Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments

Abstract

In this paper, we propose the study of an integral equation, with deviating arguments, of the type $y(t)=\mathrm{\omega }(t)-{\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}f(t,s,\mathrm{y}({\gamma }_{\mathrm{1}}(\mathrm{s})),\dots ,\mathrm{y}({\gamma }_N(\mathrm{s})))\mathrm{}ds,\mathrm{}\mathrm{}\mathrm{}\mathrm{}t\ge \mathrm{0},$ 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 $\mathrm{\infty }$ as $\mathrm{\omega }(t)$. A similar equation, but requiring a little less restrictive hypotheses, is $y(t)=\mathrm{\omega }(t)-{\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}q(t,s)\mathrm{}F(s,\mathrm{y}({\gamma }_{\mathrm{1}}(s)),\dots ,\mathrm{y}({\gamma }_N(s)))\mathrm{}ds,\mathrm{}\mathrm{}\mathrm{}\mathrm{}t\ge \mathrm{0}.$ In the case of $q(t,s)=(t-s{)}_{+}$, its solutions with asymptotic behavior given by $\mathrm{\omega }(t)$ yield solutions of the second order nonlinear abstract differential equation $y\text{'}\text{'}(t)-\omega \text{'}\text{'}(t)+F(t,\mathrm{y}({\gamma }_{\mathrm{1}}(t)),\dots ,\mathrm{y}({\gamma }_N(t)))=\mathrm{0},$ with the same asymptotic behavior at $\mathrm{\infty }$ as $\mathrm{\omega }(t)$.