Banach空间中拟线性方程周期解的取平均方法
An averaging method for periodic solutions of quasilinear equations in Banach spaces
- Université Catholique de Louvain（天主教鲁汶大学）
- Universidad de Chile（智利大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种取平均方法，用于证明Banach空间中拟线性微分方程周期解的存在性与唯一性，并应用于相对论摆的无穷系统。
AI中文摘要：
本文旨在将取平均方法推广到拟线性微分方程 $(\u03c6(t,u'))' = \varepsilon f(t,u,u',\varepsilon)$，$u(0) = u(T)$，$u'(0) = u'(T)$ 的周期问题，其中 $u$ 和 $u'$ 取值于 Banach 空间 $X$，且 $\u03c6$ 是 $X$ 中原点邻域到 $X$ 的适当同胚。该方法源于将取平均方法推广到形如 $u' = g(t,v)$，$v' = h(t,u,v)$，$u(0) = u(T)$，$v(0) = v(T)$ 的一阶系统（在 Banach 空间乘积中），该推广通过对等价问题直接应用 Banach 空间中的隐函数定理得到。我们证明：对应的平均方程的每个非退化零点，在参数足够小的非零值下，生成唯一邻近的周期解。文中给出了一个应用于具有相对论加速度的耦合受迫摆的无穷系统的实例。
英文摘要：
The aim of the paper is to provide an extension of the averaging method to the periodic problem for quasilinear differential equations $(ϕ(t,u'))' = \varepsilon f(t,u,u',\varepsilon)$, $u(0) = u(T)$, $u'(0) = u'(T)$, where $u$ and $u'$ take values in a Banach space $X$ and $ϕ$ is a suitable homeomorphism between a neighborhood of $0$ in $X$ and $X$. The approach follows from a given extension of the averaging method to first order systems of the form $u' = g(t,v)$, $v' = h(t,u,v)$, $u(0) = u(T)$, $v(0) = v(T)$, in a product of Banach spaces, obtained from a direct application of the implicit function theorem in Banach spaces to an equivalent problem. We prove that every nondegenerate zero of the corresponding averaged equation generates, for sufficiently small nonzero values of the parameter, a unique nearby periodic solution. An application is given to an infinite system of coupled forced pendulums with relativistic acceleration.
