Solutions of abstract integro‐differential equations via Poisson transformation

We study the initial value problem( * )u ( n + 1 ) − u ( n ) = A u ( n + 1 ) + ∑ k = 0 n + 1a ( n + 1 − k ) A u ( k ) , n ∈ N 0u ( 0 ) = x ,where A is closed linear operator defined on a Banach space X , x belongs to the domain of A , and the kernel a is a particular discretization of an integrable kernel a ∈ L 1 ( R + ) . Assuming that A generates a resolvent family, we find an explicit representation of the solution to the initial value problem (*) as well as for its inhomogeneous version, and then we study the stability of such solutions. We also prove that for a special class of kernels a , it suffices to assume that A generates an immediately norm continuous C 0 ‐semigroup. We employ a new computational method based on the Poisson transformation.