- Research Article
- Open Access

# The Existence of Positive Solutions for Third-Order -Laplacian -Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales

- Fuyi Xu
^{1}Email author and - Zhaowei Meng
^{1}

**2009**:169321

https://doi.org/10.1155/2009/169321

© F. Xu and Z. Meng. 2009

**Received:**25 February 2009**Accepted:**2 June 2009**Published:**5 July 2009

## Abstract

We study the following third-order -Laplacian -point boundary value problems on time scales , , , , , where is -Laplacian operator, that is, , , , . We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.

## Keywords

- Unique Solution
- Boundary Value Problem
- Fixed Point Theorem
- Iterative Scheme
- Nonlinear Boundary

## 1. Introduction

The theory of time scales was initiated by Hilger [1] as a mean of unifying and extending theories from differential and difference equations. The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example [2–6]. Recently, the boundary value problems with -Laplacian operator have also been discussed extensively in literature; for example, see [7–18]. However, to the best of our knowledge, there are not many results concerning the higher-order -Laplacian mutilpoint boundary value problem on time scales.

A time scale is a nonempty closed subset of . We make the blanket assumption that are points in . By an interval , we always mean the intersection of the real interval with the given time scale; that is .

author studied the existence of solutions for the nonlinear boundary value problem by using Krasnoselskii's fixed point theorem and Leggett and Williams fixed point theorem, respectively.

where . He obtained the existence of at least double and triple positive solutions of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

They established a corresponding iterative scheme for the problem by using the monotone iterative technique.

All the above works were done under the assumption that the nonlinear term is nonnegative. The key conditions used in the above papers ensure that positive solution is concave down. If the nonlinearity is negative somewhere, then the solution needs no longer to be concave down. As a result, it is difficult to find positive solutions of the -Laplacian equation when the nonlinearity changes sign. In particular, little work has been done on the existence of positive solutions for higher order -Laplacian -point boundary value problems with nonlinearity being nonnegative on time scales. Therefore, it is a natural problem to consider the existence of positive solution for higher order -Laplacian equations with sign changing nonlinearity on time scales. This paper attempts to fill this gap in literature.

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

## 2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in [1–5].

Definition 2.1.

for all . If , is said to be right scattered, 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 , define ; otherwise set . If has a left scattered maximum , define ; otherwise set .

Definition 2.2.

Definition 2.3.

A function is left-dense continuous (i.e., -continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in .

Definition 2.4.

Lemma 2.5.

Proof.

By caculating, we can easily get (2.7). So we omit it.

Lemma 2.6.

Proof.

where . The proof is complete.

Lemma 2.7.

Proof.

The proof is completed.

Lemma 2.8.

Proof.

This completes the proof.

Let be endowed with the ordering if for all and is defined as usual by maximum norm. Clearly, it follows that is a Banach space.

where , , and . Obviously, is a solution of the BVP(1.6) if and only if is a fixed point of operator .

Lemma 2.9.

Proof.

It is easy to see that by and Lemma 2.8. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove that operator is completely continuous.

Let be a cone in a Banach space . Let be an open bounded subset of with and . Assume that is a compact map such that for . Then the following results hold.

(2)If there exists such that for all and all , then .

(3)Let be open in such that . If and , then has a fixed point in . The same result holds if and , where denotes fixed point index.

Lemma 2.11 (see [20]).

defined above has the following properties:

Remark 2.12.

Lemma 2.13.

Proof.

This implies that for . Hence by Lemma 2.10(1) it follows that .

Lemma 2.14.

Proof.

## 3. Main Results

We now give our results on the existence of positive solutions of BVP (1.6).

Theorem 3.1.

Suppose that conditions and hold, and assume that one of the following conditions holds.

Then, the BVP (1.6) has at least one positive solution.

Proof.

This means that is a fixed point of operator .

This means that is a fixed point of operator . Therefore, the BVP (1.6) has at least one positive solution.

Theorem 3.2.

Assume that conditions and hold, and suppose that one of the following conditions holds.

There exist , and with , and such that

There exist , and with such that

Then, the BVP (1.6) has at least two positive solutions.

Proof.

This means that is a fixed point of operator . Then, the BVP (1.6) has at least two positive solutions.

