# Multiple Positive Solutions of -Point BVPs for Third-Order -Laplacian Dynamic Equations on Time Scales

- Li-Hua Bian
^{1}, - Xi-Ping He
^{1, 2}and - Hong-Rui Sun
^{1}Email author

**2009**:262857

**DOI: **10.1155/2009/262857

© Li-Hua Bian et al. 2009

**Received: **17 August 2009

**Accepted: **26 October 2009

**Published: **27 October 2009

## Abstract

This paper is concerned with the existence of multiple positive solutions for the third-order -Laplacian dynamic equation with the multipoint boundary conditions , where with . Using the fixed point theorem due to Avery and Peterson, we establish the existence criteria of at least three positive solutions to the problem. As an application, an example is given to illustrate the result. The interesting points are that not only do we consider third-order -Laplacian dynamic equation but also the nonlinear term is involved with the first-order delta derivative of the unknown function.

## 1. Introduction

The theory of dynamic equations on time scales was introduced by Stefan Hilger in 1988 [1]. This theory has attracted many researchers' attention and interest since it cannot only unify differential and difference equations but also provides accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. In addition, time-scale calculus would allow exploration of a variety of situations in economic, biological, heat transfer, stock market, and epidemic models [2, 3], and so forth.

Recently, there has been much attention paid to the existence of positive solutions for second-order nonlinear boundary value problems on time scales; see [4–10] and the references therein. On the one hand, higher-order nonlinear boundary value problems have been studied extensively; see [11–14] and the references therein. On the other hand, the boundary value problems with -Laplacian operator have also been discussed extensively in literature; for example, see [15–17]. However, very little work has been done to the third-order -Laplacian dynamic equations on time scales [18, 19].

For convenience, throughout this paper, we denote as -Laplacian operator, that is, for with and We also assume that is a closed subset of with ; an interval always means . Other types of intervals are defined similarly.

They established the existence theory for positive solutions by using various fixed point theorems [20, 21].

The main techniques are Schauder fixed point theorem and upper and lower solutions method.

By using fixed point theorems in cones, the existence criteria of multiple positive solutions are established.

They gave sufficient condition for the existence of three positive solutions by using a fixed point theorem due to Avery and Peterson [22].

where , for . By using fixed point theorem due to Avery and Peterson [22], we prove that the boundary value problems (1.5) and (1.6) have at least three positive solutions under suitable assumptions. The interesting points are that not only do we consider third-order -Laplacian dynamic equation on time scales but also the nonlinear term is involved with the first-order delta derivative of the unknown function.

Throughout this paper, it is assumed that

- (H1)
and , both and do not vanish identically on any closed subinterval of , and there exists such that hold;

- (H2)
there exist nonnegative constants and satisfying for

## 2. Preliminary

Lemma 2.1.

Proof.

So . On the other hand, it is easy to verify that if is as in (2.3), then is a solution of (2.1) and (2.2). Thus in (2.3) is the unique solution of (2.1) and (2.2).

Lemma 2.2.

Proof.

Therefore, We can choose and the proof is complete.

Lemma 2.3.

If , then for .

Proof.

The proof is complete.

The following fixed point theorem due to Avery and Peterson is fundamental in the proof of our main results.

Lemma 2.4 (see [22]).

for all . Suppose that is completely continuous and there exist positive numbers , and with such that

- (S1)
and for ;

- (S2)
for with ;

- (S3)
and for with .

## 3. Existence Results

In this section, by using the Avery-Peterson fixed point theorem, we shall give the sufficient conditions for the existence of at least three positive solutions to the BVPs (1.5) and (1.6).

Firstly, we define the nonnegative continuous concave functional , the nonnegative continuous convex functionals , and the nonnegative continuous functional on , respectively, by

Now we state and prove our main result.

Theorem 3.1.

- (A1)
for

- (A2)
for

- (A3)
for

Proof.

for . It is easy to obtain that is a completely continuous operator and every fixed point of is a solution of (1.5) and (1.6).

Thus we set out to verify that the operator satisfies Avery-Peterson fixed point theorem which will prove the existence of three fixed points of . Now the proof is divided into some steps.

Thus (3.6) holds.

So . Hence

That is, condition in Lemma 2.4 is satisfied.

for with . Hence condition in Lemma 2.4 is satisfied.

Finally, we assert that in Lemma 2.4 also holds.

Thus condition in Lemma 2.4 holds.

Therefore an application of Lemma 2.4 implies that the BVPs (1.5) and (1.6) have at least three positive solutions , and such that (3.3) holds.

## 4. Example

In this section, we present an example to explain our result.

Consequently, satisfies

- (i)
for

- (ii)
for

- (iii)
for

## Declarations

### Acknowledgment

Supported by the NNSF of China (10801065) and NSF of Gansu Province of China (0803RJZA096).

