High order approximation of implicitly defined maps (Q1060372)

The equation \(x=\Psi (u,x)\), where \(\Psi\) is a smooth mapping acting on Banach spaces is considered; the solution \(\psi\) is approximated by the use of the iterative scheme \(\psi_ n=\Psi (u,\psi_{n-1}(u))\), \(\psi_ 0(u)\equiv 0\). By means of a prolongation technique and the contraction mapping theorem, the convergence of \(\psi_ n\) to \(\psi\) in the \(C^ k\) norm is shown and the uniform, geometric rate of convergence of the derivative \(D^ j\psi_ n\) to \(D^ j\psi\), \(j=0,...,k\) is established. It is also shown that if \(\Phi\) is another mapping with \(\| \Phi -\Psi \|_{C^ k}<\eta\), then \(\psi_ n\) are approximate solutions for the equation \(x=\Phi (u,x)\) too. These results are applied to obtain an approximation y for the input-output map generated by the nonlinear control system \(\dot x=G(x)u\), where y is expressed by a linear combination of iterated integrals.