Nonexpansive retracts and weak compatible pairs in metric Spaces
1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2 Department of Mathematics, Bradley University, Peoria, IL 61625, USA
3 Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA
4 Department of Mathematics & Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Fixed Point Theory and Applications 2012, 2012:100 doi:10.1186/1687-1812-2012-100Published: 20 June 2012
In this article, we introduce a new approach to common fixed point theory for a weak compatible pair. We first introduce the concepts of R-pair and NR-pair and establish some new common fixed point theorems for a weak compatible pair in hyperconvex metric spaces and uniformly convex metric spaces. We shall also establish the well-known De Marr's theorem for a family of weak compatible pairs in a hyperconvex metric space [0,1] and show by an example that it fails to hold in general hyperconvex metric spaces.
Mathematics Subject Classification (2000): Primary 06F30, 46B20, 47E10.