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\(q^\lambda \)-hyperconvexity in quasi-metric spaces

  • Collins Amburo AgyingiEmail author
  • Yaé Ulrich Gaba
Article
  • 13 Downloads

Abstract

In this article, we introduce and study the concept of hyperconvexity (which we call \(q^\lambda \)-hyperconvexity) that is appropriate in the category of \(T_0\)-quasi-metric spaces and nonexpansive maps and this will generalize the notion of q-hyperconvexity studied by Kemajou et al. We prove a fixed point result for nonexpansive maps in \(q^\lambda \)-hyperconvex spaces and establish, among other things, that the fixed point set of nonexpansive maps on \(q^\lambda \)-hyperconvex bounded \(T_0\)-quasi-metric spaces is itself \(q^\lambda \)-hyperconvex.

Keywords

q-hyperconvexity \(q^\lambda \)-hyperconvexity Isbell-convex Isbell-complete Nonexpansive maps 

Mathematics Subject Classification

Primary 54X10 58Y30 18D35 Secondary 55Z10 

Notes

References

  1. 1.
    Agyingi, C.A., Haihambo, P., Kunzi, H.-P.A.: Tight extensions of \(T_0\)-quasi-metric spaces. Logic Comput. Hierarchies Ontos-Verlag, pp. 9–22 (2014)Google Scholar
  2. 2.
    Agyingi, C.A.: Supseparability of the space of minimal function pairs. Topol. Appl. 194, 212–227 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Agyingi, C.A.: On di-injective \(T_0\)-quasi-metric spaces. Topol. Appl. 228, 371–381 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aronszajn, N., Panitchpakdi, P.: Extensions of Uniformly continuous transformations and hyperconvex metric spaces. Pac. J. Math. 6, 405–439 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baillon, J.-B.: Nonexpansive mapping and hyperconvex spaces. In: Fixed Point Theory and its Applications, Contemporary Mathematics, vol. 72, pp. 11–19. American Mathematical Society, Providence (1988)Google Scholar
  6. 6.
    Cobzaş, S.: Asymmetric functional analysis. In: Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics. Birkhäuser, Basel (2012)Google Scholar
  7. 7.
    Grünbaum, B.: On some covering and intersection properties in Minkowski spaces. Pac. J. Math. 9, 487–494 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Grünbaum, B.: Some applications of expansion constants. Pac. J. Math. 10, 193–201 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kemajou, E., Künzi, H.-P.A., Otafudu, O.O.: The Isbell-hull of a di-space. Topol. Appl. 159, 2463–2475 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Khamsi, M.A., Knaust, H., Nguyen, N.T., O’Neill, M.D.: Lambda-hyperconvexity in metric spaces. Nonlinear Anal. 43, 21–31 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory. Willey, New York (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Künzi, H.-P.A., Otafudu, O.O.: \(q\)-hyperconvexity in quasi-pseudometric spaces and fixed point theorems. J. Funct. Spaces Appl. 765903 (2012).  https://doi.org/10.1155/2012/765903
  13. 13.
    Salbany, S.: Injective objects and morphisms, In: Categorical Topology and its Relation to Analysis, Algebra and Combinatorics (Prague, 1988), pp. 394–409. World Sci. Publ., Taeneck (1989)Google Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsNelson Mandela UniversityPort ElizabethSouth Africa
  2. 2.Institut de Mathématiques et de Sciences Physiques (IMSP)/UACPorto-NovoBenin
  3. 3.African Center for Advanced Studies (ACAS)YaoundéCameroon

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