Document Type : Research Paper


Department of Mathematics‎, ‎Bali kesir University‎, ‎10145‎, ‎Bali kesir‎, ‎Turkey


‎The aim of this paper is to establish the equivalence between the concepts‎ ‎of an $S$-metric space and a cone $S$-metric space using some topological‎ ‎approaches‎. ‎We introduce a new notion of a $TVS$-cone $S$-metric space using‎ ‎some facts about topological vector spaces‎. ‎We see that the known results on‎ ‎cone $S$-metric spaces (or $N$-cone metric spaces) can be directly obtained‎ from the studies on $S$-metric spaces‎.


Main Subjects

[1] G. Y. Chen, X. X. Huang, X. Q. Yang, Vector Optimization, Springer-Verlag, Berlin, Heidelberg, Germany, 2005.
[2] D. Dhamodharan, R. Krishnakumar, Cone S-metric space and fixed point theorems of contractive mappings, Annals of Pure. Appl. Math. 14 (2) (2017), 237-243.
[3] W. S. Du, On some nonlinear problems induced by an abstract maximal element principle, J. Math. Anal. Appl. 347 (2008), 391-399.
[4] W. S. Du, A note on cone metric fixed point theory and its equivalence, Nonlinear Anal. 72 (2010), 2259-2261.
[5] Z. Ercan, On the end of the cone metric spaces, Topology Appl. 166 (2014), 10-14.
[6] J. Fernandez, G. Modi, N. Malviya, Some fixed point theorems for contractive maps in N-cone metric spaces, Math. Sci. 9 (2015), 33-38.
[7] Chr. Gerth (Tammer), P. Weidner, Nonconvex separation theorems and some applications in vector optimization, J. Optim. Theory Appl. 67 (1990), 297-320.
[8] A. Göpfert, Chr. Tammer, C. Zalinescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces, Nonlinear Anal. 39 (2000), 909-922.
[9] A. Göpfert, Chr. Tammer, H. Riahi, C. Z˘ alinescu, Variational Methods in Partially Ordered Spaces, Springer-Verlag, New York, 2003.
[10] H. L. Guang, Z. Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007), 1468-1476.
[11] A. Gupta, Cyclic contraction on S-metric space, Int. J. Anal. Appl. 3 (2) (2013), 119-130.
[12] N. T. Hieu, N. T. Ly, N. V. Dung, A generalization of´C iri´ c quasi-contractions for maps on S-metric spaces, Thai J. Math. 13 (2) (2015), 369-380.
[13] Z. Kadelburg, S. Radenovi´ c, V. Rakoˇ cevi ´ c, A note on the eqivalence of some metric and cone metric fixed point results, Appl. Math. Lett. 24 (2011), 370-374.
[14] M. Khani, M. Pourmahdian, On the metrizability of cone metric spaces, Topology Appl. 158 (2011), 190-193.
[15] B. Khomdram, Y. Rohen, Y. M. Singh, M. S. Khan, Fixed point theorems of generalised S-β-ψ contractive type mappings, Math. Morav. 22 (1) (2018), 81-92.
[16] N. Malviya, B. Fisher, N-cone metric space and fixed points of asymptotically regular maps, Filomat., (in press).
[17] N. M. Mlaiki, Common fixed points in complex S-metric space, Adv. Fixed Point Theory. 4 (4) (2014),
[18] N. Mlaiki, α-ψ-contractive mapping on S-metric space, Math. Sci. Lett. 4 (1) (2015), 9-12.
[19] N. Y.Özgür, N. Tas, Some Generalizations of Fixed Point Theorems on S-Metric Spaces, Essays in Mathematics and Its Applications in Honor of Vladimir Arnold, New York, Springer, 2016.
[20] N. Y.Özgür, N. Tas, Some fixed point theorems on S-metric spaces, Mat. Vesnik. 69 (1) (2017), 39-52.
[21] N. Y.Özgür, N. Tas, Some new contractive mappings on S-metric spaces and their relationships with the mapping (S25), Math. Sci. 11 (1) (2017), 7-16.
[22] N. Y.Özgür, N. Tas, The Picard theorem on S-metric spaces, Acta Math. Sci. 38 (4) (2018), 1245-1258.
[23] N. Y.Özgür, N. Tas, U. Celik, New fixed-circle results on S-metric spaces, Bull. Math. Anal. Appl. 9 (2) (2017), 10-23.
[24] S. Sedghi, N. V. Dung, Fixed point theorems on S-metric spaces, Mat. Vesnik. 66 (1) (2014), 113-124.
[25] S. Sedghi, N. Shobe, A. Aliouche, A generalization of fixed point theorems in S-metric spaces, Mat. Vesnik. 64 (3) (2012), 258-266.
[26] N. Tas, Suzuki-Berinde type fixed-point and fixed-circle results on S-metric spaces, J. Linear Topol. Algebra. 7 (3) (2018), 233-244.
[27] N. Tas, Various types of fixed-point theorems on S-metric spaces, J. BAUN Inst. Sci. Technol. 20 (2) (2018), 211-223.
[28] N. Tas, N. Y.Özgür, Common fixed point results on complex valued S-metric spaces, Sahand Commun. Math. Anal. (2019), in press.
[29] N. Tas, N. Y.Özgür, Common fixed points of continuous mappings on S-metric spaces, Math. Notes. 104 (4) (2018), 587-600.