Continuous selections of solution sets to evolution equations

Abstract:

We prove the existence of a continuous selection of the multivalued map $\xi \to \mathcal {T}(\xi )$, where $\mathcal {T}(\xi )$ is the set of all weak (resp. mild) solutions of the Cauchy problem \[ \dot x(t) \in Ax(t) + F(t,x(t)),\quad x(0) = \xi \], assuming that $F$ is Lipschitzian with respect to $x$ and $- A$ is a maximal monotone map (resp. $A$ is the infinitesimal generator of a ${C_0}$-semigroup). We also establish an analog of Michael’s theorem for the solution sets of the Cauchy problem $\dot x(t) \in F(t,x(t)),\;x(0) = \xi$.