The Poisson distribution, abstract fractional difference equations, and stability

Abstract

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 , \begin{equation*} \tag {$*$} \left \{\begin {array}{rll} _C\Delta ^{\alpha } u(n) &= Au(n+1), \quad n \in \mathbb {N}_0; \\ u(0) &= u_0 \in X, \end{array}\right . \end{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 . We show how several properties for fractional differences, including their own definition, are connected with the continuous case by means of sampling using the Poisson distribution. We prove necessary conditions for stability of solutions, that are only based on the spectral properties of the operator A A in the case of Hilbert spaces.