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Accessible Unlicensed Requires Authentication Published by De Gruyter October 20, 2018

Remarks on b-Metric and metric-preserving functions

Tammatada Khemaratchatakumthorn and Prapanpong Pongsriiam
From the journal Mathematica Slovaca

Abstract

We introduce new classes of functions related to metric-preserving functions and b-metrics. We investigate their properties and compare them to those of metric-preserving functions.

  1. Communicated by Ján Borsík

Acknowledgement

We are very grateful to both referees for giving us many suggestions which improve the quality of this article. The first author receives financial support from Faculty of Science Silpakorn University, contract number SRF-PRG-2559-01. The second author currently receives financial support jointly from The Thailand Research Fund and Faculty of Science Silpakorn University, grant number RSA5980040.

References

[1] BAKHTIN, I. A.: The contraction mapping principle in quasimetric spaces, Funct. Anal. 30 (1989), 26–37.Search in Google Scholar

[2] BORSÍK, J.—DOBOŠ, J.: On metric preserving functions, Real Anal. Exchange 13 (1987–88), 285–293.Search in Google Scholar

[3] BORSÍK, J.—DOBOŠ, J.: Functions whose composition with every metric is a metric, Math. Slovaca 31 (1981), 3–12.Search in Google Scholar

[4] CORAZZA, P.: Introduction to metric-preserving functions, Amer. Math. Monthly 106(4) (1999), 309–323.Search in Google Scholar

[5] DAS, P. P.: Metricity preserving transforms, Pattern Recognition Letters 10 (1989), 73–76.Search in Google Scholar

[6] DOBOŠ, J.: Metric Preserving Functions, Online Lecture Notes available at http://web.science.upjs.sk/jozefdobos/wp-content/uploads/2012/03/mpf1.pdfSearch in Google Scholar

[7] DOBOŠ, J.: On modification of the Euclidean metric on reals, Tatra Mt. Math. Publ. 8 (1996), 51–54.Search in Google Scholar

[8] DOBOŠ, J.: A survey of metric-preserving functions, Questions Answers Gen. Topology 13 (1995), 129–133.Search in Google Scholar

[9] DOBOŠ, J.—PIOTROWSKI, Z.: When distance means money, Internat. J. Math. Ed. Sci. Tech. 28 (1997), 513–518.Search in Google Scholar

[10] DOBOŠ, J.—PIOTROWSKI, Z.: A note on metric-preserving functions, Int. J. Math. Math. Sci. 19 (1996), 199–200.Search in Google Scholar

[11] DOBOŠ, J.—PIOTROWSKI, Z.: Some remarks on metric-preserving functions, Real Anal. Exchange 19 (1993–94), 317–320.Search in Google Scholar

[12] KELLY, J.: General Topology, Springer-Verlag, 1955.Search in Google Scholar

[13] KIRK, W.A.—SHAHZAD, N.: Fixed Point Theory in Distance Spaces, Springer International Publishing Switzerland, 2014.Search in Google Scholar

[14] PETRUŞEL, A.—RUS, I. A.—ŞERBAN, M. A.: The role of equivalent metrics in fixed point theory, Topol. Methods Nonlinear Anal. 41(1), (2013), 85–112.Search in Google Scholar

[15] PIOTROWSKI, Z.—VALLIN, R. W.: Functions which preserve Lebesgue spaces, Comment. Math. Prace Mat. 43(2) (2003), 249–255.Search in Google Scholar

[16] POKORNÝ, I.: Some remarks on metric-preserving functions, Tatra Mt. Math. Publ. 2 (1993), 65–68.Search in Google Scholar

[17] PONGSRIIAM, P.—TERMWUTTIPONG, I.: On metric-preserving functions and fixed point theorems, Fixed Point Theory Appl. (2014), 2014:179, 14 pp.Search in Google Scholar

[18] PONGSRIIAM, P.—TERMWUTTIPONG, I.: Remarks on ultrametrics and metric-preserving functions, Abstr. Appl. Anal. (2014), Article ID 163258, 9 pp.Search in Google Scholar

[19] SREENIVASAN, T. K.: Some properties of distance functions, J. Indian. Math. Soc. (N.S.) 11 (1947), 38–43.Search in Google Scholar

[20] TERMWUTTIPONG, I.—OUDKAM, P.: Total boundedness, completeness and uniform limits of metric-preserving functions, Ital. J. Pure Appl. Math. 18 (2005), 187–196.Search in Google Scholar

[21] VALLIN, R. W.: Continuity and differentiability aspects of metric preserving functions, Real Anal. Exchange 25(2) (1999/2000), 849–868.Search in Google Scholar

Received: 2017-02-08
Accepted: 2017-07-02
Published Online: 2018-10-20
Published in Print: 2018-10-25

© 2018 Mathematical Institute Slovak Academy of Sciences