Maximal ℓp-regularity of multi-term fractional equations with delay.

Abstract

We provide a characterization for the existence and uniqueness of solutions in the space of vector-valued sequences $\ell_p(\mathbb{Z},X)$ for the multi-term fractional delayed model in the form: $$ \Delta^\alpha u(n)+\lambda \Delta^\beta u(n)=Au(n)+u(n-\tau)+f(n),~n\in\mathbb{Z},\alpha, \beta\in\mathbb{R}_{+},\tau\in\mathbb{Z},~\lambda\in\mathbb{R} $$ where X is a Banach space, A is a closed linear operator with domain D(A) defined on X, $f\in \ell_p(\mathbb{Z},X)$ and $\Delta^{\gamma}$ denotes the Gr\"unwald-Letkinov fractional derivative of order $\gamma>0$. We also give some conditions to ensure the existence of solutions when adding nonlinearities. Finally, we illustrate our results with an example given by a general abstract nonlinear model that includes the fractional Fisher equation with delay.