Solutions depending on something continuously

Question

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find a $u \in K$ satisfying $$ \langle A(u),v-u \rangle \geq \langle F,v-u \rangle \text{ }\text{ for all } v \in K $$

What would it mean to say ‘the solution of $u$ depends on $F$ continuously‘?Also how is this usually shown? Thanks.

Answer 1

What would it mean to say ‘the solution of $u$ depends on $F$ continuously‘

It means that for every $F$ and every $\epsilon>0$ there exists $\delta>0$ such that for every functional $G$ with $\|F-G\|<\delta$ we have $|\tilde u-u\|<\epsilon$, where $\tilde u$ is the solution of the variational inequality with $G$ instead of $F$.

how is this usually shown?

By proving that every $\tilde u$ with $\|\tilde u-u\|\ge \epsilon$ fails the inequality for some $v\in K$.