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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weak conditions for generation of cosine families in linear topological spaces
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by Michiaki Watanabe PDF
Proc. Amer. Math. Soc. 105 (1989), 151-158 Request permission

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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Additional Information
  • © Copyright 1989 American Mathematical Society
  • 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