When condition holds, the proof is similar to the above, and so we omit it here.

## 4. An Example

## Declarations

### Acknowledgment

This project was supported by the National Natural Science Foundation of China (10471075, 10771117).

## Authors’ Affiliations

## References

- Hilger S:
**Analysis on measure chains—a unified approach to continuous and discrete calculus.***Results in Mathematics*1990,**18**(1–2):18–56.MATHMathSciNetView ArticleGoogle Scholar - Agarwal RP, O'Regan D:
**Nonlinear boundary value problems on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2001,**44**(4):527–535. 10.1016/S0362-546X(99)00290-4MATHMathSciNetView ArticleGoogle Scholar - Atici FM, Guseinov GSh:
**On Green's functions and positive solutions for boundary value problems on time scales.***Journal of Computational and Applied Mathematics*2002,**141**(1–2):75–99. 10.1016/S0377-0427(01)00437-XMATHMathSciNetView ArticleGoogle Scholar - Sun H-R, Li W-T:
**Positive solutions for nonlinear three-point boundary value problems on time scales.***Journal of Mathematical Analysis and Applications*2004,**299**(2):508–524. 10.1016/j.jmaa.2004.03.079MATHMathSciNetView ArticleGoogle Scholar - Bohner M, Peterso A (Eds):
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar - Sun HR, Li WT:
**Positive solutions for nonlinear -point boundary value problems on time scales.***Acta Mathematica Sinica*2006,**49**(2):369–380.MATHMathSciNetGoogle 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 - Su Y-H, Li W-T, Sun H-R:
**Triple positive pseudo-symmetric solutions of three-point BVPs for -Laplacian dynamic equations on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(6):1442–1452.MATHMathSciNetView ArticleGoogle Scholar - He Z:
**Double positive solutions of three-point boundary value problems for -Laplacian dynamic equations on time scales.***Journal of Computational and Applied Mathematics*2005,**182**(2):304–315. 10.1016/j.cam.2004.12.012MATHMathSciNetView ArticleGoogle Scholar - He Z, Jiang X:
**Triple positive solutions of boundary value problems for -Laplacian dynamic equations on time scales.***Journal of Mathematical Analysis and Applications*2006,**321**(2):911–920. 10.1016/j.jmaa.2005.08.090MATHMathSciNetView ArticleGoogle Scholar - Xu FY:
**Positive solutions for third-order nonlinear -Laplacian -point boundary value problems on time scales.***Discrete Dynamics in Nature and Society*2008,**2008:**-16.Google Scholar - Su YH, Li S, Huang C:
**Positive solution to a singular -Laplacian BVPs with sign-changing nonlinearity involving derivative on time scales.***Advances in Difference Equations*2009,**2009:**-21.Google Scholar - Su YH, Li WT:
**Existence of positive solutions to a singular -Laplacian dynamic equations with sign changing nonlinearity.***Acta Mathematica Scientia*2009,**52:**181–196.MATHMathSciNetGoogle Scholar - Xu FY:
**Positive solutions for multipoint boundary value problems with one-dimensional -Laplacian operator.***Applied Mathematics and Computation*2007,**194**(2):366–380. 10.1016/j.amc.2007.04.118MATHMathSciNetView ArticleGoogle Scholar - Su YH, Li WT:
**Existence of positive solutions to a singular -Laplacian dynamic equations with sign changing nonlinearity.***Acta Mathematica Scientia*2008,**28:**51–60.MathSciNetGoogle Scholar - Su Y-H:
**Multiple positive pseudo-symmetric solutions of -Laplacian dynamic equations on time scales.***Mathematical and Computer Modelling*2009,**49**(7–8):1664–1681. 10.1016/j.mcm.2008.10.010MATHMathSciNetView 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 - Zhou C, Ma D:
**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:
**Green's function for a third-order generalized right focal problem.***Journal of Mathematical Analysis and Applications*2003,**288**(1):1–14. 10.1016/S0022-247X(03)00132-XMATHMathSciNetView ArticleGoogle Scholar - Lan KQ:
**Multiple positive solutions of semilinear differential equations with singularities.***Journal of the London Mathematical Society*2001,**63**(3):690–704. 10.1112/S002461070100206XMATHMathSciNetView 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

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