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: Research Article The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 586487 12 pages doi 2009 586487 Research Article The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping Jaroslaw Gornicki Department of Mathematics Rzeszow University of Technology . Box 85 35-595 Rzeszow Poland Correspondence should be addressed to Jaroslaw Gornicki gornicki@ Received 16 May 2009 Accepted 25 August 2009 Recommended by William A. Kirk The purpose of this paper is to prove by asymptotic center techniques and the methods of Hilbert spaces the following theorem. Let H be a Hilbert space let C be a nonempty bounded closed convex subset of H and let M n k 1 be a strongly ergodic matrix. If T C C is a lipschitzian mapping such that liminfn TOinfOT 0 i . SS 1 an k Tk m 2 2 then the set of fixed points FixT x e C Tx x is a retract of C. This result extends and improves the corresponding results of 7 Corollary 9 and 8 Corollary 1 . Copyright 2009 Jaroslaw Gornicki. 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 Let E be a Banach space and let C be a nonempty bounded closed convex subset of E. We say that a mapping T C C is nonexpansive if Tx - Ty x - y for every x y e C. The result of Bruck 1 asserts that if a nonexpansive mapping T C C has a fixed point in every nonempty closed convex subset of C which is invariant under T and if C is convex and weakly compact then Fix T x e C Tx x the set of fixed points is nonexpansive retract of C . there exists a nonexpansive mapping R C Fix T such that R Fix T I . A few years ago the Bruck results were extended by T. Dominguez Benavides and Lorenzo Ramirez 2 to the case of asymptotically nonexpansive mappings if the space E was sufficiently regular. On the other hand it is known that .