International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 1, Pages 23-28
doi:10.1155/S0161171286000030
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
Abstract
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.‖TIx−ITx‖≤‖Ix−Tx‖ for any x in X,and satisfy the inequality‖Tx−Ty‖≤a‖Ix−Iy‖+(1−a)max{‖Tx−Ix‖,‖Ty−Iy‖}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.