# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Asymptotic behavior and nonoscillation of Volterra integral equations and functional differential equationsHTML articles powered by AMS MathViewer

by A. F. Izé and A. A. Freiria
Proc. Amer. Math. Soc. 52 (1975), 169-177 Request permission

## 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", $x(u(s)) = ({x_1}({u_1}(s)),{x_2}({u_2}(s)), \ldots ,{x_n}({u_n}(s)))$ with ${u_i}(t) \leqslant t$ and ${\lim _{t \to \infty }}{u_i}(t) = \infty$, the solutions of the integral equation $x\left ( t \right ) = h(t) + \int _0^t {q(t,s)f(s,x(s),x(u(s)))ds}$ 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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