Abstract and Applied Analysis

A Characterization of Semilinear Dense Range Operators and Applications

H. Leiva, N. Merentes, and J. Sanchez

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Abstract

We characterize a broad class of semilinear dense range operators G H : W Z given by the following formula, G H w = G w + H ( w ) , w W , where Z , W are Hilbert spaces, G L ( W , Z ) , and H : W Z is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator G to have dense range. Second, under some condition on the nonlinear term H , we prove the following statement: If R a n g ( G ) ¯ = Z , then R a n g ( G H ) ¯ = Z and for all z Z there exists a sequence { w α Z : 0 < α 1 } given by w α = G * ( α I + G G * ) - 1 ( z - H ( w α ) ) , such that    l i m α 0 + { G u α + H ( u α ) } = z . Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: z = A z + B u ( t ) + F ( t , z , u ( t ) ) , z Z , u U , t > 0 , where Z , U are Hilbert spaces, A : D ( A ) Z Z is the infinitesimal generator of strongly continuous compact semigroup { T ( t ) } t 0 in Z , B L ( U , Z ) , the control function u belongs to L 2 ( 0 , τ ; U ) , and F : [ 0 , τ ] × Z × U Z is a suitable function. As a particular case we consider the controlled semilinear heat equation.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 729093, 11 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511777

Digital Object Identifier
doi:10.1155/2013/729093

Mathematical Reviews number (MathSciNet)
MR3035310

Zentralblatt MATH identifier
1287.47054

Citation

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


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