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

Expansion theorems, and related results, concerning nonlinear integral equations are proved, and are applied to systems of differential equations of the form
\dot{x} = f(x, u, t)
, almost all
t \geq 0, x
continuous on
[0, \infty), x(0) = x_{0}
in which the solution
x
is
n
-vector valued. In particular, we show the existence of, and show how to obtain, a locally convergent expansion for
x
in terms of
u
, 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
n
-vector valued functions defined on
[0, \infty)
, and involves terms that are sums of Volterra-like iterated integrals.