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**Generation theory for semigroups of holomorphic mappings in Banach spaces.**
*(English)*
Zbl 0945.46026

The paper is a comprehensive investigation devoted to semigroups of holomorphic mappings in Banach spaces. In the first section the authors recall some basic properties of holomorphic mappings in Banach spaces. They also include several known results in the fixed point theory of such mappings. In Section 2 they consider nonlinear semigroups of holomorphic mappings and their infinitesimal operators. They also introduce semi-plus complete vector fields and compare them with infinitesimal generators. They show, in particular, that for bounded holomorphic mappings these two notions coincide. Moreover, it follows that any strongly continuous semigroup with a bounded holomorphic generator is, in fact, continuous with respect to the topology of locally uniform convergence over \(D\). In Section 3 the authors give several geometric and analytic criteria for a Lipschitzian holomorphic mapping to be a generator.

The principal results of the paper are established in Section 4. Theorem 4.1 provides the following characterization of bounded holomorphic generators on a bounded convex domain \(D\) in a Banach space \(X\): A bounded mapping \(f\in \text{Hol}(D, X)\) generates a one-parameter semigroup of holomorphic self-mappings of \(D\) iff for each positive \(r\) its resolvent \((I+ rf)^{-1}\) exists and is a holomorphic self-mapping of \(D\). An important consequence of Theorem 4.1 is that if \(F\) is a holomorphic self-mapping of \(D\), then \(f= I-F\) is a generator of a one parameter semigroup. Thus the well-developed fixed point theory for holomorphic self-mappings can be viewed as a special case of the null point theory of semi-plus complete vector fields. The authors study this subject in Sections 5, 6 and 7. Namely, Section 5 is devoted to the structure of the null point sets of generators and their difference approximations. In Section 6 they study the spectral characteristics of null points. It is shown that such local properties can influence the global structure of the whole null point set and the asymptotic behaviour of the semigroup. Some new sufficient conditions for the existence and uniqueness of null points are presented in Section 7.

Section 8 is devoted to a global version of the implicit function theorem. In particular, Theorem 8.1 is a complete generalization of the uniform fixed point principle in [42]. In the last section several open problems are discussed.

The principal results of the paper are established in Section 4. Theorem 4.1 provides the following characterization of bounded holomorphic generators on a bounded convex domain \(D\) in a Banach space \(X\): A bounded mapping \(f\in \text{Hol}(D, X)\) generates a one-parameter semigroup of holomorphic self-mappings of \(D\) iff for each positive \(r\) its resolvent \((I+ rf)^{-1}\) exists and is a holomorphic self-mapping of \(D\). An important consequence of Theorem 4.1 is that if \(F\) is a holomorphic self-mapping of \(D\), then \(f= I-F\) is a generator of a one parameter semigroup. Thus the well-developed fixed point theory for holomorphic self-mappings can be viewed as a special case of the null point theory of semi-plus complete vector fields. The authors study this subject in Sections 5, 6 and 7. Namely, Section 5 is devoted to the structure of the null point sets of generators and their difference approximations. In Section 6 they study the spectral characteristics of null points. It is shown that such local properties can influence the global structure of the whole null point set and the asymptotic behaviour of the semigroup. Some new sufficient conditions for the existence and uniqueness of null points are presented in Section 7.

Section 8 is devoted to a global version of the implicit function theorem. In particular, Theorem 8.1 is a complete generalization of the uniform fixed point principle in [42]. In the last section several open problems are discussed.

Reviewer: Vassil Angelov (Sofia)

### MSC:

46G20 | Infinite-dimensional holomorphy |

47H20 | Semigroups of nonlinear operators |

47H10 | Fixed-point theorems |

34G20 | Nonlinear differential equations in abstract spaces |