On functions preserving regular semimetrics and quasimetrics satisfying the relaxed polygonal inequality

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas - Tập 114 Số 3 - 2020
Jacek Jachymski1, Filip Turoboś1
1Institute of Mathematics, Lodz University of Technology, 90-924 Lodz, Poland

Tóm tắt

AbstractWe obtain characterizations of non-negative functions on $$[0,+\infty )$$ [ 0 , + ∞ ) which preserve some classes of semimetrics. In particular, one of our main results says that for a non-decreasing function $$f:[0,+\infty )\rightarrow [0,+\infty )$$ f : [ 0 , + ∞ ) → [ 0 , + ∞ ) the following statements are equivalent: (i) for any semimetric space (X, d), if d satisfies the relaxed polygonal inequality, then so does $$f\circ d$$ f ∘ d ; (ii) there exist a constant $$c\geqslant 1$$ c ⩾ 1 and a subadditive function $$g:[0,+\infty ) \rightarrow [0,+\infty )$$ g : [ 0 , + ∞ ) → [ 0 , + ∞ ) such that $$g^{-1}\left( \{ 0 \} \right) = \{ 0 \}$$ g - 1 { 0 } = { 0 } and $$g\leqslant f \leqslant cg$$ g ⩽ f ⩽ c g . We also obtain a complete characterization of functions preserving regularity of a semimetric space in the sense of Bessenyei and Páles. Finally, we give another proof of the theorem of Pongsriiam and Termwuttipong on functions transforming metrics into ultrametrics.

Từ khóa


Tài liệu tham khảo

An, T.V., Dung, N.V., Kadelburg, Z., Radenović, S.: Various generalizations of metric spaces and fixed point theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109, 175–198 (2015)

An, T.V., Dung, N.V.: Answers to Kirk–Shahzad’s questions on strong $$b$$-metric spaces. Taiwan. J. Math. 20, 1175–1184 (2016)

Bessenyei, M., Páles, Z.: A contraction principle in semimetric spaces. J. Nonlinear Convex Anal. 18, 515–524 (2015)

Blumenthal, L.M.: Remarks concerning the Euclidean four-point property. Ergeb. Math. Kolloq. Wien 7, 7–10 (1936)

Borsík, J., Doboš, J.: Functions the composition with a metric of which is a metric. Math. Slovaca 31, 3–12 (1981)

Borsík, J., Doboš, J.: On metric preserving functions. Real Anal. Exch. 13, 285–293 (1988)

Chittenden, E.W.: On the equivalence of ècart and voisinage. Trans. Am. Math. Soc. 18, 161–166 (1917)

Chrza̧szcz, K., Jachymski, J., Turoboś, F.: On characterizations and topology of regular semimetric spaces. Publ. Math. Debrecen 93, 87–105 (2018)

Corazza, P.: Introduction to metric-preserving functions. Am. Math. Mon. 106, 309–323 (1999)

Doboš, J.: Metric preserving functions (1998). http://web.science.upjs.sk/jozefdobos/wp-content/uploads/2012/03/mpf1.pdf. Accessed on 1 Jan 2020

Doboš, J.: On modifications of the Euclidean metric on reals. Tatra Mt. Math. Publ. 8, 51–54 (1996)

Doboš, J., Piotrowski, Z.: Some remarks on metric preserving functions. Real Anal. Exch. 19, 317–320 (1994)

Doboš, J., Piotrowski, Z.: A note on metric preserving functions. Int. J. Math. Math. Sci. 19, 199–200 (1996)

Doboš, J., Piotrowski, Z.: When distance means money. Internat. J. Math. Ed. Sci. Tech. 28, 513–518 (1997)

Dovgoshey, O.: Combinatorial properties of ultrametrics and generalized ultrametrics (2019). arXiv:1908.08349

Dovgoshey, O.: On ultrametric-preserving functions. Math. Slovaca 70(1), 173–182 (2020)

Dung, N.V., Hang, V.T.L.: Remarks on cyclic contractions in $$b$$-metric spaces and applications to integral equations. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111, 247–255 (2017)

Dung, N.V., Hang, V.T.L.: On the metrization problem of $$\nu $$-generalized metric spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112, 1295–1303 (2018)

Fagin, R., Kumar, R., Sivakumar, D.: Comparing top k lists. SIAM J. Discrete Math. 17, 134–160 (2003)

Frink, A.H.: Distance functions and the metrization problem. Bull. Am. Math. Soc. 43, 133–142 (1937)

Khemaratchatakumthorn, T., Pongsriiam, P.: Remarks on b-metric and metric-preserving functions. Math. Slovaca 68, 1009–1016 (2018)

Khemaratchatakumthorn, T., Pongsriiam, P., Samphavat, S.: Further remarks on b-metrics, metric-preserving functions, and other related metrics. Int. J. Appl. Math. Comput. Sci. 14, 473–480 (2019)

Khemaratchatakumthorn, T., Termwuttipong, I.: Metric-preserving functions, w-distances and Cauchy w-distances. Thai J. Math. 5, 51–56 (2012)

Kirk, W.A., Shahzad, N.: Remarks on metric transforms and fixed-point theorems. Fixed Point Theory Appl. 106, 11 (2013)

Kirk, W.A., Shahzad, N.: Fixed Point Theory in Distance Spaces. Springer, Cham (2014)

Petruşel, A., Rus, I.A., Şerban, M.A.: The role of equivalent metrics in fixed point theory. Topol. Methods Nonlinear Anal. 41, 85–112 (2013)

Pongsriiam, P., Termwuttipong, I.: On metric-preserving functions and fixed point theorems. Fixed Point Theory Appl. 179, 14 (2014)

Pongsriiam, P., Termwuttipong, I.: Remarks on ultrametrics and metric-preserving functions. Abstr. Appl. Anal. Art. ID 163258, 1–9 (2014)

Samphavat, S., Khemaratchatakumthorn, T., Pongsriiam, P.: Remarks on $$b$$-metrics, ultrametrics, and metric-preserving functions. Math. Slovaca 70(1), 61–70 (2020)

Wilson, W.A.: On certain types of continuous transformations of metric spaces. Am. J. Math. 57, 62–68 (1935)

Wilson, W.A.: On semi-metric spaces. Am. J. Math. 53, 361–373 (1931)

Wu, Z.Q., Zhang, X.Z., Zhu, C.X.: Some new fixed point theorems for single and set-valued admissible mappings in Menger PM-spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 110, 755–769 (2016)

Thông tin
Thông tin xuất bản

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

Tập 114 Số 3

Thông tin tác giả