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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Singular perturbations of integrodifferential equations in Banach space


Author: James H. Liu
Journal: Proc. Amer. Math. Soc. 122 (1994), 791-799
MSC: Primary 45M05; Secondary 34E15, 34K30
MathSciNet review: 1287101
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Abstract: Let $ \varepsilon > 0$ and consider

\begin{displaymath}\begin{array}{*{20}{c}} {{\varepsilon ^2}u''(t;\varepsilon ) ... ...uad u' (0;\varepsilon ) = {u_1}(\varepsilon ),} \\ \end{array} \end{displaymath}

and

$\displaystyle w' (t) = Aw(t) + \int_0^t {K(t - s)Aw(s)\,ds + f(t)\quad t \geq 0,w(0) = {w_0},} $

in a Banach space X when $ \varepsilon \to 0$. Here A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and $ K(t)$ is a bounded linear operator for $ t \geq 0$. With some convergence conditions on initial data and $ f(t;\varepsilon )$ and smoothness conditions on $ K( \bullet )$, we prove that if $ \varepsilon \to 0$, then $ u(t;\varepsilon ) \to w(t)$ in X uniformly for $ t \in [0,T]$ for any fixed $ T > 0$. We will apply this to an equation in viscoelasticity.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1287101-0
PII: S 0002-9939(1994)1287101-0
Keywords: Singular perturbation, convergence in solutions
Article copyright: © Copyright 1994 American Mathematical Society