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Results in Journal *Russian Universities Reports. Mathematics*: 88

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Published: 1 January 2022

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.

Published: 1 January 2022

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.

Published: 1 January 2022

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-139-214-230

**Abstract:**

In this article, we consider a non-local problem with an integral condition for a fourth-order equation. The unique solvability of the problem is proved. The proof of the uniqueness of a solution is based on the a priori estimates derived in the paper. To prove the existence of a solution, the problem is reduced to two Goursat problems for second-order equations, and the equivalence of the stated problem and the resulting system of Goursat problems is proved. One of the problems of the system is the classical Goursat problem. The second problem is a characteristic problem for an integro-differential equation with a non-local integral condition on one of the characteristics. It is impossible to apply the well-known methods of substantiating the solvability of problems with conditions on characteristics to the study of this problem. The introduction of a new unknown function made it possible to reduce the second problem to an equation with a completely continuous operator, to verify, on the basis of the uniqueness theorem, that it is solvable and, by virtue of the proven equivalence of the problems, that the problem posed is solvable.

Published: 1 January 2022

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-138-143-149

**Abstract:**

The problem of covering of a given convex compact set by a homothetic image of another convex compact set with a given homothety center is considered, the coefficient of homothety is calculated. The problem has an old history and is closely related to questions about the Chebyshev center, problems about translates, and other problems of computational geometry. Polyhedral approximation methods and other approximation methods do not work in a space of already moderate dimension (more than 5 on a PC). We propose an approach based on the application of the gradient projection method, which is much less sensitive to dimension than the approximation methods. We select classes of sets for which we can prove the linear convergence rate of the gradient method, i. e. convergence with the rate of a geometric progression with a positive ratio strictly less than 1. These sets must be strongly convex and have, in a certain sense, smoothness of the boundary.

Published: 1 January 2022

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.

Published: 1 January 2022

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.

Published: 1 January 2022

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-138-136-142

**Abstract:**

In the earlier articles by the authors [A.P. Afanasiev, S.M. Dzyuba, “On new properties of recurrent motions and minimal sets of dynamical systems”, Russian Universities Reports. Mathematics, 26:133 (2021), 5–14] and [A.P. Afanasiev, S.M. Dzyuba, “New properties of recurrent motions and limit motions sets of dynamical systems”, Russian Universities Reports. Mathematics, 27:137 (2022), 5–15], there was actually established the interrelation of motions of dynamical systems in compact metric spaces. The goal of this paper is to extend these results to the case of dynamical systems in arbitrary metric spaces. Namely, let Σ, be an arbitrary metric space. In this article, first of all, a new important property is established that connects arbitrary and recurrent motions in such a space. Further, on the basis of this property, it is shown that if the positive (negative) semitrajectory of some motion f(t,p) located in Σ is relatively compact, then ω- (α-) limit set of the given motion is a compact minimal set. It follows, that in the space Σ, any nonrecurrent motion is either positively (negatively) outgoing or positively (negatively) asymptotic with respect to the corresponding minimal set.

Published: 1 January 2022

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-138-175-182

**Abstract:**

The article considers a linear matrix-differential operator of the n-th order of the form A^n. For it and for the operator (A ̃^(-1) )^n, an analytical expression is derived, for which an operator analog of the Newton binomial is obtained. A lemma on the solution of a linear equation is given. It is used in the study of the abstract Cauchy problem for an algebro-differential equation in a Banach space with the cube of the operator A at the highest derivative. The operator A has the property of having 0 as a normal eigenvalue. Conditions for the existence and uniqueness of the solution are determined; the solution is found, for which the method of cascade splitting of the equation and conditions into the corresponding equations and conditions in subspaces of lower dimensions is used. As an application, the results obtained for n=3 are used in solving a mixed problem for a fourth-order partial differential equation. These equations include the generalized shallow water wave equation and the generalized Liouville equation.

Published: 1 January 2022

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.

