Vol 27, No 139 (2022)
Articles
Antiperiodic boundary value problem for an implicit ordinary differential equation
Abstract
The paper is devoted to the investigation of the antiperiodic boundary value problem for an implicit nonlinear ordinary differential equation f t; x; x =0, x0 +x(τ)=0. We assume that the mapping f:R×Rn ×Rn →Rk defining the equation under consideration is smooth and satisfies the condition of uniform nondegeneracy of the first derivative infcovf v ' t,x,v : t,x,v ∈R×Rn ×Rn >0. Here cov A is the Banach constant of the linear operator A . The assumption of uniform nondegeneracy holds, in particular, for the mapping f defining an explicit ordinary differential equation. For implicit equations, sufficient conditions for the existence of a solution to an antiperiodic boundary value problem are obtained, and estimates for solutions are found. Corollaries for normal ordinary differential equations are formulated. To prove the main result, the original implicit equation is reduced to an explicit differential equation by applying a nonlocal implicit function theorem. Then we prove an auxiliary assertion on the solvability of the equation x + ψx =0, which is an analog of Brouwer’s fixed point theorem. It is shown that the mapping ψ, that assigns the value of the solution of the Cauchy problem at the point τ to an arbitrary initial point x 0 , is well defined and satisfies the assumptions of the auxiliary statement. This reasoning completes the proof of the existence of a solution to the boundary value problem.
Russian Universities Reports. Mathematics. 2022;27(139):205-213
205-213
A problem with a non-local condition for a fourth-order equation with multiple characteristics
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.
Russian Universities Reports. Mathematics. 2022;27(139):214-230
214-230
231-246
On stability and continuous dependence on parameter of the set of coincidence points of two mappings acting in a space with a distance
Abstract
We consider the problem of coincidence points of two mappings ψ, φ , acting from a metric space (X , ρ) into a space (Y , d), in which a distance d has only one of the properties of the metric: d( y1 , y2 )=0⇔ y1 = y2 , and is assumed to be neither symmetric nor satisfying the triangle inequality. The question of well-posedness of the equation ψx =φ(x), which determines the coincidence point, is investigated. It is shown that if x=ξ is a solution to this equation, then for any sequence of α i -covering mappings ψ i : X→Y and any sequence of β i -Lipschitz mappings φ i : X→Y , α i > β i ≥0, in the case of convergence d( φ i ( ξ), ψ i ( ξ))→0 , equation ψ i ( x)= φ i ( x) has, for any i , a solution x= ξ i such that ρ( ξ i , ξ)→0 . Further in the article, the dependence of the set Coin(t ) of coincidence points of mappings ψ(·, t ), φ(·, t ): X→Y on a parameter t , an element of the topological space T , is investigated. Assuming that the first of these mappings is α -covering and the second one is β -Lipschitz, we obtain an assertion on upper semicontinuity, lower semicontinuity, and continuity of the set-valued mapping Coin:T ⇒ X.
Russian Universities Reports. Mathematics. 2022;27(139):247-260
247-260
On a new method for obtaining a guaranteed error estimate for Numerov’s method using ellipsoids
Abstract
In this article, we consider a numerical solution of the Cauchy problem for a second-order differential equation calculated by the means of the Numerov method. A new method for obtaining a guaranteed error estimate using ellipsoids is proposed. The numerical solution is enclosed in an ellipsoid containing both the exact and the numerical solutions of the problem, which is recalculated at each step. In contrast to the previously proposed method for recalculating ellipsoids, a more accurate estimate of small terms in the difference equation for the error is proposed. This leads to a more accurate estimate of the error of the numerical solution and the applicability of the proposed method to estimating the error on longer intervals. The results of estimating the error of Numerov’s method in solving the two-body problem over a large interval are presented. This numerical experiment demonstrates the effectiveness of the proposed method.
Russian Universities Reports. Mathematics. 2022;27(139):261-269
261-269
On properties of solutions to differential systems modeling the electrical activity of the brain
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 nw jif δ v j +I it , 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 0v = 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.
Russian Universities Reports. Mathematics. 2022;27(139):270-283
270-283
On the existence of continuous selections of a multivalued mapping related to the problem of minimizing a functional
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 ϵy = x ∈ M | f ( x , y )≤ infx ∈ M fx , 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 fx ' ( 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.
Russian Universities Reports. Mathematics. 2022;27(139):284-299
284-299
Fernando Manuel Ferreira Lobo Pereira (20.10.1956 - 17.06.2022)
Russian Universities Reports. Mathematics. 2022;27(139):300
300