TIME - FRACTIONAL EQUATIONS Marko Kostić

The fractional calculus is one of the active research fields in mathematical analysis, primarily from its importance in modeling of various problems in engineering, physics, chemistry and other sciences. Presumably the first systematic exposition on abstract time-fractional equations with Caputo fractional derivatives is that of Bazhlekova [2]. In this fundamental work, the abstract time-fractional equations with Caputo fractional derivatives have been studied by converting them into equivalent abstract Volterra equations [17]. The reading of paper [7] by Kisyński served as a starting point for the genesis of this paper. We shall prove a generalization of the assertion [7, Theorem 1, (a)⇒ (b)] for abstract time-fractional equations (Theorem 2.1, Remark 2.1). The second aim of the paper is to generalize [3, Theorem 14.1] to abstract time-fractional equations (Theorem 2.2), and to clarify some classes of sequentially complete locally convex spaces in which the above-mentioned result admits a reformulation (Theorem 2.3). Throughout the paper, we assume that E is a Hausdorff sequentially complete locally convex space, SCLCS for short, and that the abbreviation ⊛ stands for the fundamental system of seminorms which defines the topology of E. By L(E) is denoted the space of all continuous linear mappings from E into E. The domain, resolvent set, spectrum and range of a closed linear operator A acting on E are