Existence of Solutions for Fractional Boundary Value Problem Involving p-Laplacian Operator

Abstract

In this paper, we investigate the question of existence of nonnegative solution for some fractional boundary value problem involving p-Laplacian operator, The results presented in this thesis are based on fixed point theorem, more precisely, Krasnosilski fixed point theorem, on the cones to prove the existence of a fixed point for a mathematics operator and that fixed point is a solution to the given fractional equation by combining some properties of the associated Green function. We will study the problem of boundary value and find the unique solution. We will present definition and properties of Riemann-Liouville derivatives, the Guo-Krasnoselskii fixed point theorem, the Green Function to find the positive solution of problem involving p-Laplacian. The main idea is to transfer the given problem to some integral equation; from this integral equation we define an operator with Green function. We prove that this operator is completely continuous and uniformly bounded which yields by Arzela Ascoli Theorem that this operator is compact. So by Krasnosilski fixed point theorem can find the result. This type of problem is very interesting in different areas such as biophysics, astronomy and others. Note that this problem can be extended to the fractional Hadamard derivative and to the Leggett Williams fixed point theorem; moreover, it can be the content of a rigorous thesis topic.