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 (∗ ₎ ∆ ^α u(n) = Au(n + 2) + f(n, u(n)), n ∈ N ₀ , 1 < α ≤ 2; u(0) = u0; u(1) = u1, 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.