Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Uniform $ L^1$ behavior in a class of linear Volterra equations


Author: Richard Noren
Journal: Quart. Appl. Math. 47 (1989), 547-554
MSC: Primary 45J05; Secondary 45D05, 45M05, 45N05
DOI: https://doi.org/10.1090/qam/1012278
MathSciNet review: MR1012278
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Abstract: We find sufficient conditions for the solution of the equation $ u'\left( t \right) + \\ \int_0^t {\sum\nolimits_{i = 1}^n {{\lambda _i}{a_i}\left( {t - s} \right)u\left( s \right)ds = 0, u\left( 0 \right) = 1} } $, to satisfy $ \int_0^\infty {{{\sup }_{{\lambda _1},...,{\lambda _n} \ge 1}}{{\left( {{\lamb... ...2}} \times \\ u'\left( {t, {\lambda _1},...,{\lambda _n}} \right)dt < \infty } $. Our results generalize the case $ n = 1$. Applications to a related equation in Hilbert space are given.


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DOI: https://doi.org/10.1090/qam/1012278
Article copyright: © Copyright 1989 American Mathematical Society


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