Fixed Point Theorems on Product of Compact Metric Spaces

Abstract

Results generalizing and unifying fixed point theorems of Edelstein, Fisher, Jungck, Matkowski, Rhoades, Seghal and others are obtained for four systems of maps on a finite product of compact metric spaces.

Co-Ordinatewise R-Weakly Commuting Maps and Fixed Point Theorem on Product Spaces

“…The purpose of this paper is to extend and unify the result of Fisher Two systems of maps (Pi,..., Pn) and (Si,..., Sn) are co-ordinatewise weakly commuting on X if and only if they are co-ordinatewise weakly commuting at every point of X. DEFINITION 3 [5]. Two systems of maps (Pi,..., Pn) and (Si,..., Sn) are co-ordinatewise asymptotically commuting or, following the terminology of Jungck [8] Two systems of maps (Pi,..., P n ) and (Si,..., S n ) are co-ordinatewise R-weakly commuting on X, if and only if they are co-ordinatewise R-weakly commuting at every point of X for any positive real R.…”

R‐weak commutativity, examples, and applications

“…In 1982 by Sessa [2], in 1989 by Fisher and Sessa [3], and in 1994 by Pant [4], the condition of commutativity was relaxed introducing the notions of weak commutativity and R ‐weak commutativity. Afterwards, various generalizations of commutativity in metric spaces were further weakened or used first under the names of compatibility by Jungck [5], weak* commutativity by Pathak [6], weak** commutativity by Pathak [7], preorbitally commutativity by Singh and Mishra [8], asymptotically commutativity by Gairola et al [9], biased by Jungck and Pathak [10], weak compatibility by Jungck and Rhoades [11], coincidentally commutativity by Dhage [12], and partially commutativity by Sastry and Murthy [13]. Such notions play a vital role developing fixed point theory in metric spaces.…”