Fixed point theorems for compatible multi-valued and single-valued mappings. (English) Zbl 0671.54023

G. Jungck [Am. Math. Mon. 83, 261-263 (1976; Zbl 0321.54023)] proved that if f: \(X\to X\) is a continuous map of a complete metric space (X,d) and g: \(X\to X\) a map which commutes with f, and if g(X)\(\subset f(X)\) and d(gx,gy)\(\leq hd(fx,fy)\) for all \(x,y\in X\) and some \(0\leq h\leq 1\), then f and g have a unique common fixed point.
The authors obtain extensions of this theorem by replacing single-valued mappings with multi-valued ones, and commutativity with an adaptation of the weak commutativity of the second author [Publ. Inst. Math., Nouv. Sér. 32(46), 146-153 (1982; Zbl 0523.54030)] to the multi-valued setting. The definitions and results are too long to state here.
Reviewer: H.Schirmer


54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology
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