Almost automorphic mild solutions to fractional partial difference-differential equations

Abadías, Luciano; Lizama, Carlos

doi:10.1080/00036811.2015.1064521

Cited by 33 publications

(47 citation statements)

References 35 publications

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“…We success solving this open problem for the following abstract model

Δ^{α} u false(n false) + λ Δ^{β} u false(n false) = A u false(n false) + f false(n false), n \in double-struckZ, α, β > 0, λ \geq 0,

where

f \in ℓ_{p} false(double-struckZ, X false), A

is a closed linear operator with domain D ( A ) defined on X and Δ γ denotes the fractional difference operator of order γ >0 as defined recently in Abadías and Lizama . Roughly speaking, it corresponds to a slight variant of the Grünwald‐Letnikov derivative.…”

Section: Introductionmentioning

confidence: 99%

“…However, none of them have considered causal solutions, ie, solutions with domain on

double-struckZ .

On the other hand, some abstract models that are less general than ours have been recently reported in the literature and analyzed from diverse perspectives. For instance, in Abadias and Lizama, it is shown that the semilinear problem

Δ^{α} u false(n false) = A u false(n + 1 false) + f false(n, u false(n false) false), n \in double-struckZ, 0 < α \leq 1,

admits almost automorphic solutions whenever the operator A is the generator of a C 0 ‐semigroup and f satisfies Lipschitz conditions of global and local type. We note that in such paper, an application to a model of population of cells is also given.…”

Section: Introductionmentioning

confidence: 99%

“…We success solving this open problem for the following abstract model Δ u(n) + Δ u(n) = Au(n) + (n), n ∈ Z, , > 0, ≥ 0, (4) where ∈ (Z, X), A is a closed linear operator with domain D(A) defined on X and Δ denotes the fractional difference operator of order > 0 as defined recently in Abadías and Lizama. 9 Roughly speaking, it corresponds to a slight variant of the Grünwald-Letnikov derivative. Compare Definition 2.1 below with Ortigueira et al 10 , Section 3.3 formula (27).…”

Section: Introductionmentioning

confidence: 99%

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Lebesgue regularity for differential difference equations with fractional damping

Lizama

Murillo‐Arcila

Leal

2018

Math Methods in App Sciences

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We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector‐valued space of sequences ℓpfalse(double-struckZ,Xfalse) for equations that can be modeled in the form normalΔαu(n)+λnormalΔβu(n)=Au(n)+G(u)(n)+f(n),n∈Z,α,β>0,λ≥0, where X is a Banach space, f∈ℓpfalse(double-struckZ,Xfalse), A is a closed linear operator with domain D(A) defined on X, and G is a nonlinear function. The operator Δγ denotes the fractional difference operator of order γ>0 in the sense of Grünwald‐Letnikov. Our class of models includes the discrete time Klein‐Gordon, telegraph, and Basset equations, among other differential difference equations of interest. We prove a simple criterion that shows the existence of solutions assuming that f is small and that G is a nonlinear term.

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“…We success solving this open problem for the following abstract model

Δ^{α} u false(n false) + λ Δ^{β} u false(n false) = A u false(n false) + f false(n false), n \in double-struckZ, α, β > 0, λ \geq 0,

where

f \in ℓ_{p} false(double-struckZ, X false), A

Section: Introductionmentioning

confidence: 99%

“…However, none of them have considered causal solutions, ie, solutions with domain on

double-struckZ .

Δ^{α} u false(n false) = A u false(n + 1 false) + f false(n, u false(n false) false), n \in double-struckZ, 0 < α \leq 1,

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lebesgue regularity for differential difference equations with fractional damping

Lizama

Murillo‐Arcila

Leal

2018

Math Methods in App Sciences

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show abstract

“…In that paper, it is shown their strong connection, by means of the Poisson distribution, with the Riemann-Liouville fractional derivative on R + . We refer the reader also to the recent papers [1] and [18] where it is shown their usefulness in different contexts of research. Definition 2.2.…”

Section: Preliminariesmentioning

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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“…Since then, this pioneer work has attracted more and more attention and has been substantially extended in several different directions. Many authors have made important contributions to this theory (see, for instance, [5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein). Especially, in [5,6], the authors gave an important overview about the theory of almost automorphic functions and their applications to differential equations.…”

Section: Introductionmentioning

confidence: 99%

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Cao

Huang²,

N’Guérékata

2018

International Journal of Differential Equations

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This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations D ( ) = ( ) + D −1 ( , ( ), ( )), ∈ R, where 1 < < 2, is a linear densely defined operator of sectorial type on a complex Banach space and is a bounded linear operator defined on , is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

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Almost automorphic mild solutions to fractional partial difference-differential equations

Cited by 33 publications

References 35 publications

Lebesgue regularity for differential difference equations with fractional damping

Lebesgue regularity for differential difference equations with fractional damping

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

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

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