We consider a regular parametric nonlinear (nonconvex) problem for constrained extremum with an operator equality constraint and a finite number of functional inequality constraints. The constraints of the problem contain additive parameters, which makes it possible to use the apparatus of the “nonlinear” perturbation method for its study. The set of admissible elements of the problem is a complete metric space, and the problem itself may not have a solution. The regularity of the problem is understood in the sense that it has a generalized Kuhn-Tucker vector. Within the framework of the ideology of the Lagrange multiplier method, a regularized nondifferential Kuhn-Tucker theorem is formulated and proved, the main purpose of which is the stable generation of generalized minimizing sequences in the problem under consideration. These minimizing sequences are constructed from subminimals (minimals) of the modified Lagrange function taken at the values of the dual variable generated by the corresponding regularization procedure for the dual problem. The construction of the modified Lagrange function is a direct consequence of the subdifferential properties of a lower semicontinuous and, generally speaking, nonconvex value function as a function of the problem parameters. The regularized Kuhn-Tucker theorem “overcomes” the instability properties of its classical counterpart, is a regularizing algorithm, and serves as a theoretical basis for creating algorithms of practical solving problems for constrained extremum.

The paper considers a hyperbolic system of two first-order partial differential equations with constant coefficients, one of which is nonlinear and contains the square of one of the unknown functions. Moreover, each equation contains two unknown functions which in turn depend on two variables. Exact solutions are found for this system: a traveling wave solution and a self-similar solution. There is also defined the type of initial-boundary conditions which allow to use the constructed general solutions in order to write out a solution of the initial-boundary value problem for the system of differential equations under consideration.

We consider the extremal problem of finding exact constants in the Jackson-Stechkin type inequalities connecting the best approximations of analytic in the unit circle U={z:|z|<1} functions by algebraic complex polynomials and the averaged values of the higher-order continuity modules of the r-th derivatives of functions in the Bergman weight space B_2,γ. The classes of analytic in the unit circle functions W^((r))_m(τ) and W^((r))_m(τ,Φ), which satisfy some specific conditions are introduced. For the introduced classes of functions, the exact values of some known n-widths are calculated. In this paper, we use the methods of solving extremal problems in normalized spaces of functions analytic in a circle and a wellknown method developed by V.M. Tikhomirov for estimating from below the n-widths of functional classes in various Banach spaces. The results obtained in the work generalize and extend the results of the works by S.B. Vakarchuk and A.N. Shchitova obtained for the classes of differentiable periodic functions to the case of analytic in the unit circle functions belonging to the Bergmann weight space.

In the present research, existence and stability of ring solutions to two-dimensional Amari neural field equation with periodic microstructure and Heaviside activation function are studied. Results on dependence of the inner and the outer radii of the ring solutions are obtained. Necessary conditions for existence and sufficient conditions for non-existence of radial travelling waves are formulated for homogeneous neural medium and neural media with mild periodic microstructure. Theoretical results obtained are illustrated with a concrete example based on a connectivity function commonly used in the neuroscience community.

The work is devoted to a survey of known results related to the study of derivationsin group algebras, bimodules and other algebraic structures, as well as to various generalizationsand variations of this problem. A review of results on derivations in L_1 (G) algebras, invon Neumann algebras, and in Banach bimodules is given. Algebraic problems are discussed,in particular, derivations in groups, (σ,τ)-derivations, and the Fox calculus. A review ofsome results related to the application to pseudodifferential operators and the constructionof the symbolic calculus is also given. In conclusion, some results related to the description ofderivations as characters on the groupoid of the adjoint action are described. Some applicationsare also described: to coding theory, the theory of ends of metric spaces, and rough geometry.

