4.—A Cauchy Problem for an Ordinary Integro-differential Equation

Abstract

SynopsisIn this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.