Existence Of Solutions For Systems Of Nonlinear Coupled Fractional Differential Equations

Nonlinear functional analysis has widespread application in applied mathematics, and the existence of solutions for fractional differential systems has been widespread concern.In this paper, we mainly study the existence of solutions for nonlinear fractional order coupled differential equations.The thesis is divided into two chapters:The chapter 1, we study the following a coupled systems of fractional differential equations with coupled integral boundary condition where D0+?, D0+? is Riemann-Liouville fractional derivative, 0<??1<??2, 0<??1 < ??2,?-?> 1, ?-?> 1, A, B is bounded variation function,?01D0+?y(s)dA(s),?01D0+?x(s)dB(s) is Riemann-Stieltjes fractional integral. And the non-linear term f1(t,x,y,z), f2(t,x,y,z) may be singular at t=0,1, x=y=z=0. By using the Guo-Krasnosel'skii fixed point theorem, we gain the existence and uniqueness of solutions. Compared with the document [1], the equations (1.1.1) consider the existence of positive solutions for fractional differential equations with nonlinear boundary condi-tions. Compared with the document [2], in equation (1.1.1) the nonlinear term contains their fractional derivative, we solve this problem by transforming it into a lower order e-quations. Compared with the document [3], the equation (1.1.1) considered the fractional differential systems.The chapter 2, we study the following a coupled system of fractional differential equations with integral boundary conditions where cDq is Caputo fractional derivative,f1, f2 ? C([0,1] x [0, +?), (-?, +?)); n-1 <??n, A, B is bounded variation function, (?S,(?) is Riemann-Stieltjes fractional integral. By using the Banach's contraction mapping principle and Leray-Schauder alternative. We gain the existence results for the given problem. Com-pared with the document [4], in the equations (2.1.1) considered the Caputo type fractional differential and the integral boundary value condition is changed to Riemann-Stieltjes in-tegral, it has become more extensive. Compared with the document [5], in the equations(2.1.1) considered the fractional differential equations and the order of the equations has become n.