On an asymptotic property of a Volterra integral equation

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.