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...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2013/729093 |
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| _version_ | 1850163876030578688 |
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| author | H. Leiva N. Merentes J. Sanchez |
| author_facet | H. Leiva N. Merentes J. Sanchez |
| author_sort | H. Leiva |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-e973a47baffc4081bc38863d8309f7df |
| institution | OA Journals |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-e973a47baffc4081bc38863d8309f7df2025-08-20T02:22:06ZengWileyAbstract and Applied Analysis1085-33751687-04092013-01-01201310.1155/2013/729093729093A Characterization of Semilinear Dense Range Operators and ApplicationsH. Leiva0N. Merentes1J. Sanchez2Universidad de los Andes, Facultad de Ciencias, Departamento de Matemática, Mérida 5101, VenezuelaUniversidad Central de Venezuela, Facultad de Ciencias, Departamento de Matemática, Caracas 1053, VenezuelaUniversidad Central de Venezuela, Facultad de Ciencias, Departamento de Matemática, Caracas 1053, VenezuelaWe 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.http://dx.doi.org/10.1155/2013/729093 |
| spellingShingle | H. Leiva N. Merentes J. Sanchez A Characterization of Semilinear Dense Range Operators and Applications Abstract and Applied Analysis |
| title | A Characterization of Semilinear Dense Range Operators and Applications |
| title_full | A Characterization of Semilinear Dense Range Operators and Applications |
| title_fullStr | A Characterization of Semilinear Dense Range Operators and Applications |
| title_full_unstemmed | A Characterization of Semilinear Dense Range Operators and Applications |
| title_short | A Characterization of Semilinear Dense Range Operators and Applications |
| title_sort | characterization of semilinear dense range operators and applications |
| url | http://dx.doi.org/10.1155/2013/729093 |
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