Abstract and Applied Analysis

Volume 2014 (2014), Article ID 676052, 6 pages

http://dx.doi.org/10.1155/2014/676052

Research Article

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

Wen Guan and Da-Bin Wang

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Received 6 June 2014; Accepted 11 July 2014; Published 4 August 2014

Academic Editor: Hui-Sheng Ding

Copyright © 2014 Wen Guan and Da-Bin Wang. 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 following third-order -Laplacian functional dynamic equation on time scales: , ,  ,  ,  , and . By applying the Five-Functional Fixed Point Theorem, the existence criteria of three positive solutions are established.

1. Introduction

Recently, much attention has been paid to the existence of positive solutions for the boundary value problems with -Laplacian operator on time scales; for example, see [1–22] and the references therein. But, to the best of our knowledge, there is not much concerning -Laplacian functional dynamic equations on time scales [6, 12–14, 19, 21, 22], especially for the third-order -Laplacian functional dynamic equations on time scales [14, 22].

In [14], Song and Gao were concerned with the existence of positive solutions for the -Laplacian functional dynamic equation on time scales:

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

is continuous; is left dense continuous (i.e., ) and does not vanish identically on any closed subinterval of , where denotes the set of all left dense continuous functions from to ; is continuous and ; is continuous, for all ; is continuous and satisfies the condition that there are such that

The existence of two positive solutions to problem (1) was obtained by using a double fixed point theorem due to Avery et al. [23] in a cone.

In [22], Wang and Guan considered the existence of positive solutions to problem (1) by applying the well-known Leggett-Williams Fixed Point Theorem.

Motivated by [14, 22], we will show that problem (1) has at least three positive solutions by means of the Five-Functional Fixed Point Theorem [24] (which is a generalization of the Leggett-Williams Fixed Point Theorem [25]). It is worth noting that the Five-Functional Fixed Point Theorem is used extensively in yielding three solutions for BVPs of differential equations, difference equations, and/or dynamic equations on time scales; see [6, 26, 27] and references therein.

Throughout this work we assume knowledge of time scales and time-scale notation, first introduced by Hilger [28]. For more on time scales, please see the texts by Bohner and Peterson [29, 30].

In the remainder of this section, we state the following theorem, which is crucial to our proof.

Let , , be nonnegative, continuous, and convex functionals on and let , be nonnegative, continuous, and concave functionals on . Then, for nonnegative real numbers , , , , and , we define the convex sets

Theorem 1 (see [24]). Let be a cone in a real Banach space . Suppose there exist positive numbers and ; nonnegative, continuous, and concave functionals and on ; and nonnegative, continuous, and convex functionals , , and on , with

for all . Suppose

is completely continuous and there exist nonnegative numbers , , , , with such that

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

Then has at least three fixed points such that

2. Existence of Three Positive Solutions

We note that is a solution of BVP (1) if and only if

Let be endowed with , so is a Banach space. Define cone by

For each , extend to with for .

Define by

We seek a point, , of in the cone . Define

Then denotes a positive solution of BVP (1).

We have the following results.

Lemma 2. Let , and then

(1) is completely continuous;(2) for ;(3) is decreasing ;(4) for and .

Proof. (1)–(3) are Lemma 3.1 of [14]. It is easy to conclude that (4) is satisfied by the concavity of .

Let be fixed such that , and set

Throughout this paper, we assume and .

We define the nonnegative, continuous, and concave functionals , and the nonnegative, continuous, and convex functionals , , on the cone , respectively, as

We observe that for each .

In addition, by Lemma 2, we have . Hence for all .

For convenience, we define

We now state growth conditions on so that BVP (1) has at least three positive solutions.

Theorem 3. Let , , and suppose that satisfies the following conditions:

, if , uniformly in , and , if , ;, if , uniformly in ;, if , uniformly in , and , if , .

Then BVP (1) has at least three positive solutions of the form

where , , and with .

Proof. Let , and then , and consequently, for . Since , so , and this implies

From , we have

Therefore

We now turn to property (i) of Theorem 1. Choosing , , it follows that

which shows that , and, for , we have

From , we have

We conclude that (i) of Theorem 1 is satisfied.

We next address (ii) of Theorem 1. If we take , , then

From this we know that . If , then

From , we have

Now we show that (iii) of Theorem 1 is satisfied. If and , then

Finally, if and , then from (4) of Lemma 2 we have

which shows that condition (iv) of Theorem 1 is fulfilled.

Thus, all the conditions of Theorem 1 are satisfied. Hence, has at least three fixed points , , satisfying

Let

which are three positive solutions of BVP (1).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees and the editors for their helpful comments and suggestions. Research was supported by the Postdoctoral Fund in China (Grant no. 2013M531717), the Excellent Young Teacher Training Program of Lanzhou University of Technology (Grant no. Q200907), and the Natural Science Foundation of Gansu Province of China (Grant no. 1310RJYA080).

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