A characterization of almost automorphic functions

Egawa, Jirō

doi:10.3792/pjaa.61.203

Cited by 5 publications

(2 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2. A Levitan almost periodic point x with relatively compact trajectory {π(t, x): t ∈ T} is also almost automorphic (see [2][3][4][5]10,22] and [35]). In other words, a Levitan almost periodic point x is almost automorphic, if and only if its trajectory {π(t, x): t ∈ T} is relatively compact.…”

Section: Recurrent Almost Periodic and Almost Automorphic Motionsmentioning

confidence: 99%

Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

Caraballo

Cheban

2009

Journal of Differential Equations

View full text Add to dashboard Cite

The well-known Favard's theorem states that the linear differential equation(1)with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equationsthere are bounded solutions which are nonseparated, sometimes almost periodic solutions do not exist (R. Johnson, R. Ortega and M. Tarallo, V. Zhikov and B. Levitan).In this paper we prove that linear differential equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equationare homoclinic to zero (i.e. lim |t|→+∞ |ϕ(t)| = 0 for all bounded solutions ϕ of (3)). If the coefficients of (1) are Bohr almost periodic and all bounded solutions of all limiting equations (2) are homoclinic to zero, then Eq. (1) admits a unique almost automorphic solution.The analogue of this result for difference equations is also given.We study the problem of existence of Bohr/Levitan almost periodic solutions of Eq. (1) in the framework of general non-autonomous dynamical systems (cocycles).

show abstract

Section: Recurrent Almost Periodic and Almost Automorphic Motionsmentioning

confidence: 99%

Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

Caraballo

Cheban

2009

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…3. A Levitan almost periodic point x with relatively compact trajectory {π(t, x) t ∈ T} is also almost automorphic (see [1]- [5], [13], [17] and [27]). In other words, a Levitan almost periodic point x is almost automorphic, if and only if its trajectory {π(t, x) t ∈ T} is relatively compact.…”

Section: Tom áS Caraballo and David Chebanmentioning

confidence: 99%

Almost periodic and almost automorphic solutions of linear differential equations

Caraballo¹,

Cheban²

2013

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

We analyze the existence of almost periodic (respectively, almost automorphic, recurrent) solutions of a linear non-homogeneous differential (or difference) equation in a Banach space, with almost periodic (respectively, almost automorphic, recurrent) coefficients. Under some conditions we prove that one of the following alternatives is fulfilled:(i) There exists a complete trajectory of the corresponding homogeneous equation with constant positive norm; (ii) The trivial solution of the homogeneous equation is uniformly asymptotically stable. If the second alternative holds, then the non-homogeneous equation with almost periodic (respectively, almost automorphic, recurrent) coefficients possesses a unique almost periodic (respectively, almost automorphic, recurrent) solution. We investigate this problem within the framework of general linear nonautonomous dynamical systems. We apply our general results also to the cases of functional-differential equations and difference equations.1. Introduction. Let (X, ρ) be a complete metric space, R (Z) be the group of real (integer) numbers. By S we will denote either R or Z and by T a sub-semigroup of S. Let (X, T, π) be a dynamical system on X, i.e., let π : T×X→X be a continuous function such that π(0, x) = x for all x ∈ X, and π(tAn m-dimensional torus is denoted by T m := R m /2πZ. Let (T m , T, σ) be an irrational winding of T m , i.e., σ(t, ν) := (ν 1 t, ν 2 t, . . . , ν m t) for all t ∈ S and ν ∈ T m and the numbers ν 1 , ν 2 , . . . , ν m are rationally independent. A point x ∈ X is called quasi-periodic with frequency ν := (ν 1 , ν 2 , . . . , ν m ) ∈ T m , if there exists a continuous function Φ : T m → X such that π(t, x) := Φ(σ(t, ω)) for 2010 Mathematics Subject Classification. 34G10,

show abstract

Levitan Almost Periodic and Almost Automorphic Solutions of V-monotone Differential Equations

Cheban

2008

J Dyn Diff Equat

View full text Add to dashboard Cite

In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.

show abstract

A characterization of almost automorphic functions

Cited by 5 publications

References 2 publications

Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

Almost periodic and almost automorphic solutions of linear differential equations

Levitan Almost Periodic and Almost Automorphic Solutions of V-monotone Differential Equations

Contact Info

Product

Resources

About