Regularity of Difference Equations on Banach Spaces

Agarwal, Ravi P.; Cuevas, Claudio; Lizama, Carlos

doi:10.1007/978-3-319-06447-5

Cited by 34 publications

(52 citation statements)

References 121 publications

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“…We immediately obtain a characterization of maximal p -regularity for Hilbert spaces, since each Hilbert space is U M D and, in such case, R-boundedness coincides with boundedness [2].…”

Section: Maximal P -Regularity Of the Linear Shifted Modelmentioning

confidence: 94%

“…Now, we recall the following Fourier multiplier theorem for operator-valued symbols given by S. Blunck [7,2]. This theorem corresponds to the discrete version of a notable result independently proven by Weis [33] and Amann [3] which provides sufficient conditions to ensure when an operator-valued symbol is a multiplier.…”

Section: R-boundedness and Blunck's Theoremmentioning

confidence: 96%

“…Now, we recall the following Fourier multiplier theorem for operator valued symbols given by S. Blunck [7,2]. Since we are dealing with unbounded operators we need to state a slight modification of Blunck's theorem concerning two Banach spaces instead of one.…”

Section: R-boundedness and Blunck's Theoremmentioning

confidence: 99%

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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 immediately obtain a characterization of maximal p -regularity for Hilbert spaces, since each Hilbert space is U M D and, in such case, R-boundedness coincides with boundedness [2].…”

Section: Maximal P -Regularity Of the Linear Shifted Modelmentioning

confidence: 94%

Section: R-boundedness and Blunck's Theoremmentioning

confidence: 96%

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Maximal regularity in l spaces for discrete time fractional shifted equations

Lizama

Murillo‐Arcila

2017

Journal of Differential Equations

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“…For a recent review of this topic in the context of discrete models on Banach spaces, see the monograph [3] and references therein. Although research in this area has been done, there are many interesting questions related to the study of fractional difference equations that remain unanswered.…”

Section: Introductionmentioning

confidence: 99%

“…We point out that characterizations of maximal regularity for evolution equations using methods of operator valued Fourier multiplier theorems has been already studied (see for example [3]). For instance, S. Bu in [11] and [12] used Fourier multipliers to characterize the Lebesgue maximal regularity of fractional evolution equations in compact intervals.…”

Section: Introductionmentioning

confidence: 99%

$\ell_{p}$ -maximal regularity for a class of fractional difference equations on UMD spaces: The case $1\lt \alpha\leq2$

Lizama¹,

Murillo‐Arcila²

2017

Banach J. Math. Anal.

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Abstract. By using Blunck's operator-valued Fourier multiplier theorem, we completely characterize the existence and uniqueness of solutions in Lebesgue spaces of sequences for a discrete version of the Cauchy problem with fractional order 1 < α ≤ 2. This characterization is given solely in spectral terms on the data of the problem, whenever the underlying Banach space belongs to the U M D-class.

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