Geodesic
Eckland and Bishop-Phelps variational principles in partially ordered spaces
Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 135, pp. 234-240
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, an assertion about the minimum of the graph of a mapping acting in partially ordered spaces is obtained. The proof of this statement uses the theorem on the minimum of a mapping in a partially ordered space from [A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy. Caristi-like condition and the existence of minima of mappings in partially ordered spaces // Journal of Optimization Theory and Applications. 2018. V. 180. Iss. 1, 48–61]. It is also shown that this statement is an analogue of the Eckland and Bishop–Phelps variational principles which are effective tools for studying extremal problems for functionals defined on metric spaces. Namely, the statement obtained in this paper and applied to a partially ordered space created from a metric space by introducing analogs of the Bishop–Phelps order relation, is equivalent to the classical Eckland and Bishop–Phelps variational principles.
Keywords: partially ordered space, Caristi-like inequality, infimum of a functional.
Mots-clés : variational principles 
@article{VTAMU_2021_26_135_a1,
     author = {Z. T. Zhukovskaya and T. V. Zhukovskaya and O. V. Filippova},
     title = {Eckland and {Bishop-Phelps} variational principles in partially ordered spaces},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {234--240},
     year = {2021},
     volume = {26},
     number = {135},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2021_26_135_a1/}
}
TY  - JOUR
AU  - Z. T. Zhukovskaya
AU  - T. V. Zhukovskaya
AU  - O. V. Filippova
TI  - Eckland and Bishop-Phelps variational principles in partially ordered spaces
JO  - Vestnik rossijskih universitetov. Matematika
PY  - 2021
SP  - 234
EP  - 240
VL  - 26
IS  - 135
UR  - http://geodesic.mathdoc.fr/item/VTAMU_2021_26_135_a1/
LA  - ru
ID  - VTAMU_2021_26_135_a1
ER  - 
%0 Journal Article
%A Z. T. Zhukovskaya
%A T. V. Zhukovskaya
%A O. V. Filippova
%T Eckland and Bishop-Phelps variational principles in partially ordered spaces
%J Vestnik rossijskih universitetov. Matematika
%D 2021
%P 234-240
%V 26
%N 135
%U http://geodesic.mathdoc.fr/item/VTAMU_2021_26_135_a1/
%G ru
%F VTAMU_2021_26_135_a1
Z. T. Zhukovskaya; T. V. Zhukovskaya; O. V. Filippova. Eckland and Bishop-Phelps variational principles in partially ordered spaces. Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 135, pp. 234-240. http://geodesic.mathdoc.fr/item/VTAMU_2021_26_135_a1/

[1] I. Ekeland, “Nonconvex minimization problems”, Bulletin of the American Mathematical Society. New Series, 1:3 (1979), 443–474 | DOI | Zbl

[2] A. Granas, J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer–Verlag, New York, 2003, 690 pp. | DOI | Zbl

[3] A. V. Arutyunov, S. E. Zhukovskiy, “Variational Principles in Nonlinear Analysis and Their Generalization”, Mathematical Notes, 103:6 (2018), 1014–1019 | DOI | Zbl

[4] A. V. Arutyunov, “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 (2015), 24–37 | DOI | Zbl

[5] J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions”, Trans. Amer. Math. Soc., 215 (1976), 241–251 | DOI | Zbl

[6] Z. T. Zhukovskaya, S. E. Zhukovskiy, “On generalizations and applications of variational principles of nonlinear analysis”, Tambov University Reports. Series: Natural and Technical Sciences, 23:123 (2018), 377–385 (In Russian)

[7] A. V. Arutyunov, B. D. Gel’man, E. S. Zhukovskiy, S. E. Zhukovskiy, “Caristi-like condition. Existence of solutions to equations and minima of functions in metric spaces”, Fixed Point Theory, 20:1 (2019), 31–58 | DOI | Zbl

[8] A. V. Arutyunov, E. S. Zhukovskiy, S. E. Zhukovskiy, “Caristi-like condition and the existence of minima of mappings in partially ordered spaces”, Journal of Optimization Theory and Applications, 180:1 (2019), 48–61 | DOI | Zbl

[9] F. Clark, Optimization and Nonsmooth Analysis, Nauka Publ., Moscow, 1988 (In Russian)