Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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.

References [Enhancements On Off] (What's this?)

  • [1] D. R. Bland, The theory of linear viscoelasticity, Pergamon Press, New York, 1960 MR 0110314
  • [2] R. W. Carr and K. B. Hannsgen, A nonhomogeneous integrodifferential equation in Hilbert space, SIAM J. Math. Anal. 10, 961-984 (1979) MR 541094
  • [3] K. B. Hannsgen, A linear integrodifferential equation for viscoelastic rods and plates, Quart. Appl. Math. 41, 75-83 (1983) MR 700662
  • [4] K. B. Hannsgen and R. L. Wheeler, A singular limit problem for an integrodifferential equation, J. of Integral Equations 5, 199-209 (1983) MR 702431
  • [5] R. D. Noren, A linear Volterra integrodifferential equation for viscoelastic rods and plates, Quart. Appl. Math. 45, 503-514 (1987) MR 910457
  • [6] R. D. Noren, A singular limit problem for a Volterra equation, SIAM J. Math. Anal., to appear MR 957669
  • [7] D. F. Shea and S. Wainger, Variants of the Wiener-Levy theorem, with applications to stability problems for some Volterra integral equations, Amer. J. Math. 97, 312-343 (1975) MR 0372521

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Additional Information

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

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