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*} {$*$} \{ {array}{rll} _CΔ α u(n) &= Au(n+1), n ∈ N_0; \\ u(0) &= u_0 ∈ 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