AI 中文总结
本文提出一种适用于Banach空间中拟线性方程的抽象平均化方法，允许主算子非线性，证明非退化零点生成邻近解并给出唯一性，涵盖线性Fredholm理论并应用于向量p-Laplacian Duffing系统等。
AI 中文摘要
我们为Banach空间中的方程$Mx=\varepsilon N(x,\varepsilon)$发展了一种抽象平均化方法。该方法适用于方程可分离为平均相容条件和非线性互补问题的情况，并允许主算子$M$为真正非线性的。我们证明了相应平均方程的每个非退化零点，对所有足够小的非零$\varepsilon$值，都会生成一个邻近解。在适当的小Lipschitz条件下，该解是局部唯一的，而在可微情形下，它属于唯一的局部$C^1$分支。该框架包含抽象线性Fredholm平均化理论作为特例，但也覆盖了主算子在参考状态没有可逆线性化的拟线性问题。我们还确定了解的渐近主轮廓。应用给出了向量$p$-Laplacian Duffing系统以及Hilbert空间中的非光滑问题。
英文摘要
We develop an abstract averaging method for equations $Mx=\varepsilon N(x,\varepsilon)$ in Banach spaces. The method applies when the equation can be separated into an averaged compatibility condition and a nonlinear complementary problem, and it allows the principal operator $M$ to be genuinely nonlinear. We prove that every nondegenerate zero of the corresponding averaged equation generates, for all sufficiently small nonzero values of $\varepsilon$, a nearby solution. Under a suitable small Lipschitz condition this solution is locally unique, while in the differentiable case it belongs to a unique local $C^1$ branch. The framework contains the abstract linear Fredholm averaging theory as a special case, but also covers quasilinear problems for which the principal operator has no invertible linearization at the reference state. We also determine the leading asymptotic profile of the solutions. Applications are given to a vector $p$-Laplacian Duffing system and to a nonsmooth problem in a Hilbert space.
