Abstract
A wide class of the operator equations F(u)=h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F′(u). It is assumed that F′(u) depends on u continuously. Existence and uniqueness of the solution to evolution equation \(\dot{u}(t)=-[F'(u(t))]^{-1}(F(u(t))-h)\), u(0)=u 0, is proved without assuming that F′(u) satisfies the Lipschitz condition. The method of the proof is new. This method is based on a novel version of the abstract inverse function theorem.
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Ramm, A.G. On the DSM Newton-type method. J. Appl. Math. Comput. 38, 523–533 (2012). https://doi.org/10.1007/s12190-011-0494-z
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DOI: https://doi.org/10.1007/s12190-011-0494-z