they can model how the number of sufferers changes during the continuous college term as well as during long breaks [2]. By using the theory on time scales we can also study insect population, biology, heat transfer, stock market, epidemic models [2, 3, 4, 5, 6], and so forth. At the same time, motivated by the wide application of boundary value problems in physical and applied mathematics, boundary value problems for dynamic equations with p-Laplacian on time scales have received lots of interest [7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

In [7], Anderson et al. considered the following three-point boundary value problem with p-Laplacian on time scales:

where Open image in new window , and Open image in new window for some positive constants Open image in new window They established the existence results for at least one positive solution by using a fixed point theorem of cone expansion and compression of functional type.

For the same boundary value problem, He in [8] using a new fixed point theorem due to Avery and Henderson obtained the existence results for at least two positive solutions.

In [9], Sun and Li studied the following one-dimensional p-Laplacian boundary value problem on time scales:

where Open image in new window is a nonnegative rd-continuous function defined in Open image in new window and satisfies that there exists Open image in new window such that Open image in new window is a nonnegative continuous function defined on Open image in new window for some positive constants Open image in new window They established the existence results for at least single, twin, or triple positive solutions of the above problem by using Krasnosel'skii's fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett-Williams fixed point theorem.

For the Sturm-Liouville-like boundary value problem, in [17] Ji and Ge investigated a class of Sturm-Liouville-like four-point boundary value problem with p-Laplacian:

where Open image in new window By using fixed-point theorem for operators on a cone, they obtained some existence of at least three positive solutions for the above problem. However, to the best of our knowledge, there has not any results concerning the similar problems on time scales.

Motivated by the above works, in this paper we consider the following multi-point boundary value problem on time scales:

where Open image in new window and we denote Open image in new window with Open image in new window

In the following, we denote Open image in new window for convenience. And we list the following hypotheses:
  1. (C1)

    Open image in new window is a nonnegative continuous function defined on Open image in new window

     
  2. (C2)
     

2. Preliminaries

In this section, we provide some background material to facilitate analysis of problem (1.4).

Let the Banach space Open image in new window is rd-continuous Open image in new window be endowed with the norm Open image in new window and choose the cone Open image in new window defined by
It is easy to see that the solution of BVP (1.4) can be expressed as
we define the operator Open image in new window by

It is easy to see Open image in new window , Open image in new window for Open image in new window and if Open image in new window then Open image in new window is the positive solution of BVP (1.4).

From the definition of Open image in new window for each Open image in new window we have Open image in new window and Open image in new window

is continuous and nonincreasing in Open image in new window Moreover, Open image in new window is a monotone increasing continuously differentiable function,
then by the chain rule on time scales, we obtain

so, Open image in new window

For the notational convenience, we denote

Lemma 2.1.

Open image in new window is completely continuous.

Proof.

First, we show that Open image in new window maps bounded set into bounded set.

Assume that Open image in new window is a constant and Open image in new window Note that the continuity of Open image in new window guarantees that there exists Open image in new window such that Open image in new window . So
That is, Open image in new window is uniformly bounded. In addition, it is easy to see

So, by applying Arzela-Ascoli Theorem on time scales, we obtain that Open image in new window is relatively compact.

Second, we will show that Open image in new window is continuous. Suppose that Open image in new window and Open image in new window converges to Open image in new window uniformly on Open image in new window . Hence, Open image in new window is uniformly bounded and equicontinuous on Open image in new window . The Arzela-Ascoli Theorem on time scales tells us that there exists uniformly convergent subsequence in Open image in new window . Let Open image in new window be a subsequence which converges to Open image in new window uniformly on Open image in new window . In addition,
Observe that
Inserting Open image in new window into the above and then letting Open image in new window , we obtain

here we have used the Lebesgues dominated convergence theorem on time scales. From the definition of Open image in new window , we know that Open image in new window on Open image in new window . This shows that each subsequence of Open image in new window uniformly converges to Open image in new window . Therefore, the sequence Open image in new window uniformly converges to Open image in new window . This means that Open image in new window is continuous at Open image in new window . So, Open image in new window is continuous on Open image in new window since Open image in new window is arbitrary. Thus, Open image in new window is completely continuous.

The proof is complete.

Lemma 2.2.

Let Open image in new window then Open image in new window for Open image in new window and Open image in new window for Open image in new window

Proof.

Since Open image in new window , it follows that Open image in new window is nonincreasing. Hence, for Open image in new window ,
from which we have

The proof is complete.

Lemma 2.3 ([18]).

is a completely continuous operator such that either

Then Open image in new window has a fixed point in Open image in new window

3. Main Results

In this section, we present our main results with respect to BVP (1.4).

For the sake of convenience, we define Open image in new window number of zeros in the set Open image in new window , and Open image in new window number of Open image in new window in the set Open image in new window

Theorem 3.1.

BVP (1.4) has at least one positive solution in the case Open image in new window and Open image in new window

Proof.

It follows that if Open image in new window then Open image in new window for Open image in new window

Set Open image in new window and Open image in new window

In other words, if Open image in new window then Open image in new window Thus by Open image in new window of Lemma 2.3, it follows that Open image in new window has a fixed point in Open image in new window with Open image in new window .

