Existence of solutions of boundary value problems for Caputo fractional differential equations on time scales

Abstract

‎In this paper‎, ‎we study the boundary-value problem of fractional‎ ‎order dynamic equations on time scales‎,<br /> ‎$$‎<br /> ‎^c{Delta}^{alpha}u(t)=f(t,u(t)),;;tin‎<br /> ‎[0,1]_{mathbb{T}^{kappa^{2}}}:=J,;;1<alpha<2‎,<br /> ‎$$‎ ‎$$‎ ‎u(0)+u^{Delta}(0)=0,;;u(1)+u^{Delta}(1)=0‎, ‎$$‎<br /> ‎where $mathbb{T}$ is a general time scale with $0,1in mathbb{T}$‎, ‎$^c{Delta}^{alpha}$ is the Caputo $Delta$-fractional derivative‎. ‎We investigate the existence and uniqueness of solution for the‎ ‎problem by Banach's fixed point theorem and Schaefer's fixed point‎ ‎theorem‎. ‎We also discuss the existence of positive solutions of the‎ ‎problem by using the Krasnoselskii theorem.