The Poisson distribution, abstract fractional difference equations, and stability

We study the initial value problem (∗) C Δ α u (n) a m p ; = A u (n + 1), n ∈ N 0 ; u (0) a m p ; = u 0 ∈ X, equation* {* \ {arrayrll C ^ u (n) \\ u (0) &= u₀ X, array. equation* when A A is a closed linear operator with domain D (A) D (A) defined on a Banach space X X. We introduce a method based on the Poisson distribution to show existence and qualitative properties of solutions for the problem (∗) (*), using operator-theoretical conditions on A A </inline-formu