Two-scale Convergence and Homogenization for a Class of Quasilinear Elliptic Equations

By use of Fourier analysis techniques, we obtain some new properties of the almost-periodic functions and extend the two-scale convergence method in the homogenization theory to the case of almost-periodic oscillations. Then, we use some new techniques to study the homogenization for quasilinear elliptic equations with almostperiodic coefficients: div a(x,x/ε, u, Du) = f(x) in Ω and obtain the weak convergence and corrector result.     

Tight estimates for general averaging applied to almost-periodic differential equations

Estimates are presented for the averaging method of ordinary differential equations. Previous results are improved by relaxing the conditions under which they hold, and by providing tight bounds for the estimations for almost-periodic differential equations and quasi-periodic differential equations.     

Global attractivity of almost-periodic solution in a model of hematopoiesis with feedback control

By means of properties of almost-periodic system, sufficient conditions are obtained for the existence and global attractivity of a positive almost-periodic solution for an almost-periodic model of hematopoiesis with feedback control.     

Spaces of piecewise-continuous almost-periodic functions and of almost-periodic sets on the line. I

In connection with the study of piecewise-continuous almost-periodic functions we introduce the notion of a countable almost-periodic number set. We investigate various properties of it: In particular, we prove that the space of almost-periodic sets is closed with respect to the operation of free union.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1613–1619, December, 1991.     

Abstract differential equations with almost-periodic solutions

In this paper we present some quite simple results concerning almost-periodic solutions of abstract differential equations. We start with a general proposition about linear equations, then we establish some new facts about bounded or relatively compact solutions which become almost-periodic; finally, we study “separation from 0 properties” of non-trivial almost-periodic solutions for equations with bounded or unbounded operator coefficients.     

Consistent solutions of linear equations in variational derivatives

In this paper, we consider the classical existence problem for consistent and uniformly consistent solutions of linear equations in variational derivatives. In particular, we generalize several results obtained by V. V. Zhikov and B. A. Shcherbakov concerning the existence of almost-periodic, consistent, and uniformly consistent solutions of ordinary differential equations.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 176–187.Original Russian Text Copyright © 2005 by A. I. Gerko.This revised version was published online in April 2005 with a corrected issue number.     

Finite-automaton transformations of strictly almost-periodic sequences

Different versions of the notion of almost-periodicity are natural generalizations of the notion of periodicity. The notion of strict almost-periodicity appeared in symbolic dynamics, but later proved to be fruitful in mathematical logic and the theory of algorithms as well. In the paper, a class of essentially almost-periodic sequences (i.e., strictly almost-periodic sequences with an arbitrary prefix added at the beginning) is considered. It is proved that the property of essential almost-periodicity is preserved under finite-automaton transformations, as well as under the action of finite transducers. The class of essentially almost-periodic sequences is contained in the class of almost-periodic sequences. It is proved that this inclusion is strict.     

Existence and exponential stability of almost-periodic delays on time scales

This paper is concerned with multidirectional associative memory neural network with distributed delays on almost-periodic time scales. Some sufficient conditions on the existence, uniqueness and the global exponential stability of almost-periodic solutions are established. An example is presented to illustrate the feasibility and effectiveness of the obtained results.     

Proof of the favard theorem on the existence of almost-periodic solution for an arbitrary Banach space

The well-known Favard-Amerio theorem on the existence of an almost-periodic solution of a linear equation is based on the geometry of a uniformly convex space, since the almost-periodic solution is found by the minimax condition. In the present note an essentially different method for finding the almost-periodic solution is developed, which enables us to prove the Favard-Amerio theorem for an arbitrary Banach space.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 121–126, January, 1978.     

双尺度收敛和拟线性抛物方程的均匀化

利用Ｆｏｕｒｉｅｒ分析的某些技巧，得到殆周期国数的一些新性质，在此基础上将均匀化理论中的双尺度收敛方法推广到殆周期振荡的情形．然后利用一些新技巧讨论了殆周期振荡的拟线性方程的均匀化问题，得到了相应的弱收敛和纠正子结果．     

On Almost-Periodic Operators in the Spaces of Sequences

We study some properties of almost-periodic operators in the spaces l ^p and the spaces of Bohr and Besicovich almost-periodic sequences.     

WEIGHTED PSEUDO ALMOST-PERIODIC SOLUTIONS OF SHUNTING INHIBITORY CELLULAR NEURAL NETWORKS WITH MIXED DELAYS

In this paper,we prove the existence and the global exponential stability of the unique weighted pseudo almost-periodic solution of shunting inhibitory cellular neural networks with mixed time-varying delays comprising different discrete and distributed time delays.Some sufficient conditions are given for the existence and the global exponential stability of the weighted pseudo almost-periodic solution by employing fixed point theorem and differential inequality techniques.The results of this paper complement the previously known ones.Finally,an illustrative example is given to demonstrate the effectiveness of our results.     

On the exponential decay for boundary layer problems

In connection with boundary layers problem, the possible decay of the solution to a Dirichlet problem on the half-space with almost-periodic boundary data is studied.     

On differential operators with Bohr-Neugebauer type property

Let B be a bounded linear operator of a Banach space X into itself. If the differential operator $\text{(}\text{d}\text{dt}\text{) ? B}$ has a property more general than Bohr-Neugebauer property for Bochner almost-periodic functions, then any Stepanov-bounded solution of the differential equation $\text{(}\text{d}\text{dt}\text{) u(t) ? Bu(t) = g(t)}$ is also almost-periodic, with g(t) being continuous and Stepanov almost-periodic.     

On Stepanov almost-periodic oscillations and their discretizations

The relationship between Carathéodory almost-periodic (a.p.) solutions and their discretizations is clarified for differential equations and inclusions in Banach spaces. Our investigation was stimulated by an old result of Meisters [Proc. Am. Math. Soc. 10 (1959), pp. 113–119] about Bohr a.p. solutions which we generalize in several directions. Unlike for functions, Stepanov and Bohr a.p. sequences are shown to coincide. A particular attention is paid to purely (i.e. non-uniformly continuous) Stepanov a.p. solutions. Many ideas are explained in detail by means of examples illustrated.     

Absolute convergence of fourier series of almost-periodic functions

We present necessary and sufficient conditions for the absolute convergence of the Fourier series of almost-periodic (in the sense of Besicovitch) functions when the Fourier exponents have limit points at infinity or at zero. The structural properties of the functions are described by the modulus of continuity or the modulus of averaging of high order, depending on the behavior of the Fourier exponents. The case of uniform almost-periodic functions of bounded variation is considered.     

Necessary and Sufficient Conditions for Existence and Uniqueness of Bounded and Almost-Periodic Solutions of Nonlinear Differential Equations

Necessary and sufficient conditions of the existence and uniqueness of bounded and almost-periodic solutions of nonlinear dif ferential equation dx/dt+f(x)=h(t), tR, are obtained. Here f: RR is continuous, h: RR is continuous and bounded.     

Unique Solvability of Parabolic Equations with Almost-Periodic Coefficients in Hölder Spaces

We obtain a criterion for invertibility in Hölder spaces of linear parabolic operators of arbitrary order with any number of spatial variables and almost-periodic coefficients.     

Holomorphic Almost-Periodic Functions

The present paper gives a survey of up-to-date results of holomorphic almost-periodic functions and mappings in one and several complex variables.