On the Asymptotic Behavior of Perturbed Volterra Integral Equations

The asymptotic behavior of the solution of the perturbed system of Volterra integral equations $x(t) = f(t) - \int _0^t a(t,s)\{ x(s) + g(s,x(s))\} ds$ is compared to that of the solution of the unperturbed system $y(t) = f(t) - \int _0^t a(t,s)y(s)ds$. We show that under suitable restrictions on th...

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Bibliographic Details
Published inSIAM journal on mathematical analysis Vol. 5; no. 2; pp. 273 - 277
Main Authors Jordan, G. S., Wheeler, Robert L.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.04.1974
Subjects
Euclidean space
Hypotheses
Integral equations
Laplace transforms
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Summary:The asymptotic behavior of the solution of the perturbed system of Volterra integral equations $x(t) = f(t) - \int _0^t a(t,s)\{ x(s) + g(s,x(s))\} ds$ is compared to that of the solution of the unperturbed system $y(t) = f(t) - \int _0^t a(t,s)y(s)ds$. We show that under suitable restrictions on the resolvent kernel $r(t,s)$, $|x(t) - y(t)| \to 0$ as $t \to \infty $ whenever $|g(t,x)| \leqq \lambda (t)(1 + |x|)$ with $\lambda $ bounded and diminishing. This establishes and generalizes a recent conjecture of J. L. Kaplan.
ISSN:0036-1410
1095-7154
DOI:10.1137/0505029