Weak conditions for generation of cosine families in linear topological spaces

Watanabe, Michiaki

doi:10.1090/S0002-9939-1989-0929411-8

Weak conditions for generation of cosine families in linear topological spaces
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by Michiaki Watanabe

Proc. Amer. Math. Soc. 105 (1989), 151-158

DOI: https://doi.org/10.1090/S0002-9939-1989-0929411-8

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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 $X$: \[ u''\left ( t \right ) = Au\left ( t \right ),\quad t \in R;\quad u\left ( 0 \right ) = {u_0},\quad u’\left ( 0 \right ) = {u_1}\] 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 $A{|_Y}$ 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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