Russian Universities Reports. Mathematics

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ISSN : 2686-9667
Total articles ≅ 88
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Mebrahtom S. Woldeab,
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-137-16-26

Abstract:
The problem of optimal extraction of a resource from the structured population consisting of individual species or divided into age groups, is considered. Population dynamics, in the absence of exploitation, is given by a system of ordinary differential equations and at certain time moments, part of the population, is extracted. In particular, it can be assumed that we extract various types of fish, each of which has a certain value. Moreover, there exist predatorprey interactions or competition relationships for food and habitat between these species. We study the properties of the average time benefit which is equal to the limit of the average cost of the resource with an unlimited increase in times of withdrawals. Conditions are obtained under which the average time benefit goes to infinity and a method for constructing a control system to achieve this value is indicated. We show that for some models of interaction between two species, this method of extracting a resource can lead to the complete extinction of one of the species and unlimited growth to the other. Therefore, it seems appropriate to study the task of constructing a control to achieve a fixed final value of the average time benefit. The results obtained here are illustrated with examples of predator-prey models and models of competition of two species and can be applied to other various models of population dynamics.
Vasiliy I. Fomin
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-138-183-197

Abstract:
A normed algebra of bounded linear complex operators acting in a complex normed space consisting of elements of the Cartesian square of a real Banach space is constructed. In this algebra, it is singled out a set of operators for each of which the real and imaginary parts commute with each other. It is proved that in this set, any operator for which the sum of squares of its real and imaginary parts is a continuously invertible operator, is invertible itself; a formula for the inverse operator is found. For an operator from the indicated set, the form of its regular points is investigated: conditions under which a complex number is a regular point of the given operator are found; a formula for the resolvent of a complex operator is obtained. The set of unbounded linear complex operators acting in the above complex normed space is considered. In this set, a subset of those operators for each of which the domains of the real and imaginary parts coincide is distinguished. For an operator from the specified subset, conditions on a complex number under which this number belongs to the resolvent set of the given operator are found; a formula for the resolvent of the operator is obtained. The concept of a semi-bounded complex operator as an operator in which one component is a bounded and the other is an unbounded operator is introduced. It is noted that the first and second resolvent identities for complex operators can be proved similarly to the case of real operators.
Jasur M. Оdinabekov
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-138-164-174

Abstract:
The main problems in the theory of singular integral operators are the problems of boundedness, invertibility, Noethericity, and calculation of the index. The general theory of multidimensional singular integral operators over the entire space E_n was constructed by S.G. Mikhlin. It is known that in the two-dimensional case, if the symbol of an operator does not vanish, then the Fredholm theory holds. For operators over a bounded domain, the boundary of this domain significantly affects the solvability of the corresponding operator equations. In this paper, we consider two-dimensional singular integral operators with continuous coefficients over a bounded domain. Such operators are used in many problems in the theory of partial differential equations. In this regard, it is of interest to establish criteria for the considered operators to be Noetherian in the form of explicit conditions on their coefficients. The paper establishes effective necessary and sufficient conditions for two-dimensional singular integral operators to be Noetherian in Lebesgue spaces L_p (D) (considered over the field of real numbers), 1<p<∞, and formulas for calculating indices are given. The method developed by R.V. Duduchava [Duduchava R. On multidimensional singular integral operators. I: The half-space case; II: The case of compact manifolds // J. Operator Theory, 1984, v. 11, 41–76 (I); 199–214 (II)]. In this case, the study of the Noetherian properties of operators is reduced to the factorization of the corresponding matrix-functions and finding their partial indices.
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-137-27-36