Published: 1 January 2022

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.

Published: 1 January 2022

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.

Published: 1 January 2022

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.

Published: 1 January 2022

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.

Published: 1 January 2022

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-137-58-79

**Abstract:**

The problem of finding a normal solution to an operator equation of the first kind on a pair of Hilbert spaces is classical in the theory of ill-posed problems. In accordance with the theory of regularization, its solutions are approximated by the extremals of the Tikhonov functional. From the point of view of the theory of problems for constrained extremum, the problem of minimizing a functional, equal to the square of the norm of an element, with an operator equality constraint (that is, given by an operator with an infinite-dimensional image) is equivalent to the classical ill-posed problem. The paper discusses the possibility of regularizing the Lagrange principle (LP) in the specified constrained extremum problem. This regularization is a transformation of the LP that turns it into a universal tool of stable solving illposed problems in terms of generalized minimizing sequences (GMS) and preserves its “general structural arrangement” based on the constructions of the classical Lagrange function. The transformed LP “contains” the classical analogue as its limiting variant when the numbers of the GMS elements tend to infinity. Both non-iterative and iterative variants of the regularization of the LP are discussed. Each of them leads to stable generation of the GMS in the original constrained extremum problem from the extremals of the regular Lagrange functional taken at the values of the dual variable generated by the corresponding procedure for the regularization of the dual problem. In conclusion, the article discusses the relationship between the extremals of the Tikhonov and Lagrange functionals in the considered classical ill-posed problem.

Published: 1 January 2022

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 ̂.

Published: 1 January 2022

Russian Universities Reports. Mathematics pp 95-104; https://doi.org/10.20310/2686-9667-2022-27-137-95-124

**Abstract:**

The questions of applying the dynamic programming (DP) apparatus to the routing problem with constraints and cost functions with the tasks list dependence are investigated. It is supposed that binary partition of the task set is given; tasks of the first task group must be fulfilled before the fulfillment of the task of the second group begins. In each of the groups, precedence conditions may be present. This setting can be applied in the case of sheet cutting on CNC machines, where two above-mentioned groups form zones planned at the cutting stage. In general case, for the optimal solution construction, the two-stage variant of DP is used. Linking two versions of DP is realized by identification of the criterion terminal component for service problem of the first group with extremum function connected with the second group. The connection of optimal solutions for above-mentioned two problems allows to construct an optimal solution for the initial joint problem. Based on the theoretical constructions algorithm realized on personal computer is constructed; computing experiment is realized.

Published: 1 January 2022

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2022-27-137-37-57

**Abstract:**

This article proposes a new method for studying differential operators with a discontinuous weight function. It is assumed that the potential of the operator is a piecewise smooth function on the segment of the operator definition. The conditions of «conjugation» at the point of discontinuity of the weight function are required. The spectral properties of a differential operator defined on a finite segment with separated boundary conditions are studied. The asymptotics of the fundamental system of solutions of the corresponding differential equation for large values of the spectral parameter is obtained. With the help of this asymptotics, the «conjugation» conditions of the differential operator in question are studied. The boundary conditions of the operator under study are investigated. As a result, we obtain an equation for the eigenvalues of the operator, which is an entire function. The indicator diagram of the eigenvalue equation, which is a regular polygon, is studied. In various sectors of the indicator diagram, the asymptotics of the eigenvalues of the investigated differential operator is found. The formula for the first regularized trace of this operator by using the found asymptotics of the eigenvalues by the Lidsky–Sadovnichy method is obtained. In the case of the passage to the limit, the resulting formula leads to the trace formula for the classical operator with a smooth potential and constant weight function.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-136-394-403

