Viability for Semilinear Differential Equations with Infinite Delay
Let X be a Banach space, A : D ( A ) ⊂ X → X the generator of a compact C 0 -semigroup S ( t ) : X → X , t ≥ 0 , D ( · ) : ( a , b ) → 2 X a tube in X, and f : ( a , b ) × B → X a function of Carathéodory type. The main result of this paper is that a neces...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2016-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | http://www.mdpi.com/2227-7390/4/4/64 |
| _version_ | 1818134992769777664 |
|---|---|
| author | Qixiang Dong Gang Li |
| author_facet | Qixiang Dong Gang Li |
| author_sort | Qixiang Dong |
| collection | DOAJ |
| description | Let X be a Banach space, A : D ( A ) ⊂ X → X the generator of a compact C 0 -semigroup S ( t ) : X → X , t ≥ 0 , D ( · ) : ( a , b ) → 2 X a tube in X, and f : ( a , b ) × B → X a function of Carathéodory type. The main result of this paper is that a necessary and sufficient condition in order that D ( · ) be viable of the semilinear differential equation with infinite delay u ′ ( t ) = A u ( t ) + f ( t , u t ) , t ∈ [ t 0 , t 0 + T ] , u t 0 = ϕ ∈ B is the tangency condition lim inf h ↓ 0 h − 1 d ( S ( h ) v ( 0 ) + h f ( t , v ) ; D ( t + h ) ) = 0 for almost every t ∈ ( a , b ) and every v ∈ B with v ( 0 ) ∈ D ( t ) . |
| first_indexed | 2024-12-11T09:17:26Z |
| format | Article |
| id | doaj.art-a7fa15ddfad445839598aa8eae706e17 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-11T09:17:26Z |
| publishDate | 2016-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a7fa15ddfad445839598aa8eae706e172022-12-22T01:13:20ZengMDPI AGMathematics2227-73902016-11-01446410.3390/math4040064math4040064Viability for Semilinear Differential Equations with Infinite DelayQixiang Dong0Gang Li1School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, ChinaSchool of Mathematical Sciences, Yangzhou University, Yangzhou 225002, ChinaLet X be a Banach space, A : D ( A ) ⊂ X → X the generator of a compact C 0 -semigroup S ( t ) : X → X , t ≥ 0 , D ( · ) : ( a , b ) → 2 X a tube in X, and f : ( a , b ) × B → X a function of Carathéodory type. The main result of this paper is that a necessary and sufficient condition in order that D ( · ) be viable of the semilinear differential equation with infinite delay u ′ ( t ) = A u ( t ) + f ( t , u t ) , t ∈ [ t 0 , t 0 + T ] , u t 0 = ϕ ∈ B is the tangency condition lim inf h ↓ 0 h − 1 d ( S ( h ) v ( 0 ) + h f ( t , v ) ; D ( t + h ) ) = 0 for almost every t ∈ ( a , b ) and every v ∈ B with v ( 0 ) ∈ D ( t ) .http://www.mdpi.com/2227-7390/4/4/64viable domaindifferential equationinfinite delaytangency condition |
| spellingShingle | Qixiang Dong Gang Li Viability for Semilinear Differential Equations with Infinite Delay Mathematics viable domain differential equation infinite delay tangency condition |
| title | Viability for Semilinear Differential Equations with Infinite Delay |
| title_full | Viability for Semilinear Differential Equations with Infinite Delay |
| title_fullStr | Viability for Semilinear Differential Equations with Infinite Delay |
| title_full_unstemmed | Viability for Semilinear Differential Equations with Infinite Delay |
| title_short | Viability for Semilinear Differential Equations with Infinite Delay |
| title_sort | viability for semilinear differential equations with infinite delay |
| topic | viable domain differential equation infinite delay tangency condition |
| url | http://www.mdpi.com/2227-7390/4/4/64 |
| work_keys_str_mv | AT qixiangdong viabilityforsemilineardifferentialequationswithinfinitedelay AT gangli viabilityforsemilineardifferentialequationswithinfinitedelay |