Invariant set criteria for \(C_ 0\) semigroups (Q1819349)

Let A be the generator of a \((C_ 0)\) semigroup \(T=\{T(t):T\geq 0\}\) of bounded linear operators on Banach space X such that \(\| T(t)\| \leq e^{\omega^ t}\) for \(t\geq 0\). Let C be a closed subset of X and \(d(z;C)=\inf \{\| z,y\|:y\in X\}\). It is well-known that (1) T(t):C\(\to C\) for all \(t\geq 0\) iff (2) \(\liminf_{h\to 0}d(T(h)x;C)=0\) for all \(x\in C\). If also C is convex, (1) iff (3) \((I-hA)^{-1}:C\to C\) for all sufficiently small \(h>0\). The author finds new criteria for invariant sets. Sample Theorem: Suppose that, for some constant \(\rho\geq 0\), \[ \liminf_{h\to 0+}(-h)^{-1}[d(x-hAx;C)-d(x;C)]\leq \rho d(x;C) \] for all \(x\in Dom(A)\setminus C\). Then (1) holds. An explicit example (observed by M. Pierre) shows that it is not sufficient to consider trajectories on Dom(A)\(\cap \partial C\) in order to guarantee invariance.