Resolvents and bounds for linear and nonlinear Volterra equations

Levin, J. J.

doi:10.1090/S0002-9947-1977-0433162-2

Resolvents and bounds for linear and nonlinear Volterra equations
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by J. J. Levin

Trans. Amer. Math. Soc. 228 (1977), 207-222

DOI: https://doi.org/10.1090/S0002-9947-1977-0433162-2

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Abstract:

The asymptotic behavior of the resolvent of a linear Volterra equation is investigated without the assumption that the kernel of the equation is in ${L^1}(0,\infty )$. A lower bound is obtained on the solutions of a related nonlinear Volterra equation. A special case of the latter result is employed in the proof of the former result.

References

Avner Friedman, On integral equations of Volterra type, J. Analyse Math. 11 (1963), 381–413. MR 158232, DOI 10.1007/BF02789991
S. I. Grossman and R. K. Miller, Nonlinear Volterra integrodifferential systems with $L^{1}$-kernels, J. Differential Equations 13 (1973), 551–566. MR 348417, DOI 10.1016/0022-0396(73)90011-9
J. J. Levin, A bound on the solutions of a Volterra equation, Arch. Rational Mech. Anal. 52 (1973), 339–349. MR 336269, DOI 10.1007/BF00247468
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). MR 306849, DOI 10.1016/0022-247X(72)90258-2
R. K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl. 22 (1968), 319–340. MR 227707, DOI 10.1016/0022-247X(68)90176-5
R. K. Miller, J. A. Nohel, and J. S. W. Wong, Perturbations of Volterra integral equations, J. Math. Anal. Appl. 25 (1969), 676–691. MR 240573, DOI 10.1016/0022-247X(69)90265-0
John A. Nohel, Asymptotic relationships between systems of Volterra equations, Ann. Mat. Pura Appl. (4) 90 (1971), 149–165. MR 310575, DOI 10.1007/BF02415046
John A. Nohel, Asymptotic equivalence of Volterra equations, Ann. Mat. Pura Appl. (4) 96 (1972), 339–347. MR 336267, DOI 10.1007/BF02414847
Komarath Padmavally, On a non-linear integral equation, J. Math. Mech. 7 (1958), 533–555. MR 0103400, DOI 10.1512/iumj.1958.7.57032
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, DOI 10.1090/coll/019
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 372521, DOI 10.2307/2373715