Skip to main content
SpringerLink
Log in

Caristi-Type Conditions in Constraint Minimisation of Mappings in Metric and Partially Ordered Spaces

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

We consider the problem of finding minima of mappings defined on metric and partially ordered spaces subject to constraints in the form of inclusions (and as a consequence in the form of equalities and/or inequalities). We introduce analogues of Caristi-type inequality proposed in the studies on minimisation of non-convex functionals in metric spaces. Statements on attainment of minima of non-convex functionals on the solutions of the corresponding inclusions are proved. The proofs are based on the construction of a mapping for which the point of unconstraint minimum is the sought-for constraint minimum for the problem under consideration. We provide conditions for stability of the constraint minima to perturbations of the minimised functionals and the constraints. We also establish connections between the obtained statements on constraint minima, namely, we demonstrate that the statements on constraint minima of functionals in partially ordered spaces are more general than the corresponding statements for functionals defined on metric 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

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 215, 241–251 (1976)

    Article  MathSciNet  Google Scholar 

  2. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley-Interscience, New York (1984)

    Google Scholar 

  3. Bao, T., Mordukhovich, B.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern. 36, 531–562 (2007)

    MathSciNet  Google Scholar 

  4. Khanh, P.Q.: On Caristi-Kirk’s theorem and Ekeland’s variational principle for Pareto extrema. Bull. Pol. Acad. Sci., Math. 37, 33–39 (1989)

    MathSciNet  Google Scholar 

  5. Mordukhovich, B., Wang, B.: Necessary suboptimality and optimality conditions via variational principles. SIAM J. Control Optim. 41, 623–640 (2002)

    Article  MathSciNet  Google Scholar 

  6. Agarwal, R.P., Khamsi, M.A.: Extension of Caristi’s fixed point theorem to vector valued metric space. Nonlinear Anal. TMA 74, 141–145 (2011)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Article  MathSciNet  Google Scholar 

  8. Jachymski, J.R.: Equivalent conditions and the Meir-Keeler type theorems. J. Math. Anal. Appl. 194, 293–303 (1995)

    Article  MathSciNet  Google Scholar 

  9. Jachymski, J.R.: Caristi’s fixed point theorem and selections of set-valued contractions. J. Math. Anal. Appl. 227, 55–67 (1998)

    Article  MathSciNet  Google Scholar 

  10. Suzuki, T.: Generalized Caristi’s fixed point theorems by Bae and others. J. Math. Anal. Appl. 302, 502–508 (2005)

    Article  MathSciNet  Google Scholar 

  11. Kirk, W.A.: Fixed points of asymptotic contractions. J. Math. Anal. Appl. 277, 645–650 (2003)

    Article  MathSciNet  Google Scholar 

  12. Suzuki, T.: Fixed-point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces. Nonlinear Anal. 64, 971–978 (2006)

    Article  MathSciNet  Google Scholar 

  13. Fabian, M.J., Preiss, D.: A generalization of the interior mapping theorem of Clarke and Pourciau. Comment. Math. Univ. Carol. 28(2), 311–324 (1987)

    MathSciNet  Google Scholar 

  14. Takahashi, W.: Minimisation theorems and fixed-point theorems. Nonlinear Anal. Math. Econ. RIMS Kokyuroku 829, 175–191 (1991)

    Google Scholar 

  15. Arutyunov, A.V., Zhukovskiy, S.E.: Variational principles in nonlinear analysis and their generalization. Math. Notes 103, 1014–1019 (2018)

    Article  MathSciNet  Google Scholar 

  16. Arutyunov, A.V., Zhukovskiy, S.E.: Variational principles in analysis and existence of minimisers for functions on metric spaces. SIAM J. Optim. 29(2), 994–1016 (2019)

    Article  MathSciNet  Google Scholar 

  17. Arutyunov, A.V., Gel’man, B.D., Zhukovskiy, E.S., et al.: Caristi-like condition. Existence of solutions to equations and minima of functions in metric spaces. Fixed Point Theory 20(1), 31–58 (2019)

    Article  MathSciNet  Google Scholar 

  18. Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Caristi-like condition and the existence of minima of mappings in partially ordered spaces. J. Optim. Theory Appl. 180(1), 48–61 (2019)

    Article  MathSciNet  Google Scholar 

  19. Zhukovskaya, Z.T., Zhukovskaya, T.V., Filippova, O.V.: Variacionnye principy Eklanda i Bishopa–Felpsa v chastichno uporyadochennyh prostranstvah [Ekeland and Bishop–Phelps variational principles in partially ordered spaces]. Vestn. Ross. Univ. Mat., Russ. Univ. Rep. Math. 26(135), 234–240 (2021). (In Russian, Abstr. in Engl.)

