On a Class of Abstract Time-Fractional Equations on Locally Convex Spaces

This paper is devoted to the study of abstract time-fractional equations of the following form: D(t)(alpha n)u(t) + Sigma(n 1)(i=1) A(i)D(t)(alpha i)u(t) = AD(t)(alpha)u(t) + f(t), t > 0, u((k))(0) = u(k), k = 0, ... , [alpha(n)] - 1, where n is an element of N \ {1}, A and A(1), ... , A(n-1) are closed linear operators on a sequentially complete locally convex space E, 0 <= alpha(1) < ... < alpha(n), 0 <= alpha < alpha(n), f(t) is an E-valued function, and D-f(a) denotes the Caputo fractional derivative of order alpha (Bazhlekova (2001)). We introduce and systematically analyze various classes of k-regularized (C-1, C-2)-existence and uniqueness (propagation) families, continuing in such a way the researches raised in (de Laubenfels (1999, 1991), Kostic (Preprint), and Xiao and Liang (2003, 2002). The obtained results are illustrated with several examples.