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Under suitable conditions on a family $(I(t))_{t\ge 0}$ of Lipschitz mappingson 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 $(S(t))_{t\ge 0}$. Thecommon idea in the present approach is to find conditions on the generatingfamily $(I(t))_{t\ge 0}$, which by iteration can be transferred to thesemigroup. The construction relies on the Lipschitz set, which is invariantunder iterations and allows to preserve Lipschitz continuity to the limit.Moreover, we provide a verifiable condition which ensures that theinfinitesimal generator of the semigroup is given by $\lim_{h\downarrow0}\tfrac{I(h)x-x}{h}$ whenever this limit exists. The results are illustratedwith several examples of nonlinear semigroups such as robustifications andperturbations of linear semigroups.

Nonlinear semigroups built on generating families and their Lipschitz sets