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: COMMON FIXED POINT AND INVARIANT APPROXIMATION RESULTS IN CERTAIN METRIZABLE TOPOLOGICAL VECTOR SPACES | COMMON FIXED POINT AND INVARIANT APPROXIMATION RESULTS IN CERTAIN METRIZABLE TOPOLOGICAL VECTOR SPACES NAWAB HUSSAIN AND VASILE BERINDE Received 27 June 2005 Revised 1 September 2005 Accepted 6 September 2005 We obtain common fixed point results for generalized I-nonexpansive R-subweakly commuting maps on nonstarshaped domain. As applications we establish noncommutative versions of various best approximation results for this class of maps in certain metrizable topological vector spaces. Copyright 2006 N. Hussain and V. Berinde. 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 and preliminaries Let X be a linear space. A p-norm on X is a real-valued function on X with 0 p 1 satisfying the following conditions i x p 0 and x p 0 x 0 ii llaxllp a p x p iii x y p llxhp IIyllp for all x y e X and all scalars a. The pair X II bp is called a p-normed space. It is a metric linear space with a translation invariant metric dp defined by dp x y x - y bp for all x y e X. If p 1 we obtain the concept of the usual normed space. It is well-known that the topology of every Hausdorff locally bounded topological linear space is given by some p-norm 0 p 1 see 9 and references therein . The spaces Ip and Lp 0 p 1 are p-normed spaces. A p-normed space is not necessarily a locally convex space. Recall that dual space X the dual of X separates points of X if for each nonzero x e X there exists f e X such that f x 0. In this case the weak topology on X is well-defined and is Hausdorff. Notice that if X is not locally convex space then X need not separate the points ofX. For example ifX Lp 0 1 0 p 1 then X 0 12 pages 36 and 37 . However there are some non-locally convex spaces X such as the p-normed spaces Ip 0 p 1 whose dual X separates the points ofX. Let X be a metric linear space and M a nonempty subset of X. .
