Study On The Existence Of Solutions For Two Kinds Of Fractional Differential Equations

In recent years,with the fractional differential equation widely used in practical prob-lems,the theoretical research and practical application of fractional differential equations have attracted the attention of scholars at home and abroad.Among them,the initial boundary value problem of fractional order differential equation and the existence prob-lem of the solution of nonlinear fractional order differential equation have become research hotspots.This paper deals with the existence of solutions for two kinds of fractional differential equations.First of all,we study the following problem that involving a class of fractional differential equations integral boundary value by using the method of convergence of fundamental columns in complete spaces:(?)Under an appropriate lipschitz condition,the existence and uniqueness of nontrivial so-lutions of above problem are obtained.Furthermore,an iterative sequence for the unique solution is also given.Where ??(3,4]is a real number and D?0+is the standard Riemann-Liouville fractional derivative of order a,f?C([0,1]x R,R).And then,the existence of positive solutions to fractional boundary value problems with semi-positive nonlinear terms is studied as follows:(?)by using the fixed index to establish two theorems for the existence of positive solutions,where ??(2,3]is a real number and D?0+ is the standard Riemann-Liouville fractional derivative of order ?,the nonlinear fi ? C([0,1]× R4+,R)(R+=[0,+?)).