= Solution
Let $S:\mathbb R^p\rightrightarrows\mathbb R^q$ be a <set-valued mapping> and let $\bar v\in S(\bar u)$. The <Aubin property> at $(\bar u,\bar v)$ means there are neighborhoods $U$ of $\bar u$, $V$ of $\bar v$, and a finite constant $\kappa\geq0$ such that
$$
\boxed{S(u)\cap V\subset S(u')+\kappa\|u-u'\|_2\mathbb B_q
\quad\text{for every }u,u'\in U}.
$$
Here $\mathbb B_q$ is the closed unit ball. Equivalently, every solution $v\in S(u)$ near $\bar v$ can be matched to a solution $v'\in S(u')$ within $\kappa\|u-u'\|_2$. The localization is on the left-hand side: the matching point need not be in $V$. The property is also called the <Lipschitz-like property>.
For <sensitivity analysis>, let $u$ represent data or perturbations and $S(u)$ the set of feasible or optimal solutions. The <Aubin property> bounds how far a nearby solution can move when the data change. Taking $u=\bar u$, $v=\bar v$ also guarantees a nearby solution for each sufficiently small perturbation $u'$. It is a stability estimate for a relation, and by itself does not imply uniqueness or differentiability. For a single-valued solution map, it reduces to local <Lipschitz continuity>.
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