Abstract and Applied Analysis

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

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 957696, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393449713

Digital Object Identifier
doi:10.1155/2013/957696

Mathematical Reviews number (MathSciNet)
MR3147793

Zentralblatt MATH identifier
07095536

Citation

González, Cristóbal; Jiménez-Melado, Antonio. Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 957696, 7 pages. doi:10.1155/2013/957696. https://projecteuclid.org/euclid.aaa/1393449713