An Operator Equation and Bounded Solutions of Integro-Differential Systems

Abstract

The main result gives conditions under which (locally) a one-to-one, bicontinuous correspondence exists between bounded solutions (or bounded solutions tending to zero as $t o + infty $) of a linear, integro-differential system of Volterra type and such solutions of perturbations of the system. The perturbations are allowed to be of any functional type which satisfy a local Lipschitz condition near the origin. Certain recently proved stability results for such systems are special cases. The results also constitute a generalization of similar results for ordinary differential equations, which motivate the approach and proofs. The proofs rely on an abstract lemma proved for a certain operator equation. In order to apply the perturbation theorems some results are also given concerning bounded solutions of linear integro-differential systems. An application is made to Volterra’s predator-prey population dynamics model with hereditary effects where it is shown, for certain specific, but reasonable hereditary kernels, that the critical (or saturation point) of the system is unstable.