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The paper is devoted to understand the large time behaviour and decay of the solution of the discrete heat equation in the one dimensional mesh $\Z$ on $\ell^p$ spaces, and its analogies with the continuous-space case. We do a deep study of the moments of the discrete gaussian kernel (which is given in terms of Bessel functions), in particular the mass conservation principle; that is reflected on the large time behaviour of solutions. We prove asymptotic pointwise and $\ell^p$ decay results for the fundamental solution. We use that estimates to get rates on the $\ell^p$ decay and large time behaviour of solutions. For the $\ell^2$ case, we get optimal decay by use of Fourier techniques., Comment: pp 25

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^\beta_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 $b\in\ell^1(\Z)$, 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., Comment: 19

In this paper we introduce a discrete fractional resolvent family $\{S_{\alpha,\beta}^n\}_{n\in\mathbb{N}_0}$ generated by a closed linear operator in a Banach space $X$ for a given $\alpha,\beta>0.$ Moreover, we study its main properties and, as a consequence, we obtain a method to study the existence and uniqueness of the solutions to discrete fractional difference equations in a Banach space., Comment: 18 pages, 2 figures