## Authors’ Affiliations

## References

- Hilger S:
*Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis*. Universität Würzburg, Würzburg, Germany; 1988.Google Scholar - Spedding V:
**Taming nature's numbers.***New Scientist*July 2003, 28–32.Google Scholar - Thomas DM, Vandemuelebroeke L, Yamaguchi K:
**A mathematical evolution model for phytoremediation of metals.***Discrete and Continuous Dynamical Systems B*2005,**5**(2):411–422.MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, Espinar VO, Perera K, Vivero DR:
**Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations.***Advances in Difference Equations*2008,**2008:**-13.Google Scholar - Agarwal RP, Espinar VO, Perera K, Vivero DR:
**Multiple positive solutions of singular Dirichlet problems on time scales via variational methods.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(2):368–381. 10.1016/j.na.2006.05.014MATHMathSciNetView ArticleGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2001:x+358.MATHView ArticleGoogle Scholar - Bohner M, Peterson A:
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHView ArticleGoogle Scholar - Li WT, Sun HR:
**Positive solutions for second-order -point boundary value problems on time scales.***Acta Mathematica Sinica*2006,**22**(6):1797–1804. 10.1007/s10114-005-0748-5MATHMathSciNetView ArticleGoogle Scholar - Wang Y, Ge W:
**Positive solutions for multipoint boundary value problems with a one-dimensional -Laplacian.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(6):1246–1256. 10.1016/j.na.2006.01.015MATHMathSciNetView ArticleGoogle Scholar - Zhao B, Sun HR:
**Multiplicity results of positive solution for nonlinear three-point boundary value problem on time scales.***Advances in Dynamical Systems and Applications*2009,**4**(2):25–43.MathSciNetGoogle Scholar - Henderson J:
**Multiple solutions for -th order Sturm-Liouville boundary value problems on a measure chain.***Journal of Difference Equations and Applications*2000,**6**(4):417–429. 10.1080/10236190008808238MATHMathSciNetView ArticleGoogle Scholar - Li SH:
**Positive solutions of nonlinear singular third-order two-point boundary value problem.***Journal of Mathematical Analysis and Applications*2006,**323**(1):413–425. 10.1016/j.jmaa.2005.10.037MATHMathSciNetView ArticleGoogle Scholar - Sun YP:
**Positive solutions of singular third-order three-point boundary value problem.***Journal of Mathematical Analysis and Applications*2005,**306**(2):589–603. 10.1016/j.jmaa.2004.10.029MATHMathSciNetView ArticleGoogle Scholar - Yaslan İ:
**Existence results for an even-order boundary value problem on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(1):483–491. 10.1016/j.na.2007.12.019MATHMathSciNetView ArticleGoogle Scholar - Su Y-H, Li W-T, Sun H-R:
**Positive solutions of singular -Laplacian BVPs with sign changing nonlinearity on time scales.***Mathematical and Computer Modelling*2008,**48**(5–6):845–858. 10.1016/j.mcm.2007.11.008MATHMathSciNetView ArticleGoogle Scholar - Sun H-R, Li W-T:
**Existence theory for positive solutions to one-dimensional -Laplacian boundary value problems on time scales.***Journal of Differential Equations*2007,**240**(2):217–248. 10.1016/j.jde.2007.06.004MATHMathSciNetView ArticleGoogle Scholar - Zhou CL, Ma DX:
**Existence and iteration of positive solutions for a generalized right-focal boundary value problem with -Laplacian operator.***Journal of Mathematical Analysis and Applications*2006,**324**(1):409–424. 10.1016/j.jmaa.2005.10.086MATHMathSciNetView ArticleGoogle Scholar - Anderson DR, Cabada A:
**Third-order right-focal multi-point problems on time scales.***Journal of Difference Equations and Applications*2006,**12**(9):919–935. 10.1080/10236190600839322MATHMathSciNetView ArticleGoogle Scholar - Han W, Kang S:
**Multiple positive solutions of nonlinear third-order BVP for a class of -Laplacian dynamic equations on time scales.***Mathematical and Computer Modelling*2009,**49**(3–4):527–535. 10.1016/j.mcm.2008.08.002MATHMathSciNetView ArticleGoogle Scholar - Deimling K:
*Nonlinear Functional Analysis*. Springer, Berlin, Germany; 1985:xiv+450.MATHView ArticleGoogle Scholar - Guo DJ, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar - Avery RI, Peterson AC:
**Three positive fixed points of nonlinear operators on ordered Banach spaces.***Computers & Mathematics with Applications*2001,**42**(3–5):313–322.MATHMathSciNetView ArticleGoogle Scholar

## Copyright

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.