On well-posedness of vector-valued fractional differential-difference equations

We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form {document}$ {equation*} (*) \{{array}{rll} Δα u(n) & = & Au(n+2) + f(n,u(n)), n ∈ N_0, \,\, 1< α ≤ 2; \\ u(0) & = & u_0;\\ u(1) & = & u_1, {array}. {equation*} document where A $ is a closed linear operator defined on a Banach space $ X $. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by $ A, $ and natural restrictions on the nonlinearity f. Finally we present some original examples to illustrate our results.