**Abstract:**

In this paper, we consider a mixed problem for a metaharmonic equation in a domain in a circular cylinder. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i. e. the function and its normal derivative are set. The other border of the cylindrical area is free. On the lateral surface of the cylindrical domain, homogeneous boundary conditions of the first kind are given. The problem is illposed and its approximate solution, stable to errors in the Cauchy data, is constructed using regularization methods. The problem is reduced to a first kind Fredholm integral equation. Based on the solution of the integral equation obtained in the form of a Fourier series by the eigenfunctions of the first boundary value problem for the Laplace equation in a circle, an explicit representation of the exact solution of the problem is constructed. A stable solution of the integral equation is obtained by the method of Tikhonov regularization. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem as a whole is constructed. A theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data, is given. The results can be used for mathematical processing of thermal imaging data in early diagnostics in medicine.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-136-372-381

**Abstract:**

We discuss the still unresolved question, posed in [S. Reich, Some Fixed Point Problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 57:8 (1974), 194–198], of existence in a complete metric space X of a fixed point for a generalized contracting multivalued map Φ: X⇉X having closed values Φ(x)⊂X for all x∈X. Generalized contraction is understood as a natural extension of the Browder–Krasnoselsky definition of this property to multivalued maps: ∀x,u∈X h(φ(x),φ(u))≤ η(ρ(x,u)), where the function η:R_+→R_+ is increasing, right continuous, and for all d>0, η(d)<d (h(•,•) denotes the Hausdorff distance between sets in the space X). We give an outline of the statements obtained in the literature that solve the S. Reich problem with additional requirements on the generalized contraction Φ. In the simplest case, when the multivalued generalized contraction map Φ acts in R, without any additional conditions, we prove the existence of a fixed point for this map.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-136-414-420

**Abstract:**

We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-136-348-362

**Abstract:**

The article is devoted to the study of implicit differential equations based on statements about covering mappings of products of metric spaces. First, we consider the system of equations Φ_i (x_i,x_1,x_2,…,x_n )=y_i, i=(1,n,) ̅ where 〖 Φ〗_i: X_i×X_1×… ×X_n→Y_i, y_i∈Y_i, X_i and Y_i are metric spaces, i=(1,n) ̅. It is assumed that the mapping 〖 Φ〗_i is covering in the first argument and Lipschitz in each of the other arguments starting from the second one. Conditions for the solvability of this system and estimates for the distance from an arbitrary given element x_0∈X to the set of solutions are obtained. Next, we obtain an assertion about the action of the Nemytskii operator in spaces of summable functions and establish the relationship between the covering properties of the Nemytskii operator and the covering of the function that generates it. The listed results are applied to the study of a system of implicit differential equations, for which a statement about the local solvability of the Cauchy problem with constraints on the derivative of a solution is proved. Such problems arise, in particular, in models of controlled systems. In the final part of the article, a differential equation of the n-th order not resolved with respect to the highest derivative is studied by similar methods. Conditions for the existence of a solution to the Cauchy problem are obtained.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-136-382-393

**Abstract:**

Conditions of negativity for the Green’s function of a two-point boundary value problem L_λ u≔u^((n) )-λ∫_0^l▒〖u(s) d_s r(x,s)=f(x), x∈[0,l], B^k (u)=α,〗 where B^k (u)=(u(0),…,u^((n-k-1) ) (0),u(l),-u^'(l) ,…,(-1)^((k-1) ) u^((k-1) ) (0) ), n≥3, 0

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-136-404-413

**Abstract:**

The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation f(t,x(t),x ̇(t))=y ̂(t), not resolved with respect to the derivative x ̇ of the required function. It is assumed that the function f satisfies the Caratheodory conditions, and the function y ̂ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide). In terms of the covering set of the function f(t,x_1,•): R→R and the Lipschitz set of the function f(t,•,x_2): R →R, conditions for the existence of solutions and their stability to perturbations of the function f generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function y ̂ and the value of the boundary condition, are obtained.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-136-341-347

