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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic behavior and nonoscillation of Volterra integral equations and functional differential equations


Authors: A. F. Izé and A. A. Freiria
Journal: Proc. Amer. Math. Soc. 52 (1975), 169-177
MSC: Primary 34K15; Secondary 45M10
MathSciNet review: 0377233
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Abstract: It is proved that if $ {q_{ij}}(t,s){\rho _j}(s){[{\rho _i}(t)]^{ - 1}}$ is bounded, $ i,j = 1,2, \ldots ,n$, and $ f(t,x,x(u(s)))$ is ``small",

$\displaystyle x(u(s)) = ({x_1}({u_1}(s)),{x_2}({u_2}(s)), \ldots ,{x_n}({u_n}(s)))$

with $ {u_i}(t) \leqslant t$ and $ {\lim _{t \to \infty }}{u_i}(t) = \infty $, the solutions of the integral equation

$\displaystyle x\left( t \right) = h(t) + \int_0^t {q(t,s)f(s,x(s),x(u(s)))ds} $

satisfy the conditions $ x(t) = h(t) + \rho (t)a(t),{\lim _{t \to \infty }}a(t) = $ constant where $ \rho (t)$ is a nonsingular diagonal matrix chosen in such a way that $ {\rho ^{ - 1}}(t)h(t)$ is bounded. The results contain, in particular, some results on the asymptotic behavior, stability and existence of nonoscillatory solutions of functional differential equations.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0377233-1
PII: S 0002-9939(1975)0377233-1
Keywords: Volterra integral equations, asymptotic properties, almost all, kernel, asymptotic behavior, uniform stability, uniform asymptotic stability, nonoscillation, functional differential equation globally bounded
Article copyright: © Copyright 1975 American Mathematical Society