Article ID : 0253 - 4827 ( 2001) 05 - 0564 - 05 FIXED POINTS ON TWO COMPLETE AND COMPACT METRIC SPACES Ξ M. Telci (Department of Mathematics , Faculty of Arts and Science , Trakya University , 22030 Ediren , Turkey) (Communicated by CHIEN Wei-zang) Abstract : By using functions , some related fixed point theorems on two metric spaces are established. These results generalize some theorems of Fisher. Key words : fixed point ; complete metric space; compact metric space CLC number : O177 . 91 Document code : A Introduction The following related fixed point theorem was proved in [ 1 ]. Theorem 1 Let ( X , d) and (Y, ) be complete metric spaces , let T be a mapping of X into Y and let S be a mapping of Y into X satisfying the inequalities ( Tx , TSy) cmax d ( x , Sy) , ( y , Tx) , ( y , TSy) , d ( S y , S Tx) cmax ( y , Tx) , d(x,Sy) , d ( x , S Tx) for all x in X and y in Y, where 0 c< 1 . Then ST has a unique fixed point z in X and TS has a unique fixed point w in Y. Further , Tz=w and Sw = z. Throughout this paper , R + stands for the non-negative reals. We will also denote by F the set of all real functions f: R 3 + R + such that : ) f is upper semi-continuous in each coordinate variable ; ) If either u f(v, 0 , u) or u f(v,u, 0) for all u,v 0 , then there exists a real constant 0 c< 1 such that u cv. 1 Fixed Points on Complete Metric Spaces We now generalize Theorem 1 as follows : Theorem 2 Let ( X , d) and (Y, ) be complete metric spaces , let T be a mapping of X into Y and let S be a mapping of Y into X satisfying the inequalities ( Tx , TSy) f(d(x,Sy), ( y , Tx) , ( y , TSy) ) , ( 1) d ( S y , S Tx) g( ( y , Tx) , d ( x , Sy) , d ( x , S Tx) ) ( 2) for all x in X and y in Y, where f,g F. Then ST has a unique fixed point z in X and TS has a unique fixed point w in Y. Further , Tz=w and Sw = z. 4 6 5 Applied Mathematics and Mechanics (English Edition, Vol 22 , No 5 , May 2001) Published by Shanghai University , Shanghai , China Ξ Received date : 2000 - 04 - 05 © 1995-2004 Tsinghua Tongfang Optical Disc Co., Ltd. All rights reserved.