On a class of abstract time-fractional equations on locally convex spaces

Abstract

This paper is devoted to the study of abstract time-fractional equations of the following form: Dtαn u(t) + =;i=1n-1 AiDtα;i u (t) = ADtαu(t) + f(t), t > 0, u(k)(0) = uk, k = 0,⋯, ⌈αn⌉ - 1, where n ε ℕ\ {1}, A and A1,⋯, An-1 are closed linear operators on a sequentially complete locally convex space E, 0 ≤ α1 ⋯ < αn, 0 ≤ α < αn, f (t) is an E -valued function, and Dtα denotes the Caputo fractional derivative of order α (Bazhlekova (2001)). We introduce and systematically analyze various classes of k -regularized (C1, C2)-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. Copyright © 2012 Marko Kostić et al.