Volterra-like expansions for solutions of nonlinear integral equations and nonlinear differential equations

Abstract

Expansion theorems, and related results, concerning nonlinear integral equations are proved, and are applied to systems of differential equations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x} = f(x, u, t)</tex> , almost all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t \geq 0, x</tex> continuous on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[0, \infty), x(0) = x_{0}</tex> in which the solution <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -vector valued. In particular, we show the existence of, and show how to obtain, a locally convergent expansion for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> in terms of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</tex> , when certain reasonable conditions are met, including the condition that an associated system of linear differential equations is bounded-input bounded-output stable. The expansion converges in a normed space of bounded continuous <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -vector valued functions defined on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[0, \infty)</tex> , and involves terms that are sums of Volterra-like iterated integrals.