Existence results for boundary value problems of fractional functional differential equations with delay

Abstract

In this paper, we study boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative with delay $$\begin{aligned} \left\{ \begin{array}{ll} ^CD^{\alpha }u(t)+f(t,u_{t})=0,\; t \in [0,T],\\ u_{0}=\varphi ,\ \ u(T)=A, \ \ u''(0)=0 \end{array} \right. \end{aligned}$$ where $$f:[0,T]\times C[-r,0]\rightarrow {\mathbb {R}}$$ is continuous function, $$\varphi \in C[-r,0]$$ and $$A \in {\mathbb {R}},$$ $$2<\alpha \le 3$$ , $$0\le r\le T.$$ We use Green function to reformulate the boundary value problems into an abstract operator equation. By means of Guo–Krasnosel’skii fixed point theorem on cone, Banach contraction principle and Schaefer’s fixed point theorem, some existence results of solutions and positive solutions are obtained, respectively. As applications, some examples are presented to illustrate the main results.