**Abstract:**

In the article, we consider a boundary value problem for a nonlinear ordinary differential equation of even order which, obviously, has a trivial solution. Sufficient conditions for the existence and uniqueness of a positive solution to this problem are obtained. With the help of linear transformations of T. Y. Na [T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Acad. Press, NY, 1979, ch. 7], the boundary value problem is reduced to the Cauchy problem, the initial conditions of which make it possible to uniquely determine the transformation parameter. It is shown that the transformations of T. Y. Na uniquely determine the solution of the original problem. In addition, based on the proof of the uniqueness of a positive solution to the boundary value problem, a sufficiently effective non–iterative numerical algorithm for constructing such a solution is obtained. A corresponding example is given.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-136-363-371

**Abstract:**

The article is devoted to investigation of integro-differential equation with the Hammerstein integral operator of the following form: ∂_t u(t,x)=-τu(t,x,x_f )+∫_(R^2)▒〖ω(x-y)f(u(t,y) )dy, t≥0, x∈R^2 〗. The equation describes the dynamics of electrical potentials u(t,x) in a planar neural medium and has the name of neural field equation.We study ring solutions that are represented by stationary radially symmetric solutions corresponding to the active state of the neural medium in between two concentric circles and the rest state elsewhere in the neural field. We suggest conditions of existence of ring solutions as well as a method of their numerical approximation. The approach used relies on the replacement of the probabilistic neuronal activation function f that has sigmoidal shape by a Heaviside-type function. The theory is accompanied by an example illustrating the procedure of investigation of ring solutions of a neural field equation containing a typically used in the neuroscience community neuronal connectivity function that allows taking into account both excitatory and inhibitory interneuronal interactions. Similar to the case of bump solutions (i. e. stationary solutions of neural field equations, which correspond to the activated area in the neural field represented by the interior of some circle) at a high values of the neuronal activation threshold there coexist a broad ring and a narrow ring solutions that merge together at the critical value of the activation threshold, above which there are no ring solutions.

Published: 1 January 2019

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2019-24-128-432-449

**Abstract:**

We oﬀer a variant of Radon transforms for a pair X and Y of hyperboloids in R^3 deﬁned by [x,x] = 1 and [y,y] = -1, y_1 ≥ 1, respectively, here [x,y] = -x_1 y_1+x_2 y_2+x_3 y_3. For a kernel of these transforms we take δ([x,y]), δ(t) being the Dirac delta function. We obtain two Radon transforms D(X) →C^∞ (Y) and D(Y) →C^∞ (X). We describe kernels and images of these transforms. For that we decompose a sesqui-linear form with the kernel δ([x,y]) into inner products of Fourier components.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-135-250-270

**Abstract:**

In this paper, we study a periodic boundary value problem for a class of semilinear differential inclusions of fractional order in a Banach space for which the multivalued nonlinearity satisfies the regularity condition expressed in terms of measures of noncompactness. To prove the existence of solutions to the problem, we first construct the corresponding Green function. Then we introduce into consideration a multivalued resolving operator in the space of continuous functions and reduce the posed problem to the existence of fixed points of the resolving multioperator. To prove the existence of a fixed point, a generalized theorem of B.N. Sadovskii type for a condensing multivalued map is used.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-135-305-314

**Abstract:**

For a multivalued mapping F:[a; b] × R^m → comp(R^m), the problem of superpositional measurability and superpositional selectivity is considered. As it is known, for superpositional measurability it is sufficient that the mapping F satisfies the Caratheodory conditions, for superpositional selectivity it is sufficient that F(•,x) has a measurable section and F(t; •) is upper semicontinuous. In this paper, we propose generalizations of these conditions based on the replacement, in the definitions of continuity and semicontinuity, of the limit of the sequence of coordinates of points in the images of multivalued mappings to a one-sided limit. It is shown that under such weakened conditions the multivalued mapping F possesses the required properties of superpositional measurability / superpositional selectivity. Illustrative examples are given, as well as examples of the significance of the proposed conditions. For single-valued mappings, the proposed conditions coincide with the generalized Caratheodory conditions proposed by I.V. Shragin (see [Bulletin of the Tambov University. Series: natural and technical sciences, 2014, 19:2, 476–478]).

