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A fixed point theorem for Matkowski contractions. (English) Zbl 1151.47054
In this short note, the authors prove a fixed point theorem for a Matkowski contraction. They prove that, if \(K\) is a nonempty closed subset of a complete metric space \((X,\rho)\) and \(T : K \rightarrow X\) satisfies \(\rho (Tx, Ty) \leq \phi (\rho (x,y))\) for all \(x,y \in K\), where \(\phi : [0,\infty) \rightarrow [0,\infty)\) is increasing and satisfies \(\lim_{n \rightarrow \infty} \phi^n (t) = 0\) for all \(t > 0\) and, finally, if there is a nonempty bounded subset \(K_0 \subset K\) such that for each natural number \(n\), there exists \(x_n \in K_0\) such that \(T^n x_n\) is defined, then \(T\) has a unique fixed point in \(K\). With this result, the authors successfully extend the known Banach fixed point theorem to a non-self mapping.

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54H25 Fixed-point and coincidence theorems (topological aspects)
54E50 Complete metric spaces