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Contractive type non-self mappings on metric spaces of hyperbolic type. (English) Zbl 1089.54019
A metric space \((X,d)\) is said to be of hyperbolic type if all points \(x,y\in X\) are endpoints of a metric segment (isometric image of a real segment), denoted by \(\text{seg}[x,y]\), such that for any \(u,x,y\in X\) and \(z\in \text{seg}[x,y]\) satisfying \(d(x,z)=\lambda d(x,y)\) for some \(\lambda \in [0,1]\) the following inequality \(d(u,z)\leq (1-\lambda )d(u,x)+\lambda d(u,y)\) holds. In this paper the author obtains several fixed point theorems for mappings \(T:K\rightarrow X\) defined on a closed subset \(K\) of a complete metric space of hyperbolic type \(X\) that satisfy \(T(\partial K)\subset K\) and various contraction-type conditions. Some examples are given to show that the obtained theorems are genuine generalizations of some known results from this area.

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
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