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-135-315-336

**Abstract:**

The k[S] -hierarchy and its strict version are two deformations of the commutative algebra k[S], k=R or C; in the N×N-matrices, where S is the matrix of the shift operator. In this paper we show first of all that both deformations correspond to conjugating k[S] with elements from an appropriate group. The dressing matrix of the deformation is unique in the case of the k[S]-hierarchy and it is determined up to a multiple of the identity in the strict case. This uniqueness enables one to prove directly the equivalence of the Lax form of the k[S]-hierarchy with a set of Sato-Wilson equations. The analogue of the Sato-Wilson equations for the strict k[S]-hierarchy always implies the Lax equations of this hierarchy. Both systems are equivalent if the setting one works in, is Cauchy solvable in dimension one. Finally we present a Banach Lie group G(S_2), two subgroups P_+ (H) and U_+ (H) of G(S_2), with U_+ (H)⊂P_+ (H), such that one can construct from the homogeneous spaces G(S_2 )/P_+ (H) resp. G(S_2)/U_+ (H) solutions of respectively the k[S]-hierarchy and its strict version.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-135-271-295

**Abstract:**

We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the existence of a special class of solutions that we call 1-bump periodic solutions. We then analyze the stability of these solutions by studying the spectrum of the Frechet derivative of the corresponding Hammerstein operator. We prove that the spectrum of this operator agrees up to zero with the spectrum of a block Laurent operator. We show that the non-zero spectrum consists of only eigenvalues and obtain an analytical expression for the eigenvalues and the eigenfunctions. The results are illustrated by multiple examples.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-135-296-304

**Abstract:**

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-135-234-240

**Abstract:**

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.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-135-225-233

**Abstract:**

We consider the Cauchy problem for the implicit differential equation of order n g(t,x,(x,) ̇…,x^((n)) )=0,t ∈ [0; T],x(0)= A. It is assumed that A=(A_0,…,A_(n-1) )∈R^n, the function g:[0,T] × R^(n+1)→ R is measurable with respect to the first argument t∈[0,T], and for a fixed t, the function g(t,∙)×R^(n+1)→ R is right continuous and monotone in each of the first n arguments, and is continuous in the last n+1-th argument. It is also assumed that for some sufficiently smooth functions η,ν there hold the inequalities ν^((i) ) (0)≥ A_i ≥ η^((i) ) (0),i= (0,n-1,) ̅ ν^((n) ) (t)≥ η^((n) ) (t),t∈[0; T]; g(t; ν(t),ν ̇(t),...,ν^((n) ) (t) )≥ 0,g(t,η(t),η ̇(t),…,η^((n)) (t))≤ 0,t∈[0; T]. Sufficient conditions for the solvability of the Cauchy problem are derived as well as estimates of its solutions. Moreover, it is shown that under the listed conditions, the set of solutions satisfying the inequalities η^((n) ) (t)≤x^((n) ) (t)≤ν^((n) ) (t), is not empty and contains solutions with the largest and the smallest n -th derivative. This statement is similar to the classical Chaplygin theorem on differential inequality. The proof method uses results on the solvability of equations in partially ordered spaces. Examples of applying the results obtained to the study of second-order implicit differential equations are given.

Published: 1 January 2020

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-135-241-249

**Abstract:**

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-133-44-54

**Abstract:**

Consider the equation G(x)=(y,) ̃ where the mapping G acts from a metric space X into a space Y, on which a distance is defined, y ̃ ∈ Y. The metric in X and the distance in Y can take on the value ∞, the distance satisfies only one property of a metric: the distance between y,z ∈Y is zero if and only if y= z. For mappings X → Y the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space X of solutions of the considered equation to changes of the mapping G and the element y ̃. This assertion is applied to the study of the integral equation f(t,∫_0^1▒K (t,s)x(s)ds,x(t))= y ̃(t),t ∈[0,1], with respect to an unknown Lebesgue measurable function x: [0,1] ∈R. Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions f,K,(y.) ̃

