On w-Isbell-convexity





Modular pseudometric, Isbell-convexity, $w$-Isbell-convexity


Chistyakov introduced and developed a concept of modular metric for an arbitrary set in order to generalise the classical notion of modular on a linear space. In this article, we introduce the theory of hyperconvexity in the setting of modular pseudometric that is herein called w-Isbell-convexity. We show that on a modular set, w-Isbell-convexity is equivalent to hyperconvexity whenever the modular pseudometric is continuous from the right on the set of positive numbers.


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Author Biographies

Olivier Olela Otafudu, North-West University

School of Mathematical and Statistical Sciences

Katlego Sebogodi, University of Johannesburg

Department of Mathematics and Applied Mathematics


A. A. N. Abdou, Fixed points of Kannan maps in modular metric spaces, AIMS Maths 5 (2020), 6395-6403. https://doi.org/10.3934/math.2020411

A. A. N. Abdou and M. A. Khamsi, Fixed point results of pointwise contractions in modular metric spaces, Fixed Point Theory Appl. 2013 (2013):163. https://doi.org/10.1186/1687-1812-2013-163

C. Alaca, M. E. Ege and C. Park, Fixed point results for modular ultrametric spaces, J. Comput. Anal. Appl. 20 (2016), 1259-1267.

A. H. Ansari, M. Demma, L. Guran, J. R. Lee and C. Park, Fixed point results for C-class functions in modular metric spaces. J. Fixed Point Theory Appl. 20, no. 3 (2018), Paper No. 103. https://doi.org/10.1007/s11784-018-0580-z

V. V. Chistyakov, A fixed point theorem for contractions in modular metric spaces, arXiv:1112.5561.

V. V. Chistyakov, Metric modular spaces: Theory and applications, SpringerBriefs in Mathematics, Springer, Switzerland, 2015. https://doi.org/10.1007/978-3-319-25283-4

V. V. Chistyakov, Modular metric spaces generated by F-modulars, Folia Math. 15 (2008), 3-24.

V. V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72 (2010), 1-14. https://doi.org/10.1016/j.na.2009.04.057

S. Cobzas, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics, Springer, Basel, 2012. https://doi.org/10.1007/978-3-0348-0478-3

M. E. Ege and C. Alaca, Fixed point results and an application to homotopy in modular metric spaces, J. Nonlinear Sci. Appl. 8 (2015), 900-908. https://doi.org/10.22436/jnsa.008.06.01

M. E. Ege and C. Alaca, Some properties of modular S-metric spaces and its fixed point results, J. Comput. Anal. Appl. 20 (2016), 24-33.

M. E. Ege and C. Alaca, Some results for modular b-metric spaces and an application to system of linear equations, Azerb. J. Math. 8 (2018), 3-14.

R. Espínola and M. A. Khamsi, Introduction to hyperconvex spaces, in: Handbook of Metric Fixed Point Theory, Kluwer Academic, Dordrecht, The Netherlands (2001), pp. 39135. https://doi.org/10.1007/978-94-017-1748-9_13

A. Gholidahneh, S. Sedghi, O. Ege, Z. D. Mitrovic and M. de la Sen, The Meir-Keeler type contractions in extended modular b-metric spaces with an application, AIMS Math. 6 (2021), 1781-1799. https://doi.org/10.3934/math.2021107

H. Hosseinzadeh and V. Parvaneh, Meir-Keeler type contractive mappings in modular and partial modular metric spaces, Asian-Eur. J. Math. 13 (2020): 2050087. https://doi.org/10.1142/S1793557120500874

E. Kemajou, H.-P. Künzi and O. Olela Otafudu, The Isbell-hull of di-space, Topology Appl. 159 (2012), 2463-2475. https://doi.org/10.1016/j.topol.2011.02.016

M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, Wiley-Interscience, New York, NY, USA, 2001. https://doi.org/10.1002/9781118033074

H.-P. Künzi, An introduction to quasi-uniform spaces, Contemp. Math. 486 (2009), 239-304. https://doi.org/10.1090/conm/486/09511

H.-P. Künzi and O. Olela Otafudu, q-hyperconvexity in quasipseudometric spaces and fixed point theorems, J. Funct. Spaces Appl. 2012 (2012): Art. ID 765903. https://doi.org/10.1155/2012/765903

N. Kumar and R. Chugh, Convergence and stability results for new three step iteration process in modular spaces, Aust. J. Math. Anal. Appl. 14 (2017): 14.

Y. Mutemwa, O. Olela Otafudu and H. Sabao, On gluing of quasi-pseudometric spaces, Khayyam J. Math. 6 (2020), 129-140.

O. Olela Otafudu, On one-local retract in quasi-metric spaces, Topology Proc. 45 (2015), 271-281.

O. Olela Otafudu and H. Sabao, Set-valued contractions and $q$-hyperconvex spaces, J. Nonlinear Convex Anal. 18 (2017), 1609-1617. https://doi.org/10.4995/agt.2017.5818

R. C. Sine, On nonlinear contraction semigroups in sup norm spaces, Nonlinear Anal. 3 (1979), 885-890. https://doi.org/10.1016/0362-546X(79)90055-5

H. Sabao and O. Olela Otafudu, On soft quasi-pseudometric spaces, Appl. Gen. Topol. 22 (2021), 17-30. https://doi.org/10.4995/agt.2021.13084

S. Salbany, Injective objects and morphisms, in: Categorical Topology and Its Relation to Analysis, Algebra and Combinatorics, Prague, 1988, World Sci. Publ., Teaneck, NJ, 1989, pp. 394-409.

S. Yamamuro, On conjugate space of Nakano space, Trans. Amer. Math. Soc. 90 (1959), 291-311. https://doi.org/10.1090/S0002-9947-1959-0132378-1

C. I. Zhu, J. Chen, X. J. Huang and J. H. Chen, Fixed point theorems in modular spaces with simulation functions and altering distance functions with applications, J. Nonlinear Convex Anal. 21 (2020), 1403-1424.




How to Cite

O. Olela Otafudu and K. Sebogodi, “On w-Isbell-convexity”, Appl. Gen. Topol., vol. 23, no. 1, pp. 91–105, Apr. 2022.



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