    Google Scholar 

  20. Hai, L.P., Khanh, P.Q.: An induction theorem and Ekeland’s variational principle in partial metric spaces with applications. Optimization 69, 1481–1511 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Clarke, F.H.: Optimisation and Nonsmooth Analysis. Classics in Applied Mathematics, vol. 5. SIAM, Philadelphia (1990)

    Book  Google Scholar 

  23. Martinet, L.E., Fiddyment, G., Madsen, J.R., et al.: Human seizures couple across spatial scales through travelling wave dynamics. Nat. Commun. 8, 14896 (2017)

    Article  Google Scholar 

  24. Muller, L., Chavane, F., Reynolds, J., et al.: Cortical travelling waves: mechanisms and computational principles. Nat. Rev. Neurosci. 5, 255–268 (2018)

    Article  Google Scholar 

  25. Goldenberg, M.M.: Overview of drugs used for epilepsy and seizures: etiology, diagnosis, and treatment. P. T. 35(7), 392–415 (2010)

    Google Scholar 

  26. Meisel, C., Schulze-Bonhage, A., Freestone, D., et al.: Intrinsic excitability measures track antiepileptic drug action and uncover increasing/decreasing excitability over the wake/sleep cycle. Proc. Natl. Acad. Sci. USA 112(47), 14694–14699 (2015)

    Article  Google Scholar 

  27. Bressloff, P.C., Carroll, S.R.: Laminar neural field model of laterally propagating waves of orientation selectivity. PLoS Comput. Biol. 11(10), e1004545 (2015)

    Article  Google Scholar 

  28. Burlakov, E., Verkhlyutov, V., Malkov, I., et al.: Assessment of cortical travelling waves parameters using radially symmetric solutions to neural field equations with microstructure. Stud. Comput. Intell. 925, 51–57 (2020)

    Article  Google Scholar 

  29. Burlakov, E., Verkhlyutov, V., Ushakov, V.: A simple human brain model reproducing evoked MEG based on neural field theory. Stud. Comput. Intell. 1008, 109–116 (2022)

    Article  Google Scholar 

  30. Pinto, D., Ermentrout, B.: Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses. SIAM J. Appl. Math. 62(1), 206–225 (2001)

    Article  MathSciNet  Google Scholar 

  31. Amari, S.: Dynamics of pattern formation in lateral-inhibition type neural fields. Biol. Cybern. 27, 77–87 (1977)

    Article  MathSciNet  Google Scholar 

  32. Coombes, S., Owen, M.R.: Evans functios for integral neural field equations with Heaviside firing rate function. SIAM J. Appl. Dyn. Syst. 34, 574–600 (2004)

    Article  Google Scholar 

  33. Castaing, C.: Sur les equations differentielles multivoques [On multivalued differential equations]. C.R. Acad. Sci. Paris 263, 63–66 (1966)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous reviewers for their constructive comments that helped to improve the paper.

Funding

The results in Sects. 2, 4 and 5 were supported by the Russian Science Foundation (Project No 20-11-20131, https://rscf.ru/en/project/20-11-20131/) and obtained at V.A. Trapeznikov Institute of Control Sciences of RAS. The results in Sect. 3 were supported by the Russian Science Foundation (Project No 23-11-20020, https://rscf.ru/en/project/23-11-20020/) and obtained at Derzhavin Tambov State University.

Author information

Authors and Affiliations

  1. Research Institute of Mathematics, Physics and Computer Sciences, Derzhavin Tambov State University, 33 Internacionalnaya st., Tambov, 392000, Russia

    Evgeny Zhukovskiy & Ivan Malkov

  2. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 65 Profsoyuznaya st., Moscow, 117997, Russia

    Evgeny Zhukovskiy & Evgenii Burlakov

  3. X-Bio Institute, University of Tyumen, 6 Volodarskogo st., Tyumen, 625003, Russia

    Evgenii Burlakov & Ivan Malkov

Authors
  1. Evgeny Zhukovskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evgenii Burlakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ivan Malkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evgeny Zhukovskiy.

Ethics declarations

Competing Interests

The authors have no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhukovskiy, E., Burlakov, E. & Malkov, I. Caristi-Type Conditions in Constraint Minimisation of Mappings in Metric and Partially Ordered Spaces. Set-Valued Var. Anal 31, 35 (2023). https://doi.org/10.1007/s11228-023-00697-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11228-023-00697-w

Keywords

Mathematics Subject Classification

Search

Navigation