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-134-121-129

**Abstract:**

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-134-216-220

**Abstract:**

We consider a two-point (including periodic) boundary value problem for the following system of differential equations that are not resolved with respect to the derivative of the desired function: f_i (t,x,x ̇,(x_i ) ̇ )=0,i= (1,n) ̅. Here, for any i= (1,n) ̅ the function f_i:[0,1]×R^n×R^n×R→R is measurable in the first argument, continuous in the last argument, right-continuous, and satisfies the special condition of monotonicity in each component of the second and third arguments. Assertions about the existence and two-sided estimates of solutions (of the type of Chaplygin’s theorem on differential inequality) are obtained. Conditions for the existence of the largest and the smallest (with respect to a special order) solution are also obtained. The study is based on results on abstract equations with mappings acting from a partially ordered space to an arbitrary set (see [S. Benarab, Z.T. Zhukovskaya, E.S. Zhukovskiy, S.E. Zhukovskiy. On functional and differential inequalities and their applications to control problems // Differential Equations, 2020, 56:11, 1440–1451]).

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-133-5-14

**Abstract:**

The article presents a new property of recurrent motions of dynamical systems. Within a compact metric space, this property establishes the relation between motions of general type and recurrent motions. Besides, this property establishes rather simple behaviour of recurrent motions, thus naturally corroborating the classical definition given in the monograph [V.V. Nemytskii, V.V. Stepanov. Qualitative Theory of Differential Equations. URSS Publ., Moscow, 2004 (In Russian)]. Actually, the above-stated new property of recurrent motions was announced, for the first time, in the earlier article by the same authors [A.P. Afanas’ev, S. M. Dzyuba. On recurrent trajectories, minimal sets, and quasiperoidic motions of dynamical systems // Differential Equations. 2005, v. 41, № 11, p. 1544–1549]. The very same article provides a short proof for the corresponding theorem. The proof in question turned out to be too schematic. Moreover, it (the proof) includes a range of obvious gaps. Some time ago it was found that, on the basis of this new property, it is possible to show that within a compact metric space α- and ω-limit sets of each and every motion are minimal. Therefore, within a compact metric space each and every motion, which is positively (negatively) stable in the sense of Poisson, is recurrent. Those results are of obvious significance. They clearly show the reason why, at present, there are no criteria for existence of non-recurrent motions stable in the sense of Poisson. Moreover, those results show the reason why the existing attempts of creating non-recurrent motions, stable in the sense of Poisson, on compact closed manifolds turned out to be futile. At least, there are no examples of such motions. The key point of the new property of minimal sets is the stated new property of recurrent motions. That is why here, in our present article, we provide a full and detailed proof for that latter property. For the first time, the results of the present study were reported on the 28th of January, 2020 at a seminar of Dobrushin Mathematic Laboratory at the Institute for Information Transmission Problems named after A. A. Kharkevich of the Russian Academy of Sciences.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-134-109-120

**Abstract:**

We propose multicompartmental models of infectious diseases dynamics for numerical study of the spread parameters of the new coronavirus infection SARS-CoV-2, which take into account the delay effects associated with the presence of the latent period of the infection, as well as the possibility of an asymptomatic course of the disease. The dynamics of the spread of COVID-19 in the Russian Federation was investigated, using these models with distributed parameters that formalize the interactions of the models’ compartments. The paper provides numerical estimates of the spread dynamics of the new coronavirus infection in various age groups of the population. We also investigate possible consequences of the mask regime and quarantine measures. We obtain an explicit estimate allowing to assess the necessary scope of these measures for the epidemy extinction.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-133-15-25

