A Tauberian problem for a Volterra integral operator

Gripenberg, Gustaf

doi:10.1090/S0002-9939-1981-0614881-8

A Tauberian problem for a Volterra integral operator

Author: Gustaf Gripenberg
Journal: Proc. Amer. Math. Soc. 82 (1981), 576-582
MSC: Primary 45D05; Secondary 40E05
DOI: https://doi.org/10.1090/S0002-9939-1981-0614881-8
MathSciNet review: 614881
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Abstract: The following question is studied: For which (nonintegrable) kernels does ${\lim _{t \to \infty }}\int _0^tA(t - s)x(s)ds = 0$ imply that ${\lim _{t \to \infty }}x(t) = 0$ when is bounded and satisfies a Tauberian condition.

[1] W. F. Donoghue, Distributions and Fourier transforms, Academic Press, New York, 1969.
[2] Gustaf Gripenberg, A Volterra equation with nonintegrable resolvent, Proc. Amer. Math. Soc. 73 (1979), no. 1, 57–60. MR 512058, https://doi.org/10.1090/S0002-9939-1979-0512058-9
[3] Gustaf Gripenberg, Integrability of resolvents of systems of Volterra equations, SIAM J. Math. Anal. 12 (1981), no. 4, 585–594. MR 617717, https://doi.org/10.1137/0512051
[4] G. S. Jordan and Robert L. Wheeler, A generalization of the Wiener-Lévy theorem applicable to some Volterra equations, Proc. Amer. Math. Soc. 57 (1976), no. 1, 109–114. MR 0405023, https://doi.org/10.1090/S0002-9939-1976-0405023-0
[5] J. J. Levin and D. F. Shea, On the asymptotic behavior of the bounded solutions of some integral equations. I, II, III, J. Math. Anal. Appl. 37 (1972), 42–82; ibid. 37 (1972), 288–326; ibid. 37 (1972), 537–575. MR 0306849, https://doi.org/10.1016/0022-247X(72)90258-2
[6] Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142
[7] Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043
[8] Daniel F. Shea and Stephen Wainger, Variants of the Wiener-Lévy theorem, with applications to stability problems for some Volterra integral equations, Amer. J. Math. 97 (1975), 312–343. MR 0372521, https://doi.org/10.2307/2373715
[9] O. J. Staffans, On the integrability of the resolvent of a Volterra equation, Report-HTKK-MAT-A103, Helsinki University of Technology, Espoo, Finland, 1977.
[10] D. V. Widder, The Laplace transform, Princeton Univ. Press, Princeton, N.J., 1946.