Journal of Applied Mathematics

Volume 2013 (2013), Article ID 736583, 7 pages

http://dx.doi.org/10.1155/2013/736583

Research Article

Existence of Positive Solutions for -Laplacian Dynamic Equations with Derivative on Time Scales

Jinjun Fan and Liqing Li

School of Mathematical Science, Shandong Normal University, Jinan 250014, China

Received 26 November 2012; Accepted 16 January 2013

Academic Editor: Yansheng Liu

Copyright © 2013 Jinjun Fan and Liqing Li. 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 consider the existence of positive solutions of nonlinear p-Laplacian dynamic equations with derivative on time scales. Applying the Avery-Peterson fixed point theorem, we obtain at least three positive solutions to the problem. An example is also presented to illustrate the applications of the obtained results.

1. Introduction

In this paper we consider the existence of positive solutions of nonlinear p-Laplacian dynamic equations with derivative on time scales

where is a time scale, is a delta derivative, is a nabla derivative, , , , , , and other types of intervals are defined similarly. Throughout this paper, we denote the p-Laplacian operator by ; that is, ,, and .

The theory of dynamic equations on time scales was introduced firstly by Stefan Hilger in 1988. Since then, more and more scholars are interested in this area. The main reason is that the time scales theory can not only unite continuous and discrete dynamic equations but also have important applications, for example, in the study of insect population models, neural networks, heat transfer, economic, stock market, and epidemic models.

In recent years, there were a lot of attention focused on the study of boundary value problems of nonlinear dynamic equations. Hong in [1] studied the existence of positive solutions for the boundary value problem of p-Laplacian dynamic equations

Yaslan in [2] considered nonlinear second-order three-point boundary value problems for dynamic equations on time scales

P. G. Wang and Y. Wang in [3] studied the following boundary value problem:

Yaslan in [4] considered the existence of solutions of the following m-point boundary value problem on time scales

They obtained the existence of at least three nonnegative solutions by using fixed point theorems in cones. The authors in [5–10] dealt with some other classes of differential equations on time scales.

However, the nonlinear terms in the above mentioned literatures [1–4] are not involving a derivative. As far as we know, there is no paper to study the existence of positive solutions for second-order p-Laplacian dynamic equations with derivative on time scales. This paper is to fill this gap. By constructing a special cone and using the Avery-Peterson fixed point theorem, we obtain the existence of at least three nonnegative solutions.

The rest of this paper is organized as follows. In Section 2, we prove several preliminary results which are needed later. In Section 3, conditions for the existence of at least three nonnegative solutions to the BVP (1) are discussed by using the Avery-Peterson fixed point theorem. In Section 4, we give one example to illustrate our results.

2. Preliminaries and Some Lemmas

In this section, we first introduce some concepts on time scales and several lemmas.

Let be a time scale (an arbitrary nonempty closed subset of the real numbers ). For we define the forward jump operator by

while the backward jump operator is defined by

We set , if has a maximum , and , if has a minimum . If , we say that is right scattered, while we say that is left scattered, if . Also, if and , then is called right dense, and if and , then is called left dense. If has a right-scattered minimum m, we define ; otherwise set . If has a left-scattered maximum M, we define ; otherwise set .

For and , we define the delta derivative of , to be the number (provided it exists) with the property that, for any , there is a neighborhood of such that

for all . For and , we define the nabla derivative of , to be the number (provided it exists) with the property that, for any , there is a neighborhood of such that

for all .

A function is left-dense continuous or ld-continuous provided it is continuous at left-dense points in , and its right-sided limits exist (finite) at right-dense points in .

For more details about the subject of concepts and calculations on time scales, please see the book by Bohner and Peterson [11].

Consider the Banach space ; that is,

is differential on , and is ld-continuous on with the norm

Let

It is easy to see that is a cone in .

For convenience, we list the following assumptions.

does not vanish identically on any closed subinterval of . The function is left-dense continuous, and .

A function is said to be a solution of the problem (1) provided that is delta differential, and are continuous on , and u satisfies the problem (1).

