{"title":"Cocycles for equations with infinite delay and hyperbolicity","authors":"","doi":"10.1016/j.nonrwa.2024.104221","DOIUrl":null,"url":null,"abstract":"<div><p>We show that the hyperbolicity of a linear delay-difference equation with <em>infinite delay</em>, expressed in terms of the existence of an exponential dichotomy, can be completely characterized by the hyperbolicity of a linear cocycle obtained from the solutions of the equation. As an application of this characterization, we obtain several consequences: the extension of hyperbolicity to all equations in the invariant hull; the robustness of the existence of hyperbolicity for all equations in this hull under sufficiently small linear perturbations; the equality of all spectra in the invariant hull; and a characterization of hyperbolicity for all equations in the invariant hull in terms of an admissibility property taking bounded perturbations to bounded solutions.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Real World Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1468121824001603","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We show that the hyperbolicity of a linear delay-difference equation with infinite delay, expressed in terms of the existence of an exponential dichotomy, can be completely characterized by the hyperbolicity of a linear cocycle obtained from the solutions of the equation. As an application of this characterization, we obtain several consequences: the extension of hyperbolicity to all equations in the invariant hull; the robustness of the existence of hyperbolicity for all equations in this hull under sufficiently small linear perturbations; the equality of all spectra in the invariant hull; and a characterization of hyperbolicity for all equations in the invariant hull in terms of an admissibility property taking bounded perturbations to bounded solutions.
期刊介绍：
Nonlinear Analysis: Real World Applications welcomes all research articles of the highest quality with special emphasis on applying techniques of nonlinear analysis to model and to treat nonlinear phenomena with which nature confronts us. Coverage of applications includes any branch of science and technology such as solid and fluid mechanics, material science, mathematical biology and chemistry, control theory, and inverse problems.
The aim of Nonlinear Analysis: Real World Applications is to publish articles which are predominantly devoted to employing methods and techniques from analysis, including partial differential equations, functional analysis, dynamical systems and evolution equations, calculus of variations, and bifurcations theory.
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