Advertisement

Caristi-Like Condition and the Existence of Minima of Mappings in Partially Ordered Spaces

  • Aram V. Arutyunov
  • Evgeny S. Zhukovskiy
  • Sergey E. ZhukovskiyEmail author
Article
  • 58 Downloads

Abstract

In this paper, we study mappings acting in partially ordered spaces. For these mappings, we introduce a condition, analogous to the Caristi-like condition, used for functions defined on metric spaces. A proposition on the achievement of a minimal point by a mapping of partially ordered spaces is proved. It is shown that a known result on the existence of the minimum of a lower semicontinuous function defined on a complete metric space follows from the obtained proposition. New results on coincidence points of mappings of partially ordered spaces are obtained.

Keywords

Partially ordered space Caristi-like condition Coincidence point Orderly covering mapping 

Mathematics Subject Classification

06A06 65K10 

Notes

Acknowledgements

The publication was supported by a grant from the Russian Science Foundation (Project No. 17-11-01168). Authors are grateful to anonymous referees for useful comments and remarks.

References

  1. 1.
    Arutyunov, A.V.: Caristi’s condition and existence of a minimum of a lower bounded function in a metric space. Applications to the theory of coincidence points. Proc. Steklov Inst. Math. 291, 24–37 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arutyunov, A.V., Zhukovskiy, S.E.: Variational principles in nonlinear analysis and their generalization. Math. Notes 103, 1014–1019 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arutyunov, A.V., Gel’man, B.D., Zhukovskiy, E.S., Zhukovskiy, S.E.: Caristi-like condition. Existence of solutions to equations and minima of functions in metric spaces. Fixed Point Theory (2019, to appear)Google Scholar
  4. 4.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)zbMATHGoogle Scholar
  5. 5.
    Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Agarwal, R.P., El-Gebeily, M.A., O’Reagan, D.: Generalized contractions in partially ordered metric spaces. Appl. Anal. 87, 109–116 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Turinici, M.: Contraction maps in ordered metrical structures. In: Pardalos, P., Rassias, T. (eds.) Mathematics Without Boundaries, pp. 533–575. Springer, New York (2014)Google Scholar
  9. 9.
    Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Coincidence points principle for mappings in partially ordered spaces. Topol. Appl. 179, 13–33 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Coincidence points principle for set-valued mappings in partially ordered spaces. Topol. Appl. 201, 330–343 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Coincidence points of set-valued mappings in partially ordered spaces. Dokl. Math. 88, 727–729 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: On coincidence points of mappings in partially ordered spaces. Dokl. Math. 88, 710–713 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Caristi, J.: Fixed point theorems form mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241–251 (1976)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Dover Publications Inc., New York (1999)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Peoples’ Friendship University of RussiaMoscowRussia
  2. 2.Institute for Information Transmission Problem of the Russian Academy of Sciences (Kharkevich Institute)MoscowRussia
  3. 3.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia
  4. 4.Tambov State University named after G.R. DerzhavinTambovRussia
  5. 5.Moscow Institute of Physics and TechnologyDolgoprudny, Moscow regionRussia

Personalised recommendations