Existence of Positive Solutions of Boundary Value Problem For FractionalOrder Differential Equation

Abstract

Recently boundary value problems for differential equations of non-integral order have studied in many papers ( see [1,2] ). Zaho etal [ 1 ] studied the following boundary value problem of fractional differential equations. Where denotes the Rimann-Liouville fractional derivative equation of order . By using the lower and upper solution method and fixed point theorem. Liang and Zhang [3] studied the non-linear fractional differential boundary value problem Where is a real number . is the Rimann-Liouville fractional differential operator of order . By means of fixed point theorems , they obtained results on the existence of positive solutions for boundary value problem of fractional differential equations. In this paper , we deal with some existence of positive solution of the following non-linear fractional differential equation. Where is a real number. denotes Rimann-Liouville fractional derivative of order . Our work based on Banach contraction mapping and Krasnoel'skii fixed point theorems to investigate the existence of positive solution. Finally , we suggest studing the existence solutions for the following Integrodifferential equation with boundary value conditions Where H is a nonlinear integral operator given as