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: ${\mathbf{D}}_t^{{\alpha }_n}u(t)+{\sum }_{i=1}^{n-1}{A}_i{\mathbf{D}}_t^{{\alpha }_i}u(t)=A{\mathbf{D}}_t^{\alpha }u(t)+f(t)$, $t>0$, $u^{(k)}(0)=u_k$, $k=0,\mathrm{...},\lceil {\alpha }_n\rceil -1$, where $n\in \mathbb{N}\setminus \{1\}$, $A$ and $A_1,\mathrm{...},A_{n-1}$ are closed linear operators on a sequentially complete locally convex space , , $f(t)$ is an $E$-valued function, and ${\mathbf{\text{D}}}_t^{\alpha }$ 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), Kostić (Preprint), and Xiao and Liang (2003, 2002). The obtained results are illustrated with several examples.