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 $(X, \preceq)$ be a partially ordered space, the mapping $\Phi: X \times X \to \{0,1\}$ satisfies the following condition: for any $x \in X $ there exists $x'\in X$ such that $x' \preceq x$ and $\Phi(x', x) = 1.$ The predicate $F(x)=\Phi(x, x)$ 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.
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