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

Cristóbal González and Antonio Jiménez-Melado

Abstract and Applied Analysis, 2013, vol. 2013, issue 1

Abstract: 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).

Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:957696

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