Positive Solutions for Discrete Boundary Value Problems to One-Dimensional p-Laplacian with Delay

We study the existence of positive solutions for discrete boundary value problems to one-dimensional -Laplacian with delay. The proof is based on the Guo-Krasnoselskii fixed-point theorem in cones. Two numerical examples are also provided to illustrate the theoretical results.

In the real world, some processes are more reasonably described as -Laplacian differential equations with delay [2,8,9].The reason is that the differential of the unknown solutions depends not only on the values of the unknown solutions at the current time but also on the values prior to that.Such equations, to a certain extent, reflect much more exactly the physical reality than the equations without delay.
In reality, ( 5) is applied together with some boundary value conditions, see for example, [1,4,[8][9][10][11].By using the Guo-Krasnoselskii fixed-point theorem, Jin and Yin [9] proved the existence of one positive solutions for the following boundary value problem of one-dimensional -Laplacian with delay Based on a fixed-point approach, Bai and Xu [8] obtained the existence of positive solutions for problem (6) with (, ) having some singularities.
The main motivation of our work is twofold.The one is to prove the existence of positive solutions for discrete boundary value problems with delay.It is of interest to note here that the existence of single and multiple positive solutions for discrete boundary value problems to one-dimensional -Laplacian have been studied in great detail in the literature [12][13][14][15][16][17][18] and the references therein.However, there are few papers dealing with the existence of positive solutions for discrete boundary value problems to one-dimensional -Laplacian with delay.
The other motivation is coming from the numerical solutions of problem (6).In order to get the numerical solutions of problem (6), we can apply the standard Euler method to discretize problem (6) and approximate its solutions numerically.An immediate and natural question arises if the corresponding difference equation together with boundary conditions has positive solutions.
Motivated by above, our purpose in this paper is to show the existence of positive solutions for the discretization equations of problem (6).Namely, we will prove the existence of positive solutions of the following problem: Δ (  (Δ ( − 1))) +  ()  (,  ( −  0 )) = 0,  ∈ {1, 2, . . ., } ,  ( + 1) = 0,  () = 0,  ∈ {− 0 , − 0 + 1, . . ., 0} , where Δ() = ( + 1) − () and  0 is a positive integer.This paper is organized as follows.In Section 2, we introduce some basic definitions and then we state the Guo-Krasnoselskii fixed-point theorem.In Section 3, we write a representation for a solution to problem (7) in terms of the fixed point of an appropriate operator.Then we prove that problem (7) has a positive solution with  belonging to an open interval by employing the fixed-point theorem.In Section 4, two numerical examples are presented to illustrate the theoretical results.

Preliminaries
In this section, we introduce some basic definitions and recall the Guo-Krasnoselskii fixed-point theorem which plays a fundamental role in our subsequent analysis.
We collect some properties of the function   (⋅).

Existence of Positive Solutions
In this section, we study the existence of positive solutions for the following equation with delay: subject to the boundary condition We assume that (A1)  : continuous as a map of the topological space Z(1,  + 1) × R into the topological space R + .Throughout this paper, the topology on Z(1,  + 1) will be the discrete topology.
Proof.For any  ∈ , it is easily to check According to Lemma 3.2 in [1], we have ()() ≥ ()‖‖.Thus, () ⊂ .Moreover,   and  are both continuous, and  is a finite space.This implies that the operator  is a completely continuous.This completes the proof.
For convenience, we introduce some notations: If  is a positive solution of problems ( 8) and ( 9), then it is a fixed point of .So the existence of positive solution is transformed into the existence of fixed point.The following theorem is the main result of this paper.Theorem 6. Suppose that (A1) and (A2) hold.Assume that the delay time is appropriately small, say  0 is a positive integer not bigger than [/2].If  ∞ > 0,  0 < ∞, then there exists at least one positive solution to the boundary value problems (8) and (9) for  ∈ (max{ 1 ,  2 }, ), where while if  0 > 0,  ∞ < ∞, then for any  ∈ (max{ 1 ,  2 }, ), where problems ( 8) and ( 9) admit at least one positive solution  ∈ .

Numerical Illustration
In this section, we present two numerical experiments to illustrate our results.

Figure 1 :
Figure 1: The figures of the numerical solutions.