Maximal regularity in $l_p$ spaces for discrete time fractional shifted equations

Abstract

In this paper, we are presenting a new method based on operator-valued Fourier multipliers to \- characterize the existence and uniqueness of $\ell_p$-solutions for discrete time fractional models in the form $$ \Delta^{\alpha}u(n,x) = Au(n ,x) + \sum_{j=1}^k \beta_j u(n-\tau_j,x) +f(n,u(n,x)),\,\,\, n \in \mathbb{Z}, x \in \Omega \subset \mathbb{R}^N, \beta_j\in\mathbb{R}\hspace{0.1cm}\mbox{and}\hspace{0.1cm} \tau_j \in \mathbb{Z}, $$ where $A$ is a closed linear operator defined on a Banach space $X$ and $\Delta^{\alpha}$ denotes the Gr\"unwald-Letnikov fractional derivative of order $\alpha>0.$ If $X$ is a $UMD$ 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.