Some Problems Of Solutions Of Fractional Differential Equations

Fractional differential equations have received considerable attention because of it's widely applications in various fields, such as in physics, mechanics, geology and automation, etc.In this paper, firstly we are concerned with the weighted Cauchy-type problem and proved that solutions of the nonlinear differential equations decay as a power function. Then we discussed the same equation with Cauchy initial conditions under different definitions of the operator and discussed the existence of solution and the continuous dependence of solutions on the order of equations. We also studied the existence of solution by applying contraction mapping methods to these equations.Finally we analyzed the existence of positive solutions to a class of system of sub-linear fractional differential equations in which the nonlinear terms subject to a power function. The main method is the application of Krasnosel skii fixed point theorem of cone expansion and compression type.