Questions tagged [set-valued-analysis]

I have a question about closed operator. Operator $F:X \to Y$ is closed if $(x_n)$ is a sequence in $D(F)$ that is convergent in $X$ and the sequence $(F(x_n))$ is convergent in $Y$, then we have $\lim x_n \in D(F)$ and $F(\lim x_n)= \lim F(x_n)$.…

I have to prove that: $$ F\text{ - l.s.c. in } x_0\Rightarrow\forall (x_n)_n\subseteq X\text{ s.t. }x_n\rightarrow x_0,\forall y_0\in F(x_0),\exists(y_n)_n\subseteq Y \text{ s.t. } y_n\rightarrow y_0 \text{ with } y_n\in F((x_n)_n) $$ where:…

Let $F:\Omega\to 2^X\setminus\emptyset$, $\Omega$ is compact. If $F$ has bounded values then they are uniformly bounded. Please help me prove this! Thanks!