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Some notes on fixed points of quasi-contraction maps. (English) Zbl 1206.54061
A self map $$T:X\to X$$ such that for some $$\lambda\in(0,1)$$ and for every $$x,y\in X$$ there exists
$u\in C(T,x,y)=\{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}$
such that
$d(Tx,Ty)\leq\lambda u,$
is said to be a quasi-contraction. It is proved that every quasi-contraction defined on a complete cone metric space has a unique fixed point. Moreover, every quasi-contraction defined on a cone metric space possesses the property $$(P)$$, that is $$F(T)=F(T^n)$$ for all $$n\geq 1$$, where $$F(T)$$ denotes the set of all fixed points of the mapping $$T:X\to X$$.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E35 Metric spaces, metrizability
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##### References:
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