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Common fixed points for commuting and compatible maps on compacta. (English) Zbl 0661.54043
Selfmaps f, g of a metric space (X,d) are said to be compatible iff \(\lim_{n}d(fgx_ n,gfx_ n)=0\) when \((x_ n)\) is a sequence such that \(\lim_{n}fx_ n=\lim_{n}gx_ n=t\) for some \(t\in X.\)
This generalizes the notion of commuting maps and weakly commuting maps introduced by S. Sessa [Publ. Inst. Math., Nouv. Sér. 32(46), 149-153 (1982; Zbl 0523.54030)]. The author proves that two continuous selfmaps on a compact metric space are compatible iff they commute on their set of coincidence points. (This is a corollary from a more general result for introduced by the author “proper maps”). Then he proves several fixed point theorems for compatible and commuting selfmaps of a metric space which generalize many earlier results.
Reviewer: J.Matkowski

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
54E45 Compact (locally compact) metric spaces
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