## 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