International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 1, Pages 75-87

Some fixed point theorems for Hardy-Rogers type mappings

B. E. Rhoades,1 S. Sessa,2 M. S. Khan,3 and M. D. Khan3

1Department of Mathematics, Indiana University, Bloomington 47405, IN, USA
2Universita di Napoli, Faculta di Architettura, Instituto Di Matematica, Via Monteoliveto 3, Napoli 80134, Italy
3Department of Mathematics, Aligarh Muslim University, Aligarh 20200, India

Received 10 December 1983

Copyright © 1984 B. E. Rhoades et al. 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.


The first result establishes a fixed point theorem for three maps of a complete metric space. The contractive definition is a generalization of that of Hardy and Rogers, and the commuting condition of Jungck is replaced by the concept of weakly commuting. The other results are extensions of some theorems of Kannan.