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:Dtαnu(t)+∑i=1n−1AiDtαiu(t)=ADtαu(t)+f(t),t>0,u(k)(0)=uk,k=0,...,⌈αn⌉−1, wheren∈ℕ∖{1},AandA1,...,An−1are closed linear operators on a sequentially complete locally convex spaceE,0≤α1<⋯<αn,0≤α<αn,f(t)is anE-valued function, andDtαdenotes the Caputo fractional derivative of orderα(Bazhlekova (2001)). We introduce and systematically analyze various classes ofk-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.