MOHAMED AKKOUCHI
A common fixed point theorem connected to a result of B. Fisher
Pages: 9-17
Received: 29 October 2001
Revised: 8 March 2002
Mathematics Subject Classification (2000): 47H10 - 54H25
Abstract: Let (M,d) be a complete metric space, let 0 ≤ a < 1, and let S,T be two selfmappings of M. We suppose that S belongs to the class B(T,a) (i.e. the condition (B) below is satisfied). Our main result is Theorem 1.1, in which we prove that S, T have a unique common fixed point. Although we do not suppose any continuity assumption neither for T nor for S, we conclude some regularity properties. Indeed, we show that S and TS must be continuous at the unique common fixed point and that the mapping FS:x |--> d(x,Sx) is an r.g.i. mapping. We establish four equivalent properties characterizing the existence and uniqueness of the common fixed point for S,T, and give sequences of points approximating this fixed point. In particular, we show that all the Picard sequences defined by S converge to this common fixed point. This paper provides improvements to a well known result of B. Fisher (see [4]).