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.     

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.     

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.     

Almost-periodic solutions for Riccati equations of stochastic control

We consider an average quadratic cost criteria for affine stochastic differential equations with almost-periodic coefficients. Under stabilizability and detectability conditions we show that the Riccati equation associated with the quadratic control problem has a unique almost-periodic solution. In the periodic case the corresponding result is proved in [4].     

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.     

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.     

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．     

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.     

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.     

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 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.     

Existence of dichotomies and invariant splittings for linear differential systems I

This paper is concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy for a class of equations which includes those with Bohr almost-periodic coefficients. The problem is treated in the context of linear skew-product flows, where it becomes clear how to generalize to the case of fiber-preserving flows on vector bundles. Both continuous and discrete flows are treated and the results apply to the linearized variational equation for a time-varying vector field on a manifold as well as the linearization of a diffeomorphism acting on a manifold. Sufficient conditions are given for a diffeomorphism on a manifold to be an Anosov diffeomorphism. For linear skew-product flows arising from ordinary differential equations our theory is a partial generalization of Floquet theory to the almost-periodic case.     

Reduced bodies in Minkowski space

Summary We study the existence of almost-periodic solutions for a second order abstract Cauchy problem defined in a Banach space.     

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.     

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 the absolute summability of Fourier series of almost-periodic besicovitch functions

For almost-periodic Besicovitch functions whose spectrum has a limit point only at infinity, we establish criteria for the absolute Cesàro summability of their Fourier series of order greater than –1.     

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.     

Summability of Fourier series for almost-periodic functions on locally compact Abelian groups

Some results concerning the summability of Fourier series of continuous 2π-periodic functions are generalized for the case of almost-periodic functions defined on locally compact Abelian groups.     

Complete Asymptotic Expansion of the Integrated Density of States of Multidimensional Almost-Periodic Pseudo-Differential Operators

We obtain a complete asymptotic expansion of the integrated density of states of operators of the form ${H = (-\Delta)^w+ B}$ in ${\mathbb{R}^d}$ . Here w >  0 and B belong to a wide class of almost-periodic self-adjoint pseudo-differential operators of order less than 2w. In particular, we obtain such an expansion for magnetic Schrödinger operators with either smooth periodic or generic almost-periodic coefficients.