Theorems on perturbations of covering mappings in spaces with a distance and in spaces with a binary relation

S. Benarab ; E. S. Zhukovskiy ; W. Merchela

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 52-63

Résumé

Statements on the existence of solutions of special-type equations in spaces with a distance and in spaces with a binary relation are derived. The results obtained generalize the well-known theorems on coincidence points of a covering and a Lipschitz mappings and on Lipschitz perturbations of covering mappings in metric spaces as well as the theorems on coincidence points of a covering and an isotonic mappings and on antitone perturbations of covering mappings in partially ordered spaces. In the first part of the paper, we consider a mapping $F\colon X\times X \to Y$, where $X$ is a metric space and $Y$ is equipped with a distance satisfying only the identity axiom. “Weakened analogs” of the notions of covering and Lipschitz mappings from $X$ to $Y$ are defined. Under the assumption that $F$ is covering in the first argument and Lipschitz in the second argument (in the sense of the definitions of these properties given in the paper), the existence of a solution $x$ to the equation $F(x,x)=y$ is established. It is shown that this statement yields conditions for the existence of a coincidence point of a covering and a Lipschitz mappings acting from $X$ to $Y$. In the second part of the paper, similar results are obtained in the case when $X$ is a partially ordered space and $Y$ is equipped with a reflexive binary relation (which is neither transitive nor antisymmetric). “Weakened analogs” of the notions of ordered covering and monotonicity of mappings from $X$ to $Y$ are defined. Under the assumption that $F$ is covering in the first argument and antitone in the second argument (in the sense of the definitions of these properties given in the paper), the existence of a solution $x$ to the equation $F(x,x)=y$ is established and conditions for the existence of a coincidence point of a covering and an isotone mappings acting from $X$ to $Y$ are deduced from this statement. In the third part, a connection between the obtained statements is established. Namely, it is proved that the theorem on the solvability of an operator equation in spaces with a binary relation implies a similar theorem in spaces with a distance and, accordingly, the statements on coincidence points.

Détail

Keywords: metric space, ordered space, covering mapping, Lipschitz mapping, monotone mapping.

Bibliographie

[1] Avakov E.R., Arutyunov A.V., Zhukovskii E.S., “Nakryvayuschie otobrazheniya i ikh prilozheniya k differentsialnym uravneniyam, ne razreshennym otnositelno proizvodnoi”, Differents. uravneniya, 45:5 (2009), 613–634 | MR | Zbl

[2] Arutyunov A.V., “Nakryvayuschie otobrazheniya v metricheskikh prostranstvakh i nepodvizhnye tochki”, Dokl. AN, 416:2 (2007), 151–155 | Zbl

[3] Arutyunov A.V., Greshnov A.V., “$(q_1,q_2)$-kvazimetricheskie prostranstva. Nakryvayuschie otobrazheniya i tochki sovpadeniya”, Izv. RAN. Ser. matematicheskaya, 82:2 (2018), 3–32 | DOI | MR | Zbl

[4] Arutyunov A.V., Zhukovskiy E.S., Zhukovskiy S.E., “Coincidence points principle for mappings in partially ordered spaces”, Topology Appl., 179:1 (2015), 13–33 | DOI | MR | Zbl

[5] Benarab S., Zhukovskii E.S., “Ob usloviyakh suschestvovaniya tochek sovpadeniya otobrazhenii v chastichno uporyadochennykh prostranstvakh”, Vest. Tambov. un-ta. Seriya: Estestvennye i tekhnicheskie nauki, 23:121 (2018), 10–16 | DOI

[6] Benarab S., Zhukovskii E.S., Merchela V., “Rasprostranenie teorem o vozmuscheniyakh nakryvayuschikh otobrazhenii”, Ustoichivost, upravlenie, differentsialnye igry (SCDG2019), materialy Mezhdunar. konf., posvyaschennoi 95-letiyu so dnya rozhdeniya akademika N.N. Krasovskogo, IMM UrO RAN, Ekaterinburg, 2019, 67–71

[7] Dmitruk A.V., Milyutin A.A., Osmolovskii N.P., “Teorema Lyusternika i teoriya ekstremuma”, Uspekhi mat. nauk, 35:6 (216) (1980), 11–46 | MR | Zbl

[8] Zhukovskii E.S., “Ob uporyadochenno nakryvayuschikh otobrazheniyakh i integralnykh neravenstvakh tipa Chaplygina”, Algebra i analiz, 30:1 (2018), 96–127

[9] Zhukovskii E.S., “O tochkakh sovpadeniya mnogoznachnykh vektornykh otobrazhenii metricheskikh prostranstv”, Mat. zametki, 100:3 (2016), 344–362 | DOI | MR

[10] Zhukovskii E.S., “O vozmuscheniyakh vektorno nakryvayuschikh otobrazhenii i sistemakh uravnenii v metricheskikh prostranstvakh”, Sib. mat. zhurn., 57:2 (2016), 297–311 | DOI | MR | Zbl

[11] Merchela V., “K teoreme Arutyunova o tochkakh sovpadeniya dvukh otobrazhenii metricheskikh prostranstv”, Vest. Tambov. un-ta. Ser.: Estestvennye i tekhnicheskie nauki, 23:121 (2018), 65–73 | DOI

[12] Pluzhnikova E.A., Zhukovskaya T.V., Moiseev Yu.A., “O mnozhestvakh metricheskoi regulyarnosti otobrazhenii v prostranstvakh s vektornoznachnoi metrikoi”, Vest. Tambov. un-ta. Ser.: Estestvennye i tekhnicheskie nauki, 23:123 (2018), 547–554

[13] Bishop E., Phelps R.R., “The support functionals of a convex set”, Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, v. 7, 27–35 | DOI | MR

[14] DeMarr R., “Partially ordered spaces and metric spaces”, Am. Math. Mon., 72:6 (1965), 628–631 | DOI | MR | Zbl