This article is devoted to the study of the algebro-differential equation Ad^2u/dt^2=Bdu/dt+Cu(t)+f(t), where A, B, C are closed linear operators acting from a Banach space E_1 into a Banach space E_2 whose domains are everywhere dense in E_1. A is a Fredholm operator with zero index (hereinafter, Fredholm), the function f(t) takes values in E_2; t∈[0; T]. The kernel of the operator A is assumed to be one-dimensional. For solvability of the equation with respect to the derivative, the method of cascade splitting is applied, consisting in the stepwise splitting of the equation and conditions to the corresponding equations and conditions in subspaces of lower dimensions. One-step and two-step splitting are considered, theorems on the solvability of the equation are obtained. The theorems are used to obtain the existence conditions for a solution to the Cauchy problem. In order to illustrate the results obtained, a homogeneous Cauchy problem with given operator coefficients in the space R^2 is solved. For this, it is considered the second-order differential equation in the finite-dimensional space C^m d^2u/dt^2=Hdu/dt+Ku(t). The characteristic equation M(λ):=det(λ^2I-λH-K)-0 is studied. For the polynomial M(λ) in the cases m=2, m=3, the Maclaurin formulas are obtained. General solution of the equation is defined in the case of the unit algebraic multiplicity of the characteristic equation.

We investigate population dynamics models given by difference equations with stochastic parameters. In the absence of harvesting, the development of the population at time points k=1,2,... is given by the equation X(k+1)=f(X(k)), where X(k) is amount of renewable resource, f(x) is a real differentiable function. It is assumed that at times k=1,2,... a random fraction ω∈[0,1] of the population is harvested. The harvesting process stops when at the moment k the share of the collected resource becomes greater than a certain value u(k)∈[0,1), in order to save a part of the population for reproduction and to increase the size of the next harvest. In this case, the share of the extracted resource is equal to l(k)=min{ω(k), u(k)}, k=1,2,.... Then the model of the exploited population has the form X(k+1)=f((1-l(k))X(k)), k=1,2,..., where X(1)=f(x(0)). For the stochastic population model, we study the problem of choosing a control u=(u(1), …, u(k),…), that limits at each time moment k the share of the extracted resource and under which the limit of the average time profit function H((l)̅, x(0))≐lim┬(n→∞)⁡〖∑^n_k=1▒〖X(k)l(k), где 〗〗(l)̅≐(l(1),…, l(k),…) exists and can be estimated from below with probability one by as a large number as possible. If the equation X(k+1)=f(X(k)) has a solution of the form X(k)≡x^*, then this solution is called the equilibrium position of the equation. For any k=1,2,..., we consider random variables A(k+1,x)=f((1-l(k))A(k,x)), B(k+1,x^*)=f((1-l(k))B(k,x^*)); here A(1,x)=f(x), B(1,x^*)=x^*. It is shown that when certain conditions are met, there exists a control u under which there holds the estimate of the average time profit 1/m∑^m_k=1▒〖M(A(k,x)l(k))≤H((l)̅, x(0))≤1/m∑^m_k=1▒〖M(B(k,x^*)l(k)),〗〗 where M denotes the mathematical expectation. In addition, the conditions for the existence of control u are obtained under which there exists, with probability one, a positive limit to the average time profit equal to H((l)̅, x(0))=lim┬(k→∞)⁡〖MA(k,x)l(k)=〗lim┬(k→∞)MB(k,x^*)l(k).

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(y_1,y_2)=0⇔y_1=y_2, and is assumed to be neither symmetric nor satisfying the triangle inequality. The question of well-posedness of the equation ψ(x)=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.

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.

For the parabolic equation with the Bessel operator ∂u/∂t=(∂^2 u)/(∂x^2 )+k/x ∂u/∂x in the rectangular domain 0<x<l, 0<t≤T, we consider a boundary value problem with the non-local integral condition of the first kind ∫_0^l▒〖u(x,t)xdx=0, 0≤t≤T.〗 This problem is reduced to an equivalent boundary value problem with mixed boundary conditions of the first and third kind. It is shown that the homogeneous equivalent boundary value problem has only a trivial zero solution, and hence the original inhomogeneous problem cannot have more than one solution. This proof uses Gronwall’s lemma. Then, by the method of spectral analysis, the existence theorem for a solution to an equivalent problem is proved. This solution is defined explicitly in the form of a Dini series. Sufficient conditions with respect to the initial condition are obtained. These conditions guarantee the convergence of the constructed series in the class of regular solutions.