Abstract and Applied Analysis
- Abstr. Appl. Anal.
- Volume 2013 (2013), Article ID 729093, 11 pages.
A Characterization of Semilinear Dense Range Operators and Applications
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.
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 729093, 11 pages.
First available in Project Euclid: 27 February 2014
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Leiva, H.; Merentes, N.; Sanchez, J. A Characterization of Semilinear Dense Range Operators and Applications. Abstr. Appl. Anal. 2013 (2013), Article ID 729093, 11 pages. doi:10.1155/2013/729093. https://projecteuclid.org/euclid.aaa/1393511777