Lebesgue regularity for differential difference equations with fractional damping

Abstract

We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector-valued space of sequences $ \ell_p(\Z,X)$ for equations that can be modeled in the form $$ \Delta^{\alpha}u(n)+\lambda \Delta^{\beta}u(n)=Au(n) + G(u)(n) + f(n),\, n \in \Z, \, \alpha, \beta >0,\, \lambda \geq 0,$$ where $X$ is a Banach space, $f\in\ell_p(\Z,X),$ $A$ is a closed linear operator with domain $D(A)$ defined on $X$ and $G$ is a nonlinear function. The operator $\Delta^{\gamma}$ denotes the fractional difference operator of order $\gamma> 0$ in the sense of Gr\"unwald-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.