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

A self map T:XX such that for some λ(0,1) and for every x,yX there exists

uC(T,x,y)={d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)}

such that

d(Tx,Ty)λu,

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 n1, where F(T) denotes the set of all fixed points of the mapping T:XX.


MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54E35Metric 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(T)=F(Tn), Fixed point theory and applications 6, 71-105 (2007)
[7]Jeong, G. S.; Rhoades, B. E.: More maps for which F(T)=F(Tn), 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)