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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
DOI: https://doi.org/10.1090/S0002-9939-1971-0275078-0
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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DOI: https://doi.org/10.1090/S0002-9939-1971-0275078-0
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