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Entire solutions of the abstract Cauchy problem. (English) Zbl 0746.47018
In the past several generalizations of strongly continuous one parameter semigroups and their infinitesimal generators were introduced in connection with Cauchy-problems, e.g. polynomials of generators, integrated semigroups, operator matrices and \(C\)- semigroups. A \(C\)- semigroup, \(C\) a bounded injective linear operator on a Banach space, is a continuous family \((W(t),\;t\geq 0)\), such that \(W(0)=C\) and fulfilling the functional equation \(W(t)W(s)=C\cdot W(t+s)\). The author defines in a similar way \(C\)-groups and entire \(C\)-groups in order to obtain a new and unified approach to Cauchy problems.
The first chapters are concerned with definitions, general properties and examples of entire \(C\)-groups and their generators.
Chapter III is concerned with first order problems and \(C\)-groups defined by polynomials of generators of holomorphic semigroups.
In Chapter IV the author considers second order problems, matrices of operators and representation of the solutions by \(C\)-groups.
In Chapter V models for elastic systems with damping are treated and Chapter VI is concerned with examples (heat equation, Laplace equation) which can be treated in this unified way.
Reviewer: W.Hazod (Dortmund)

MSC:
47D06 One-parameter semigroups and linear evolution equations
47D09 Operator sine and cosine functions and higher-order Cauchy problems
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