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<p>In this paper, without the assumption of bounded growth, we prove that if an evolution family is sufficiently close to one which has an exponential dichotomy, then it also has an exponential dichotomy. It was proved by Henry (1981) in the case where the evolution family has bounded growth. The difficulty caused by the absence of bounded growth, which is an essential condition in Henry’s proof, is resolved by proving a result on system-dependence of extended Green functions. We also extend this result to exponential dichotomies on a half line ℝ<sup>+</sup> or ℝ<sup>−</sup> as well as on any nonempty closed interval of ℝ.</p>

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Roughness of the exponential dichotomy for evolution families in Banach spaces without bounded growth

  • 期刊论文

  • 发表日期：

    2026年7月22日

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  • Weinian Zhang,
  • Linfeng Zhou

摘要

In this paper, without the assumption of bounded growth, we prove that if an evolution family is sufficiently close to one which has an exponential dichotomy, then it also has an exponential dichotomy. It was proved by Henry (1981) in the case where the evolution family has bounded growth. The difficulty caused by the absence of bounded growth, which is an essential condition in Henry’s proof, is resolved by proving a result on system-dependence of extended Green functions. We also extend this result to exponential dichotomies on a half line ℝ+ or ℝ− as well as on any nonempty closed interval of ℝ.