On approximate constraint satisfaction

We consider an abstract problem on fulfilling asymptotic constraints. We propose a very general approach to constructing “nonsequential” attraction sets in the space of generalized elements formalizable as finitely additive measures. We study the existence and the structure of the asymptote universal in the range of “asymptotic constraints” not requiring the compactifiability of the space of ordinary solutions.     

Ultrafilters of measurable spaces as generalized solutions in abstract attainability problems

We consider problems of asymptotic analysis that arise, in particular, in the formalization of effects related to an approximate observation of constraints. We study nonsequential (generally speaking) variants of asymptotic behavior that can be formalized in the class of ultrafilters of an appropriate measurable space. We construct attraction sets in a topological space that are realized in the class of ultrafilters of the corresponding measurable space and specify conditions under which ultrafilters of a measurable space are sufficient for constructing the “complete” attraction set corresponding to applying ultrafilters of the family of all subsets of the space of ordinary solutions. We study a compactification of this space that is constructed in the class of Stone ultrafilters (ultrafilters of a measurable space with an algebra of sets) such that the attraction set is realized as a continuous image of the compact set of generalized solutions; we also study the structure of this compact set in terms of free ultrafilters and ordinary solutions that observe the constraints of the problem exactly. We show that, in the case when there are no exact ordinary solutions, this compact set consists of free ultrafilters only; i.e., it is contained in the remainder of the compactifier (an example is given showing that the similar property may be absent for other variants of the extension of the original problem).     

Asymptotics of the optimal time in a singular perturbed linear time-optimal problem

We consider an abstract attainability problem under constraints of asymptotic character and describe a general approach to constructing “nonsequential” attraction sets in the space of generalized elements formalized as finitely additive measures. We also study the existence and structure of an asymptotic formula universal in the range of “asymptotic constraints” and topologies of the space of generalized elements in the case when the space of ordinary solutions is not necessarily compactifiable.     

Ultrafilters in the constructions of attraction sets: Problem of compliance to constraints of asymptotic character

We consider the properties of generalized elements in the problem of compliance to constraints of asymptotic character; these elements are identified with ultrafilters of special families of sets in the space of ordinary solutions.     

Asymptotic stability of solutions to Volterra-renewal integral equations with space maps

In this paper we consider linear Volterra-renewal integral equations (VIEs) whose solutions depend on a space variable, via a map transformation. We investigate the asymptotic properties of the solutions, and study the asymptotic stability of a numerical method based on direct quadrature in time and interpolation in space. We show its properties through test examples.     

Solution Estimates for Nonlinear Nonautonomous Evolution Equations in Spaces with Normalizing Mappings

Nonlinear nonautonomous evolution equations in a space with a normalizing mapping (a generalized norm) are considered. Solution estimates are established. In particular cases these estimates generalize the Wazewski and Lozinskii estimates from the theory of ordinary differential equations. By the obtained estimates, the following problems are investigated: asymptotic stability, boundedness of solutions, input-output stability, existence of periodic solutions. Applications to integro-differential equations are discussed.     

Asymptotically recurrent solutions ofβ-differential equations

In the paper methods from the theory of extensions of dynamical systems are used to studyβ-differential equations whose solutions possess the uniqueness property and depend continuously on the initial data and on the right-hand side of the equation. The Zhikov-Bronshtein theorems concerning asymptotically almost periodic solutions of ordinary differential equations are extended toβ-differential equations (in particular, to total differential equations). Along with asymptotic almost periodicity, we also consider asymptotic recurrence, weak asymptotic distality, and asymptotic distality. To the equations we associate dynamical systems generated by the space of the right-hand sides and the spaces of the solutions and of the initial data of solutions of the equation. Generally, the phase semigroups of the dynamical systems are not locally compact. Translated fromMatermaticheskie Zametki, Vol. 67, No. 6, pp. 837–851, June, 2000.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. III

We describe the technique of normalization based on the method of asymptotic decomposition in the space of representation of a finite-dimensional Lie group. The main topics of the theory necessary for understanding the method are outlined. Models based on the Van der Pol equation are investigated by the method of asymptotic decomposition in the space of homogeneous polynomials (the space of representation of a general linear group in a plane) and in the space of representation of a rotation group on a plane (ordinary Fourier series). The comparison made shows a dramatic decrease in the necessary algebraic manipulations in the second case. We also discuss other details of the technique of normalization based on the method of asymptotic decomposition.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1627–1646, December, 1994.This research was partially supported by the International Scientific Foundation, grant No. UB2000.     