Thus, we let Open image in new window so that Open image in new window for Open image in new window

Now suppose Open image in new window is unbounded. From condition Open image in new window it is easy to know that there exists Open image in new window such that Open image in new window for Open image in new window If Open image in new window with Open image in new window then by using (3.8) we have
Consequently, in either case we take

so that for Open image in new window we have Open image in new window Thus by (ii) of Lemma 2.3, it follows that Open image in new window has a fixed point Open image in new window in Open image in new window with Open image in new window

The proof is complete.

Theorem 3.2.

Suppose Open image in new window , and the following conditions hold,

furthermore, Open image in new window Then BVP (1.4) has at least one positive solution Open image in new window such that Open image in new window lies between Open image in new window and Open image in new window

Proof.

Without loss of generality, we may assume that Open image in new window

which yields
Hence by condition Open image in new window we can get

Consequently, in view of Open image in new window (3.16), and (3.19), it follows from Lemma 2.3 that Open image in new window has a fixed point Open image in new window in Open image in new window Moreover, it is a positive solution of (1.4) and Open image in new window

The proof is complete.

For the case Open image in new window or Open image in new window we have the following results.

Theorem 3.3.

Suppose that Open image in new window and Open image in new window hold. Then BVP (1.4) has at least one positive solution.

Proof.

It is easy to see that under the assumptions, the conditions Open image in new window and Open image in new window in Theorem 3.2 are satisfied. So the proof is easy and we omit it here.

Theorem 3.4.

Suppose that Open image in new window and Open image in new window hold. Then BVP (1.4) has at least one positive solution.

Proof.

Since Open image in new window for Open image in new window there exists a sufficiently small Open image in new window such that
Thus, if Open image in new window , then we have
by the similar method, one can get if Open image in new window then

So, if we choose Open image in new window then for Open image in new window we have Open image in new window which yields condition Open image in new window in Theorem 3.2.

Next, by Open image in new window for Open image in new window there exists a sufficiently large Open image in new window such that

where we consider two cases.

Case 1.

Suppose that Open image in new window is bounded, say

In this case, take sufficiently large Open image in new window such that Open image in new window then from (3.24), we know Open image in new window for Open image in new window which yields condition Open image in new window in Theorem 3.2.

Case 2.

Suppose that Open image in new window is unbounded. it is easy to know that there is Open image in new window such that
Since Open image in new window then from (3.23) and (3.25), we get

Thus, the condition Open image in new window of Theorem 3.2 is satisfied.

Hence, from Theorem 3.2, BVP (1.4) has at least one positive solution.

The proof is complete.

From Theorems 3.3 and 3.4, we have the following two results.

Corollary 3.5.

Suppose that Open image in new window and the condition Open image in new window in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.

Corollary 3.6.

Suppose that Open image in new window and the condition Open image in new window in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.

Theorem 3.7.

Suppose that Open image in new window and Open image in new window hold. Then BVP (1.4) has at least one positive solution.

Proof.

In view of Open image in new window similar to the first part of Theorem 3.1, we have
Since Open image in new window for Open image in new window there exists a sufficiently small Open image in new window such that
Similar to the proof of Theorem 3.2, we obtain

The result is obtained, and the proof is complete.

Theorem 3.8.

Suppose that Open image in new window and Open image in new window hold. Then BVP (1.4) has at least one positive solution.

Proof.

Since Open image in new window similar to the second part of Theorem 3.1, we have Open image in new window for Open image in new window

By Open image in new window similar to the second part of proof of Theorem 3.4, we have Open image in new window where Open image in new window Thus BVP (1.4) has at least one positive solution.

The proof is complete.

From Theorems 3.7 and 3.8, we can get the following corollaries.

Corollary 3.9.

Suppose that Open image in new window and the condition Open image in new window in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.

Corollary 3.10.

Suppose that Open image in new window and the condition Open image in new window in Theorem 3.2 hold. Then BVP (1.4) has at least one positive solution.

Theorem 3.11.

Suppose that Open image in new window and the condition Open image in new window of Theorem 3.2 hold. Then BVP (1.4) has at least two positive solutions Open image in new window such that Open image in new window

Proof.

By using the method of proving Theorems 3.1 and 3.2, we can deduce the conclusion easily, so we omit it here.

Theorem 3.12.

Suppose that Open image in new window and the condition Open image in new window of Theorem 3.2 hold. Then BVP (1.4) has at least two positive solutions Open image in new window such that Open image in new window

Proof.

Combining the proofs of Theorems 3.1 and 3.2, the conclusion is easy to see, and we omit it here.

4. Applications and Examples

In this section, we present a simple example to explain our result. When Open image in new window ,

where, Open image in new window

It is easy to see that the condition Open image in new window and Open image in new window are satisfied and

So, by Theorem 3.1, the BVP (4.1) has at least one positive solution.

Notes

Acknowledgments

This research is supported by the Natural Science Foundation of China (60774004), China Postdoctoral Science Foundation Funded Project (20080441126), Shandong Postdoctoral Funded Project (200802018), the Natural Science Foundation of Shandong (Y2007A27, Y2008A28), and the Fund of Doctoral Program Research of University of Jinan (B0621, XBS0843).

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Copyright information

© Meng Zhang et al. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.School of ScienceUniversity of JinanJinanChina
  2. 2.School of Control Science and EngineeringShandong UniversityJinanChina

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