Viability for semilinear differential equations of retarded type

Bull. Korean Math. Soc. 2007 Vol. 44, No. 4, 731-742 Published online December 1, 2007

Qixiang Dong and Gang Li Yangzhou University, Yangzhou University

Abstract : Let $X$ be a Banach space, $A:D(A)\subset X\rightarrow X$ the generator of a compact $C_0$-semigroup $S(t):X\rightarrow X, t\geq 0$, $D$ a locally closed subset in $X$, and $f:(a,b)\times C([-q,0];X)\rightarrow X$ a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order that $D$ be a viable domain of the semilinear differential equation of retarded type \begin{equation*} u'(t)=Au(t)+f(t,u_t),t\in [t_0,t_0+T],u_{t_0}=\phi\in C([-q,0];X) \end{equation*} is the tangency condition \begin{equation*} \liminf_{h\downarrow 0}h^{-1}d(S(h)v(0)+hf(t,v);D)=0 \end{equation*} for almost every $t\in(a,b)$ and every $v\in C([-q,0];X)$ with $v(0)\in D$.