# Positive Solutions for Third-Order -Laplacian Functional Dynamic Equations on Time Scales

- Changxiu Song
^{1}Email author and - Xuejun Gao
^{1}

**2011**:279752

**DOI: **10.1155/2011/279752

© C. Song and X. Gao. 2011

**Received: **31 March 2010

**Accepted: **9 December 2010

**Published: **15 December 2010

## Abstract

The authors study the boundary value problems for a -Laplacian functional dynamic equation on a time scale, , , , , , . By using the twin fixed-point theorem, sufficient conditions are established for the existence of twin positive solutions.

## 1. Introduction

Let be a closed nonempty subset of , and let have the subspace topology inherited from the Euclidean topology on . In some of the current literature, is called a time scale (or measure chain). For notation, we shall use the convention that, for each interval of of , will denote time scales interval, that is, .

where is the -Laplacian operator, that is, , , , where ; and

the function is continuous,

the function is left dense continuous (i.e., and does not vanish identically on any closed subinterval of . Here, denotes the set of all left dense continuous functions from to ,

is continuous and ,

is continuous, for all ,

-Laplacian problems with two-, three-, m-point boundary conditions for ordinary differential equations and finite difference equations have been studied extensively, for example see [1–4] and references therein. However, there are not many concerning the -Laplacian problems on time scales, especially for -Laplacian functional dynamic equations on time scales.

The motivations for the present work stems from many recent investigations in [5–8] and references therein. Especially, Kaufmann and Raffoul [8] considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions. In this paper, we apply the twin fixed-point theorem to obtain at least two positive solutions of boundary value problem (BVP for short) (1.1) when growth conditions are imposed on . Finally, we present two corollaries, which show that under the assumptions that is superlinear or sublinear, BVP (1.1) has at least two positive solutions.

The following twin fixed-point lemma due to [9] will play an important role in the proof of our results.

Lemma 1.1.

and

(i) for ,

(ii) for ,

(iii) and for .

## 2. Positive Solutions

Let be endowed with the norm and is concave and nonnegative valued on , and .

Clearly, is a Banach space with the norm and is a cone in . For each , extend to with for .

Then, denotes a positive solution of BVP (1.1).

It follows from (2.2) that

Lemma 2.1.

Let be defined by (2.2). If , then

(i) .

(ii) is completely continuous.

(iii) , .

(iv) is decreasing on .

The proof is similar to the proofs of Lemma 2.3 and Theorem 3.1 in [7], and is omitted.

Throughout this paper, we assume and .

Theorem 2.2.

Assume satisfies the following conditions:

for , uniformly in ,

for , uniformly in .

where , and , .

Proof.

By the definition of operator and its properties, it suffices to show that the conditions of Lemma 1.1 hold with respect to .

First, we verify that implies .

Secondly, we prove that implies .

which are twin positive solutions of BVP (1.1). The proof is complete.

In analogy to Theorem 2.2, we have the following result.

Theorem 2.3.

Assume satisfies the following conditions:

for , uniformly in ,

Now, we give theorems, which may be considered as the corollaries of Theorems 2.2 and 2.3.

From above, we deduce that .

Theorem 2.4.

If the following conditions are satisfied:

, , uniformly in ,

Proof.

We get now , and then the conditions in Theorem 2.2 are all satisfied. By Theorem 2.2, BVP (1.1) has at least two positive solutions. The proof is complete.

Theorem 2.5.

If the following conditions are satisfied:

, uniformly in ; ,

The proof is similar to that of Theorem 2.4 and we omitted it.

The following Corollaries are obvious.

Corollary 2.6.

If the following conditions are satisfied:

, , uniformly in ,

Corollary 2.7.

If the following conditions are satisfied:

, uniformly in , ;

## 3. Example

Example 3.1.

Let , , , , , .

where and ; , .

Consequently, and satisfies

for , uniformly in ,

Then all conditions of Theorem 2.3 hold. Thus, with Theorem 2.3, the BVP (3.1) has at least two positive solutions.

## Declarations

### Acknowledgment

This paper is supported by Grants nos. (10871052) and (10901060) from the NNSF of China, and by Grant (no. 10151009001000032) from the NSF of Guangdong.

## Authors’ Affiliations

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