Asymptotic behavior of solutions to a vector integral equation with deviating arguments (Q2319312)

Summary: In this paper, we propose the study of an integral equation, with deviating arguments, of the type \(y(t) = \omega(t) - \int_0^{\infty} f(t, s, y(\gamma_1(s)), \ldots, y(\gamma_N(s))) d s\), \(t \geq 0\), in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at \(\infty\) as \(\omega(t)\). A similar equation, but requiring a little less restrictive hypotheses, is \(y(t) = \omega(t) - \int_0^{\infty} q(t, s) F(s, y(\gamma_1(s)), \ldots, y(\gamma_N(s))) d s, t \geq 0 \). In the case of \(q(t, s) = (t - s)_+\), its solutions with asymptotic behavior given by \(\omega(t)\) yield solutions of the second order nonlinear abstract differential equation \(y''(t) - \omega''(t) + F(t, y(\gamma_1(t)), \ldots, y(\gamma_N(t))) = 0\), with the same asymptotic behavior at \(\infty\) as \(\omega(t)\).