Conditions of a solubility and assessment of solutions of an implicit differential equation with autoadjustable (that is depending on required function) argument deviation are received. Results about the covering displays of partially ordered spaces are used.

For a multivalued mapping F:[a; b] × R^m → comp(R^m), the problem of superpositional measurability and superpositional selectivity is considered. As it is known, for superpositional measurability it is sufficient that the mapping F satisfies the Caratheodory conditions, for superpositional selectivity it is sufficient that F(•,x) has a measurable section and F(t; •) is upper semicontinuous. In this paper, we propose generalizations of these conditions based on the replacement, in the definitions of continuity and semicontinuity, of the limit of the sequence of coordinates of points in the images of multivalued mappings to a one-sided limit. It is shown that under such weakened conditions the multivalued mapping F possesses the required properties of superpositional measurability / superpositional selectivity. Illustrative examples are given, as well as examples of the significance of the proposed conditions. For single-valued mappings, the proposed conditions coincide with the generalized Caratheodory conditions proposed by I.V. Shragin (see [Bulletin of the Tambov University. Series: natural and technical sciences, 2014, 19:2, 476–478]).

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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].

Исследуется двухточечная краевая задача для неявного дифференциального уравнения с отклоняющимся аргументом. Получена теорема о существовании и оценке решения, аналогичная теореме Чаплыгина о дифференциальном неравенстве. Используются результаты об уравнениях с накрывающими и монотонными отображениями в частично упорядоченных пространствах, а также условия упорядоченного накрывания оператора Немыцкого в пространстве измеримых существенно ограниченных функций.