Ramm DSM of Newton type

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. MSC: 47J05; 47J06; 47J35. Dynamical systems method (DSM); nonlinear operator equations; monotone operators; discrepancy principle. to this paper should be made as follows: Hoang, N. S. and Ramm, A. G. (2010) ‘DSM of Newton-type for solving operator equations F (u) = f with minimal smoothness assumptions on F .’, Int. J. Computing Science and Mathematics, Vol. x, No. x, pp.xxx–xxx. Nguyen S. Hoang is currently a PhD student at Kansas State University under the supervision of Professor Alexander G. Ramm. He is an author and a co-author of more than 20 papers. His fields of interest are numerical analysis, applied mathematics, inverse and ill-posed problems, image processing, differential and integral equations, operator theory. Prof. Alexander G. Ramm is an author of more than 590 papers, 2 patents, 12 monographs, an editor of 3 books. He is an associte editor of several Journals. He gave more than 140 addresses at various Conferences, visited many Universities in Europe, Asia, Australia and America. He won Khwarizmi International Award in Mathematics, was a London Mathematical Society speaker, distinguished HKSTAM speaker, CNRS research professor, Fulbright professor in Israel, distinguished Foreign Professor in Mexico and Egypt. His research interests include many areas of analysis, numerical analysis and mathematical physics.