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Metric domains, holomorphic mappings and nonlinear semigroups. (English) Zbl 0993.47040

This article deals with nonlinear semigroups of holomorphic mappings in a certain domain \(D\) in a complex Banach space \(X\). The authors prove the existence of the infinitesimal generator for such a semigroup and that this generator is a holomorphic mapping from \(D\) into \(X\), present some natural exponential formulas connecting a semigroup and its generator, describe conditions under which a given operator \(f\) is the generator of a semigroup of holomorphic mappings, investigate relations between properties of a semigroup of holomorphic mappings and the corresponding Cauchy problem, construct holomorphic retractions onto the stationary point sets of such semigroups.

MSC:

47H20 Semigroups of nonlinear operators
46G20 Infinite-dimensional holomorphy
34G20 Nonlinear differential equations in abstract spaces
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