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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: A FIXED POINT THEOREM FOR A CLASS OF DIFFERENTIABLE STABLE OPERATORS IN BANACH SPACES | A FIXED POINT THEOREM FOR A CLASS OF DIFFERENTIABLE STABLE OPERATORS IN BANACH SPACES VADIM AZHMYAKOV Received 31 January 2005 Accepted 10 October 2005 We study Frechet 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. 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. 1. Introduction The basic inspiration for studying stable and strongly stable operators in a real Banach space X is the operator equation of the form A x a a G X 1.1 where A X X is a nonlinear operator. We consider a single-valued mapping A whose domain of definition is X and whose range R A is contained in X. Throughout this paper the terms mapping function and operator will be used synonymously. We start by recalling some basic concepts and preliminary results see e.g. 29 . Definition 1.1. An operator A X X is called stable if A x1 _ A x2 g x1 _ Cl Vx1 x2 G X 1.2 where g R R is a strictly monotone increasing and continuous function with g 0 0 lim g t . 1.3 t The function g is called a stabilizing function of the operator A. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006 Article ID 92429 Pages 1-17 DOI 10.1155 FPTA 2006 92429 2 A fixed point theorem Let H be a real Hilbert space. By we denote the inner product of H. The Hilbert space H will be identified with the dual space H . It is easy to see that Definition 1.1 is closely related to the concept of a coercive operator .
