Continuity with Respect to the Data for Delay Equations Subjected to Nonlocal Conditions

Daniela Rosu*, Monica-Dana Burlica and Ioan Vrabie

We prove the continuity of the $C^0$-solution with respect to the right-hand side $f$ and the nonlocal constraint $g$ to the 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 $X$ is a real Banach space, $\tau>0$, $A:D(A)\subseteq X\leadsto X$ is an $m$-dissipative operator, $f:\mathbb{R}_+\times C([\,-\tau,0\,];\overline{D(A)})\to X$ is jointly continuous and Lipschitz with respect to its second argument and $g:C_b([\,-\tau,+\infty);\overline{D(A)})\to C([\,-\tau,0\,];\overline{D(A)})$ is nonexpansive.

Mathematics Subject Classification: 35K30 34G25 47H06

Keywords: Differential delay evolution equation; nonlocal delay initial condition

Minisymposion: Analysis and Control of Evolution Equations and Inclusions