Skip to main content
SpringerLink
Log in

Noncompactness Measure and Fixed Points for Multi-Valued Functions on Uniform Spaces

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Some concepts, such as non-compactness measure and condensing operators, defined on metric spaces are extended to uniform spaces. Such extensions allow us to locate, in the context of uniform spaces, some classical results existing in nonlinear analysis. Applications of our results are given for multi-valued operators defined on locally convex spaces. The main aim of this work is to unify some well-known results existing in complete metric and vector topological spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amini, A., Fakhar, J., Zafarani, J.: Fixed point theorems for the class S-KKM mappings in abstract convex spaces. Nonlinear Anal. Theory Methods Appl. 66(1), 14–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atkin, C.J.: Boundedness in uniform spaces, topological groups, and homogeneous spaces. Acta Mathematica Hugarica 57(3–4), 213–232 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Banach, S.: Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundamenta Mathematicae 3(1), 133–181 (1922)

    Article  MATH  Google Scholar 

  4. Bourbaki, N.: Elements of Mathematics, General Topology. Part 1. Hermann, Paris (1966)

  5. Bourbaki, N.: Elements of Mathematics, General Topology. Part 2. Hermann, Paris (1966)

  6. Brøndsted, A.: On a lemma of Bishop and Phelps. Pac. J. Math. 55(2), 335–341 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brouwer, L.E.J.: Über abbildungen von mannigfaltigkeiten. Mathematische Annalen 70, 161–115 (1912)

    Article  Google Scholar 

  8. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  9. Castaing, Ch.: Sur les multi-applications mesurables. Reveu française d’informatique et de recherche opérationnelle 1(1), 91–126 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Darbo, G.: Punti uniti in transformazioni a codominio non compacto. Rendiconti del Seminario Matematico della Università di Padova, Tome 24, 84–92 (1955)

    MathSciNet  MATH  Google Scholar 

  11. Dhompongsa, S., Inthakon, W., Kaewkhao, A.: Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 350(1), 12–17 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Espínola, R., Kirk, W.A.: Fixed points and approximate fixed points in product spaces. Taiwan. J. Math. 5(2), 405–416 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fang, J.X.: The variational principle and fixed point theorems in certain topological spaces. J. Math. Anal. Appl. 202(2), 398–4120 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fierro, R., Martínez, C., Orellana, E.: Weak conditions for existence of random fixed points. Fixed Point Theory 12(1), 83–90 (2011)

    MathSciNet  MATH  Google Scholar 

  15. Furi, M., Vignoli, A.: Fixed points for densifying mappings. Rendiconti Accademia Nazionale Lincei 47(6), 465–467 (1969)

    MathSciNet  MATH  Google Scholar 

  16. Hamel, A.H.: Equivalents to Ekeland’s variational principle in uniform spaces. Nonlinear Anal. Theory Methods Appl. 62(5), 913–924 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hamel, A.H., Löhne, A.: A minimal point theorem in uniform spaces. In: Agarwal, R.P., O’Regan, D. (eds.) Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, vol. 1, pp. 577–593. Kluwer Academic Publisher, Dordrecht (2003)

    Google Scholar 

  18. Hejcman, J.: Boundedness in uniform spaces and topological groups. Czechoslov. Math. J. 9(4), 544–563 (1959)

    MathSciNet  MATH  Google Scholar 

  19. Himmelberg, C.J., Porter, J.R., Van Vleck, F.S.: Fixed point theorems for condensing multifunctions. Proc. Am. Math. Soc. 23, 635–641 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kakutani, S.: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 8(3), 457–459 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kuratowski, C.: Sur les espaces complets. Fundamenta Mathematicae 15, 301–309 (1930)

    Article  MATH  Google Scholar 

  22. Mizoguchi, N.: A generalization of Brøndsted’s results and its applications. Proc. Am. Math. Soc. 108(3), 707–714 (1990)

    MATH  Google Scholar 

  23. Mizoguchi, N., Takahashi, W.: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1), 177–188 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  24. Reich, S., Zaslavski, A.J.: Approximate fixed points of nonexpansive mappings in unbounded sets. J. Fixed Point Theory Appl. 13(2), 627–632 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sadovskii, B.N.: A fixed point principle. J. Funct. Anal. Appl. 1(2), 151–153 (1977)

    Article  MathSciNet  Google Scholar 

  26. Saint-Raymond, J.: Topologie sur l’ensemble des compacts non vides d’un espace topologique séparé. Séminaire Choquet, Tome 9 (2, exp No. 21):1–6 (1969/70)

  27. Schauder, J.: Der fixpunktsatz in funktionalrumen. Studia Math. 2, 171–180 (1930)

    Article  MATH  Google Scholar 

  28. Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340(2), 1088–1095 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tarafdar, E.: An approach to fixed-point theorems on uniform spaces. Trans. Am. Math. Soc. 74, 209–225 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tychonoff, A.: Ein fixpunktsatz. Mathematische Annalen 111, 767–776 (1935)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaiso, Chile

    Raúl Fierro

Authors
  1. Raúl Fierro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raúl Fierro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fierro, R. Noncompactness Measure and Fixed Points for Multi-Valued Functions on Uniform Spaces. Mediterr. J. Math. 15, 95 (2018). https://doi.org/10.1007/s00009-018-1140-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-018-1140-x

Mathematics Subject Classification

Keywords

Search

Navigation