Continuity with respect to the data for a delay evolution equation with nonlocal initial conditions

Abstract

We prove the continuity of the $C^0$-solution with respect to the right-hand side and the initial nonlocal condition to the nonlinear delay differential evolution equation
$$\left\{\begin{array}{ll}
\displaystyle u'(t)\in Au(t)+f(t,u_t),&\quad t\in \mathbb{R}_+,
\\[1mm]
u(t)=g(u)(t),&\quad t\in [\,-\tau,0\,],
\end{array}\right.$$
where $\tau>0$, $X$ is a real Banach space, $A$ is an $m$-dissipative operator, $f:\mathbb{R}_+\times C([\,-\tau,0\,];\overline{D(A)})\to X$ is Lipschitz continuous with respect to its second argument and
$g:C_b([\,-\tau,+\infty);\overline{D(A)})\to C([\,-\tau,0\,];\overline{D(A)})$, is nonexpansive.