Existence Of Solutions Of The Boundary Value Problems For Some Fractional Differential Equations

This paper is concerned with the existence of positive solutions of the boundary value problem for fractional differential equations,the first one is where CD0?+ is the standard Caputo fractional derivative , a ? (n — 1,n], n > 3, n? N, 0 < ? < 1, 0 < r) < 1, f: [G, 1] x [0,?) ? [0, ?) and the / meet the conditions of Caratheodory .The second one is where D0?+ is the standard Rimann—Liouville fractional derivative, a ? (1,2], 0 < ? < 1, 0 < ? < 1, ???-?-2 ? 1-?, a-?-1? 0, / : [0,?)? [0,?) ,and the / meet the conditions of Caratheodory. q : (0,1) —> [0, oo)is continuous, and JQ q(t)dt > 0.In chapter 2, through transfer differential equation for the corresponding inte-gral equation,and obtained the green function , then discuss the characters of green function, using the principle of Banach compressed image, cone - stretching theo-rem, nonlinear alternative theorems and Leggett — Williams fixed point theorem to research the first kind of fractional differential equation boundary value problem for the existence and uniqueness of positive solutions.In chapter 3, through transfer differential equation for the corresponding inte-gral equation,and obtained the green function , then discuss the characters of green function, using the principle of Banach compressed image, cone - stretching the-orem, Leggett — Williams fixed point theorem and topological degree theory to research the second kind of fractional differential equation boundary value problem for the existence and uniqueness of positive solutions.