**Abstract:**

A structured population the individuals of which are divided into n age or typical groups x_1,…,x_n. is considered. We assume that at any time moment k, k = 0,1,2… the size of the population x(k) is determined by the normal autonomous system of difference equations x(k+1)=F(x(k)), where F(x)=col(f_1 (x),…,〖 f〗_n (x) ) are given vector functions with real non-negative components f_i (x), i=1,…n. We investigate the case when it is possible to influence the population size by means of harvesting. The model of the exploited population under discussion has the form x(k+1)=F((1-u(k) )x(k) ), where u(k)= (u_1 (k),…,u_n (k))∈〖[0; 1]〗^n is a control vector, which can be varied to achieve the best result of harvesting the resource. We assume that the cost of a conventional unit of each of n classes is constant and equals to C_i≥0, i=1,…,n. To determine the cost of the resource obtained as the result of harvesting, the discounted income function is introduced into consideration. It has the form H_α (u ̅,x(0))=∑_(j=0)^∞▒〖∑_(i=1)^n▒〖C_i x_i (j) u_i (j) e^(-αj) 〗,〗 where α>0 is the discount coefficient. The problem of constructing controls on finite and infinite time intervals at which the discounted income from the extraction of a renewable resource reaches the maximal value is solved. As a corollary, the results on the construction of the optimal harvesting mode for a homogeneous population are obtained (that is, for n = 1).

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-133-77-104

**Abstract:**

Linked and maximal linked systems (MLS) on π -systems of measurable (in the wide sense) rectangles are considered (π-system is a family of sets closed with respect to finite intersections). Structures in the form of measurable rectangles are used in measure theory and probability theory and usually lead to semi-algebra of subsets of cartesian product. In the present article, sets-factors are supposed to be equipped with π-systems with “zero” and “unit”. This, in particular, can correspond to a standard measurable structure in the form of semialgebra, algebra, or σ-algebra of sets. In the general case, the family of measurable rectangles itself forms a π -system of set-product (the measurability is identified with belonging to a π - system) which allows to consider MLS on a given π -system (of measurable rectangles). The following principal property is established: for all considered variants of π -system of measurable rectangles, MLS on a product are exhausted by products of MLS on sets-factors. In addition, in the case of infinity product, along with traditional, the “box” variant allowing a natural analogy with the base of box topology is considered. For the case of product of two widely understood measurable spaces, one homeomorphism property concerning equipments by the Stone type topologies is established.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-133-55-67

**Abstract:**

In the work, the stability conditions for a solution of an evolutionary hyperbolic system with distributed parameters on a graph describing the oscillating process of continuous medium in a spatial network are indicated. The hyperbolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral identity which determines the variational formulation for the initial-boundary value problem. The basic idea, that has determined the content of this work, is to present a weak solution in the form of a generalized Fourier series and continue with an analysis of the convergence of this series and the series obtained by its single termwise differentiation. The used approach is based on a priori estimates of a weak solution and the construction (by the Fayedo–Galerkin method with a special basis, the system of eigenfunctions of the elliptic operator of a hyperbolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems of oscillations of netset-like industrial constructions which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-134-182-215

**Abstract:**

Maximal linked systems (MLS) of sets on widely understood measurable spaces (MS) are considered; in addition, every such MS is realized by equipment of a nonempty set with a π-system of its subsets with «zero» and «unit» (π-system is a nonempty family of sets closed with respect to finite intersections). Constructions of the MS product connected with two variants of measurable (in wide sense) rectangles are investigated. Families of MLS are equipped with topologies of the Stone type. The connection of product of above-mentioned topologies considered for box and Tychonoff variants and the corresponding (to every variant) topology of the Stone type on the MLS set for the MS product is studied. The properties of condensation and homeomorphism for resulting variants of topological equipment are obtained.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-134-151-171

