Existence and uniqueness of mild and classical solutions of impulsive evolution equations
We consider the non-linear impulsive evolution equation $$displaylines{ u'(t)=Au(t)+f(t,u(t),Tu(t),Su(t)), quad 0<t<T_0, ; t eq t_i,cr u(0) =u_0,cr Delta u(t_i) =I_i(u(t_i)),quad i=1,2,3,dots,p. }$$ in a Banach space $ X$, where $ A $ is the infinitesimal generator of a $C_0 $ semigroup....
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Texas State University
2005-10-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2005/111/abstr.html |
| _version_ | 1818958906262552576 |
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| author | Annamalai Anguraj Mani Mallika Arjunan |
| author_facet | Annamalai Anguraj Mani Mallika Arjunan |
| author_sort | Annamalai Anguraj |
| collection | DOAJ |
| description | We consider the non-linear impulsive evolution equation $$displaylines{ u'(t)=Au(t)+f(t,u(t),Tu(t),Su(t)), quad 0<t<T_0, ; t eq t_i,cr u(0) =u_0,cr Delta u(t_i) =I_i(u(t_i)),quad i=1,2,3,dots,p. }$$ in a Banach space $ X$, where $ A $ is the infinitesimal generator of a $C_0 $ semigroup. We study the existence and uniqueness of the mild solutions of the evolution equation by using semigroup theory and then show that the mild solutions give rise to a classical solutions. |
| first_indexed | 2024-12-20T11:33:11Z |
| format | Article |
| id | doaj.art-ddfe5ed32c5c46fbb340883c84e14dcf |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-20T11:33:11Z |
| publishDate | 2005-10-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-ddfe5ed32c5c46fbb340883c84e14dcf2022-12-21T19:42:13ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912005-10-01200511118Existence and uniqueness of mild and classical solutions of impulsive evolution equationsAnnamalai AngurajMani Mallika ArjunanWe consider the non-linear impulsive evolution equation $$displaylines{ u'(t)=Au(t)+f(t,u(t),Tu(t),Su(t)), quad 0<t<T_0, ; t eq t_i,cr u(0) =u_0,cr Delta u(t_i) =I_i(u(t_i)),quad i=1,2,3,dots,p. }$$ in a Banach space $ X$, where $ A $ is the infinitesimal generator of a $C_0 $ semigroup. We study the existence and uniqueness of the mild solutions of the evolution equation by using semigroup theory and then show that the mild solutions give rise to a classical solutions.http://ejde.math.txstate.edu/Volumes/2005/111/abstr.htmlSemigroupsevolution equationsimpulsive conditions. |
| spellingShingle | Annamalai Anguraj Mani Mallika Arjunan Existence and uniqueness of mild and classical solutions of impulsive evolution equations Electronic Journal of Differential Equations Semigroups evolution equations impulsive conditions. |
| title | Existence and uniqueness of mild and classical solutions of impulsive evolution equations |
| title_full | Existence and uniqueness of mild and classical solutions of impulsive evolution equations |
| title_fullStr | Existence and uniqueness of mild and classical solutions of impulsive evolution equations |
| title_full_unstemmed | Existence and uniqueness of mild and classical solutions of impulsive evolution equations |
| title_short | Existence and uniqueness of mild and classical solutions of impulsive evolution equations |
| title_sort | existence and uniqueness of mild and classical solutions of impulsive evolution equations |
| topic | Semigroups evolution equations impulsive conditions. |
| url | http://ejde.math.txstate.edu/Volumes/2005/111/abstr.html |
| work_keys_str_mv | AT annamalaianguraj existenceanduniquenessofmildandclassicalsolutionsofimpulsiveevolutionequations AT manimallikaarjunan existenceanduniquenessofmildandclassicalsolutionsofimpulsiveevolutionequations |