- Research Article
- Open Access

- S. Gulsan Topal
^{1}Email author, - O. Batit Ozen
^{1}and - Erbil Cetin
^{1}

**2009**:123565

https://doi.org/10.1155/2009/123565

© S. Gulsan Topal et al. 2009

**Received:**16 March 2009**Accepted:**20 July 2009**Published:**19 August 2009

## Abstract

We are concerned with proving the existence of positive solutions of a nonlinear second-order four-point boundary value problem with a -Laplacian operator on time scales. The proofs are based on the fixed point theorems concerning cones in a Banach space. Existence result for -Laplacian boundary value problem is also given by the monotone method.

## Keywords

- Banach Space
- Dynamic Equation
- Convex Subset
- Fixed Point Theorem
- Continuous Operator

## 1. Introduction

Let be any time scale such that be subset of . The concept of dynamic equations on time scales can build bridges between differential and difference equations. This concept not only gives us unified approach to study the boundary value problems on discrete intervals with uniform step size and real intervals but also gives an extended approach to study on discrete case with non uniform step size or combination of real and discrete intervals. Some basic definitions and theorems on time scales can be found in [1, 2].

In recent years, the existence of positive solutions for nonlinear boundary value problems with
-Laplacians has received wide attention, since it has led to several important mathematical and physical applications [3, 4]. In particular, for
or
is linear, the existence of positive solutions for nonlinear singular boundary value problems has been obtained [5, 6].
-Laplacian problems with two-, three-, and *m*-point boundary conditions for ordinary differential equations and difference equations have been studied in [7–9] and the references therein. Recently, there is much attention paid to question of positive solutions of boundary value problems for second-order dynamic equations on time scales, see [10–13]. In particular, we would like to mention some results of Agarwal and O'Regan [14], Chyan and Henderson [5], Song and Weng [15], Sun and Li [16], and Liu [17], which motivate us to consider the
-Laplacian boundary value problem on time scales.

The aim of this paper is to establish some simple criterions for the existence of positive solutions of the -Laplacian BVP (1.1)-(1.2). This paper is organized as follows. In Section 2 we first present the solution and some properties of the solution of the linear -Laplacian BVP corresponding to (1.1)-(1.2). Consequently we define the Banach space, cone and the integral operator to prove the existence of the solution of (1.1)-(1.2). In Section 3, we state the fixed point theorems in order to prove the main results and we get the existence of at least one and two positive solutions for nonlinear -Laplacian BVP (1.1)-(1.2). Finally, using the monotone method, we prove the existence of solutions for -Laplacian BVP in Section 4.

## 2. Preliminaries and Lemmas

Lemma 2.1.

is a positive continuous function, therefore, has a minimum on , hence one supposes that there exists such that for .

Proof.

It is easily seen that is continuous on .

Then, from condition (H2), we have that the function is strictly monoton nondecreasing on and , the function is strictly monoton nonincreasing on and , which implies

Lemma 2.2.

Proof.

Lemma 2.3.

Proof.

which is a contradiction.

Throughout this paper, we assume that .

Lemma 2.4.

Proof.

we get , for and , for , thus the operator is monotone increasing on and monotone decreasing on and also is the maximum point of the operator . So the operator is concave on and . Therefore, .

Lemma 2.5.

Suppose that the conditions (H1)–(H3) hold. is completely continuous.

Proof.

By the Arzela-Ascoli theorem, we can easily see that is completely continuous operator.

In order to follow the main results of this paper easily, now we state the fixed point theorems which we applied to prove Theorems 3.1–3.4.

Theorem 2.6 (see [18] (Krasnoselskii fixed point theorem)).

hold. Then has a fixed point in .

Theorem 2.7 (see [19] (Schauder fixed point theorem)).

Let be a Banach space, and let be a completely continuous operator. Assume is a bounded, closed, and convex set. If , then has a fixed point in .

Theorem 2.8 (see [20] (Avery-Henderson fixed point theorem)).

for all . Suppose that there exist positive numbers such that for all and

## 3. Main Results

In this section, we will prove the existence of at least one and two positive solution of -Laplacian BVP (1.1)-(1.2). In the following theorems we will make use of Krasnoselskii, Schauder, and Avery-Henderson fixed point theorems, respectively.

Theorem 3.1.

Assume that (H1)–(H3) are satisfied. In addition, suppose that satisfies

where and . Then the -Laplacian BVP (1.1)-(1.2) has a positive solution such that .

Proof.

We define two open subsets and of such that and .

Then, has a fixed point . Obviously, is a positive solution of the -Laplacian BVP (1.1)-(1.2) and .

Existence of at least one positive solution is also proved using Schauder fixed point theorem (Theorem 2.7). Then we have the following result.

Theorem 3.2.

then the -Laplacian BVP (1.1)-(1.2) has at least one positive solution.

Proof.

which implies . The compactness of the operator follows from the Arzela-Ascoli theorem. Hence has a fixed point in .

Corollary 3.3.

If is continuous and bounded on , then the -Laplacian BVP (1.1)-(1.2) has a positive solution.

