Some notes on fixed points of quasi-contraction maps. (English) Zbl 1206.54061

Rezapour, Sh.; Haghi, R. H.; Shahzad, N.

Appl. Math. Lett. 23, No. 4, 498-502 (2010).

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 \(T:X\to X\).

Reviewer: Dariusz Bugajewski (Baltimore)

MSC:

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

Keywords:

cone metric space; fixed point; orbitally continuous; quasi-contraction map

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