Vol 28, No 144 (2023)

This work is devoted to the study of the covering property of linear and nonlinear mappings of Banach spaces. We consider linear continuous operators acting from one Banach space to another. For a given operator, it is shown that for any point $y_0$ from the relative interior of the image of a given convex closed cone there exists a conical neighborhood of , with respect to which the given operator has the covering property at zero with a covering constant depending on the point $y_0$: We provide an example showing that for a linear continuous operator the covering property with respect to the image of a given cone at zero may fail, i. e. the statement of Banach’s theorem on an open mapping may not hold for restrictions of linear continuous operators to closed convex cones. We obtain a corollary of the obtained theorem for the case when the target space is finite-dimensional. Moreover, nonlinear twice differentiable mappings of Banach spaces are considered. For them, conditions for local covering along a certain curve with respect to a given cone are presented. The corresponding sufficient conditions are formulated in terms of 2 -regular directions. They remain meaningful even in the case of degeneracy of the first derivative of the mapping under consideration at a given point.

The Hopfield-type model of the dynamics of the electrical activity of the brain which is a system of differential equations of the form
\begin{equation*}
\dot{v}_{i}= -\alpha v_{i}+\sum_{j=1}^{n}w_{ji}f_{\delta}(v_{j})+I_{i}( t), \quad i=\overline{1,n}, \quad t\geq 0,
\end{equation*}
is under discussion.
  The model parameters are assumed to be given: $\alpha>0,$ $\tau_{ii}=0,$ $w_{ii}= 0,$ $\tau_{ji}\geq 0$ and $w_{ji}>0$ at $i\neq j,$  $I_{i}(t)\geq 0$ at $t\geq 0.$ Activation function $f_{\delta}$ ($\delta$ --- the time of the transition of a neuron to the state of activity) is considered of two types:
$$
\delta=0 \ \Rightarrow \
f_{0}(v)=\left\{
\begin{array}{ll}
0, &v\leq\theta,\\
1, &v>\theta;
\end{array}\right. \ \ \ \ \ \ \
\delta> 0 \ \Rightarrow \ f_{\delta}(v)=\left\{
\begin{array}{ll}
0, & v\leq \theta,\\
{\delta}^{-1}( v-\theta), & \theta < v \leq \theta+\delta,\\
1, &v>\theta+\delta.
\end{array}\right.$$
For the system of differential equations under consideration, a boundary value problem with the conditions  ${v_{i}(0)-v_{i}(T)=\gamma_{i},}$ $i=\overline{1,n},$ is studied. In both cases
 $\delta= 0$ (discontinuous function $f_{0}$) and $\delta > 0$ ($f_{0}$ continuous function), a solution exists, and if
${\delta} > \frac{T|W|_{\mathbb{R}^{n}\to \mathbb{R}^{n}}}{1 - e^{-\alpha T}},$
\quad \mbox{where} \quad W=(w_{ij})_{n\times n}, the problem has a unique solution. The work also provides estimates for the solution and its derivative. Theorems on fixed points of continuous mappings of metric and normed spaces and on fixed points of monotonic mappings of partially ordered spaces are used. The results obtained are applied to the study of periodic solutions of the differential system under consideration.

This paper proposes an approach to the identification of a nonstationary linear dynamic system. Its input-output mathematical model is presented as a Volterra equation of the first kind. The problem of nonparametric identification of Volterra kernels is solved on the basis of an active experiment using test piecewise linear signals (that have a rising front). The problem statement is based on the conditions for modeling the dynamics of technical devices in the energy and power industry. The choice of an admissible family of input signals is driven by the complexity of generating piecewise-constant type signals for real energy objects. The original problem is reduced to solving Volterra integral equations of the first kind with two variable integration limits. A formula for the inversion of the integral equations under study is constructed. Sufficient conditions are obtained for the solvability of the corresponding equations with respect to Volterra kernels in the class of continuous functions.

The Cauchy problem for the first-order algebraic differential equation is considered
\begin{equation*}
A\frac{du}{dt}=(B+\varepsilon C+\varepsilon^2 D)u(t,\varepsilon),
\end{equation*}
\begin{equation*}
u(t_0,\varepsilon)=u^0(\varepsilon)\in E_1,
\end{equation*}
where $A,B,C,D$ are closed linear operators acting from a Banach space $E_1$ to a Banach space $E_2$ with domains everywhere dense in $E_1,$ $u^0$ is a holomorphic function at the point $\varepsilon=0,$ $\varepsilon$ is a small parameter, $t\in[t_0;t_{max}].$ Such equations describe, in particular, the processes of filtration and moisture transfer, transverse vibrations of plates, vibrations in DNA molecules, phenomena in electromechanical systems, etc. The operator $A$ is the Fredholm operator with zero index. The aim of the work is to study the boundary layer phenomenon caused by the presence of a small parameter. The necessary information and statements are given. A~bifurcation equation is obtained. Two cases are considered: a) boundary layer functions of one type, b) boundary layer functions of two types. Newton's diagram is used to solve the bifurcation equation. In both, the conditions under which boundary layer phenomenon arises are obtained --- these are the conditions for the regularity of degeneracy. Case a) is illustrated by an example of the Cauchy problem with certain operator coefficients acting in the space $\mathbb{R}^4.$