Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

Bu, Shangquan

doi:10.1007/s00020-011-1895-y

Cited by 17 publications

(11 citation statements)

References 17 publications

(46 reference statements)

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“…Some other authors continued this avenue of research. See for instance [4], [10], [11], [12] and [31]. More recently, methods from operator-valued Fourier multipliers were used in [23], [24], and [25] to successfully characterize the existence and uniqueness of p -solutions for discrete time fractional models.…”

Section: Introductionmentioning

confidence: 99%

Well posedness for semidiscrete fractional Cauchy problems with finite delay

Lizama¹,

Murillo‐Arcila²

2018

Journal of Computational and Applied Mathematics

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We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delaywhere T is a bounded linear operator defined on a Banach space X (typically a space of functions like L p (Ω), 1 < p < ∞) and ∆ α corresponds to the time discretization of the continuous Riemann-Liouville fractional derivative by means of the Poisson distribution. We characterize the existence and uniqueness of solutions in vector-valued Lebesgue spaces of sequences of the model (0.1) in terms of boundedness of the operator-valued symbolwhenever 0 < α ≤ 1 and X satisfies a geometrical condition. For this purpose, we use methods from operator-valued Fourier multipliers and resolvent operator families associated to the homogeneous problem. We apply this result to show a practical and computational criterion in the context of Hilbert spaces.

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Section: Introductionmentioning

confidence: 99%

Well posedness for semidiscrete fractional Cauchy problems with finite delay

Lizama¹,

Murillo‐Arcila²

2018

Journal of Computational and Applied Mathematics

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“…They are called U M D-spaces. See also [11], [10] and [9] for more information on this topic and related work.…”

Section: Introductionmentioning

confidence: 99%

Maximal regularity in l spaces for discrete time fractional shifted equations

Lizama

Murillo‐Arcila

2017

Journal of Differential Equations

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In this paper, we are presenting a new method based on operator-valued Fourier multipliers to characterize the existence and uniqueness of p -solutions for discrete time fractional models in the formwhere A is a closed linear operator defined on a Banach space X and ∆ α denotes the Grünwald-Letnikov fractional derivative of order α > 0. If X is a U M D space, we provide this characterization only in terms of the R-boundedness of the operator-valued symbol associated to the abstract model. To illustrate our results, we derive new qualitative properties of nonlinear difference equations with shiftings, including fractional versions of the logistic and Nagumo equations.

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“…We shall refer to the seminal paper [7] as a standard reference for maximal regularity. We also refer the reader to [5,6,3] for the L p -maximal regularity of second-order Cauchy problems, to [1,2] for maximal regularity for non-autonomous problems, to [14,17] for integro-differential equations and to [4,13] for fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

On maximal regularity for a class of evolutionary equations

Picard

Trostorff

Waurick³

2017

Journal of Mathematical Analysis and Applications

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The issue of so-called maximal regularity is discussed within a Hilbert space framework for a class of evolutionary equations. Viewing evolutionary equations as a sums of two unbounded operators, showing maximal regularity amounts to establishing that the operator sum considered with its natural domain is already closed. For this we use structural constraints of the coefficients rather than semi-group strategies or sesqui-linear form methods, which would be difficult to come by for our general problem class. Our approach, although limited to the Hilbert space case, complements known strategies for approaching maximal regularity and extends them in a different direction. The abstract findings are illustrated by re-considering some known maximal regularity results within the framework presented.

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Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

Cited by 17 publications

References 17 publications

Well posedness for semidiscrete fractional Cauchy problems with finite delay

Well posedness for semidiscrete fractional Cauchy problems with finite delay

Maximal regularity in l spaces for discrete time fractional shifted equations

On maximal regularity for a class of evolutionary equations

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