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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 $\lambda \in \left(0,1\right)$ and for every $x,y\in X$ there exists

$u\in C\left(T,x,y\right)=\left\{d\left(x,y\right),d\left(x,Tx\right),d\left(y,Ty\right),d\left(x,Ty\right),d\left(y,Tx\right)\right\}$

such that

$d\left(Tx,Ty\right)\le \lambda u,$

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 $\left(P\right)$, that is $F\left(T\right)=F\left({T}^{n}\right)$ for all $n\ge 1$, where $F\left(T\right)$ denotes the set of all fixed points of the mapping $T:X\to X$.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54E35 Metric spaces, metrizability
##### References:
 [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 · doi:10.1016/j.jmaa.2005.03.087 [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 · doi:10.1016/j.jmaa.2008.04.049 [3] Ćirić, Lj.B.: A generalization of Banach’s contraction principle, Proc. amer. Math. soc. 45, 267-273 (1974) · Zbl 0291.54056 · doi:10.2307/2040075 [4] Ilić, D.; Rakočević, V.: Quasi-contracion on a cone metric space, Appl. math. Lett. 22, 728-731 (2009) · Zbl 1179.54060 · doi:10.1016/j.aml.2008.08.011 [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 $F\left(T\right)=F\left(Tn\right)$, Fixed point theory and applications 6, 71-105 (2007) [7] Jeong, G. S.; Rhoades, B. E.: More maps for which $F\left(T\right)=F\left(Tn\right)$, Demonstratio math. 40, No. 3, 671-680 (2007) · Zbl 1147.47041 [8] Rhoades, B. E.: Some maps for which periodic and fixed points coincide, Fixed point theory 4, No. 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)