Tatiana V. Zhukovskaia

Implicit nonautonomous first-order difference equations in spaces endowed with distance are considered. The distance is only required to be non-negative (the distance need not possess any of the standard properties of a metric). Analogs of classical definitions of stability with respect to initial data (Lyapunov, asymptotic, and exponential stability) are introduced, taking into account the possible non-uniqueness of the solution to the Cauchy problem. Under the assumption that the mapping of two arguments generating the equation under consideration is covering with respect to one argument and Lipschitz with respect to the other, statements are obtained on the stability with respect to the initial data of a particular solution to the Cauchy problem and of all solutions to the equation. As a corollary, the stability of a stationary solution to an implicit autonomous equation is investigated. Conditions for the existence of a stationary solution for such an autonomous equation are also obtained. It is demonstrated how the presented results imply well-known conditions for the exponential stability of equilibria of explicit difference equations in linear normed spaces. A scheme for applying the obtained results to the study of the stability of some functional equations with delay is proposed. An example of a study using this scheme of a specific functional equation in the space of Lebesgue measurable essentially bounded functions is given.

The properties of the f–quasimetric space (X,ρ) are studied. In a space as such, the distance ρ satisfies the identity axiom and the generalized triangle inequality: ρ(x,z) ≤ f(ρ(x,y),ρ(y; z)) for any x,y,z∈X. Here the function f is positive for positive arguments, continuous at the point (0,0), and f(0,0)=0. The symmetry of the distance is not assumed. The topology on X generated by the distance ρ is defined in the standard way. The properties of convergent sequences and sequentially compact sets are studied. Conditions are obtained under which the convergence in itself (mutual convergence) is necessary for the convergence of a sequence. The relationship between the rates of convergence of a fundamental sequence and its convergence in itself is considered. The concept of a sequentially precompact set is introduced. Conditions are obtained under which the closure of a sequentially precompact set is sequentially compact.

The article considers a partial differential equation of the form ∂u/∂t=f(t,x,y,u,∂u/∂x,∂u/∂y,(∂^2 u)/(∂x^2 ),(∂^2 u)/(∂y^2 ),(∂^2 u)/∂x∂y),(x,y)∈D⊂R^2,t≥0, with respect to an unknown function u, defined in a domain D of spatial variables x,y and for t≥0. A method for finding an approximate solution is proposed. The equation under consideration is replaced by an approximate one by introducing the shift operator S:D→D, which allows replacing at each step of the calculations the unknown values of the function u(x,y,t) on the right side with the values u(S(x,y),t), obtained at the previous step. The idea of the proposed method goes back to the idea of the Tonelli method, known for differential equations with respect to functions of one variable (with ordinary, not partial derivatives). The advantages of the proposed method are the simplicity of the obtained iteration relation and the possibility of application to a wide class of equations and boundary conditions. In the article, iteration formulas are obtained for solving a boundary value problem with the Dirichlet condition for spatial variables and with an initial or boundary condition for the variable t. Based on the proposed method, an approximate solution is obtained for a specific initial-boundary value problem for the heat conductivity equation in a square domain.

We consider the differential equation with delay x ̇(t)=f(t,x(h(t) ) ), t≥0, x(s)=φ(s), s0, then the Cauchy problem for this equation is uniquely solvable and any solution can be extended to the semiaxis R_+. At the same time, the Cauchy problem for the corresponding differential equation x ̇(t)=f(t,x(t) ), t≥0, may have infinitely many solutions, and the maximum interval of existence of solutions may be finite. In the article, we investigate which of the listed properties a delay equation possesses (i.e. has a unique solution or infinitely many solutions, has finite or infinite maximum interval of existence of solutions), if the function h has only one “critical’’ point t_0≥0, a point for which the measure of the set {t∈(t_0-ε,t_0+ε)∩R_+:h(t)>t-ε} is positive for any ε>0. It turns out that for such a delay function, the properties of solutions are close to those of solutions of an ordinary differential equation. In addition, we consider the problem of the dependence of solutions of a delay equation on the function h.

We consider the problem of coincidence points of two mappings ψ,φ, acting from a metric space (X,ρ) into a space (Y,d), in which a distance d has only one of the properties of the metric: d(y_1,y_2)=0⇔y_1=y_2, and is assumed to be neither symmetric nor satisfying the triangle inequality. The question of well-posedness of the equation ψ(x)=φ(x), which determines the coincidence point, is investigated. It is shown that if x=ξ is a solution to this equation, then for any sequence of α_i-covering mappings ψ_i:X→Y and any sequence of β_i-Lipschitz mappings φ_i:X→Y, α_i>β_i≥0, in the case of convergence d(φ_i (ξ),ψ_i (ξ))→0, equation ψ_i (x)=φ_i (x) has, for any i, a solution x=ξ_i such that ρ(ξ_i,ξ)→0. Further in the article, the dependence of the set "Coin"(t) of coincidence points of mappings ψ(•,t),φ(•,t):X→Y on a parameter t, an element of the topological space T, is investigated. Assuming that the first of these mappings is α-covering and the second one is β-Lipschitz, we obtain an assertion on upper semicontinuity, lower semicontinuity, and continuity of the set-valued mapping "Coin":T⇒X.

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.

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].

Let 𝑋 be a space with ∞-metric 𝜌 (a metric with possibly infinite value) and 𝑌 a space with ∞-distance 𝑑 satisfying the identity axiom. We consider the problem of coincidence point for mappings 𝐹,𝐺:𝑋→𝑌, i.e. the problem of existence of a solution for the equation 𝐹(𝑥)=𝐺(𝑥). We provide conditions of the existence of coincidence points in terms of a covering set for the mapping 𝐹 and a Lipschitz set for the mapping 𝐺 in the space 𝑋×𝑌. An 𝛼-covering set (𝛼>0) of the mapping 𝐹 is a set of (𝑥,𝑦) such that ∃𝑢∈𝑋 𝐹(𝑢)=𝑦, 𝜌(𝑥,𝑢)≤𝛼−1𝑑(𝐹(𝑥),𝑦), 𝜌(𝑥,𝑢)<∞, and a 𝛽 - Lipschitz set (𝛽≥0) for the mapping 𝐺 is a set of (𝑥,𝑦) such that ∀𝑢∈𝑋 𝐺(𝑢)=𝑦⇒𝑑(𝑦,𝐺(𝑥))≤𝛽𝜌(𝑢,𝑥). The new results are compared with the known theorems about coincidence points.

We consider multivalued mappings acting in spaces with a vector-valued metric. A vector-valued metric is understood as a mapping satisfying the axioms “of an ordinary metric” with values in the cone of a linear normed space. The concept of the regularity set of a multivalued mapping is defined. A set of regularity is used in the study of inclusions in spaces with a vector-valued metric.