Department of Mathematics, Faculty of Science of Masaryk University, Brno
Abstract
summary:Let K be a closed convex subset of a complete convex metric space X and T,I:K→K two compatible mappings satisfying following contraction definition: Tx,Ty)≤(Ix,Iy)+(1−a)max{Ix.Tx),Iy,Ty)} for all x,y in K, where 0<a<1/2p−1 and p≥1. If I is continuous and I(K) contains [T(K)] , then T and I have a unique common fixed point in K and at this point T is continuous. This result gives affirmative answers to open questions set forth by Diviccaro, Fisher and Sessa in connection with necessarity of hypotheses of linearity and non-expansivity of I in their Theorem [3] and is a generalisation of that Theorem. Also this result generalizes theorems of Delbosco, Ferrero and Rossati [2], Fisher and Sessa [4], Gregus [5], G. Jungck [7] and Mukherjee and Verma [8]. Two examples are presented, one of which shows the generality of this result