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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 729093, 11 pages
http://dx.doi.org/10.1155/2013/729093
Research Article

A Characterization of Semilinear Dense Range Operators and Applications

1Universidad de los Andes, Facultad de Ciencias, Departamento de Matemática, Mérida 5101, Venezuela
2Universidad Central de Venezuela, Facultad de Ciencias, Departamento de Matemática, Caracas 1053, Venezuela

Received 13 October 2012; Revised 25 November 2012; Accepted 22 January 2013

Academic Editor: Valery Y. Glizer

Copyright © 2013 H. Leiva et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We characterize a broad class of semilinear dense range operators given by the following formula, , where , are Hilbert spaces, , and is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator to have dense range. Second, under some condition on the nonlinear term , we prove the following statement: If , then and for all there exists a sequence given by , such that . Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: , where , are Hilbert spaces, is the infinitesimal generator of strongly continuous compact semigroup in , the control function belongs to , and is a suitable function. As a particular case we consider the controlled semilinear heat equation.