生成族上的非线性半群及其Lipschitz集
Nonlinear semigroups built on generating families and their Lipschitz sets
摘要：
在完全度量空间上Lipschitz映射族$(I(t))_{t\ge 0}$上，在适当的条件下，证明了直到一个子序列，强极限$S(t):=\lim_{n\to\infty}(I(t 2^{-n}))^{2^n}$对所有并元时间点$t$都存在，并推广到强连续半群$(S(t))_{t\ge 0}$ .本文方法的共同思想是寻找生成族$(I(t))_{t\ge 0}$上的条件，通过迭代可以将其转移到半群上。构造依赖于Lipschitz集，它在迭代下是不变的，并且允许将Lipschitz连续性保持到极限。此外，我们提供了一个可验证的条件，确保该半群的无穷小生成元是由$\lim_{h\downarrow 0}\tfrac{I(h)x-x}{h}$，只要这个极限存在。最后用几个非线性半群的例子说明了所得结果，如线性半群的鲁棒性和扰动性。
英文摘要：
Under suitable conditions on a family $(I(t))_{t\ge 0}$ of Lipschitz mappings on a complete metric space, we show that up to a subsequence the strong limit $S(t):=\lim_{n\to\infty}(I(t 2^{-n}))^{2^n}$ exists for all dyadic time points $t$, and extends to a strongly continuous semigroup $(S(t))_{t\ge 0}$. The common idea in the present approach is to find conditions on the generating family $(I(t))_{t\ge 0}$, which by iteration can be transferred to the semigroup. The construction relies on the Lipschitz set, which is invariant under iterations and allows to preserve Lipschitz continuity to the limit. Moreover, we provide a verifiable condition which ensures that the infinitesimal generator of the semigroup is given by $\lim_{h\downarrow 0}\tfrac{I(h)x-x}{h}$ whenever this limit exists. The results are illustrated with several examples of nonlinear semigroups such as robustifications and perturbations of linear semigroups.
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