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Cubo (Temuco)

versão On-line ISSN 0719-0646

Cubo vol.14 no.1 Temuco  2012

http://dx.doi.org/10.4067/S0719-06462012000100006 

CUBO A Mathematical Journal Vol.14, N° 01, (55-79). March 2012

Bounded and Periodic Solutions of Integral Equations

T. A. Burton and Bo Zhang

Northwest Research Institute 732 Caroline Street, Port Angeles, WA 98362 email: taburton@olypen.com

Department of Mathematics and Computer Science Fayetteville State University Fayetteville, NC 28301 email: bzhang@uncfsu.edu


ABSTRACT

In this paper we introduce a new method for obtaining boundedness of solutions of integral equations. From the integral equation we form an integrodifferential equation by computing x' + kx to which we apply a Liapunov functional. This can be far more effective than the usual technique of differentiating the equation. The qualitative properties derived from that equation are then transferred to a majorizing function for the integral equation. Schaefer's fixed point theorem is used to conclude that there is a periodic solution. Three kinds of integral equations are studied and they are shown to be related through two examples.

Keywords and Phrases: Integral Equations, Boundedness, Periodic Solutions, Liapunov Functions.


RESUMEN

En este artículo presentamos un nuevo método para obtener acotación de soluciones de ecuaciones integrales. A partir de la ecuación integral, formamos una ecuación integrales diferencial calculando x' + kx mediante la aplicación de un funcional de Liapunov. Ello puede resultar bastante más efectivo que la técnica usual de diferenciación de la ecuación. Las propiedades cualitativas derivadas de la ecuación son entonces transferidas a la función mayorante para la ecuación integral. El teorema del punto fijo de Schaefer es usado para concluir que hay una solución periódica. Se estudia tres tipos de ecuaciones integrales y se muestra que ellas están relacionadas a través de dos ejemplos.

2010 AMS Mathematics Subject Classification: 45D05, 45D20, 45M15.


 

References

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Received: April 2011. Revised: May 2011.