A comparison of various noncommuting conditions in metric fixed point theory and their applications
It is well known that transforming a fixed point result for a single-valued mapping \(f:X\to X\) under some contractive-type condition into a common fixed point for two mappings \(f,g:X\to X\) usually requires some commutativity-type condition for these mappings. The article under review is a very good survey of various conditions of this kind that have appeared in the literature during more than three decades. Namely, the authors list and discuss more than 40 conditions, starting from the notion of weakly commuting mappings of \textit{S. Sessa} [Publ. Inst. Math., Nouv. Sér. 32(46), 149--153 (1982; Zbl 0523.54030)]. In most cases, examples are provided showing the difference between introduced concepts. Also, the (non)-usefulness of these concepts in coincidence and common fixed point results is discussed. The (far from complete) list of 144 references shows that the related problems attracted a lot of researchers. Finally, some of many possible applications of coincidence and common fixed point results (here, to differential equations and approximation theory) are presented as illustrations.
\(R\)-weakly commuting mappings, compatible mappings
scientific article; zbMATH DE number 6417858
scientific article