Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multivalued | Nanan and Dhompongsa Fixed Point Theory and Applications 2011 2011 54 http content 2011 1 54 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multivalued Narawadee Nanan 1 and Sompong Dhompongsa1 2 Correspondence sompongd@ department of Mathematics Faculty of Science Chiang Mai University Chiang Mai 50200 Thailand Full list of author information is available at the end of the article Springer Abstract Bruck Pac. J. Math. 53 59-71 1974 Theorem 1 proved that for a nonempty closed convex subset E of a Banach space X if E is weakly compact or bounded and separable and suppose that E has both FPP and CFPP then for any commuting family S of nonexpansive self-mappings of E the set F S of common fixed points of S is a nonempty nonexpansive retract of E. In this paper we extend the above result when one of its elements in S is multivalued. The result extends previously known results on common fixed points of a pair of single valued and multivalued commuting mappings to infinite number of mappings and to a wider class of spaces. 2000 Mathematics Subject Classification 47H09 47H10 Keywords Common fixed point Nonexpansive retract Property D Kirk-Massa condition 1 Introduction For a pair t T of nonexpansive mappings t E E and T E 2X defined on a bounded closed and convex subset E of a convex metric space or a Banach space X we are interested in finding a common fixed point of t and T. In 1 Dhompongsa et al. obtained a result for the CAT 0 space setting Theorem . 1 Theorem Let E be a nonempty bounded closed and convex subset of a complete CAT 0 space X and let t E E and T E 2X be nonexpan-sive mappings with T x a nonempty compact convex subset of X. Assume that for some p e Fix t ap 1 a Tx is convex for x e E a e 0 1 . If t and T are commuting then Fix t n Fix T 0. Shahzad and Markin 2 improved .
