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Continuous selections of solution sets to evolution equations

Author: Vasile Staicu
Journal: Proc. Amer. Math. Soc. 113 (1991), 403-413
MSC: Primary 49J24; Secondary 34A60, 47H04, 54C65
MathSciNet review: 1076580
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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

$\displaystyle \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 $.

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