2020/11/27 by Jonas Blessing, Blessing, Jonas, Michael Kupper +1 · 2 citations
Computer Science · Engineering · Mathematics · #35B20 #35K55 #47J25 #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Optimization and Variational Analysis #Primary 47H20 #Probability (math.PR) #Secondary 47J35 #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2011.13664
openalex publication_date 2020/11/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Under suitable conditions on a family (I(t))t≥ 0 of Lipschitz mappings on a complete metric space, we show that up to a subsequence the strong limit S(t):=limn→∞(I(t 2-n))2n exists for all dyadic time points t, and extends to a strongly continuous semigroup (S(t))t≥ 0. The common idea in the present approach is to find conditions on the generating family (I(t))t≥ 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 limh\downarrow 0\tfracI(h)x-xh whenever this limit exists. The results are illustrated with several examples of nonlinear semigroups such as robustifications and perturbations of linear semigroups.