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 Nonlinear Mean Ergodic Theorems for Semigroups in Hilbert Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 73246 9 pages doi 2007 73246 Research Article Nonlinear Mean Ergodic Theorems for Semigroups in Hilbert Spaces Seyit Temir and Ozlem Gul Received 26 December 2006 Accepted 4 April 2007 Recommended by Nan-Jing Huang Let K be a nonempty subset not necessarily closed and convex of a Hilbert space and let r T t t 0 be a semigroup on K and let . 0 to - K be an almost orbit of r. In this paper we prove that every almost orbit of r is almost weakly and strongly convergent to its asymptotic center. Copyright 2007 S. Temir and O. Gul. 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 K be a nonempty subset of a Hilbert space GX where K is not necessarily closed and convex. A family r T t t 0 of mappings T t is called a semigroup on K if 51 T t is a mapping from K into itself for t 0 52 T 0 x x and T t s x T t T s x for x G K and t s 0 53 for each x G K T - x is strongly measurable and bounded on every bounded subinterval of 0 to . Let r be a semigroup on K. Then F x G K T t x x t 0 is said to be fixed-points set of r. We state a condition introduced by Miyadera 1 . If for every bounded set B G K v G K and s 0 there exists a Ss B v 0 with lims to Ss B v 0 such that T s u - T s v u - v Ss B v for u G B then r is said to be an asymptotically nonexpansive semigroup. 2 Fixed Point Theory and Applications Definition . A function a 0 to K is called almost-orbit of r if a 0 to K is strongly measurable and bounded on every bounded subinterval of 0 to and if lim sup a s 1 - Tk s a t I p 0. t to s 0 Using these conditions we prove that every almost-orbit of r is weakly and strongly convergent to its asymptotic center see 1 . Xu 2 studied strong asymptotic behavior of almost-orbits of both of .
