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

Authors: N.S. Hoang, A.G. Ramm

Addresses: Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA. ' Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA

Abstract: This paper is a review of the authors| results on the Dynamical Systems Method (DSM) 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 Frechet 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 f_δ are given, ||f − f_δ|| ≤ δ.

Keywords: DSM; dynamical systems method; nonlinear operator equations; monotone operators; discrepancy principle; smoothness assumptions.

DOI: 10.1504/IJCSM.2010.033926

International Journal of Computing Science and Mathematics, 2010 Vol.3 No.1/2, pp.3 - 55

Published online: 04 Jul 2010 *

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