Operational Calculi for Nonlocal Cauchy Problems in Resonance Cases

The class of the nonlocal Cauchy problems is a natural extension of the class of initial value Cauchy problems for LODEs with constant coefficients. A simple nonlocal Cauchy problem is the problem of determining the periodic solutions with a given period T of a LODE with constant coefficients. For a given linear functional Phi, the corresponding nonlocal Cauchy problem for a LODE with constant coefficients is determined by BVCs of the form Phi{y((k))} = 0, k = 0, 1, 2,..., n. Such problems arise naturally as problems for determining mean- periodic solutions of LODEs with constant coefficients. Two classes of nonlocal Cauchy problems are distinguished: non- resonance and resonance. For effective solution of both classes of problems Mikusi ' nski type operational calculi are used. They are based on a non- classical convolution proposed by one of the authors in 1974. Compared with previous publications of the authors, this paper is focussed on the resonance case.