Some equivalences between cone \(b\)-metric spaces and \(b\)-metric spaces. (English) Zbl 1292.54027

Summary: We introduce a \(b\)-metric on cone \(b\)-metric spaces and then prove some equivalences between them. As applications, we show that fixed point theorems on cone \(b\)-metric spaces can be obtained from fixed point theorems on \(b\)-metric spaces.


54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI


[1] Bakhtin, I. A., The contraction mapping principle in quasimetric spaces, Functional Analysis, 30, 26-37 (1989) · Zbl 0748.47048
[2] Czerwik, S., Contraction mappings in \(b\)-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1, 1, 5-11 (1993) · Zbl 0849.54036
[3] Akkouchi, M., A common fixed point theorem for expansive mappings under strict implicit conditions on \(b\)-metric spaces, Acta Universitatis Palackianae Olomucensis, Facultas Rerum Naturalium. Mathematica, 50, 1, 5-15 (2011) · Zbl 1246.54035
[4] Olatinwo, M. O.; Imoru, C. O., A generalization of some results on multi-valued weakly Picard mappings in \(b\)-metric space, Fasciculi Mathematici, 40, 45-56 (2008) · Zbl 1177.54028
[5] Păcurar, M., A fixed point result for \(\varphi \)-contractions on \(b\)-metric spaces without the boundedness assumption, Fasciculi Mathematici, 43, 127-137 (2010) · Zbl 1196.54079
[6] Jovanović, M.; Kadelburg, Z.; Radenović, S., Common fixed point results in metric-type spaces, Fixed Point Theory and Applications, 2010 (2010) · Zbl 1207.54058
[7] Hussain, N.; Dorić, D.; Kadelburg, Z.; Radenović, S., Suzuki-type fixed point results in metric type spaces, Fixed Point Theory and Applications, 2012, article 126 (2012) · Zbl 1274.54128
[8] Hussain, N.; Shah, M. H., KKM mappings in cone \(b\)-metric spaces, Computers & Mathematics with Applications, 62, 4, 1677-1684 (2011) · Zbl 1231.54022
[9] Huang, H.; Xu, S., Fixed point theorems of contractive mappings in cone \(b\)-metric spaces and applications, Fixed Point Theory and Applications, 2013, article 112 (2013) · Zbl 1423.54081
[10] Shi, L.; Xu, S., Common fixed point theorems for two weakly compatible self-mappings in cone \(b\)-metric spaces, Fixed Point Theory and Applications, 2013, article 120 (2013) · Zbl 1429.54059
[11] Huang, L. G.; Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, 332, 2, 1468-1476 (2007) · Zbl 1118.54022
[12] Du, W. S., A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis: Theory, Methods & Applications, 72, 5, 2259-2261 (2010) · Zbl 1205.54040
[13] Feng, Y.; Mao, W., The equivalence of cone metric spaces and metric spaces, Fixed Point Theory, 11, 2, 259-264 (2010) · Zbl 1221.54055
[14] Harandi, A. A.; Fakhar, M., Fixed point theory in cone metric spaces obtained via the scalarization method, Computers & Mathematics with Applications, 59, 11, 3529-3534 (2010) · Zbl 1197.54055
[15] Kaekhao, A.; Sintunavarat, W.; Kumam, P., Common fixed point theorems of \(c\)-distance on cone metric spaces, Journal of Nonlinear Analysis and Application, 2012 (2012)
[16] Khani, M.; Pourmahdian, M., On the metrizability of cone metric spaces, Topology and Its Applications, 158, 2, 190-193 (2011) · Zbl 1206.54026
[17] Du, W. S.; Karapinar, E., A note on cone \(b\)-metric and its related results: generalizations or equivalence?, Fixed Point Theory and Applications, 2013, article 210 (2013) · Zbl 1293.54021
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.