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Common fixed points for set-valued mappings. (English) Zbl 0573.54039

Let F and G be mappings of a complete metric space (X,d) into the hyperspace of bounded subsets B(X) satisfying the inequality \[ \delta (Fx,Gy)\leq c.\max \{\delta (x,Gy),\delta (y,Fx),d(x,y)\} \] for all x,y in X, where \(0\leq c<1\) and \(\delta (A,B)=\sup \{d(a,b):\) \(a\in A\), \(b\in B\}\). Then F and G have a unique common fixed point z. Further \(Fz=Gz=\{z\}\) and z is the unique fixed point of F and G.

MSC:

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