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Single-valued and multi-valued f-contractions. (English) Zbl 0568.54031
Let (X,d) be a complete metric space and CB(X) the family of all closed bounded subsets of X. The main result is as follows. Theorem 1: Let $$T: X\to CB(X)$$ and let $$f: X\to X$$ be a continuous mapping which commutes with T and T(X)$$\subseteq f(X)$$. Suppose that there exists $$h\in (0,1)$$ such that $$H(T(x),T(y))\leq hd(f(x),f(y))$$ for each x,y$$\in X$$. Moreover one of the following holds: either $$(i)\quad f(x)\neq f^ 2(x)$$ implies $$f(x)\cap T(x)=\emptyset$$ or $$(ii)\quad f(x)\in T(x)$$ implies that $$\lim_{n\to \infty}f^ n(x)$$ exists. Then T has a fixed point (which is also a fixed point of f).
Reviewer: K.Chung

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects)
##### Keywords:
complete metric space