Solutions Of Abstract Integro-Differential Equations Via Poisson Transformation

(*)u(n+1)-u(n)=Au(n+1)+ n-ary sumation k=0n+1a(n+1-k)Au(k),n is an element of N0u(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 is an element of L1(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