%0 Journal Article 
%A Leal, C.
%A Lizama, Carlos
%A Murillo Arcila, Marina
%T Lebesgue regularity for differential difference equations with fractional damping
%D 2018
%@ 1099-1476

%U http://hdl.handle.net/10498/35418
%X 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.
%K Delay
%K Differential difference equations
%K Fractional differences
%K Lebesgue maximal regularity
%~ Universidad de Cádiz