具有分段常数参数的微分方程的光滑不变流形和叶状

IF 0.9 3区数学 Q2 MATHEMATICS, APPLIED

Bulletin des Sciences Mathematiques Pub Date : 2025-01-22 DOI:10.1016/j.bulsci.2025.103579

Weijie Lu , Donal O'Regan , Yonghui Xia

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引用次数: 0

摘要

本文建立了一类广义分段常数微分方程的光滑不变流形和光滑不变叶状理论。假设线性depcag允许α-指数二分，我们得到了基于分段常数参数Lyapunov-Perron积分和其他非平凡技巧（如分段常数参数二分不等式）的Lipschitz稳定（不稳定）不变流形和Lipschitz稳定（不稳定）不变叶形的存在性。在此基础上，利用纤维收缩定理，给出并证明了这些流形和叶形的c1 -光滑性。

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Smooth invariant manifolds and foliations for the differential equations with piecewise constant argument

In this work, we establish the theory of smooth invariant manifolds and smooth invariant foliations for the differential equations with piecewise constant argument of a generalized type (DEPCAGs). Suppose that the linear DEPCAGs admits a α-exponential dichotomy, we obtain the existence of Lipschitz stable (unstable) invariant manifolds and Lipschitz stable (unstable) invariant foliations, which are based on the Lyapunov-Perron integrals with piecewise constant argument and other non-trivial techniques (such as, dichotomy inequalities with piecewise constant argument). Furthermore, we formulate and prove the