Abstract:
The article deals with an inclusion in which a multivalued mapping acts from a metric space (X,ρ) into a set Y with distance d. This distance satisfies only the first axiom of the metric: d(y_1,y_2 ) is equal to zero if and only if y_1=y_2. The distance does not have to be symmetric or to satisfy the triangle inequality. For the space (Y,d), the simplest concepts (of a ball, convergence, the distance from a point to a set) are defined, and for a multivalued map G:X⇉Y, the sets of covering, Lipschitz and closedness are introduced. In these terms (allowing us to adapt the classical conditions of covering, Lipschitz property and closedness of mappings of metric spaces to the maps with values in (Y,d) and to weaken such conditions), a theorem on solvability of the inclusion F(x,x)∋y ̂ is formulated, and an estimate for the deviation in the space (X,ρ) of the set of solutions from a given element x_0∈X is given. The main conditions of the obtained statement are the following: for any x from some ball, the pair (x,y ̂) belongs to the α-covering set of the mapping F(•,x) and to the β-Lipschitz set of the mapping F(x,∙), where α>β. The proof of the corresponding statement is based on the construction of the sequences {x_n}⊂X and {y_n}⊂Y satisfying the relations y_n∈F(x_n,x_n ), y ̂∈F(x_(n+1),x_n ), αρ(x_(n+1),x_n)≤d(y ̂,y_n)≤βρ(x_n,x_(n-1)). Also, in the paper, we obtain sufficient conditions for the stability of solutions of the considered inclusion to changes in the multivalued mapping F and in the element y ̂.
Rafik A. Khachatryan
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-139-284-299

Abstract:
The article considers a parametric problem of the form f(x,y)→"inf",x∈M, where M is a convex closed subset of a Hilbert or uniformly convex space X, y is a parameter belonging to a topological space Y. For this problem, the set of ϵ-optimal points is given by a_ϵ (y)={x∈M|f(x,y)≤〖"inf" 〗┬(x∈M)⁡〖f(x,y)+ϵ〗 }, where ϵ>0. Conditions for the semicontinuity and continuity of the multivalued mapping a_ϵ are discussed. Using gradient projection and linearization methods, we obtain theorems on the existence of continuous selections of the multivalued mapping a_ϵ. One of the main assumptions of these theorems is the convexity of the functional f(x,y) with respect to the variable x on the set M and continuity of the derivative f_x^' (x,y) on the set M×Y. Examples that confirm the significance of the assumptions made are given, as well as examples illustrating the application of the obtained statements to optimization problems.
Anastasia S. Lanina, Elena A. Pluzhnikova
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-139-270-283

Abstract:
The Hopfield-type model of the dynamics of the electrical activity of the brain, which is a system of differential equations of the form v ̇_i=-αv_i+∑_(j=1)^n▒〖w_ji f_δ (v_j )+I_i (t), i=(1,n) ̅ 〗, t≥0. is investigated. The model parameters are assumed to be given: α>0, w_ji>0 for i≠j and w_ii=0, I_i (t)≥0. The activation function f_δ (δ is the time of the neuron transition to the state of activity) of two types is considered: δ=0⟹f_0 (v)={■(0,&v≤θ,@1,&v>θ;) δ>0⟹f_δ (v)={■(0,&v≤θ,@δ^(-1) (v-θ),&θθ+δ.)┤ ┤ In the case of δ>0 (the function f_δ is continuous), the solution of the Cauchy problem for the system under consideration exists, is unique, and is non-negative for non-negative initial values. In the case of δ=0 (the function f_0 is discontinuous at the point θ), it is shown that the set of solutions of the Cauchy problem has the largest and the smallest solutions, estimates for the solutions are obtained, and an example of a system for which the Cauchy problem has an infinite number of solutions is given. In this study, methods of analysis of mappings acting in partially ordered spaces are used. An improved Hopfield model is also investigated. It takes into account the time of movement of an electrical impulse from one neuron to another, and therefore such a model is represented by a system of differential equations with delay. For such a system, both in the case of continuous and in the case of discontinuous activation function, it is shown that the Cauchy problem is uniquely solvable, estimates for the solution are obtained, and an algorithm for analytical finding of solution is described.
Gusen E. Abduragimov
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-138-129-135

