Equations in Banach spaces with a degenerate operator under a fractional derivative

We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter-Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input. Mathematics 2019, 7, 735 2 of 15 homogeneous ( f ≡ 0) Equation (1). The existence of a unique solution of the Cauchy problem and of the Showalter-Sidorov problem

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“…Hence we can study problem (25), (26) and (28) with α ∈ (0, 1] analogously. Analogously, we can obtain the next assertion by the obvious way.…”

Section: Application To An Initial-boundary Value Problemmentioning

confidence: 99%

Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

et al. 2019

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“…The basic aim is to introduce the study of inverse problems related to degenerate fractional integro-differential equations, extending the previous results of Al Horani and Favini [1], Al Horani et al [2][3][4][5] and Favaron et al [6]. Completely different methods were used by Fedorov and Ivanova [7], Sviridyuk and Fedorov [8] together with many papers from their school, see References [7][8][9][10][11][12][13], see also [14][15][16][17][18][19][20][21] and the monograph of Bazhlekova [22]. Let us also remind, in particular [23,24] where the authors considered equations of Sobolev type, with nonlocal conditions, of the form D q (Bu(t)) = Au(t)…”

Section: Introductionmentioning

confidence: 99%

Inverse Problems for Degenerate Fractional Integro-Differential Equations

et al. 2020

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This paper deals with inverse problems related to degenerate fractional integro-differential equations in Banach spaces. We study existence, uniqueness and regularity of solutions to the problem, claiming to extend well known studies for the case of non-fractional equations. Our method is based on transforming the inverse problem to a direct problem and identifying the conditions under which this direct problem has a unique solution. The conditions under which the unique strict solution can be compared with the case of a mild solution, obtained in previous studies under quite restrictive requirements, are on the underlying functions. Applications from partial differential equations are given to illustrate our abstract results.

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“…Set, for short, E α (z) := E α,1 (z), z ∈ C. For more details about the Mittag-Leffler functions, the Wright functions, fractional calculus and fractional differential equations, one may refer e.g. to [3], [12], [15]- [17], [19]- [21], [27]- [28] and [31].…”

Section: Introductionmentioning

confidence: 99%

Abstract degenerate Volterra integro-differential equations: inverse generator problem

Kostić¹

2019

Preprint

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In this paper, we investigate the inverse generator problem for abstract degenerate Volterra integro-differential equations in locally convex spaces. The classes of degenerate (a, k)-regularized C-resolvent families and degenerate mild (a, k)-regularized (C 1 , C 2 )-existence and uniqueness families play an important role in our analysis. We provide several illustrative applications of obtained theoretical results, primarily to abstract degenerate fractional differential equations with Caputo derivatives.

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Equations in Banach spaces with a degenerate operator under a fractional derivative

Cited by 31 publications

References 5 publications

Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

Inverse Problems for Degenerate Fractional Integro-Differential Equations

Abstract degenerate Volterra integro-differential equations: inverse generator problem

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