Abstract:
The questions of existence of solutions of equations and attainability of minimum values of functions are considered. All the obtained statements are united by the idea of existence for any approximation to the desired solution or to the minimum point of the improved approximation. The relationship between the considered problems in metric and partially ordered spaces is established. It is also shown how some well-known results on fixed points and coincidence points of mappings of metric and partially ordered spaces are derived from the obtained statements. Further, on the basis of analogies in the proofs of all the obtained statements, we propose a method for obtaining similar results from the theorem being proved on the satisfiability of a predicate of the following form. Let (X,⪯) be a partially ordered space, the mapping Φ:X×X→{0,1} satisfies the following condition: for any x∈X there exists x′∈X such that x′⪯x and Φ(x′,x)=1. The predicate F(x)=Φ(x,x) is considered, sufficient conditions for its satisfiability, that is, the existence of a solution to the equation F(x)=1. This result was announced in [Zhukovskaya T.V., Zhukovsky E.S. Satisfaction of predicates given on partially ordered spaces // Kolmogorov Readings. General Control Problems and their Applications (GCP–2020). Tambov, 2020, 34-36].
Keywords:
fixed point, coincidence point, minimum of function, partially ordered space, satisfiable predicate.
Section 1 was written by the first author, section 2 by the third author with the support of the Russian Science Foundation (project no. 20-11-20131), section 3 by the second author with the support of the Russian Foundation for Basic Research (project no. 20-04-60524_Вирусы).
Received: 27.08.2020
Document Type:
Article
UDC:517.988.38, 515.126.4
Language: Russian
Citation:
T. V. Zhukovskaya, E. S. Zhukovskiy, I. D. Serova, “Some questions of the analysis of mappings of metricand partially ordered spaces”, Russian Universities Reports. Mathematics, 25:132 (2020), 345–358
\Bibitem{ZhuZhuSer20}
\by T.~V.~Zhukovskaya, E.~S.~Zhukovskiy, I.~D.~Serova
\paper Some questions of the analysis of mappings of metricand partially ordered spaces
\jour Russian Universities Reports. Mathematics
\yr 2020
\vol 25
\issue 132
\pages 345--358
\mathnet{http://mi.mathnet.ru/vtamu203}
\crossref{https://doi.org/10.20310/2686-9667-2020-25-132-345-358}
Linking options:
https://www.mathnet.ru/eng/vtamu203
https://www.mathnet.ru/eng/vtamu/v25/i132/p345
This publication is cited in the following 3 articles:
E. S. Zhukovskiy, E. A. Panasenko, “The method of comparison with a model equation in the study of inclusions in vector metric spaces”, Proc. Steklov Inst. Math. (Suppl.), 325, suppl. 1 (2024), S239–S254
S. Benarab, “Dvustoronnie otsenki reshenii kraevykh zadach dlya neyavnykh differentsialnykh uravnenii”, Vestnik rossiiskikh universitetov. Matematika, 26:134 (2021), 216–220
S. Benarab, “O teoreme Chaplygina dlya neyavnogo differentsialnogo uravneniya n-go poryadka”, Vestnik rossiiskikh universitetov. Matematika, 26:135 (2021), 225–233