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 frac- tional order dynamic equations on time scales, cΔαu(t) = f(t; u(t)); t 2 [0; 1]Tκ2 := J; 1 < α < 2; u(0) + uΔ(0) = 0; u(1) + uΔ(1) = 0; where 享 is a general time scale with 0; 1 ∈ 享, c?α is the Caputo ?- fractional derivative. We investigate the existence and uniqueness of so- lution 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. [ABSTRACT FROM AUTHOR]