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Weak conditions for generation of cosine families in linear topological spaces


Author: Michiaki Watanabe
Journal: Proc. Amer. Math. Soc. 105 (1989), 151-158
MSC: Primary 47D05; Secondary 34G10, 35L99
DOI: https://doi.org/10.1090/S0002-9939-1989-0929411-8
MathSciNet review: 929411
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Abstract: Let $ A$ be a closed linear operator in a Banach space $ X$. Weak conditions are found under which (I) the abstract Cauchy problem in $ X$:

$\displaystyle u''\left( t \right) = Au\left( t \right),\quad t \in R;\quad u\left( 0 \right) = {u_0},\quad u'\left( 0 \right) = {u_1}$

has a unique solution for each $ {u_0}$ and $ {u_1}$ given in a dense subset $ Y$ of $ X$, and (II) the set $ Y$ becomes a linear topological space where $ A{\vert _Y}$ generates a continuous cosine family.

The conditions are satisfied for example by the generator of a strongly continuous or holomorphic semigroup in $ X$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0929411-8
Keywords: Evolution equation of second order, cosine family, abstract Gevrey space
Article copyright: © Copyright 1989 American Mathematical Society

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