Abstract
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):=lim _{nto infty }(I(t 2^{-n}))^{2^{n}} 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 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 _{hdownarrow 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.
Highlights
Let X be a complete metric space and (I (t))t≥0 a family of Lipschitz continuous mappings I (t) : X → X
We provide a detailed study how a modified version of the Chernoff approximation can be used for the construction of nonlinear semigroups
If (I (t))t≥0 is a family of linear contractions, it was shown in [8, Theorem 3.7] that the infinitesimal generator of (S(t))t≥0 is an extension of the derivative I (0)
Summary
Let X be a complete metric space and (I (t))t≥0 a family of Lipschitz continuous mappings I (t) : X → X. Formulas of this type are called Chernoff approximation In his monograph [8], Chernoff generalized his previous work [7] and the results by Trotter [33, 34]: under additional stability conditions on the approximating sequence (I (2−nt)2n x)n∈N and the assumption that (I (t))t≥0 is strongly continuous, it was shown in [8, Theorem 2.5.3] that the family (S(t))t>0 is a strongly continuous semigroup of Lipschitz continuous mappings. Compared to the Nisio semigroup, the construction based on the norm convergence as in Eq (1.2) does not rely at all on monotonicity and we do not require (I (t))t≥0 to be the supremum over a family of monotone linear semigroups
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