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


54H25 Fixed-point and coincidence theorems (topological aspects)