The function is positive and continuous on . Therefore, has a minimum on . Hence we suppose there exists such that

We observe here that, for every , and from Lemma 2.2, . Also, for ,

Theorem 3.4.

- (i)
- (ii)
- (iii)

Proof.

Define the cone as in (2.5). From Lemmas 2.2 and 2.3 and the conditions (H1) and (H2), we can obtain . Also from Lemma 2.5, we see that is completely continuous.

We now show that the conditions of Theorem 2.8 are satisfied.

Thus we have and condition (i) of Theorem 2.8 holds. Next we will show condition (ii) of Theorem 2.8 is satisfied. If , then .

So condition (ii) of Theorem 2.8 holds.

## 4. Monotone Method

In this section, we will prove the existence of solution of -Laplacian BVP (1.1)-(1.2) by using upper and lower solution method. We define the set

Definition 4.1.

We will prove when the lower and the upper solutions are given in the well order, that is, , the -Laplacian BVP (1.1)-(1.2) admits a solution lying between both functions.

Theorem 4.2.

Assume that (H1)–(H3) are satisfied and and are, respectively, lower and upper solutions for the -Laplacian BVP (1.1)-(1.2) such that on . Then the -Laplacian BVP (1.1)-(1.2) has a solution on .

Proof.

which is a contradiction and thus cannot be an element of .

which is a contradiction.

which is a contradiction. Thus we have on .

Similarly, we can get on . Thus is a solution of -Laplacian BVP (1.1)-(1.2) which lies between and .

## Authors’ Affiliations

## References

- Bohner M, Peterson A:
*Dynamic Equations on Time Scales, An Introduction with Application*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Bohner M, Peterson A:
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHView ArticleGoogle Scholar - Bandle C, Kwong MK:
**Semilinear elliptic problems in annular domains.***Journal of Applied Mathematics and Physics*1989,**40**(2):245-257. 10.1007/BF00945001MATHMathSciNetView ArticleGoogle Scholar - Wang H:
**On the existence of positive solutions for semilinear elliptic equations in the annulus.***Journal of Differential Equations*1994,**109**(1):1-7. 10.1006/jdeq.1994.1042MATHMathSciNetView ArticleGoogle Scholar - Chyan CJ, Henderson J:
**Twin solutions of boundary value problems for differential equations on measure chains.***Journal of Computational and Applied Mathematics*2002,**141**(1-2):123-131. 10.1016/S0377-0427(01)00440-XMATHMathSciNetView ArticleGoogle Scholar - Gatica JA, Oliker V, Waltman P:
**Singular nonlinear boundary value problems for second-order ordinary differential equations.***Journal of Differential Equations*1989,**79**(1):62-78. 10.1016/0022-0396(89)90113-7MATHMathSciNetView ArticleGoogle Scholar - Avery R, Henderson J:
**Existence of three positive pseudo-symmetric solutions for a one dimensional****-Laplacian.***Journal of Mathematical Analysis and Applications*2001,**42:**593-601.MathSciNetGoogle Scholar - Cabada A:
**Extremal solutions for the difference****-Laplacian problem with nonlinear functional boundary conditions.***Computers & Mathematics with Applications*2001,**42**(3-5):593-601. 10.1016/S0898-1221(01)00179-1MATHMathSciNetView ArticleGoogle Scholar - He XM:
**The existence of positive solutions of****-Laplacian equation.***Acta Mathematica Sinica*2003,**46**(4):805-810.MATHMathSciNetGoogle Scholar - Anderson DR:
**Solutions to second-order three-point problems on time scales.***Journal of Difference Equations and Applications*2002,**8**(8):673-688. 10.1080/1023619021000000717MATHMathSciNetView 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 - Akin E:
**Boundary value problems for a differential equation on a measure chain.***Panamerican Mathematical Journal*2000,**10**(3):17-30.MATHMathSciNetGoogle Scholar - Kaufmann ER: Positive solutions of a three-point boundary-value problem on a time scale. Electronic Journal of Differential Equations 2003, (82):1-11.Google Scholar
- Agarwal RP, O'Regan D:
**Triple solutions to boundary value problems on time scales.***Applied Mathematics Letters*2000,**13**(4):7-11. 10.1016/S0893-9659(99)00200-1MathSciNetView ArticleGoogle Scholar - Song C, Weng P:
**Multiple positive solutions for****-Laplacian functional dynamic equations on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(1):208-215. 10.1016/j.na.2006.10.043MATHMathSciNetView 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 - Liu B:
**Positive solutions of three-point boundary value problems for the one-dimensional****-Laplacian with infinitely many singularities.***Applied Mathematics Letters*2004,**17**(6):655-661. 10.1016/S0893-9659(04)90100-0MATHMathSciNetView 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 - Krasnosel'skii MA:
*Positive Solutions of Operator Equations*. P. Noordhoff, Groningen, The Netherlands; 1964:381.Google Scholar - Avery RI, Henderson J:
**Two positive fixed points of nonlinear operators on ordered Banach spaces.***Communications on Applied Nonlinear Analysis*2001,**8**(1):27-36.MATHMathSciNetGoogle 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.