Vol 30, No 152 (2025)

In this paper, we present a theorem on a coincidence point of mappings which extends the Arutyunov theorem. The original version of the Arutyunov theorem guaranteed the existence of a coincidence point for two mappings acting in metric spaces, one of which is $\alpha$-covering and the other is $\beta$-Lipschitz, where $\alpha > \beta.$ This theorem was then extended to mappings acting in $(q_1, q_2)$-quasimetric spaces. In this paper, the problem of the existence of a coincidence point is solved for mappings acting from a $(q_1, q_2)$-quasimetric space to a set equipped with a distance satisfying only the identity condition (the distance vanishes if and only if the points coincide). Under conditions similar to those of the Arutyunov theorem, the existence of a coincidence point is proved. In addition, the questions of convergence of sequences of coincidence points of mappings $\psi_n, \varphi_n$ to the coincidence point $\xi$ of mappings $\psi, \varphi$ are investigated under the convergences $\psi_n(\xi)\to \psi(\xi),$ $\varphi_n(\xi)\to \varphi(\xi).$

A property is presented that characterizes quite fully the interrelation of motions of a dynamical system $g^t$ defined in a Hausdorff sequentially compact topological space $\Gamma.$ It is noted that in the space $\Gamma$ with an invariant (with respect to $g^t$) Lebesgue measure $\mu,$ a direct analogue of the Poincare--Caratheodory recurrence theorem for sets is valid. In addition, it is shown that if $\bar{\mathcal{M}}$ is the closure of the union $\mathcal{M}$ of all minimal sets of the space $\Gamma,$ then $\mu\bar{\mathcal{M}}=\mu\Gamma,$ and through each point $p\notin\mathcal{M}$ there passes a motion $f(t,p)$ that is both positively and negatively asymptotic with respect to the compact minimal sets $\Omega_p\subset\mathcal{M}$ and $\mathrm{A}_p\subset\mathcal{M}.$ If $\Gamma$ satisfies the second axiom of countability, then $\mu\mathcal{M}=\mu\Gamma,$ i.~e. in $\Gamma,$ there is an important addition to the Poincare-Caratheodory theorem on the points recurrence.

We study the initial value problem $u(x,0)=\varphi(x),$   $u_t(x,0)=0$ for the wave equation $u_{xx}(x,t)=u_{tt}(x,t)$ for $x\in\Gamma\setminus J$ and $t>0,$   where $\Gamma$ is a geometric graph (according to Yu. V. Pokornyi) with straight-line edges and without boundary vertices ($\partial\Gamma=\varnothing$), $J$ is the set of all internal vertices of $\Gamma,$   and the function $\varphi$ is given; the transmission conditions that close the problem are, in addition to the continuity of the function $u(\,\cdot\,,t)$ at the interior vertices, the smoothness conditions for it, the essence of which is that for each $t\geqslant0$ at each interior vertex $a\in J$ the sum of the right derivatives of the function $u(\,\cdot\,,t)$ in all admissible directions is 0. It is proved that if $G^\ast$ is a generalized Green's function (according to M. G. Zavgorodniy, 2019) for the boundary value problem $-y''(x)=f(x),$   $x\in\Gamma\setminus J,$   under smooth transmission conditions (here $y$ is the desired function, continuous at the points of $J,$   and $f$ is a given function, uniformly continuous on each edge of $\Gamma$), then the classical solution $u$ of the initial value problem is representable in form:
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u(x,t)=\langle\varphi\rangle-\int\limits_\Gamma g^\ast(x,t,s)\varphi''(s)\,ds,

where $\langle\varphi\rangle$ is the average of $\varphi$ over $\Gamma,$   and $g^\ast(x,t,s)=[\mathcal C(t)G^\ast(\,\cdot\,,s)](x),$   where, in turn, $\mathcal C$ is an operator function finitely described only through the metric and topological characteristics of $\Gamma.$    The approach to obtaining this representation of $u$ is similar to the approach implemented by the author earlier (2006) in the case where $\partial\Gamma\ne\varnothing$ and Dirichlet conditions are imposed at the points of $\partial\Gamma.$

Abstract problems about attainability in topological space (TS) under constraints of asymptotic nature (CAN) realized by nonempty family of sets in the space of usual solutions (controls) are considered. As analog of the attainability set defined by image of the target operator (TO) with values in TS, attraction set (AS) in classes of filters or directednesses of usual solutions is considered. Questions connected with the AS dependence under cange in the family of sets of usual solutions generating CAN are investigated. The special attention is paid to the case when this family is a filter (every AS is either generated by a filter used as CAN or is empty set). At the same time, AS under CAN generated by ultrafilters (u/f) that is by maximal filter under unrestrictive conditions on TS and TO there is a singleton, which allows to enter attraction operator which, in the case of regular TS, is continuous under equipment of the set of all ultrafilters on the set of usual solutions with Stone topology. On this basis, it is possible to give a practically exhaustive representation of constructions connected with AS in a regular TS in the class of u/f with their natural factorization based on TO. A whole range of obtained properties extend to the case of TO with values in a Hausdorff TS. Some questions connected with topology weakening of the space in which AS is realized are investigated.