A method for constructing asymptotic expansions of bisingularly perturbed problems

In this paper we propose an analog of the method of boundary functions for constructing uniform asymptotic expansions of solutions to bisingularly perturbed problems. With the help of this method we construct uniform asymptotic expansions of solutions to the Dirichlet problem for bisingularly perturbed ordinary differential equations and elliptic equations of the second order. By the use of the maximum principle we obtain estimates for the remainder terms.     

Estimation of solutions of nonautonomous systems

We study the problem of using the direct Lyapunov method to get estimates for solutions of a system of ordinary differential system in general form. Theorems on asymptotic stability and behavior of solutions are proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 242–248, February, 1995.     

具有代数衰减的边界层问题

本文研究了不满足Tikhnov定理中稳定性要求的一类常微分方程奇摄动边值问题.利用边界层函数法以及微分不等式理论,分别构造了渐进解的形式和证明了解的存在性和渐近解一致有效性并进行了余项估计,得出了该类问题边界层代数式衰减的结论.     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.     

A new approach to the method of nonlinear variation of parameters

A nonlinear variation of parameters method for ordinary differential equations is obtained, which generalizes existing techniques. The versatility of the method is demonstrated by studying the asymptotic behavior of solutions of perturbed equations.     

The Painlevé-Kowalevski and Poly-Painlevé Tests for Integrability

The characteristic feature of the so-called Painlevé test for integrability of an ordinary (or partial) analytic differential equation, as usually carried out, is to determine whether all its solutions are single-valued by local analysis near individual singular points of solutions. This test, interpreted flexibly, has been quite successful in spite of various evident flaws. We review the Painleve test in detail and then propose a more robust and generally more appropriate definition of integrability: a multivalued function is accepted as an integral if its possible values (at any given point in phase space) are not dense. This definition is illustrated and justified by examples, and a widely applicable method (the poly-Painlevé method) of testing for it is presented, based on asymptotic analysis covering several singularities simultaneously.     

The initial development of the WKB solutions of linear second order ordinary differential equations and their use in the connection problem

In this paper we trace the early development of a method for finding the approximate solutions —called the WKB solutions—for a class of ordinary differential equations of second order. We also analyze the attempts made by the various contributors to this method to substantiate their results. These approximating solutions were subsequently shown to be asymptotic in the sense of Poincaré. Also presented and examined are the several methods used to deal with the “connection problem” which arises in the use of the WKB solutions.     

非线性扰动Klein-Gordon方程初值问题的渐近理论

在二维空间中研究一类非线性扰动Klein-Gordon方程初值问题解的渐近理论. 首先利用压缩映象原理,结合一些先验估计式及Bessel函数的收敛性,根据Klein-Gordon方程初值问题的等价积分方程,在二次连续可微空间中得到了初值问题解的适定性;其次,利用扰动方法构造了初值问题的形式近似解,并得到了该形式近似解的渐近合理性;最后给出了所得渐近理论的一个应用,用渐近近似定理分析了一个具体的非线性Klein-Gordon方程初值问题解的渐近近似程度．     

Some Remarks on Left-Definite Hamiltonian Systems in the Regular Case

In the paper, a new method of constructing asymptotic solutions of differential equations on manifolds with singularities is presented. This method allows not only to widen essentially the space of asymptotics but also to obtain explicit formulas for asymptotic expansions, in particular, in the case when in a neighborhood of a singular point there exist strata of different dimensions.     

关于含双参数的非线性常微分方程的奇异摄动*

本文应用微分不等式方法研究含双小参数的非线性常微分方程的边值问题,作出渐近解并对余项进行估计.     

Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms

This paper is concerned with the asymptotic behavior of solutions of a stochastic nonlinear wave equation with dispersive and dissipative terms defined on an unbounded domain. It is proved that the random dynamical system generated by the equation has a random attractor in a Sobolev space. To overcome the difficulty caused by the non-compactness of Sobolev embeddings on unbounded domains, a cut-off method and a decomposition trick are combined to prove the asymptotic compactness of the solutions.