Vol 27, No 138 (2022)
Articles
On the interrelation of motions of dynamical systems
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.
Russian Universities Reports. Mathematics. 2022;27(138):136-142
136-142
On the existence and uniqueness of a positive solution to a boundary value problem for one nonlinear functional differential equation of fractional order
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.
Russian Universities Reports. Mathematics. 2022;27(138):129-135
129-135
Embedding of a homothete in a convex compactum: an algorithm and its convergence
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.
Russian Universities Reports. Mathematics. 2022;27(138):143-149
143-149
Inner product and Gegenbauer polynomials in Sobolev space
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.
Russian Universities Reports. Mathematics. 2022;27(138):150-163
150-163
On the Noethericity conditions and the index of some two-dimensional singular integral operators
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
Russian Universities Reports. Mathematics. 2022;27(138):164-174
164-174
Properties of one higher order matrix-differential operator
Abstract
The article considers a linear matrix-differential operator of the n n-th order of the form An . 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.
Russian Universities Reports. Mathematics. 2022;27(138):175-182
175-182
About a complex operator resolvent
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.
Russian Universities Reports. Mathematics. 2022;27(138):183-197
183-197
Gerrit van Dijk (14.08.1939 - 16.04.2022)
Russian Universities Reports. Mathematics. 2022;27(138):198
198
In memory of professor of mathematics Gerrit van Dijk
Russian Universities Reports. Mathematics. 2022;27(138):199-200
199-200