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Open AccessArticle

Viability for Semilinear Differential Equations with Infinite Delay

by Qixiang Dong *,† and Gang Li
School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editor: Johnny Henderson
Mathematics 2016, 4(4), 64; https://doi.org/10.3390/math4040064
Received: 1 September 2016 / Revised: 25 October 2016 / Accepted: 2 November 2016 / Published: 8 November 2016
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 ) . View Full-Text
Keywords: viable domain; differential equation; infinite delay; tangency condition viable domain; differential equation; infinite delay; tangency condition
MDPI and ACS Style

Dong, Q.; Li, G. Viability for Semilinear Differential Equations with Infinite Delay. Mathematics 2016, 4, 64.

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