Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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A singular limit problem for a linear Volterra equation


Author: Richard Noren
Journal: Quart. Appl. Math. 46 (1988), 169-179
MSC: Primary 45D05; Secondary 45J05, 45M99, 73F99
DOI: https://doi.org/10.1090/qam/934690
MathSciNet review: 934690
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Abstract: We study the dependence on $ {c_1}$ and $ {c_2}$ of the solution $ u\left( {t, {c_1}, {c_2}} \right)$ of the equation

$\displaystyle u'\left( t \right) + \int_0^t A \left( {t - s, {c_1}, {c_2}} \right)u\left( s \right)ds = 0, \qquad u\left( 0 \right) = 1,$

where the conditions on $ A$ are stated in terms of its Fourier transform. We obtain sufficient conditions and (weaker) necessary conditions for

$\displaystyle \int_0^\infty {\mathop {\sup }\limits_{0 \le {c_i} \le 1} \left\vert {u\left( {t, {c_1}, {c_2}} \right)} \right\vert dt} < \infty , i = 1,2$

and for

$\displaystyle \int_0^\infty {\mathop {\sup }\limits_{0 \le {c_1},{c_2} \le 1} } \left\vert {u\left( {t, {c_1}, {c_2}} \right)} \right\vert dt < \infty $

The kernel $ A$ is a combination of nonnegative nonincreasing convex functions and arises in the linear theory of viscoelastic rods and plates.

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DOI: https://doi.org/10.1090/qam/934690
Article copyright: © Copyright 1988 American Mathematical Society


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