Degenerate multi-term fractional differential equations in locally convex spaces

Abstract

We investigate, in the setting of sequentially complete locally convex spaces, degenerate multi-term fractional differential equations with Caputo derivatives. The obtained theoretical results are illustrated with some examples. [Projekat Ministarstva nauke Republike Srbije, br. 174024]

Degenerate abstract Volterra equations in locally convex spaces

“…, in contrast with the assertions of Theorem 4.2-Theorem 4.3 below, which can be applied only in the case that λ > 0 (cf. [22] for more details); as observed by G. A. Sviridyuk, this equation is important in evolution modeling of some problems appearing in the theory of liquid filtration, see e.g. [11, p. 6].…”

“…It should be also observed that (G α (t)) t≥0 is an exponentially equicontinuous ( α , C)-regularized resolvent family generated by P 1 (A), P 2 (A) (cf. [22] for the notion and more details), and that for each f ∈ D(P 1 (A)) ∩ D(P 2 (A)), the function u(t) := R α (t)x, t ≥ 0 is a unique solution of the following Cauchy problem:…”

“…A similar result holds for the second order degenerate equations considered in the next subsection, and for the first order degenerate equations, in the case that the requirements of Theorem 4.3 (the part (ii) of this remark, or Remark 4.5 below) hold; cf. [22].…”

“…The main purpose of this paper is to provide the basic information about degenerate exponentially equicontinuous (a, k)-regularized C-resolvent families and their applications in the study of abstract Cauchy problem (1). In our previous research studies [22]- [23], we have investigated degenerate abstract multi-term fractional differential equations with Caputo fractional derivatives, as well as their hypercyclic and topologically mixing properties.…”

Degenerate k-regularized (C1, C2)-existence and uniqueness families

“…The theory of abstract degenerate (multi-term) fractional differential equations is at its beginning stage and we can freely say that it is a still-undeveloped subject. The most important qualitative properties of abstract degenerate (multi-term) fractional differential equations have been recently considered in the papers [19]- [20], [22], [27]- [30] and [34]. The existence and uniqueness of solutions of the Cauchy and Showalter problems for a class of degenerate fractional evolution systems have been analyzed by V. E. Fedorov and A. Debbouche in [19], while the necessary and sufficient conditions for the relative p-boundedness of a pair of operators have been obtained by V. E. Fedorov and D. M. Gordievskikh in [20].…”