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

In this paper, we propose the study of an integral equation, with deviating arguments, of the type y(t)=ω(t)-∫0∞f(t,s,y(γ1(s)),…,y(γN(s)))ds,t≥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 ∞ as ω(t). A similar equation, but requiring a little less restrictive hypotheses, is y(t)=ω(t)-∫0∞q(t,s)F(s,y(γ1(s)),…,y(γN(s)))ds,t≥0. In the case of q(t,s)=(t-s)+, its solutions with asymptotic behavior given by ω(t) yield solutions of the second order nonlinear abstract differential equation y''(t)-ω''(t)+F(t,y(γ1(t)),…,y(γN(t)))=0, with the same asymptotic behavior at ∞ as ω(t)