Asymptotically dichotomic almost periodic differential equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Campos, Juan¹  Tarallo, Massimo² 

[1] Univ Granada, E-18071 Granada, Spain

[2] Univ Milan, Via Saldini 50, I-20133 Milan, Italy

关键词: Almost periodic solutions;    Exponential dichotomies;    Contractions and Fixed points;    Positive operators;    Spectral radius;    Haar measure;   

DOI  :  10.1016/j.jde.2017.03.014

来源: Elsevier

【 摘 要 】

Consider a non-linear differential equation in RN which asymptotically behaves as a linear equation admitting an exponential dichotomy. We wonder if almost periodic solutions exist when we add to the equation an almost periodic forcing term, large enough and not vanishing too much. A positive answer has been given in [3] for the scalar case N = 1 and our aim is to extend that result to higher dimensions. We discover that the extension seems to be driven by a new ingredient, namely the type of the exponential dichotomy: besides the pure stable types, the mixed hyperbolic type is now possible and leads to a weaker than expected extension. An example shows that a stronger extension cannot be obtained by the same method. The approach is blended and mixes methods of differential equations and functional analysis, especially when estimating norm and spectral radius of some crucial positive but non-compact linear integral operators. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free