- Research Article
- Open Access

# Multiple Positive Solutions for Second-Order -Laplacian Dynamic Equations with Integral Boundary Conditions

- Yongkun Li
^{1}Email author and - Tianwei Zhang
^{1}

**Received:**13 July 2010**Accepted:**25 November 2010**Published:**6 December 2010

## Abstract

We are concerned with the following second-order -Laplacian dynamic equations on time scales , , with integral boundary conditions , . By using Legget-Williams fixed point theorem, some criteria for the existence of at least three positive solutions are established. An example is presented to illustrate the main result.

## Keywords

- Dynamic Equation
- Existence Result
- Fixed Point Theorem
- Heat Transfer
- Epidemic Model

## 1. Introduction

Boundary value problems with -Laplacian have received a lot of attention in recent years. They often occur in the study of the -dimensional -Laplacian equation, non-Newtonian fluid theory, and the turbulent flow of gas in porous medium [1–7]. Many works have been carried out to discuss the existence of solutions or positive solutions and multiple solutions for the local or nonlocal boundary value problems.

On the other hand, the study of dynamic equations on time scales goes back to its founder Stefan Hilger [8] and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete equations. Further, the study of time scales has led to several important applications, for example, in the study of insect population models, neural networks, heat transfer, and epidemic models, we refer to [8–10]. In addition, the study of BVPs on time scales has received a lot of attention in the literature, with the pioneering existence results to be found in [11–16].

where is positive parameter, for with and , is the delta derivative, is the nabla derivative, is a time scale which is a nonempty closed subset of with the topology and ordering inherited from , 0 and are points in , an interval , with for all , , with , and where .

The main purpose of this paper is to establish some sufficient conditions for the existence of at least three positive solutions for BVPs (1.1)-(1.2) by using Legget-Williams fixed point theorem. This paper is organized as follows. In Section 2, some useful lemmas are established. In Section 3, by using Legget-Williams fixed point theorem, we establish sufficient conditions for the existence of at least three positive solutions for BVPs (1.1)-(1.2). An illustrative example is given in Section 4.

## 2. Preliminaries

In this section, we will first recall some basic definitions and lemmas which are used in what follows.

Definition 2.1 (see [8]).

The point is called left-dense, left-scattered, right-dense, or right-scattered if , , and or , respectively. Points that are right-dense and left-dense at the same time are called dense. If has a left-scattered maximum , defined ; otherwise, set . If has a right-scattered minimum , defined ; otherwise, set .

Definition 2.2 (see [8]).

Definition 2.3 (see [8]).

A function is rd-continuous provided it is continuous at each right-dense point in and has a left-sided limit at each left-dense point in . The set of rd-continuous functions will be denoted by . A function is left-dense continuous (i.e., ld-continuous) if is continuous at each left-dense point in and its right-sided limit exists (finite) at each right-dense point in . The set of left-dense continuous functions will be denoted by .

Definition 2.4 (see [8]).

Lemma 2.5 (see [8]).

Lemma 2.6.

Proof.

So Lemma 2.6 is proved.

Lemma 2.7.

Proof.

The proof of sufficiency is complete.

The proof of Lemma 2.7 is complete.

for all . Obviously, for all .

Lemma 2.8.

Proof.

It is easily obtained from the second part of the proof in Lemma 2.7. The proof is complete.

Lemma 2.9.

Proof.

So is equicontinuous for any . Using Arzela-Ascoli theorem on time scales [17], we obtain that is relatively compact. In view of Lebesgue's dominated convergence theorem on time scales [18], it is easy to prove that is continuous. Hence, is complete continuous. The proof of this lemma is complete.

and introduce two assumptions with regard to the functionals , as follows:

(H1) there exists such that for all ;

The following fixed point theorem duo to Bai and Ge is crucial in the arguments of our main result.

Lemma 2.10 (see [19]).

Let be Banach space, a cone, and , . Assume that and are nonnegative continuous convex functionals satisfying (H1) and (H2), is a nonnegative continuous concave functional on such that for all , and is a complete continuous operator. Suppose

## 3. Main Result

In this section, we will give sufficient conditions for the existence of at least three positive solutions to BVPs (1.1)-(1.2).

Theorem 3.1.

Suppose that there are positive numbers , , and with , and such that the following conditions are satisfied.

Then BVPs (1.1)-(1.2) have at least three positive solutions.

Proof.

By the definition of the operator and its properties, it suffices to show that the conditions of Lemma 2.10 hold with respect to the operator .

Then it is easy to see that and (H1)-(H2) hold.

As in the argument above, we can get that . Thus, condition (C2) of Lemma 2.10 holds.

Therefore, for . So condition (C1) in Lemma 2.10 is satisfied.

Thus by Lemma 2.10 and the assumption that on , BVPs (1.1)-(1.2) have at least three positive solutions. The proof is complete.

Theorem 3.2.

Suppose that there are positive numbers , , and with , , and such that (H3)-(H4) and the following condition are satisfied.

Then BVPs (1.1)-(1.2) have at least three positive solutions.

Proof.

Therefore, for . So condition (C1) in Lemma 2.10 is satisfied. Using a similar proof to Theorem 3.1, the other conditions in Lemma 2.10 are satisfied. By Lemma 2.10, BVPs (1.1)-(1.2) have at least three positive solutions. The proof is complete.

