Tauberian theorems for a positive definite form, with applications to a Volterra equation

Staffans, Olof J.

doi:10.1090/S0002-9947-1976-0422936-9

Tauberian theorems for a positive definite form, with applications to a Volterra equation
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Trans. Amer. Math. Soc. 218 (1976), 239-259 Request permission

Abstract:

We study the relation between the condition \[ \sup \limits _{T > 0} \int _{[0,T]} {\bar \varphi (t)} \int _{[t - T,t]} {\varphi (t - s)\;dv (s)\;dt < \infty } \] and the asymptotic behavior of the bounded function $\varphi$ when $\nu$ is a positive definite measure. Earlier we have proved that if $\nu$ is strictly positive definite and $\varphi$ satisfies a tauberian condition, then $\varphi (t) \to 0$ as $t \to \infty$. Here we characterize the spectrum of the limit set of $\varphi$ in the case when $\nu$ is not strictly positive definite. Applying this theory to a nonlinear Volterra equation we get some new results on the asymptotic behavior of its bounded solutions.

References

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
A. Halanay, On the asymptotic behavior of the solutions of an integro-differential equation, J. Math. Anal. Appl. 10 (1965), 319–324. MR 176304, DOI 10.1016/0022-247X(65)90126-5
Kenneth B. Hannsgen, On a nonlinear Volterra equation, Michigan Math. J. 16 (1969), 365–376. MR 249984
Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
J. J. Levin, On a nonlinear Volterra equation, J. Math. Anal. Appl. 39 (1972), 458–476. MR 304994, DOI 10.1016/0022-247X(72)90217-X
Jacob J. Levin, On some geometric structures for integrodifferential equations, Advances in Math. 22 (1976), no. 2, 146–186. MR 438063, DOI 10.1016/0001-8708(76)90152-3
J. J. Levin and J. A. Nohel, Note on a nonlinear Volterra equation, Proc. Amer. Math. Soc. 14 (1963), 924–929. MR 157201, DOI 10.1090/S0002-9939-1963-0157201-7
J. J. Levin and J. A. Nohel, On a nonlinear delay equation, J. Math. Anal. Appl. 8 (1964), 31–44. MR 163142, DOI 10.1016/0022-247X(64)90080-0
J. J. Levin and J. A. Nohel, Perturbations of a nonlinear Volterra equation, Michigan Math. J. 12 (1965), 431–447. MR 182854, DOI 10.1307/mmj/1028999427
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
Stig-Olof Londen, The qualitative behavior of the solutions of a nonlinear Volterra equation, Michigan Math. J. 18 (1971), 321–330. MR 293354
Stig-Olof Londen, On a nonlinear Volterra integral equation, J. Differential Equations 14 (1973), 106–120. MR 340995, DOI 10.1016/0022-0396(73)90080-6
Stig-Olof Londen, On the variation of the solutions of a nonlinear integral equation, J. Math. Anal. Appl. 52 (1975), no. 3, 430–449. MR 420177, DOI 10.1016/0022-247X(75)90070-0
Richard C. MacCamy, Nonlinear Volterra equations on a Hilbert space, J. Differential Equations 16 (1974), 373–393. MR 377605, DOI 10.1016/0022-0396(74)90021-7
R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc. 164 (1972), 1–37. MR 293355, DOI 10.1090/S0002-9947-1972-0293355-X
J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations, Advances in Math. 22 (1976), no. 3, 278–304. MR 500024, DOI 10.1016/0001-8708(76)90096-7
H. L. Royden, Real analysis, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1963. MR 0151555
Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. MR 0344043
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
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
Olof J. Staffans, Nonlinear Volterra integral equations with positive definite kernels, Proc. Amer. Math. Soc. 51 (1975), 103–108. MR 370081, DOI 10.1090/S0002-9939-1975-0370081-8
Olof J. Staffans, Positive definite measures with applications to a Volterra equation, Trans. Amer. Math. Soc. 218 (1976), 219–237. MR 458086, DOI 10.1090/S0002-9947-1976-0458086-5