TY - JOUR

T1 - Geometric properties of the nonlinear resolvent of holomorphic generators

AU - Elin, Mark

AU - Shoikhet, David

AU - Sugawa, Toshiyuki

N1 - Funding Information:
T. S. was supported in part by JSPS KAKENHI Grant Number JP17H02847.
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2020/3/15

Y1 - 2020/3/15

N2 - Let f be the infinitesimal generator of a one-parameter semigroup of holomorphic self-mappings of the open unit disk Δ. Our main purpose is to study properties of the family R of non-linear resolvents (I+rf)−1:Δ→Δ,r≥0, in the spirit of classical geometric function theory. To make a connection with this theory, we mostly consider the case where f(0)=0 and f′(0) is a positive real number. We found, in particular, that R forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of R satisfies the Noshiro-Warschawski condition. This, in turn, implies that each element of R is also the infinitesimal generator of a one-parameter semigroup on Δ. We consider also quasiconformal extensions of elements of R. Finally we study the existence of repelling fixed points of this family.

AB - Let f be the infinitesimal generator of a one-parameter semigroup of holomorphic self-mappings of the open unit disk Δ. Our main purpose is to study properties of the family R of non-linear resolvents (I+rf)−1:Δ→Δ,r≥0, in the spirit of classical geometric function theory. To make a connection with this theory, we mostly consider the case where f(0)=0 and f′(0) is a positive real number. We found, in particular, that R forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of R satisfies the Noshiro-Warschawski condition. This, in turn, implies that each element of R is also the infinitesimal generator of a one-parameter semigroup on Δ. We consider also quasiconformal extensions of elements of R. Finally we study the existence of repelling fixed points of this family.

KW - Boundary regular fixed point

KW - Hyperbolically convex

KW - Inverse Löwner chain

KW - Nonlinear resolvent

KW - Semigroup generator

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U2 - 10.1016/j.jmaa.2019.123614

DO - 10.1016/j.jmaa.2019.123614

M3 - Article

AN - SCOPUS:85075375970

SN - 0022-247X

VL - 483

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

M1 - 123614

ER -