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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.

We introduce a two-kernel dependent family of strong continuous operators defined in a Banach space, which allows us to consider in an unified treatment the notions of, among others, C -semigroups of operators, cosine families, n-times 0 integrated semigroups, resolvent families and k-generalized solutions.The results are applied to the study of existence and uniqueness of solutions for the Volterra equation of convolution type u s f q a) Au, in the case A is not necessarily densely defined. Examples for equations defined in L p spaces are also given. ᮊ

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Let T be a bounded linear operator defined on a U M D Banach space X . We introduce an operator theoretical method for linear fractional difference equations based on the notion of α-resolvent sequence of bounded and linear operators. Then, we define and characterize the l p -maximal regularity of solutions for the problemsolely in terms of the R-boundedness of the set