Robustness of Exponential Dichotomy in a Class of Generalised Almost Periodic Linear Differential Equations in Infinite Dimensional Banach Spaces

In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily sma...

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Published in:	Journal of dynamics and differential equations Vol. 34; no. 4; pp. 2841 - 2865
Main Authors:	Rodrigues, H. M., Caraballo, T., Nakassima, G. K.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.12.2022 Springer Nature B.V
Subjects:	Applications of Mathematics Banach spaces Dichotomies Differential equations Mathematics Mathematics and Statistics Ordinary Differential Equations Partial Differential Equations Periodic functions Perturbation Robustness (mathematics) Stability Nonautonomous ordinary differential equations Generalized almost period functions Exponential dichotomy
ISSN:	1040-7294, 1572-9222
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Abstract	In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (Dichotomies in stability theory. Lecture notes in mathematics, Springer, Berlin, 1970). Based in Rodrigues (Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo, São Carlos, 1970) and in Kloeden and Rodrigues (Nonlinear Anal 74:2695–2719, 2011), Rodrigues et al. (Stability problems in non autonomous linear differential equations in infinite dimensions. arXiv:1906.04642, 2019) we use the class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.
AbstractList	In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (Dichotomies in stability theory. Lecture notes in mathematics, Springer, Berlin, 1970). Based in Rodrigues (Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo, São Carlos, 1970) and in Kloeden and Rodrigues (Nonlinear Anal 74:2695–2719, 2011), Rodrigues et al. (Stability problems in non autonomous linear differential equations in infinite dimensions. arXiv:1906.04642, 2019) we use the class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results.
Author	Rodrigues, H. M. Caraballo, T. Nakassima, G. K.
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References	DaleckĭiJLKreinMGStability of Solutions of Differential Equations in Banach Space, Translation of Mathematical Monographs1974ProvidenceAmerican Mathematical Society Rodrigues, H.M., Solà-Morales, J., Nakassima, G.K.: Stability problems in non autonomous linear differential equations in infinite dimensions (2019). arXiv:1906.04642 HenryDGeometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics1981BerlinSpringer10.1007/BFb0089647 CarvalhoANLangaJARobinsonJCAttractors of Infinite Dimensional Nonautonomous Dynamical Syustems2011BerlinSpringer KloedenPERodriguesHMDynamics of a class of ODEs more general than almost periodicNonlinear Anal.20117426952719277651910.1016/j.na.2010.12.0251217.34101 Rodrigues, H.M.: Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo, São Carlos (1970) CoppelWADichotomies in Stability Theory1970BerlinSpringer629 D Henry (9854_CR4) 1981 JL Daleckĭi (9854_CR3) 1974 WA Coppel (9854_CR2) 1970 PE Kloeden (9854_CR5) 2011; 74 AN Carvalho (9854_CR1) 2011 9854_CR7 9854_CR6
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SubjectTerms	Applications of Mathematics Banach spaces Dichotomies Differential equations Mathematics Mathematics and Statistics Ordinary Differential Equations Partial Differential Equations Periodic functions Perturbation Robustness (mathematics) Stability
Title	Robustness of Exponential Dichotomy in a Class of Generalised Almost Periodic Linear Differential Equations in Infinite Dimensional Banach Spaces
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