Flow invariance for differential delay equations

Abstract:

The flow invariance problem for the functional differential equation $u’(t)\in Au(t)+F(u_t)$ for $t\geq 0$ with initial condition $u_0=\phi \in \frak {D}$ is solved in a Banach space $X$, where $A$ is a quasi-dissipative operator in $X$ and $F$ is a continuous operator from a closed set $\frak {D}$ in the so-called initial-history space $\frak {X}$ into $X$ satisfying a dissipativity condition in the following sense: There exists $\omega _F\geq 0$ such that $[\phi (0)-\hat {\phi }(0),~F(\phi )-F(\hat {\phi })]_{+}\leq \omega _F\|\phi -\hat {\phi }\|_{\frak {X}}$ for $\phi , \hat {\phi }\in \frak {D}$ satisfying that $\|\phi -\hat {\phi }\|_{\frak {X}}\leq \|\phi (0)-\hat {\phi }(0)\|_X$, where $[x,\xi ]_{+}=\lim _{h\to 0+}(\|x+h\xi \|_X-\|x\|_X)/h$ for $x,\xi \in X$.