Existence Of Solutions For The Initial And Boundary Value Problems Of Fractional Differential Equations

In this paper,we first study a class of initial value problems for fractional functional differential equations.By employing Darbo fixed point theorem as well as properties of measure compact sets in L1 space,a sufficient condition is obtained to guarantee the existence of solution in L1 space for this kind of problems.Noting that most of the works concerning the initial and boundary value problems for fractional differential equations are discussed in the continuous functions space,in the second part of the paper,we study the existence of solutions for a kind of periodic boundary value problem of nonlinear fractional differential equations in Lp space.Our analysis in this section relies on Shauder fixed point theorem and Kolmogorov-Riesz Theorem.Furthermore,we are concerned with the existence of solutions for a class of frac-tional functional differential equation with boundary value conditions.Some sufficient conditions are obtained to acquire the existence of positive solutions by means of fixed point theorems.