High order approximation of implicitly defined maps

Approximations for the function φ{symbol} implicitly defined by φ{symbol}(u)=Φ(u, φ{symbol}(u)) are obtained via the iterative scheme φ{symbol}_n(u)=Φ(u, φ{symbol}_n-1(u)). In this paper the uniform convergence of high order derivatives of φ{symbol}_n to the corresponding derivatives of φ{symbol} is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.