Abstract time-fractional equations: Existence and growth of solutions

Abstract

We contribute to the existence theory of abstract time-fractional equations by stating the sufficient conditions for generation of not exponentially bounded α-times C-regularized resolvent families (α > 1) in sequentially complete locally convex spaces. We also consider the growth order of constructed solutions.MSC 2010 : Primary 47D06: Secondary 47D09, 47D60, 47D62, 47D99 Key Words and Phrases: abstract time-fractional equations, α-times C-regularized resolvent families IntroductionThroughout this 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. Henceforth A is a closed linear operator acting on E, L(E) C is an injective operator which satisfies CA ⊆ AC, and the convolution like mapping * is given by f * g(t) : Then (S α (t)) t≥0 is said to be locally equicontinuous, resp. exponentially equicontinuous, if for each T > 0, the family {S α (t) : t ∈ [0, T ]} is equicontinuous, resp. if there exists ω ≥ 0 such that the family {e −ωt S α (t) : t ≥ 0} is equicontinuous; if C = I, then it is also said that (S α (t)) t≥0 is an α-times regularized resolvent family having A as a subgenerator.It is worthwhile to mention that the local equicontinuity of (S α (t)) t≥0 automatically holds if E is a barrelled space ([9]).Let α > 0, let β > 0 and let the Mittag-Leffler functionThen it is well known ([2], [5]) that, for every α > 1, there exists c α ≥ 1 such that:

A note on the existence of solutions for some boundary value problems of fractional differential inclusions

“…In the literature there are several papers devoted to the study of the existence of solutions of boundary value problems associated to fractional differential equations [1,2,3,6,8,21] etc. The existence results are based, as in [4], mainly on nonlinear alternative of Leray-Schauder type and CovitzNadler contraction principle for set-valued maps.…”

On a class of time-fractional differential equations

“…See also her recent work [5]. For the fractional differential equations governed by some concrete partial differential operators, we refer to [20,21,22,23,25]. Let us recall the definition of the α-times resolvent families for (1.2).…”

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