Generalization of Edelstein's fixed point theorem

Abstract

Let i = 1 , . . . , n be metric spaces. Let T.,, df i = 1 , . . . , n be transformations mapping of B = X^ * . . . * XQ i n t o X^. itor any p o s i t i v e number a we define (cf. also [3]) Za = »x n ) 6 B s d i ^ i t 1 ! ^ » ' " i x n ) ] < a» i = 1 , . . . , n | . In [ l ] the following f i x e d point theorem has been proved, generalizing the Banach p r i n c i p l e for contraction maps ( c f . [4]): Let E be a metric space and T an operator which transforms E into i t s e l f . Suppose that d[T(x), T(y)] < d(x,y), x ^ y, x,y e E. Assume t h a t there e x i s t s x e E such that the sequence at i t e r a t e s {Tm(x)| contains a subsequence m I T (x)j convergent to a point u e E. Then u is a unique f i x e d point of T. The purpose of the present paper i s to prove (using the n o t a t i o n of t h e s e t s Za„ ) a f i x e d point theorem which generalize s the E d e l s t e i n ' s theorem and the r e s u l t in [5] (Fragment tekstu).