- Research Article
- Open Access
© S. Gulsan Topal et al. 2009
- Received: 16 March 2009
- Accepted: 20 July 2009
- Published: 19 August 2009
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.
- Banach Space
- Dynamic Equation
- Convex Subset
- Fixed Point Theorem
- Continuous Operator
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 , Chyan and Henderson , Song and Weng , Sun and Li , and Liu , 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.
which is a contradiction.
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  (Krasnoselskii fixed point theorem)).
Theorem 2.7 (see  (Schauder fixed point theorem)).
Theorem 2.8 (see  (Avery-Henderson fixed point theorem)).
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.
Existence of at least one positive solution is also proved using Schauder fixed point theorem (Theorem 2.7). Then we have the following result.
We now show that the conditions of Theorem 2.8 are satisfied.
So condition (ii) of Theorem 2.8 holds.
which is a contradiction.
- 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
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