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. hleiva@ula.ve, jmatute@ula.ve.
^** Escuela de Matemáticas, Universidad Central de Venezuela, Caracas. Venezuela. nmerucv@gmail.com, casanay085@hotmail.com.
¹ corresponding author

ABSTRACT
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.

RESUMEN
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.

References
[1] G. Astorga y L. Barbanti; Ecuaciones de evolución como ecuaciones integrales, Revista de la Facultad de Ingeniería. www.ingenieria.uda.cl . 22 (2008) 46-51. [in Spanish] .
[2] L. A. Azocar, H. Leiva, J. Matkowski and N. Merentes; Controllability of semilinear Volterra- Stieltjes equation in the space of regulated functions, J. Control Theory Appl. 2012, 10(1) 123- 127.
[3] L. Barbanti; Introduçao a teoria do controle para equacoes integrais de Fredholm-Stieltjes lineares, MAT-INE-USP, 1981. [in Portuguese] .
[4] L. Barbanti; Densidade de conjuntos de atingibilidade em equaçoes integrais de Volterra- Stieltjes controladas, IME/USP, 1984. [in Portuguese] .
[5] L. Barbanti; Controllability and approximate controllability for linear integral Volterra-Stieltjes equations, Mathematical and statiscal Institute, University of São Paulo, São Paulo, Brazil, 1989.
[6] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011.
[7] D. Bugajewska; On the superposition operator in the space of functions of bounded variation, revisited, Mathematical and Computer Modelling 52 (2010) 791-796.
[8] D. Bugajeswska, D. Bugajewski, and H. Hudzik; BV_φ-solutions of nonlinear integral equations, J. Math. Anal. Appl. 287 (2003) 265-278.
[9] D. Bugajewska and D. O′Regan; On nonlinear integral equations and Λ-bounded variation, Acta Math. Hung., 107 (4) (2005), 295-306.
[10] D. Bugajewski; On the existence of weak solutions of integral equations in Banach spaces, Comment. Math. Univ. Carolin., 35,1 (1994) 35-41.
[11] D. Bugajewski; On BV-solutions of some nonlinear integral equations, Integr. Equa. Oper. Theory 46 (2003), 387-398.
[12] T. A. Burton; A fixed-point theorem of Krasnselskii, Appl. Math. Lett. Vol. 11, No. 1, pp. 85-88 (1998).
[13] D. N. Chalishajar, R. K. George; Exact controllability of generalized Hammerstein type integral equations and applications, Electronic Journal of Differential Equations, Vol. 2006(2006), No. 142, pp.1-15.
[14] H. G. Heuser; Functional Analysis, Wiley-Interscience Publications, © 1982.
[15] C. S. Hönig; Volterra-Stieltjes integral equations with linear constraints and discontinuous solutions, Bulletin of the American mathematical Society. Volume 81, Number 3, May 1975.
[16] C. S. Hönig; Volterra-Stieltjes Integral Equations, North-Holland/American Elseiver, North- Holland Mathematics Studies 16, North-Holland Publishing Company-1975, printed in The Netherlands.
[17] O. A. Ilhan; Solvability of some integral equations in Banach Space and their applications to the theory of viscoelasticity, Abstract and Applied Analysis, Volume 2012, Article ID 717969, 13 pages.
[18] G. E. Ladas and V. Lakshmikantham; Differential Equations in Abstract Spaces, Academic Press, 1972.
[19] M. H. Noori, H. R. Erfabian, A. V. Kamyad; A new approach for a class of optimal control problems of Volterra integral equations, Intelligent Control and Automation, 2011, 2, 121-125.
[20] A. D. Polyanin and A. V. Manzhirov; Handbook of Integral Equations, Second Edition, Chapman Hall/CRC, © 2008.
[21] Š. Schwabik, M. Tvrdý, O. Vejvoda; Differential and Integral Equations. Boundary Value Problems and Adjoints, D. Reidel Publishing Co., DordrechtBoston, Mass.London, 1979.
[22] G. F. Webb; Asymtotic stability in the α-norm for an abstract nonlinear Volterra integral equation, Stability of Dynamical Systems, Theory and Applications, volume 28, Chapter 19, Lectures Notes in Pure and Applied Mathematics, Edited by John R. Graef, Marcel Dekker Inc., New York and Basel.

Received: September 2013. Accepted: February 2015.