A Note on The Equivalence of Some Metric and Non-Newtonian Metric Results

Year 2017, Volume: 7, 56 - 58, 19.12.2017

Bahri Turan , Cüneyt Çevik

Abstract

In this short note is on the equivalence between non-Newtonian metric
    (particularly multiplicative metric) and metric. We present a different
    proof the fact that the notion of a non-Newtonian metric space is not more
    general than that of a metric space. Also, we emphasize that a lot of fixed
    point results in the non-Newtonian metric setting can be directly obtained
    from their metric counterparts.

Keywords

Arithmetic, non-Newtonian metric, multiplicative metric, fixed point

References

• Abbas, M., Ali, B., Suleiman, Y.I., Common fixed points of locally contractive mappings in multiplicative metric spaces with application, Int. J. Math. Math. Sci., 2015, Art. ID 218683, 7 pp.
• Abbas, M., De la Sen, M., Nazir, T., Common fixed points of generalized rational type cocyclic mappings in multiplicative metric spaces, Discrete Dyn. Nat. Soc., 2015, Art. ID 532725, 10 pp.
• Binbaşıoğlu, D., Demiriz, S., Türkoğlu, D., Fixed points of non-Newtonian contraction mappings on non-Newtonian metric spaces, J. Fixed Point Theory Appl. 2015, doi: 10.1007/s11784-015-0271-y, 12 pp.
• Çakmak, A.F., Başar, F., Some new results on sequence spaces with respect to non-Newtonian calculus, J. Inequal. Appl. 2012, doi:10.1186/1029-242X-2012-228, 17 pp.
• Deshpande, B., Sheikh, S.A., Common fixed point theorems of Meir-Keeler type on multiplicative metric spaces, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math., 23(2016), no. 2, 131–143.
• Grossman, M., Katz, R., Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972.
• Gu, F., Cho, Y.J., Common fixed point results for four maps satisfying \phi-contractive condition in multiplicative metric spaces, Fixed Point Theory Appl., 2015, 2015:165, 19 pp.
• He, X., Song, M., Chen, D., Common fixed points for weak commutative mappings on a multiplicative metric space, Fixed Point Theory Appl., 2014, 2014:48, 9 pp.
• Jiang, Y., Gu, F., Common coupled fixed point results in multiplicative metric spaces and applications, J. Nonlinear Sci. Appl., 10(2017), no. 4, 1881–1895.
• Kirişci, M., Topological structures of non-Newtonian metric spaces, Electron. J. Math. Anal. Appl., 5(2017), no. 2, 156–169.
• Mongkolkeha, C., Sintunavarat, W., Best proximity points for multiplicative proximal contraction mapping on multiplicative metric spaces, J. Nonlinear Sci. Appl., 8(2015), no. 6, 1134-1140.
• Özavşar, M., Çevikel, A.C., Fixed points of multiplicative contraction mappings on multiplicative metric spaces, arXiv:1205.5131v1, 2012.
• Yamaod, O., Sintunavarat, W., Some fixed point results for generalized contraction mappings with cyclic (\alpha ,\beta )-admissible mapping in multiplicative metric spaces, J. Inequal. Appl., 2014, 2014:488, 15 pp.