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Generators of nonlinear semigroups and Lie-Trotter formula. (Générateur des semi-groupes non linéaires et la formule de Lie-Trotter.) (French) Zbl 0565.47046

Let S(t) be a continuous semigroup of nonlinear contractions on a Banach space X. We show that S(t) has a graph infinitesimal generator that is an m-accretive operator A with dense domain such that \[ \forall y\in Ax,\quad \exists (x_ t)_{t>0}\subset X,\quad \lim_{t\to 0}\| x_ t-x\| +\| \frac{x_ t-S(t)x_ t}{t} (break?) \| =0 \] if and only if for any \(B: X\to X\) continuous accretive, the Lie-Trotter formula \(\lim_{n\to \infty}(S(\frac{t}{n})S^ B(\frac{t}{n}))^ nx\) converges for any \(x\in X\) uniformly for \(t\geq 0\) bounded, where \(S^ B(t)\) is the semigroup generated by B.

MSC:

47H20 Semigroups of nonlinear operators
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References:

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