## 4. An Example

Example 4.1.

Then BVPs (4.1)-(4.2) have at least three positive solutions.

Proof.

Moreover, we have

Therefore, conditions (H3)–(H5) in Theorem 3.1 are satisfied. Further, it is easy to verify that the other conditions in Theorem 3.1 hold. By Theorem 3.1, BVPs (4.1)-(4.2) have at least three positive solutions. The proof is complete.

## Declarations

### Acknowledgment

This work is supported the by the National Natural Sciences Foundation of China under Grant no. 10971183.

## Authors’ Affiliations

## References

- O'Regan D: Some general existence principles and results for
,
.
*SIAM Journal on Mathematical Analysis*1993, 24(3):648–668. 10.1137/0524040View ArticleMathSciNetMATHGoogle Scholar - del Pino M, Drábek P, Manásevich R: The Fredholm alternative at the first eigenvalue for the one-dimensional
*p*-Laplacian.*Journal of Differential Equations*1999, 151(2):386–419. 10.1006/jdeq.1998.3506View ArticleMathSciNetMATHGoogle Scholar - Cabada A, Pouso RL: Existence results for the problem
with nonlinear boundary conditions.
*Nonlinear Analysis: Theory, Methods & Applications*1999, 35(2):221–231. 10.1016/S0362-546X(98)00009-1View ArticleMathSciNetMATHGoogle Scholar - Lü H, Zhong C: A note on singular nonlinear boundary value problems for the one-dimensional
*p*-Laplacian.*Applied Mathematics Letters*2001, 14(2):189–194. 10.1016/S0893-9659(00)00134-8View ArticleMathSciNetMATHGoogle Scholar - Feng W, Webb JRL: Solvability of three point boundary value problems at resonance.
*Nonlinear Analysis: Theory, Methods & Applications*1997, 30(6):3227–3238. 10.1016/S0362-546X(96)00118-6View ArticleMathSciNetMATHGoogle Scholar - Gupta CP: A non-resonant multi-point boundary-value problem for a
*p*-Laplacian type operator. In*Proceedings of the 5th Mississippi State Conference on Differential Equations and Computational Simulations (Mississippi State, MS, 2001), Electron. J. Differ. Equ. Conf.*.*Volume 10*. Southwest Texas State University; 2003:143–152.Google Scholar - Tian Y, Ge W: Periodic solutions of non-autonomous second-order systems with a
*p*-Laplacian.*Nonlinear Analysis: Theory, Methods & Applications*2007, 66(1):192–203. 10.1016/j.na.2005.11.020View ArticleMathSciNetMATHGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Applications*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Bohner M, Peterson A (Eds):
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar - Agarwal RP, Bohner M, Li W-T:
*Nonoscillation and Oscillation: Theory for Functional Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 267*. Marcel Dekker, New York, NY, USA; 2004:viii+376.Google Scholar - Sun H-R, Tang L-T, Wang Y-H: Eigenvalue problem for
*p*-Laplacian three-point boundary value problems on time scales.*Journal of Mathematical Analysis and Applications*2007, 331(1):248–262. 10.1016/j.jmaa.2006.08.080View ArticleMathSciNetMATHGoogle Scholar - Jiang L, Zhou Z: Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(4):1376–1388. 10.1016/j.na.2007.06.034View ArticleMathSciNetMATHGoogle Scholar - Tian Y, Ge W: Existence and uniqueness results for nonlinear first-order three-point boundary value problems on time scales.
*Nonlinear Analysis: Theory, Methods & Applications*2008, 69(9):2833–2842. 10.1016/j.na.2007.08.054View ArticleMathSciNetMATHGoogle Scholar - Aykut Hamal N, Yoruk Fulya: Positive solutions of nonlinear
*m*-point boundary value problems on time scales.*Journal of Computational and Applied Mathematics*2009, 231(1):92–105. 10.1016/j.cam.2009.02.003View ArticleMathSciNetMATHGoogle Scholar - Sun H-R: Triple positive solutions for
*p*-Laplacian*m*-point boundary value problem on time scales.*Computers & Mathematics with Applications*2009, 58(9):1736–1741. 10.1016/j.camwa.2009.07.083View ArticleMathSciNetMATHGoogle Scholar - Yang Y, Meng F: Positive solutions of the singular semipositone boundary value problem on time scales.
*Mathematical and Computer Modelling*2010, 52(3–4):481–489. 10.1016/j.mcm.2010.03.045View ArticleMathSciNetMATHGoogle Scholar - Agarwal RP, Bohner M, Rehák P: Half-linear dynamic equations. In
*Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday*.*Volume 1*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:1–57.Google Scholar - Aulbach B, Neidhart L: Integration on measure chains. In
*Proceedings of the 6th International Conference on Difference Equations, Boca Raton, Fla, USA*. CRC Press; 2004:239–252.Google Scholar - Bai Z, Ge W: Existence of three positive solutions for some second-order boundary value problems.
*Computers & Mathematics with Applications*2004, 48(5–6):699–707. 10.1016/j.camwa.2004.03.002View ArticleMathSciNetMATHGoogle 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.