- Research Article
- Open Access

# Existence of Positive Solutions in Generalized Boundary Value Problem for -Laplacian Dynamic Equations on Time Scales

- Wenyong Zhong
^{1, 2}and - Wei Lin
^{1}Email author

**2009**:848191

https://doi.org/10.1155/2009/848191

© W. Zhong and W. Lin. 2009

**Received:**31 March 2009**Accepted:**7 May 2009**Published:**14 June 2009

## Abstract

We analytically establish the conditions for the existence of at least two or three positive solutions in the generalized -point boundary value problem for the -Laplacian dynamic equations on time scales by means of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem. Furthermore, we illustrate the possible application of our analytical results with a concrete and nontrivial dynamic equation on specific time scales.

## Keywords

- Dynamic Equation
- Point Theorem
- Positive Real Number
- Real Banach Space
- Nonnegative Real Number

## 1. Introduction

Since the seminal work by Stefan Hilger in 1988, there has been a rapid development in the research of dynamic equations on time scales. The gradually maturing theory of dynamic equations not only includes the majority of the existing analytical results on both differential equations and difference equations with uniform time-steps but also establishes a solid foundation for the research of hybrid equations on different kinds of time scales. More importantly, with this foundation and those ongoing investigations, concrete applications of dynamic equations on time scales in mathematical modeling of real processes and phenomena, such as population dynamics, economic evolutions, chemical kinetics, and neural signal processing, have been becoming fruitful [1–8].

Recently, among the topics in the research of dynamic equations on time scales, the investigation of the boundary value problems for some specific dynamic equations on time scales has become a focal one that attained a great deal of attention from many mathematicians. In fact, systematic framework has been established for the study of the positive solutions in the boundary value problems for the second-order equations on time scales [9–15]. In particular, some results have been analytically obtained on the existence of positive solutions in some specific boundary value problems for the -Laplacian dynamic equations on time scales [16–19].

Here and throughout, is supposed to be a time scale, that is, is any nonempty closed subset of real numbers in with order and topological structure defined in a canonical way. The closed interval in is defined as . Accordingly, the open interval and the half-open interval could be defined, respectively. In addition, it is assumed that , , , , and for some positive constants and . Moreover, is supposed to be the -Laplacian operator, that is, and , in which and . With these configurations and with the aid of the five functionals fixed point theorem [20], they established the criteria for the existence of at least triple positive solutions of the above boundary value problem.

In the following discussion, we impose the following three hypotheses.

(H1) for , , and , where

(H2) is left dense continuous ( -continuous), and there exists a such that . is continuous.

Note that the definition of the -continuous function will be described in Definition 2.3 of Section 2. Also note that, together with conditions (1.5) and the above hypotheses (H )–(H ), the dynamic equation (1.4) with conditions (1.5) not only includes the above-mentioned specific boundary value problems in literature but also nontrivially extends the situation to a much wider class of boundary value problems on time scales. A question naturally appears: "can we still establish some criteria for the existence of at least double or triple positive solutions in the generalized boundary value problems (1.4) and (1.5)?" In this paper, we will give a positive answer to this question by virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem. Particularly, those obtained criteria will significantly extend the results in literature [19, 21, 23].

The rest of paper is organized as follows. In Section 2, we preliminarily import some definitions and properties of time scales and introduce some useful lemmas which will be utilized in the following discussion. In Section 3, we analytically present a criteria for the existence of at least two positive solutions in the boundary value problems (1.4) and (1.5) by virtue of the Avery-Henderson fixed point theorem. In Section 4, we provide some sufficient conditions for the existence of at least three positive solutions in light of the five functionals fixed point theorem. Finally, we further provides concrete and nontrivial example to illustrate the possible application of the obtained analytical results on dynamic equations on time scales in Section 5.

## 2. Preliminaries

### 2.1. Time Scales

For the sake of self-consistency, we import some necessary definitions and lemmas on time scales. More details can be found in [4] and reference therein. First of all, a time scale is any nonempty closed subset of real numbers with order and topological structure defined in a canonical way, as mentioned above. Then, we have the following definition of the categories of points on time scales.

Definition 2.1.

Then, the graininess operator is defined as In addition, if , is said to be right scattered, and if , is said to be left scattered. If , is said to be right dense, and if , is said to be left dense. If has a right scattered minimum , denote by ; otherwise, set . If has a left scattered maximum , denote by ; otherwise, set .

The following definitions describe the categories of functions on time scales and the basic computations of integral and derivative.

Definition 2.2.

for all . Then is said to be the nabla derivative of at .

