Existence of Positive Solutions for Higher Order -Laplacian Two-Point Boundary Value Problems

We derive sufficient conditions for the existence of positive solutions to higher order -Laplacian two-point boundary value problem, , , , , , , , ; , , , , , and , where are continuous functions from to , and . We establish the existence of at least three positive solutions for the two-point coupled system by utilizing five-functional fixed point theorem. And also, we demonstrate our result with an example. 1. Introduction The goal of differential equations is to understand the physical phenomena of nature by developing mathematical models. Among all, a class of differential equations governed by nonlinear differential operators, which have wide applications and interest, has been developed to study such type of equations. In this theory, the most investigated operator is the classical -Laplacian, given by with . These problems have a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design. Due to the importance in both theory and applications, -Laplacian boundary value problems have created a great deal of interest in recent years; we mention a few [1–11]. Recently, Prasad and Murali [12] established the existence of positive solutions of -Laplacian singular boundary value problem on time scale, by assuming suitable conditions on . Till now in the literature of boundary value problems, the theory was not developed to the system of higher order boundary value problems with -Laplacian. Mainly, this type of problems arises in radar invention models and microatom invention models. Due to our interest in the literature, in this paper, we consider two-point higher order )-Laplacian boundary value problem (BVP) where are continuous functions from to ,？？ and . ？ If we take in the above problem then it reduces to -Laplacian problem. To obtain a solution of the BVP (2), we construct Green's functions for the corresponding homogeneous BVPs. For , let be Green's function of the BVP, and it is given by Let be Green's function of the BVP and it can be recursively defined as where is Green's function of and is given by Then can be expressed in the form Since , we have For , For , let be Green’s function of the BVP and it is given by Let be Green's function of the BVP and it can be recursively defined as where is Green’s function of and is given by Then can be expressed in the form Since , we have For , Further, it is easily seen that , , and , all are nonnegative on . A solution of the BVP (2) is a function such that , and satisfies the BVP (2). A positive solution of the BVP (2) is a solution of