© 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.
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.
2. Preliminaries and Lemmas
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)).
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.
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.
4. Monotone Method
which is a contradiction.
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