Fundamental solutions for semidiscrete evolution equations via Banach algebras

Abstract

We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups and cosine functions. We present the linear and algebraic structures (in particular, factorization properties) and their norms and spectra in the Lebesgue space of summable sequences. We identify fractional powers of these generators and apply to them the subordination principle. We also give some applications and consequences of our results.

Well-posedness for Cauchy fractional problem involving discrete convolution operators

“…e tA 1 = e ta 1 . See [12] and [21] for more information about the Banach Algebra ℓ 1 (Z) and result for discrete convolution operators. We refer [10,27] for more information about the semigroup theory.…”

“…This topic has been studied by the branch of mathematical analysis, using techniques from PDEs, functional and harmonic analysis, among others. We refer [8,11,12,18,19,20,22,36,37,38] and references therein.…”

“…In regards to fractional powers of difference operators, the discrete fractional Laplacian (−∆ d ) α and fractional backward /forward Euler operator ∆ α /(−∇) α have received special attention, due to the numerous applications, mainly diffusive and transport nature. We refer recent references [2,9,11,12,20,23].…”

“…A general method for the study of the system (1.1) was presented by González-Camus et. al in [12], where the solution (unique) was obtained explicitly in terms of Mittag-Leffler on the Banach Algebra ℓ 1 (Z), as well as an integral representation, based in subordination principles and functional calculus presented in [3] and [6], respectively. The explicit representation through Mittag-Leffler function is given by…”

Discrete Hölder spaces, their characterization via semigroups associated to the discrete Laplacian and kernels estimates

“…To that aim, we recall the definition of the fractional powers of the discrete Laplacian, by using the semigroup method, see [7,28,29]. For other works considering fractional powers of the discrete Laplacian see for instance [12,20].…”