Existence Of Solutions For Boundary Value Problems Of Several Class Of Nonlinear Fractional Differential Equations

By using the method of topological degree, the fixed point theory on cone, and the partially ordered method, the existence of solutions to several class of nonlinear fractional differential equations boundary value problems was considered in this thesis.The thesis is divided into four chapters.In the first chapter, the basic definitions and theories and some impor-tant fixed point theories of nonlinear functional analysis and fractional differential equations was introduced.In the second chapter, on the basis of [22] [23], the following boundary value problem for nonlinear fractional differential equation[D0?+u(t) + f(t, u(t)) = 0, t ? (0,1), 1 < ? ? 2(4)u(0)=0,u'(1)=0.was considered. By using the Barnach contraction mapping principle, we obtain the uniqueness of its positive solution, the existence of solution was considered by using Leray - Schauder alternative theorem, at the same time, by using theLeggett- Williams theorem, we obtain the multiplicity results.In the third chapter, on the basis of [24] [25], by using the Guo -Krasnoselskill's fixed point theory, the following problem D?x(t)+f(t,x(t-?)) = 0,t?(0, 1)\{?},x(t) =?(t), t ? [-?,0], (5)x'(0) = x"(0)=x"(1)=0.was discussed. We get the existence of positive solution to a singular boundary value problem for fractional differential equation with chang-ing sign nonlinearity.In the fourth chapter, on the basis of [31] [32] [33] [34], we study the following fractional differential equation with integral boundary condi-tion D0? +u(t)f(t,u(t))=0,0<t<1,(6)u(0) = u(1) = 0, u(1) = ?01g(s)u(s)ds.The uniqueness of positive solution was considered by using the fixed point theory, Barnach contraction mapping principle, the first eigenvalue and the first characteristic function of the operator.