Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces

Continuous and holomorphic semicocycles in Banach spaces

“…A bounded subset D * ⊂ D is said to lie strictly inside D if it is bounded away from the boundary ∂D, that is, inf x∈D * dist(x, ∂D) > 0. One of the surprising features of infinite-dimensional holomorphy is that the inclusion f ∈ Hol(D, Y ) does not imply that f is bounded on all subsets D * strictly inside D (see [13,19,17,18]).…”

“…One of the deep questions in semigroup theory is whether all semigroups are differentiable with respect to the parameter t. In general, the answer is negative. Reich and Shoikhet (see [19,Theorems 6.8-6.9]) proved the following criterion for differentiability. Theorem 2.1.…”

“…It follows from the Cauchy inequality (see [19,Proposition 2.3 ]) that if M is bounded on each subset strictly inside D then the same holds for M ′ . Therefore the last formula implies that B also is bounded on each subset strictly inside D. Hence by Theorem 2.3, {Γ t } t≥0 is T -continuous.…”

“…In this paper we study semicocycles over semigroups of holomorphic selfmappings of a domain in a complex Banach space (the books [19,11] and references therein can be used as good sources for the state of the art on semigroups of holomorphic mappings). While, under an assumption of uniform joint continuity (see [9]) such semicocycles can be defined as solutions of certain nonautonomous differential equations which cannot be explicitly solved, if a semicocycle is cohomolgous to a constant one it has an explicit representation.…”

“…In multi-dimensional settings, the situation is inherently more complicated, as is also the case with respect to the question of conjugacy of semigroups (see, for example, [2,10,19]). We obtain some simple sufficient conditions for such a linearization to exist, in terms of the semicocycle generator.…”