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

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Expansion theorems, and related results, concerning nonlinear integral equations are proved, and are applied to systems of differential equations of the formdot{x} = f(x, u, t), almost allt geq 0, xcontinuous on[0, infty), x(0) = x_{0}in which the solutionxisn-vector valued. In particular, we show the existence of, and show how to obtain, a locally convergent expansion forxin terms ofu, 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 continuousn-vector valued functions defined on[0, infty), and involves terms that are sums of Volterra-like iterated integrals.

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Circuits and Systems, IEEE Transactions on  (Volume:30 ,  Issue: 2 )