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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On an asymptotic property of a Volterra integral equation

Author: A. F. Izé
Journal: Proc. Amer. Math. Soc. 28 (1971), 93-99
MSC: Primary 45.13; Secondary 34.00
MathSciNet review: 0275078
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Abstract: It is proved that if $q(t - s)$ is bounded and $f(t,x)$ is “small,” the solutions of the integral equation $x(t) = a(t) + \int _0^t {q(t - s)f(s,x(s))ds}$ satisfies the conditions $x(t) = h(t) + \rho (t)a(t),{\lim _{t \to \infty }}a(t) = a$ constant where $\rho (t)$ is a nonsingular diagonal matrix chosen in such a way that $\rho (t)h(t)$ is bounded. The results are extended to the more general integral equation $x(t) = h(t) + \int _0^t {F(t,s,x(s))ds}$ and contain, in particular, some results on the boundedness, asymptotic behavior and existence of nonoscillatory solution of differential equations.

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Keywords: Volterra integral equations, asymptotic properties, measurable set, summable, almost all, global existence, kernel, asymptotic behavior asymptotic equilibrium
Article copyright: © Copyright 1971 American Mathematical Society