Nonlinear semigroups built on generating families and their Lipschitz sets

TL;DR

This paper establishes conditions under which nonlinear semigroups can be constructed from Lipschitz generating families on metric spaces, ensuring strong continuity and providing a way to identify their infinitesimal generators.

Contribution

It introduces verifiable conditions for constructing nonlinear semigroups from Lipschitz families and characterizes their generators, extending the theory of nonlinear semigroup generation.

Findings

Existence of strong limits for dyadic time points

Construction of strongly continuous nonlinear semigroups

Conditions for identifying the infinitesimal generator

Abstract

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.