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 |
| Summary: | 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 ) . |
|---|---|
| ISSN: | 2227-7390 |