The Existence of Solutions for Boundary Value Problem of Fractional Functional Differential Equations with Delay

A class of boundary value problem for fractional functional differential equation with delay C D σ ω ( t ) = f ( t , ω t ) , t ∈ [ 0 , ζ ] , ω ( 0 ) = 0 , ω ′ ( 0 ) = 0 , ω ″ ( ζ ) = 1 , $ \left\{ {\begin{array}{*{20}c} {^{C} D^{\sigma } \omega (t) = f(t,\omega _{t} ),t \in [0,\zeta ],} \\ {\omega (0) = 0,\,\omega ^{\prime}(0) = 0,\,\omega ^{\prime\prime}(\zeta ) = 1,} \\ \end{array} } \right. $ is studied, where 2 < σ ≤ 3 , c D σ $ 2 < \sigma \le 3,\,\,^{c} D^{\sigma } $ devote standard Caputo fractional derivative. In this article, three new criteria on existence and uniqueness of solution are obtained by Banach contraction mapping principle, Schauder fixed point theorem and nonlinear alternative theorem.