Nonlinear Semigroups Built on Generating Families and their Lipschitz Sets

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 $S(t):=\lim _{n\to \infty }(I(t 2^{-n}))^{2^{n}}$ S ( t ) : = lim n → ∞ ( 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 $\lim _{h\downarrow 0}\tfrac {I(h)x-x}{h}$ lim h ↓ 0 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.

Convex monotone semigroups and their generators with respect to $Γ$-convergence

“…However, in several examples, the symmetric Lipschitz set is invariant and can, in contrast to L S + and L S , be determined explicitly. This leads to regularity results for the associated semigroup, see [7,8] and Subsection 5.1.…”

“…The next statement is a consequence of the results in [8] and Appendix C. For the reader's convenience, we provide a proof in Appendix D. Furthermore, S satisfies Assumption 2.4 and is a strongly continuous convex monotone semigroup on L S . It holds L I ⊂ L S .…”

“…Furthermore, they can be used in order to construct nonlinear semigroups, see [8]. In this case, we call I a generating family and S an associated semigroup.…”

“…However, condition (vi), which implies continuity from above, also guarantees the semigroup property. Another possibility to ensure the semigroup property is to assume norm convergence or monotone convergence, see [8,23,34]. Remark 4.2.…”

“…Furthermore, under mild assumptions, one can show that S(t)f = sup π∈Pt I(π)f , where P t consists of all finite partitions of the interval [0.t]. For details we refer to [8,Lemma 2.15].…”