On a Class of Abstract Time-Fractional Equations on Locally Convex Spaces. (11th November 2012) - Kostić, Marko
Abstract : This paper is devoted to the study of abstract time-fractional equations of the following form: D t α n u ( t ) + ∑ i = 1 n − 1 A i D t α i u ( t ) = A D t α u ( t ) + f ( t ), t > 0, u ( k ) ( 0 ) = u k, k = 0, ..., ⌈ α n ⌉ − 1, where n ∈ ℕ ∖ { 1 }, A and A 1, ..., A n − 1 are closed linear operators on a sequentially complete locally convex space E, 0 ≤ α 1 < ⋯ < α n, 0 ≤ α < α n, f ( t ) is an E -valued function, and D t α denotes the Caputo fractional derivative of order α (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), Kostić (Preprint), and Xiao and Liang (2003, 2002). The obtained results are illustrated with several examples.