Well-posedness for Cauchy fractional problems involving discrete convolution operators

This work is focused on establishing sufficient conditions to guarantee the well-posedness of the following nonlinear fractional semidiscrete model \begin{equation*} \begin{cases} \mathbb D^β_t u(n,t)= B u(n,t) + f(n-ct,u(n,t)),, &n\in$\mathbb{Z}$, ;t>0, u(n,0)=\varphi(n),; &n\in$\mathbb{Z}$, \end{cases} \end{equation*} under the assumptions that $\beta \in (0,1]$, $c>0$ some constant, $B$ is a discrete convolution operator with kernel $b\in {\mathrm{\ell }}^1(\mathbb{Z})$, which is the infinitesimal generator of the Markovian $C_0$-semigroup and suitable nonlinearity $f$. We present results concerning the existence and uniqueness of solution, as well as establishing a comparison principle of solutions according to respective initial values.