Vol 30, No 151 (2025)

We consider two mappings acting between metric spaces and such that one of them is covering and the other satisfies the enhanced Lipschitz property. It is assumed here that the covering constant and the Lipschitz constant of these mappings are equal. We prove the result of the existence of a coincidence point of single-valued mappings in the case when the series of iterations of the function that provides execution of the enhanced Lipschitz property converges. We prove the similar result for set-valued mappings. We provide examples of functions for which the series of their iterations converges or diverges.

The properties of the $f$–quasimetric space $(X,\rho)$ are studied. In a space as such, the distance $\rho$  satisfies the identity axiom and the generalized triangle inequality: $\rho(x,z) \leq f(\rho(x,y),\rho(y,z))$ for any $x,y,z\in X.$ 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 $\rho$ 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.

We consider models of homogeneous or structured (by type, age, or other charac\-te\-ris\-tic) populations, the dynamics of which, in the absence of exploitation, is given by a system of difference equations
$x(k+1) = F\big(k, x(k)\big),$
where $x(k) = \big(x_1(k), \ldots, x_n(k)\big),$ $x_i(k),$\linebreak $i=1,\ldots,n$ is the amount of the $i$-th type or age class of the population at a time\linebreak $k=0,1,2,\ldots;$ $F(k,x)=\bigl(F_1(k,x), \ldots, F_n(k,x)\bigr),$ $F_i(k,x)$ are real functions that are defined and continuous on the set$\mathbb{R}^n_+ \doteq\big\{x\in\mathbb{R}^n : x_1\geqslant0, \ldots, x_n\geqslant0\big\}.$

It is assumed that at times $k=1, 2, \ldots$ the population is exposed to harvesting $u(k)=(u_1(k),\ldots,u_n(k))\in[0, 1]^n.$ Then the model of the exploited population is investigated, given by a system of difference equations
$<$
X(k+1) = F\bigl(k,(1-u(k))X(k)\bigr), \quad k=1, 2, \ldots,

where $X(k)=\big(X_1(k),\ldots,X_n(k)\big),$ $(1-u(k))X(k)=\big((1-u_1(k))X_1(k),\ldots,(1-u_n(k))X_n(k)\big),$ $X_i(k)$ and
$(1-u_i(k))X_i(k)$ is the amount of the resource of the $i$ type before and after harvesting at the time $k$ respectively, $i=1,\ldots,n.$

The problem of optimal harvesting of a renewable resources for an unlimited period of time under periodic operation mode, in which the highest values of collection characteristics are achieved, is investigated. The first of these characteristics is the average time profit given by the limit at $k\to\infty$ of the arithmetic mean of the cost of the resource over $k$ harvesting. Another characteristic is the harvesting effciency equal to the limit at $k\to\infty$ of the ratio of the cost of the resource gathered in $k$ harvestings to the amount of applied control (collection efforts). The results of the work are illustrated by examples of a homogeneous exploited population, given by a discrete logistic equation, and a structured population of two species.

The regularization of the Lagrange principle (LP) and the Kuhn-Tucker theorem (KTT) in a non-differential form is considered in a nonlinear (non-convex) optimal control problem for a system of ordinary differential equations with a pointwise state equality constraint. The existence of a solution to the problem is not assumed a priori. The equality constraint contains an additive parameter, which makes it possible to use the "nonlinear version" of the perturbation method to study the problem. The main purpose of the regularized LP and KTT is to stably generate generalized minimizing sequences (GMS) in the problem under consideration. They can be interpreted as GMS-forming (regularizing) operators that associate with each set of initial data of the problem a subminimal (minimal) of its regular augmented Lagrangian (AL) functional corresponding to this set, the dual variable in which is generated in accordance with the Tikhonov stabilization procedure of the dual problem. The construction of the AL is completely determined by the form of the "nonlinear" subdifferentials (proximal subgradient, Frechet subdifferential) of the lower semicontinuous function of values as a function of the problem parameter. Regularized LP and KTT "overcome" the properties of ill-posedness of classical analogs, thus giving a theoretical basis for creating stable methods for solving nonlinear optimal control problems. In the particular case when the problem is regular in the sense of the existence of a generalized Kuhn-Tucker vector in it, and its initial data depend affinely on the control, the limit transition in the relations of the regularized KTT leads to optimality conditions in the form of the corresponding non-differential KTT and Pontryagin's maximum principle.