A Volterra equation with square integrable solution

Staffans, Olof J.

doi:10.1090/S0002-9939-1980-0550496-7

A Volterra equation with square integrable solution
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by Olof J. Staffans

Proc. Amer. Math. Soc. 78 (1980), 213-217

DOI: https://doi.org/10.1090/S0002-9939-1980-0550496-7

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Abstract:

We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation \[ x’(t) + \int _0^t {a(t - s)g(x(s))\;ds\; = f(t)\quad (t \in {R^ + }).} \] Here ${R^ + } = [0,\infty ),a,g$ and f are given real functions, and x is the unknown solution. In particular, we give sufficient conditions which imply that x and x’ are square integrable.

References

R. C. MacCamy, An integro-differential equation with application in heat flow, Quart. Appl. Math. 35 (1977/78), no. 1, 1–19. MR 452184, DOI 10.1090/S0033-569X-1977-0452184-2
R. C. MacCamy, A model for one-dimensional, nonlinear viscoelasticity, Quart. Appl. Math. 35 (1977/78), no. 1, 21–33. MR 478939, DOI 10.1090/S0033-569X-1977-0478939-6
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
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
Olof J. Staffans, An inequality for positive definite Volterra kernels, Proc. Amer. Math. Soc. 58 (1976), 205–210. MR 500049, DOI 10.1090/S0002-9939-1976-0500049-0
Olof J. Staffans, On the asymptotic spectra of the bounded solutions of a nonlinear Volterra equation, J. Differential Equations 24 (1977), no. 3, 365–382. MR 463855, DOI 10.1016/0022-0396(77)90006-7
Olof J. Staffans, Boundedness and asymptotic behavior of solutions of a Volterra equation, Michigan Math. J. 24 (1977), no. 1, 77–95. MR 463861
Olof J. Staffans, A nonlinear Volterra integral equation with square integrable solutions, Volterra equations (Proc. Helsinki Sympos. Integral Equations, Otaniemi, 1978) Lecture Notes in Math., vol. 737, Springer, Berlin, 1979, pp. 281–286. MR 551047
Olof J. Staffans, A nonlinear Volterra equation with rapidly decaying solutions, Trans. Amer. Math. Soc. 258 (1980), no. 2, 523–530. MR 558188, DOI 10.1090/S0002-9947-1980-0558188-X
Olof J. Staffans, On a nonlinear hyperbolic Volterra equation, SIAM J. Math. Anal. 11 (1980), no. 5, 793–812. MR 586908, DOI 10.1137/0511071