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DSM of Newton type for solving operator equations F (u) =f with minimal smoothness assumptions on F

By N. S. Hoang and A. G. Ramm

Abstract

This paper is a review of the authors’ results on the DSM (Dynamical Systems Method) for solving operator equation (*) F (u) = f. It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of F. It is assumed that F is continuously Fréchet differentiable, but no smoothness assumptions on F ′ (u) are imposed. The DSM for solving equation (*) is developed. Under weak assumptions global existence of the solution u(t) is proved, the existence of u(∞) is established, and the relation F (u(∞)) = f is obtained. The DSM is developed for a stable solution of equation (*) when noisy data fδ are given, ‖f − fδ ‖ ≤ δ

Year: 2010
OAI identifier: oai:CiteSeerX.psu:10.1.1.186.159
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