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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.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems
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##### References:
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