Summary. Nonlinear stationary fixed point iterations in R" are considered. The Perron-Ostrowski theorem [23] guarantees convergence if the iteration function G possesses an isolated fixed point u. In this paper a sufficient condition for convergence is given if G possesses a manifold of fixed points.As an application, convergence of a nonlinear extension of the method of Kaczmarz is proved. This method is applicable to underdetermined equations; it is appropriate for the numerical treatment of large and possibly ill-conditioned problems with a sparse, nonsquare Jacobian matrix. A practical example of this type (nonlinear image reconstruction in ultrasound tomography) is included.

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