**Abstract:**

The paper is devoted to the regularization of the classical optimality conditions (COC) — the Lagrange principle and the Pontryagin maximum principle in a convex optimal control problem for a parabolic equation with an operator (pointwise state) equality-constraint at the final time. The problem contains distributed, initial and boundary controls, and the set of its admissible controls is not assumed to be bounded. In the case of a specific form of the quadratic quality functional, it is natural to interpret the problem as the inverse problem of the final observation to find the perturbing effect that caused this observation. The main purpose of regularized COCs is stable generation of minimizing approximate solutions (MAS) in the sense of J. Warga. Regularized COCs are: 1) formulated as existence theorems of the MASs in the original problem with a simultaneous constructive representation of specific MASs; 2) expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions; 3) are sequential generalizations of the COCs and retain the general structure of the latter; 4) “overcome” the ill-posedness of the COCs, are regularizing algorithms for solving optimization problems, and form the theoretical basis for the stable solving modern meaningful ill-posed optimization and inverse problems.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-133-68-76

**Abstract:**

An initial-boundary value problem for a system of third-order partial differential equations is considered. Equations and systems of equations with the highest mixed third derivative describe heat exchange in the soil complicated by the movement of soil moisture, quasi-stationary processes in a two-component semiconductor plasma, etc. The system is reduced to a differential equation with a degenerate operator at the highest derivative with respect to the distinguished variable in a Banach space. This operator has the property of having 0 as a normal eigenvalue, which makes it possible to split the original equations into an equation in subspaces. The conditions are obtained under which a unique solution to the problem exists; the analytical formula is found.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-133-26-34

**Abstract:**

The article discusses a number of aspects of the application of i -smooth analysis in the development of numerical methods for solving functional differential equations (FDE). The principle of separating finite- and infinite-dimensional components in the structure of numerical schemes for FDE is demonstrated with concrete examples, as well as the usage of different types of prehistory interpolation, those by Lagrange and Hermite. A general approach to constructing Runge–Kutta-like numerical methods for nonlinear neutral differential equations is presented. Convergence conditions are obtained and the order of convergence of such methods is established.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-134-172-181

**Abstract:**

The rigidity of a dynamical system described by a first-order differential equationwith an irreversible operator at the highest derivative is investigated. The system is perturbed by an operator addition of the order of the second power of a small parameter. Conditions under which the system is robust with respect to these disturbances are determined as well as conditions under which the influence of disturbances is significant. For this, the bifurcation equation is derived. It is used to set the type of boundary layer functions. As an example, we investigate the initial boundary value problem for a system of partial differential equations with a mixed second partial derivative which occurs in the study of the processes of sorption anddesorption of gases, drying processes, etc.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-134-130-142

**Abstract:**

A system of first-order partial differential-algebraic equations in a Banach space with constant degenerate operators in the case of a regular operator pencil is considered. In this case, under some additional condition, the original system splits into two subsystems in disjoint subspaces in order to search for the projections of the original unknown function in the subspaces. The matching conditions for the parameters of the systems are identified. A solution of the considered system of differential-algebraic equations is constructed.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-133-35-43

**Abstract:**

In this paper, we consider a mixed problem for the Laplace equation in a region in a circular cylinder. On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i.e. a function and its normal derivative are given. The other border of the cylindrical area is free. This problem is ill-posed, and to construct its approximate solution in the case of Cauchy data known with some error it is necessary to use regularizing algorithms. In this paper, the problem is reduced to a Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained in the form of a Fourier series with the eigenfunctions of the first boundary value problem for the Laplace equation in a circle. A stable solution of the integral equation is obtained by the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem in the whole is constructed. The theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data is given. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

Published: 1 January 2021

Russian Universities Reports. Mathematics; https://doi.org/10.20310/2686-9667-2021-26-134-143-150

**Abstract:**

It is shown that the stochastic counterpart of the classical fixed point theorem for continuous maps in a finite dimensional Euclidean space (“Brouwer’s theorem”) is not, in general, true. This result implies, in particular, that a careful choice of invariant sets in the stochastic version of Brouwer’s theorem is necessary in the theory of stochastic nonlinear operators.