2013 Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments
Cristóbal González, Antonio Jiménez-Melado
Abstr. Appl. Anal. 2013(SI28): 1-7 (2013). DOI: 10.1155/2013/957696

## 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 $\mathrm{\omega }(t)$.

## Citation

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Cristóbal González. Antonio Jiménez-Melado. "Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments." Abstr. Appl. Anal. 2013 (SI28) 1 - 7, 2013. https://doi.org/10.1155/2013/957696

## Information

Published: 2013
First available in Project Euclid: 26 February 2014

zbMATH: 07095536
MathSciNet: MR3147793
Digital Object Identifier: 10.1155/2013/957696

Rights: Copyright © 2013 Hindawi

JOURNAL ARTICLE
7 PAGES

Vol.2013 • No. SI28 • 2013