Advertisement

Extended Antipodal Theorems

  • Viacheslav V. Kalashnikov
  • Adolphus J. J. Talman
  • Lilia Alanís-López
  • Nataliya I. Kalashnykova
Article
  • 97 Downloads

Abstract

Since 1909 when Brouwer proved the first fixed-point theorem named after him, the fixed-point results in various settings play an important role in the optimization theory and applications. This technique has proven to be indispensable for the proofs of multiple results related to the existence of solutions to numerous problems in the areas of optimization and approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, mathematical economics, etc. It is also worthwhile to mention that the majority of problems of finding solutions (zero-points) of functions (operators) can be easily reduced to that of discovering of fixed points of properly modified mappings. Not only theoretical but also practical (algorithmic) developments are based on the fixed-point theory. For instance, the well-known simplicial (triangulation) algorithms help one to find the desired fixed points in a constructive way. That approach allows one to investigate the solvability of complicated problems arising in theory and applications. In this paper, making use of the triangulation technique, we extend some antipodal and fixed-point theorems to the case of nonconvex, more exactly, star-shaped sets. Also, similar extensions are made for set-valued mappings defined over star-shaped sets.

Keywords

Antipodal theorems Star-shaped subsets Triangulation techniques Set-valued mappings 

Mathematics Subject Classification

90C33 49J53 58C30 

Notes

Acknowledgements

The research activity of the first author was financially supported by the R&D Department “Optimization and Data Science” of the Tecnológico de Monterrey (ITESM), Campus Monterrey, as well as by the SEP-CONACYT Projects CB-2013-01-221676 and FC-2016-01-1938 (Mexico).

References

  1. 1.
    Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford (1964)Google Scholar
  2. 2.
    Herings, P.J.-J.: Static and Dynamic Aspects of General Equilibrium Theory. Kluwer Academic Publishers, Boston (1996)CrossRefzbMATHGoogle Scholar
  3. 3.
    Yang, Z.F.: Computing Equilibria and Fixed Points. Kluwer Academic Publishers, Boston (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Boston (2002)zbMATHGoogle Scholar
  5. 5.
    van der Laan, G., Talman, A.J.J.: A class of simplicial restart fixed point algorithms without an extra dimension. Math. Program. 20, 33–48 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    van der Laan, G.: Existence and approximation of zeros. Math. Program. 28, 1–24 (1984)CrossRefzbMATHGoogle Scholar
  7. 7.
    Todd, M.J.: Improving the convergence of fixed point algorithms. Math. Program. Study 7, 151–179 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Munkres, J.R.: Topology. Prentice Hall, Upper Saddle River (2000)zbMATHGoogle Scholar
  9. 9.
    Searcóid, M.Ó.: Metric Spaces, Springer Undergraduate Mathematics Series, vol. 106. Springer, Berlin (2007)Google Scholar
  10. 10.
    Isac, G., Bulavsky, V.A., Kalashnikov, V.V.: Complementarity, Equilibrium, Efficiency and Economics. Kluwer Academic Publishers, Boston (2002)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Tecnológico de Monterrey (ITESM), Campus MonterreyMonterreyMexico
  2. 2.Central Economics and Mathematics Institute (CEMI)MoscowRussian Federation
  3. 3.Sumy State University (SumDU)SumyUkraine
  4. 4.School of Economics and ManagementTilburg UniversityTilburgThe Netherlands
  5. 5.Universidad Autónoma de Nuevo León (UANL)San Nicolás de los GarzaMexico

Personalised recommendations