International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 1, Pages 23-28

On a fixed point theorem of Greguš

Brian Fisher1 and Salvatore Sessa2

1Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
2Istituto Matematico, Facolta' Di Architettura, Universita' Di Napoli, Via Monteoliveto 3, Naples 80134, Italy

Received 18 May 1984

Copyright © 1986 Brian Fisher and Salvatore Sessa. 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.


We consider two selfmaps T and I of a closed convex subset C of a Banach space X which are weakly commuting in X, i.e.TIxITxIxTxforanyxinX,and satisfy the inequalityTxTyaIxIy+(1a)max{TxIx,TyIy}for all x, y in C, where 0<a<1. It is proved that if I is linear and non-expansive in C and such that IC contains TC, then T and I have a unique common fixed point in C.