Abstract:
In this article, we consider a two-point boundary value problem for a nonlinear functional differential equation of fractional order with weak nonlinearity on the interval [0,1] with zero Dirichlet conditions on the boundary. The boundary value problem is reduced to an equivalent integral equation in the space of continuous functions. Using special topological tools (using the geometric properties of cones in the space of continuous functions, statements about fixed points of monotone and concave operators), the existence of a unique positive solution to the problem under consideration is proved. An example is given that illustrates the fulfillment of sufficient conditions that ensure the unique solvability of the problem. The results obtained are a continuation of the author’s research (see [Results of science and technology. Ser. Modern mat. and her appl. Subject. review, 2021, vol. 194, pp. 3–7]) devoted to the existence and uniqueness of positive solutions of boundary value problems for non-linear functional differential equations.
Mohamed Ahmed Boudref
Russian Universities Reports. Mathematics pp 150-165; https://doi.org/10.20310/2686-9667-2022-27-138-150-163

Abstract:
In this paper we consider the system of functions G_(r,n)^α (x) (r∈N,n=0,1,…) which is orthogonal with respect to the Sobolev-type inner product on (-1,1) and generated by orthogonal Gegenbauer polynomials. The main goal of this work is to study some properties related to the system {φ_(k,r) (x)}_(k≥0) of the functions generated by the orthogonal system {G_(r,n)^α (x)} of Gegenbauer functions. We study the conditions on a function f(x) given in a generalized Gegenbauer orthogonal system for it to be expandable into a generalized mixed Fourier series of the form f(x)~∑_(k=0)^(r-1)▒〖f^((k) ) (-1) (x+1)^k/k!+∑_(k=r)^∞▒〖G_(r,k)^α (f) 〗〗 φ_(r,k)^α (x), as well as the convergence of this Fourier series. The second result of this paper is the proof of a recurrence formula for the system {φ_(k,r) (x)}_(k≥0). We also discuss the asymptotic properties of these functions, and this represents the latter result of our contribution.
Aleksandr P. Afanas’Ev, Sergei M. Dzyuba
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-137-5-15

Abstract:
In the earlier article by the authors [A.P. Afanas’ev, S. M. Dzyuba “About new properties of recurrent motions and minimal sets of dynamical systems”, Russian Universities Reports. Mathematics, 26:133 (2021), 5–14] a connection between general motions and recurrent motions in a compact metric space is established, and a very simple behavior of recurrent motions is proved. Based on these results, we introduce here a new definition of recurrent motion which, in contrast to the one widely used in modern literature, provides fairly complete information about the structure of a recurrent motion as a function of time and, therefore, is more illustrative. At the same time, we show that in an abstract metric space, the proposed definition is equivalent to Birkhoff’s definition and is equivalent to the generally accepted modern definition in a complete metric space. Necessary and sufficient conditions for recurrence (in the sense of the definition proposed in the article) of a motion in a compact metric space are obtained. It is proved that α- and ω-limit sets of any motion are minimal in a compact metric space (this assertion was announced in an earlier paper by the authors). From the minimality of α- and ω-limit sets, it is deduced that in a compact metric space, each positively (negatively) Poisson-stable point lies on the trajectory of a recurrent motion, i.e. is a point of a minimal set, and thus, in a compact metric space with a finite positive invariant measure almost all points are points of minimal sets.
Vyacheslav V. Provotorov, Van N. Hoang
Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-137-80-94

Abstract:
In the paper, the stability conditions of a three-layer symmetric differential-difference scheme with a weight parameter in the class of functions summable on a network-like domain are obtained. To analyze the stability of the differential-difference system in the space of feasible solutions H, a composite norm is introduced that has the structure of a norm in the space H^2=H⊕H. Namely, for Y={Y_1,Y_2}∈H^2, Y_l∈H (l=1,2), 〖∥Y∥〗_H^2 = 〖∥Y_1∥〗_(1,H)^2+〖∥Y_2∥〗_(2,H)^2, where 〖∥•∥〗_(1,H)^2 〖∥•∥〗_(2,H)^2 are some norms in H. The use of such a norm in the description of the energy identity opens the way for constructing a priori estimates for weak solutions of the differential-difference system, convenient for practical testing in the case of specific differentialdifference schemes. The results obtained can be used to analyze optimization problems that arise when modeling network-like transfer processes with the help of formalisms of differentialdifference systems.
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