ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL/DIFFERENCE EQUATIONS WITHOUT FAVARD’S SEPARATION CONDITION. II
ALMOST PERIODIC AND ALMOST AUTOMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL/DIFFERENCE EQUATIONS WITHOUT FAVARD’S SEPARATION CONDITION. II
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DOI：
10.1016/j.jde.2008.07.025
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Tomás Caraballo;D. Cheban
中科院分区：
文献类型：
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作者：
Tomás Caraballo;D. Cheban
本文继续了前一篇文章的研究，证明了具有Levitan概周期系数的线性微分方程存在唯一的Levitan概周期解，如果它至少有一个有界解，且齐次方程的有界解同宿于零（即lim|不|→+∞| φ（t）|=0的所有有界解）。如果方程（1）的系数是玻尔概周期的，且方程（1）的所有有界解都是周期的，则方程（1）的所有有界解都是周期的。(2)同宿于零，则Eq. (1)承认一个唯一的几乎自守的解决方案。在这第二部分中，我们首先概括了这一结果的线性泛函微分方程（FDES）的形式，以及中性FDES。对有限时滞泛函差分方程和某些类型的偏微分方程也给出了类似的结果。本文在一般半群非自治动力系统（上循环）的背景下研究了（3）型微分方程的Bohr-Levitan概周期解的存在性问题，与第一部分中所考虑的群非自治动力系统框架不同.
In this paper we continue the research started in a previous paper, where we proved that the linear differential equation 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 equation are homoclinic to zero (i.e. lim|t|→+∞|φ(t)|=0 for all bounded solutions φ of (2)). If the coefficients of (1) are Bohr almost periodic and all bounded solutions of Eq. (2) are homoclinic to zero, then Eq. (1) admits a unique almost automorphic solution. In this second part we first generalise this result for linear functional differential equations (FDEs) of the form as well as for neutral FDEs. Analogous results for functional difference equations with finite delay and some classes of partial differential equations are also given. We study the problem of existence of Bohr–Levitan almost periodic solutions of differential equations of type (3) in the context of general semi-group non-autonomous dynamical systems (cocycles), in contrast with the group non-autonomous dynamical systems framework considered in the first part.