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Time periodic solutions of one-dimensional forced Kirchhoff equations with x -dependent coefficients under spatial periodic conditions
Analysis and Mathematical Physics ( IF 1.7 ) Pub Date : 2019-08-26 , DOI: 10.1007/s13324-019-00339-1
Mu Ma , Shuguan Ji

In this paper, we consider the one-dimensional Kirchhoff equation with x-dependent coefficients under the spatial periodic conditions, which models the forced vibrations of an inhomogeneous string in presence of a time periodic external forcing with period \(2\pi /\omega \) and amplitude \(\epsilon >0\). By using the Lyapunov–Schmidt reduction and the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period \(2\pi /\omega \) and order \(\epsilon \). Such results hold for parameters \((\omega ,\epsilon )\) in a positive measure Cantor set that has asymptotically full measure as the amplitude \(\epsilon \) goes to zero.

中文翻译:

空间周期条件下具有x依赖系数的一维强迫Kirchhoff方程的时间周期解

在本文中,我们考虑了在空间周期条件下具有x依赖系数的一维Kirchhoff方程,该方程可模拟存在周期为((2 \ pi / \ omega)的时间周期外强迫作用下不均匀弦的强迫振动\)和幅度\(\ epsilon> 0 \)。通过使用Lyapunov–Schmidt约简和Nash–Moser迭代技术,我们获得了周期为((2 \ pi / \ omega \)和阶为((epsilon \))的时间周期解的存在性,规则性和局部唯一性。这样的结果适用于正量度Cantor集中的参数\((\ omega,\ epsilon)\),其渐近完全量度为振幅\(\ epsilon \)为零。
更新日期:2019-08-26
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