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Journal of Applied Mathematics and Computing

, Volume 38, Issue 1–2, pp 523–533 | Cite as

On the DSM Newton-type method

  • A. G. RammEmail author
Article

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.

Keywords

Inverse function theorem Newton’s method DSM (dynamical systems method) Evolution equations 

Mathematics Subject Classification (2000)

47J05 47J07 58C15 

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Copyright information

© Korean Society for Computational and Applied Mathematics 2011

Authors and Affiliations

  1. 1.Department of MathematicsKansas State UniversityManhattanUSA

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