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 \(T:X\to X\).


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


[1] Huang, L. G.; Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332, 1468-1476 (2007) · Zbl 1118.54022
[2] Rezapour, Sh.; Hamlbarani, R., Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”, J. Math. Anal. Appl., 345, 719-724 (2008) · Zbl 1145.54045
[3] Ćirić, Lj. B., A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc., 45, 267-273 (1974) · Zbl 0291.54056
[4] Ilić, D.; Rakočević, V., Quasi-contracion on a cone metric space, Appl. Math. Lett., 22, 728-731 (2009) · Zbl 1179.54060
[5] Kadelburg, Z.; Radenović, S.; Rakočević, V., Remarks on quasi-contracion on a cone metric space, Appl. Math. Lett. (2009)
[6] Jeong, G. S.; Rhoades, B. E., Maps for which \(F(T) = F(T^n)\), (Fixed Point Theory and Applications, vol. 6 (2007), Nova Sci. Publ.: Nova Sci. Publ. New York), 71-105 · Zbl 1147.47041
[7] Jeong, G. S.; Rhoades, B. E., More maps for which \(F(T) = F(T^n)\), Demonstratio Math., 40, 3, 671-680 (2007) · Zbl 1147.47041
[8] Rhoades, B. E., Some maps for which periodic and fixed points coincide, Fixed Point Theory, 4, 2, 173-176 (2003) · Zbl 1062.47057
[9] Pathak, H. K.; Shahzad, N., Fixed point results for generalized quasi-contraction mappings in abstract metric spaces, Nonlinear Anal. (2009) · Zbl 1189.54036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.