Continuous selections of solution sets to evolution equations

Abstract

We prove the existence of a continuous selection of the multivalued map £ —»& ~(Ç), where ^"(i) is the set of all weak (resp. mild) solutions of the Cauchy problem x(t)€Ax(t) + F(t,x(t)), x(0)=i, assuming that F is Lipschitzian with respect to x and -A is a maximal monotone map (resp. A is the infinitesimal generator of a C0-semigroup). We also establish an analog of Michael's theorem for the solution sets of the Cauchy problem x(t) € F(t, x(t)), x(0) = £, .