Some Results on the Theory of Physical Systems Governed Nonlinear Functional Equations

Abstract

The main purpose of this paper is to present a reasonably complete picture of the results of the first phase of some recent research on the properties of solutions of nonlinear functional equations that frequently arise in the study of physical systems. We consider in detail the properties of a vector nonlinear Volterra integral equation of the second kind, and some conditions are presented for the norm boundedness of solutions of a functional equation of similar type defined on an abstract space. More specifically, concerning the Volterra equation, conditions are presented under which the solutions (a) approach zero as t → ∞, (b) approach zero exponentially as t → ∞, (c) are uniformly bounded on t ≧ 0, (d) are square integrable on [0, ∞), or (e) are ultimately periodic. On the basis of these results, it appears that an input-output stability theory of a large class of time-varying nonlinear systems of engineering interest is well within sight.