To obtain our main results, we make use of the following lemmas.

Lemma 1. Assume that and are satisfied. Then is the solution of the BVP (1) if and only if

Proof. Firstly, we prove the necessity. Let be the solution of the BVP (1). -integrating on (1) from to , we have

By using the second boundary condition, we get

∆-integrating from to , we get

Therefore,

Submitting into the first boundary condition of the problem (1), we obtain

Thus,

Secondly, we prove the sufficiency. Let be the solution of the problem (12). Then,

Taking the -differentiating for (20), we obtain

that is,

By (12), we have

Hence,

The proof is complete.

Define operator by

By using Lemma 1, is the solution of the BVP (1) if and only if .

Lemma 2. Assume that and hold. Then is a completely continuous operator.

Proof. Firstly we prove .

For , by Lemma 1, , and , we obtain

which implies .

Secondly, maps a bounded set into a bounded set. Let , where . Because is continuous, there exists a such that

Hence, for , we have

Consequently, is bounded.

For , we get

Thus is relatively compact by using the Arzela-Ascoli theorem.

Finally we prove that is continuous. Let satisfying

This means that

Note that

Using the Lebesgue dominated convergence theorem on time scales, this together with (28) implies that

From the definition of , we know that

Therefore is continuous on . The proof is complete.

Lemma 3. If , then

(i), ;(ii), and .

Proof. First, prove (i). Let . From , , and , we have .

Second, prove (ii). Since is a concave function, for , we have

From , it follows that . The proof is complete.

Lemma 4. Let . Then there exists a real number such that , where .

Proof. Since , , and , we have

By Lemma 3, we obtain

The proof is complete.

In the rest of this section, we state the main tool used in this paper.

Let be functionals on cone in a real Banach space, where are nonnegative continuous convex functionals, is a nonnegative continuous concave functional, is a nonnegative continuous functional, and ; define the following sets:

Lemma 5 (see [1], the Avery-Peterson fixed point theorem). Let be a cone in a real Banach space and , and    defined as above; moreover, satisfies  for , and

for some positive numbers and . Suppose that is a completely continuous operator and that there exist positive real numbers ,  and    with such that the following conditions are satisfied:

(i), and for ;(ii) for with ;(iii) and for with .

Then has at least three fixed points ,  and   such that

3. Main Results

Define the nonnegative continuous convex functionals , nonnegative continuous concave functional , and nonnegative continuous functional as follows:

By Lemma 3, we have

Define

Theorem 6. Assume that and hold. In addition, suppose that there exist constants such that and the function satisfies the following conditions:

;;.

Then BVP (1) has at least three nonnegative solutions , and satisfying

(i);(ii), , and .

Proof. It is sufficient to show that has at least three fixed points. To this purpose, we show that all conditions of Lemma 5 are fulfilled. Now we divide this proof into three steps.

Step 1. Consider, .

For , we have . By Lemma 4, we get . From , it follows that

Applying this and Lemma 2, we have that is completely continuous.

Step 2. (i) and (ii) in Lemma 5 hold.

Let . Clearly , and

So .

From

we have , where .

By , for , we know

So (i) in Lemma 5 is satisfied.

For with , since , by (ii) in Lemma 3, we can get

So

Then (ii) in Lemma 5 holds.

Step 3. (iii) in Lemma 5 holds.

Clearly , so . If with , by , we obtain

From above, the hypotheses of the Avery-Peterson theorem are satisfied. Therefore the problem (1) has at least three nonnegative solutions , and satisfying

(i);(ii), .

The proof is complete.

4. Application

Let . We consider the following BVP:

where , and

By a simple calculation, we obtain . Choose . Thus satisfies

(1), (2), (3).

Then all conditions of Theorem 6 hold. Therefore, the problem (51) has at least three nonnegative solutions , and such that

Acknowledgments

The authors express their heartful thanks to reviewer(s) comments. This research is supported by Innovation Project for Graduate Education of Shandong Province (SDYY10058).

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