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, ‎$$‎ ‎^c{\Delta}^{\alpha}u(t)=f(t,u(t)),\;\;t\in‎ ‎[0,1]_{\mathbb{T}^{\kappa^{2}}}:=J,\;\;1 < \alpha < 2‎, ‎$$‎ ‎$$‎ ‎u(0)+u^{\Delta}(0)=0,\;\;u(1)+u^{\Delta}(1)=0‎, ‎$$‎ ‎where $\mathbb{T}$ is a general time scale with $0,1\in \mathbb{T}$‎, ‎$^c{\Delta}^{\alpha}$ is the Caputo $\Delta$-fractional derivative‎. ‎We investigate the existence and uniqueness of solution for the‎ ‎problem by Banachʹs fixed point theorem and Schaeferʹs fixed point‎ ‎theorem‎. ‎We also discuss the existence of positive solutions of the‎ ‎problem by using the Krasnoselskii theorem.