On the convergence of solutions of Volterra equations to almost-periodic functions
Author:
Gustaf Gripenberg
Journal:
Quart. Appl. Math. 39 (1981), 363-373
MSC:
Primary 45G10; Secondary 45D05
DOI:
https://doi.org/10.1090/qam/636241
MathSciNet review:
636241
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Abstract: The rate of convergence of solutions of a certain Volterra integral equation and a system of two Volterra equations to almost-periodic limit functions is studied. The equations considered arise from some diffusion problems with nonlinear and almost-periodic boundary conditions.
- C. Corduneanu, Almost periodic functions, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1968. With the collaboration of N. Gheorghiu and V. Barbu; Translated from the Romanian by Gitta Bernstein and Eugene Tomer; Interscience Tracts in Pure and Applied Mathematics, No. 22. MR 0481915
- William F. Donoghue Jr., Distributions and Fourier transforms, Pure and Applied Mathematics, vol. 32, Academic Press, New York, 1969. MR 3363413
- Avner Friedman, Periodic behavior of solutions of Volterra integral equations, J. Analyse Math. 15 (1965), 287–303. MR 187035, DOI https://doi.org/10.1007/BF02787698
- Gustaf Gripenberg, On Volterra equations of the first kind, Integral Equations Operator Theory 3 (1980), no. 4, 473–488. MR 595747, DOI https://doi.org/10.1007/BF01702311
- I. I. Hirschman and D. V. Widder, The convolution transform, Princeton University Press, Princeton, N. J., 1955. MR 0073746
- M. J. Leitman and V. J. Mizel, Asymptotic stability and the periodic solutions of $x(t)+\int ^t_{-\infty }a(t-s)g(s,x(s))ds=f(t)$, J. Math. Anal. Appl. 66 (1978), no. 3, 606–625. MR 517750, DOI https://doi.org/10.1016/0022-247X%2878%2990257-3
N. Levinson, A nonlinear Volterra equation arising in the theory of superfluidity, J. Math. Anal. Appl. 1, 1–11 (1960)
C. C. Lin, Hydrodynamics of liquid helium II, Phys. Rev. Letters 2, 245–246 (1959)
- Richard K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. Mathematics Lecture Note Series. MR 0511193
- R. K. Miller, Almost-periodic behavior of solutions of a nonlinear Volterra system, Quart. Appl. Math. 28 (1971), 553–570. MR 271670, DOI https://doi.org/10.1090/S0033-569X-1971-0271670-5
- R. K. Miller, Asymptotically almost periodic solutions of a nonlinear Volterra system, SIAM J. Math. Anal. 2 (1971), 435–444. MR 296618, DOI https://doi.org/10.1137/0502042
- Dennis G. Weis, Asymptotic behavior of some nonlinear Volterra integral equations, J. Math. Anal. Appl. 49 (1975), 59–87. MR 367596, DOI https://doi.org/10.1016/0022-247X%2875%2990162-6
- Dennis G. Weis, Asymptotic behavior of integral equations using monotonicity, J. Math. Anal. Appl. 54 (1976), no. 1, 49–58. MR 493208, DOI https://doi.org/10.1016/0022-247X%2876%2990234-1
C. Corduneanu, Almost periodic functions, Interscience Publishers, New York, 1968
W. F. Donoghue, Distributions and Fourier transforms, Academic Press, New York, 1969
A. Friedman, Periodic behavior of solutions of Volterra integral equations, J. Analyse Math. 15, 287–303 (1965)
G. Gripenberg, On Volterra equations of the first kind, Integ. Eqs. Operator Th. (to appear)
I. I. Hirschman and D. V. Widder, The convolution transform, Princeton University Press, Princeton, 1955
M. J. Leitman and V. J. Mizel, Asymptotic stability and the periodic solutions of $x\left ( t \right ) + \\ \int _{ - \infty }^t {a\left ( {t - s} \right )g\left ( {s, x\left ( s \right )} \right ) ds = f\left ( t \right )}$, J. Math. Anal. Appl. 66, 606–625 (1978)
N. Levinson, A nonlinear Volterra equation arising in the theory of superfluidity, J. Math. Anal. Appl. 1, 1–11 (1960)
C. C. Lin, Hydrodynamics of liquid helium II, Phys. Rev. Letters 2, 245–246 (1959)
R. K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Menlo Park, 1971
R. K. Miller, Almost periodic behavior of solutions of a nonlinear Volterra system, Quart. Appl. Math. 28, 553–570 (1971)
R. K. Miller, Asymptotically almost periodic solutions of a nonlinear Volterra system, SIAM J. Math. Anal. 2, 435–444 (1971)
D. G. Weis, Asymptotic behavior of some nonlinear Volterra integral equations, J. Math. Anal. Appl. 49, 59–87 (1975)
D. G. Weis, Asymptotic behavior of integral equations using monotonicity, J. Math. Anal. Appl. 54, 49–58 (1976)
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Article copyright:
© Copyright 1981
American Mathematical Society