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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.

##### MSC:
 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