Resolving operators of degenerate evolution equations with fractional derivative with respect to time

Fedorov, Vladimir E.; Gordievskikh, Dmitriy M.

doi:10.3103/s1066369x15010065

Cited by 29 publications

(9 citation statements)

References 3 publications

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“…Hence, the solution exists also for equation (10) and it is of the form ( ) = ,1 ( ) 1 0 = ,1 ( ) 0 . The statement of the theorem on the phase space of equation (11) can be proved in the same way.…”

Section: Homogeneous Degenerate Equationmentioning

confidence: 83%

“…There are also works [5]- [10], in which fractional differential equations were studied in locally convex spaces. The present work provides the generalizations of results of work [11], in which the solvability of Cauchy problem (2) for homogeneous equation (1) in a Banach space was studied by using the conditions of ( , )-boundedness of the operator [12] introduced in studying a degenerate equation of order = 1. In the present work we employ the notion of ( , )-regular operator used before in [13] for studying first order degenerate equation in locally convex spaces.…”

Section: Introductionmentioning

confidence: 98%

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Degenerate fractional differential equations in locally convex spaces with a $\sigma$-regular pair of operators

Kostić¹,

Fedorov²

2016

Ufimskii Matematicheskii Zhurnal

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We consider a degenerate fractional order differential equation () = () in a Hausdorff sequentially complete locally convex space. Under the-regularity of the operator pair (,), we find the phase space of the equation and the family of its resolving operators. We show that the identity image of the latter coincides with the phase space. We prove an unique solvability theorem and obtain the form of the solution to the Cauchy problem for the corresponding inhomogeneous equation. We give an example of application the obtained abstract results to studying the solvability of the initial boundary value problems for the partial differential equations involving entire functions on an unbounded operator in a Banach space, which is a specially constructed Frechét space. It allows us to consider, for instance, a periodic in a spatial variable problem for the equation with a shift along and with a fractional order derivative with respect to time .

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Section: Homogeneous Degenerate Equationmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 98%

Degenerate fractional differential equations in locally convex spaces with a $\sigma$-regular pair of operators

Kostić¹,

Fedorov²

2016

Ufimskii Matematicheskii Zhurnal

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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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“…It is worth noting that first‐order equations degenerating not only on the kernel but also on the M ‐adjoint vectors of operator L were researched in . Note also the papers concerning some classes of the linear degenerate fractional differential equations of the form with N ≡ 0 and the works devoting to the controllability of the equations in a Banach space that are not resolved with respect to the fractional order derivative.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear equations with degenerate operator at fractional Caputo derivative

Плеханова

2016

Math Methods in App Sciences

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At first, the existence of a unique solution for the Cauchy problem to nondegenerate fractional differential equation was proved. These results were used for research of the unique solvability for the initial Cauchy and Showalter–Sidorov problems to differential equations in Banach spaces with degenerate operator at fractional Caputo derivative in linear and nonlinear cases. Abstract results are applied to the research of an initial boundary value problem for time‐fractional order Oskolkov system of equations. Copyright © 2016 John Wiley & Sons, Ltd.

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Resolving operators of degenerate evolution equations with fractional derivative with respect to time

Cited by 29 publications

References 3 publications

Degenerate fractional differential equations in locally convex spaces with a $\sigma$-regular pair of operators

Degenerate fractional differential equations in locally convex spaces with a $\sigma$-regular pair of operators

Inverse Problems for Degenerate Fractional Integro-Differential Equations

Nonlinear equations with degenerate operator at fractional Caputo derivative

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