Definition 2.3.

A function is left dense continuous ( -continuous) provided that it is continuous at all left dense points of , and its right side limits exists (being finite) at right dense points of . Denote by the set of all left dense continuous functions on .

Definition 2.4.

### 2.2. Main Lemmas

Lemma 2.5.

Proof.

which consequently leads to the completion of the proof, that is, specified in (2.8) is the unique solution of the problems (2.6) and (2.7).

Lemma 2.6.

Proof.

which consequently completes the proof.

Lemma 2.7.

for with

Proof.

This completes the proof.

Naturally, we denote by and by . With these settings and notations, we are in a position to have the following properties.

Lemma 2.8.

If then (i) for any ; (ii) for any pair of with .

for any . Here, . Thus, we obtain the following properties on this map.

Lemma 2.9.

Assume that the hypotheses are all fulfilled. Then, and is completely continuous.

Proof.

for all . Thus, the latter inequality implies that is decreasing on . This implies that for . Consequently, we complete the proof of the first part of the conclusion that for any .

This validates the equicontinuity of the elements in the set . Therefore, according to the Arzelà-Ascoli theorem on time scales [2], we conclude that is relatively compact. Now, let with . Then for all and Also, is uniformly valid on . These, with the uniform continuity of on the compact set , leads to a conclusion that is uniformly valid on . Hence, it is easy to verify that as tends toward positive infinity. As a consequence, we complete the whole proof.

## 3. At Least Two Positive Solutions in Boundary Value Problems

This section aims to prove the existence of at least two positive solutions in the boundary value problems (1.4) and (1.5) in light of the well-known Avery-Henderson fixed point theorem. Firstly, we introduce the Avery-Henderson fixed point theorem as follows.

Theorem 3.1 ([24]).

and (i) for all ; (ii) for all ; (iii) and for all . Then, the operator has at least two fixed points, denoted by and , belonging to and satisfying with and with .

Then, we arrive at the following results.

Theorem 3.2.

In addition, suppose that satisfies the following conditions:

(B1) for and ;

(B2) for and ;

(B3) for and

Proof.

Obviously, for each .

for each . Also, notice that for and . Furthermore, from Lemma 2.9, it follows that the operator is completely continuous.

Next, we are to verify the validity of all the conditions in Theorem 3.1 with respect to the operator .

Therefore, condition (i) in Theorem 3.1 is satisfied.

which consequently leads to the validity of condition (ii) in Theorem 3.1.

which shows the validity of condition (iii) in Theorem 3.1.

Now, in the light of Theorem 3.1, we consequently arrive to the conclusion that the boundary value problems (1.4) and (1.5) admit at least two positive solutions, denoted by and , satisfying with , and with , respectively.

## 4. At Least Three Positive Solutions in Boundary Value Problems

By means of the five functionals fixed point theorem which is attributed to Avery [20], this section is to analytically prove the existence of at least three positive solutions in the boundary value problems (1.4) and (1.5).

Theorem 4.1 ([20]).

- (i)
- (ii)
- (iii)
- (iv)

Then the operator admits at least three fixed points , , satisfying , , and with , respectively.

In the light of this theorem, we can have the following result on the existence of at least three solutions in the boundary value problems (1.4) and (1.5).

Theorem 4.2.

Furthermore, let satisfies the following conditions:

(C1) for and ;

(C2) for and ;

(C3) for and .

Proof.

for . In what follows, we aim to show the validity of all the conditions in Theorem 4.1 with respect to the operator .

So, according to Lemma 2.9, we have the complete continuity of the operator .

Clearly, we verify the validity of condition ( ) in Theorem 4.1.

Therefore, we further verify the validity of condition ( ) in Theorem 4.1.

Accordingly, both conditions ( ) and ( ) in Theorem 4.1 are satisfied. Now, in light of Theorem 4.1, the boundary value problems (1.4) and (1.5) have at least three positive solutions circumscribed on satisfying , , and with .

## 5. An Illustrative Example

This section will provide a nontrivial example to clearly illustrate the feasibility of the above-established analytical results on the dynamic equations on time scales.

possesses at least three positive solutions defined on satisfying , , and with .

## Declarations

### Acknowledgments

The authors are grateful to the two anonymous referees and Professor Alberto Cabada for their significant suggestions on the improvement of this paper. This work was supported by the NNSF of China (Grant nos. 10501008 and 60874121) and by the Rising-Star Program Foundation of Shanghai, China (Grant no. 07QA14002).

## Authors’ Affiliations

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