A fixed point theorem for a class of
 differentiable stable  operators  in banach
spaces

Azhmyakov, Vadim

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Fixed Point Theory and Applications
Volume 2006 (2006), Article ID 92429, 17 pages
doi:10.1155/FPTA/2006/92429

A fixed point theorem for a class of differentiable stable operators in banach spaces

Vadim Azhmyakov

Fakultät IV--Elektrotechnik und Informatik, Institut für Energie und Automatisierungstechnik, Technische Universität Berlin, Sekretariat EN 11, Einsteinufer 17, Berlin 10587, Germany

Received 31 January 2005; Accepted 10 October 2005

Copyright © 2006 Vadim Azhmyakov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study Frèchet differentiable stable operators in real Banach spaces. We present the theory of linear and nonlinear stable operators in a systematic way and prove solvability theorems for operator equations with differentiable expanding operators. In addition, some relations to the theory of monotone operators in Hilbert spaces are discussed. Using the obtained solvability results, we formulate the corresponding fixed point theorem for a class of nonlinear expanding operators.