Two Tauberian theorems for nonconvolution Volterra integral operators
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- by Gustaf Gripenberg PDF
- Proc. Amer. Math. Soc. 89 (1983), 219-225 Request permission
Abstract:
Two sets of sufficient conditions on the kernel $k(t,s)$ are given so that one can prove that if $x$ is a bounded function such that \[ \lim \limits _{\begin {array}{*{20}{c}} {t \to \infty } \\ {\tau \to 0} \\ \end {array} } \left | {x(t + \tau ) - x(t)} \right | = 0\quad {\text {and}}\quad \lim \limits _{t \to \infty } \int _0^t {k(t,s)x(s)ds} \] exists, then ${\lim _{t \to \infty }}x(t)$ exists.References
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G. Gripenberg, On Volterra equations with nonconvolution kernels, Report-HTKK-MAT-A118, Helsinki Univ. of Technology, 1978.
- Gustaf Gripenberg, A Tauberian problem for a Volterra integral operator, Proc. Amer. Math. Soc. 82 (1981), no. 4, 576–582. MR 614881, DOI 10.1090/S0002-9939-1981-0614881-8
- Richard K. Miller, Nonlinear Volterra integral equations, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. MR 0511193
- David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 89 (1983), 219-225
- MSC: Primary 45D05; Secondary 40E05
- DOI: https://doi.org/10.1090/S0002-9939-1983-0712626-6
- MathSciNet review: 712626