Operational Calculi for Nonlocal Cauchy Problems in Resonance Cases

Abstract

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 Φ, the corresponding nonlocal Cauchy problem for a LODE with constant coefficients is determined by BVCs of the form Φ{ 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 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.