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Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.17 no.2 Temuco June 2015 

On a Type of Volterra Integral Equation in the Space of Continuous Functions with Bounded Variation valued in Banach Spaces


Hugo Leiva* & Jesús Matute*,1 y Nelson Merentes** & José Sánchez**.
* Dpto. de Matemáticas, Universidad de Los Andes, La Hechicera. Mérida 5101. Venezuela.,
** Escuela de Matemáticas, Universidad Central de Venezuela, Caracas. Venezuela.,
1 corresponding author

In this paper we prove existence and uniqueness of the solutions for a kind of Volterra equation, with an initial condition, in the space of the continuous functions with bounded variation which take values in an arbitrary Banach space. Then we give a parameters variation formula for the solutions of certain kind of linear integral equation. Finally, we prove exact controllability of a particular integral equation using that formula. Moreover, under certain condition, we find a formula for a control steering of a type of system which is studied in the current work, from an initial state to a final one in a prescribed time.

En este trabajo probamos existencia y unicidad de las soluciones para una ecuación de Volterra, con condición inicial, en el espacio de funciones continuas con variación acotada y valores en un espacio de Banach arbitrario. Damos una formula de variación de parámetros para las soluciones de cierta clase de ecuación lineal integral. Finalmente probamos la controlabilidad exacta de una ecuación integral particular usando esa formula. Más aún, bajo cierta condición, encontramos una formula para una dirección de control de un tipo de Sistema que se estudia en el presente trabajo, desde un estado inicial a uno final en un tiempo prescrito.

Keywords and Phrases: Existence and uniqueness of solutions of integral equations in Banach spaces; continuous functions; bounded variation norm; parameters variation formula; controllability.
2010 AMS Mathematics Subject Classification: 26B30, 34A12, 45D99, 45N05.


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Received: September 2013. Accepted: February 2015.

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