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

‎In this paper‎, ‎we study the boundary-value problem of fractional‎ ‎order dynamic equations on time scales‎, ‎‎cΔαu(t)=f(t,u(t)),t∈‎‎[0,1]Tκ2:=J,1 α 2‎,‎ ‎‎u(0)+uΔ(0)=0,u(1)+uΔ(1)=0‎,‎ ‎where T is a general time scale with 0,1∈T‎, ‎cΔα is the Caputo Δ-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.