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Some new common fixed point theorems for converse commuting multi-valued mappings in symmetric spaces with applications. (English) Zbl 1080.47044
The notion of converse commuting mappings introduced by [Z.-X. Lü, Acta Anal. Funct. Appl. 4, No. 3, 226–228 (2002; Zbl 1024.54027)] is generalized to multivalued maps, and some corresponding common fixed point results are obtained in the case of symmetric spaces, i.e., for any set \(X\) with a function \(d:X\times X\to \mathbb{R}_+\) satisfying \(d(x,y)=0\) iff \(x=y\), and \(d(x,y)=d(y,x)\). As an application, the author considers a probabilistic space where he defines a symmetry, and proves a common fixed point theorem for a pair of converse commuting single-valued mappings.

47H10 Fixed-point theorems
47H04 Set-valued operators
54H25 Fixed-point and coincidence theorems (topological aspects)