Definition:Weak Extremum

Definition
Let $S$ be a set of mappings.

Let $y,\hat y\in S:\R\to\R$ be real functions.

Let $J\sqbrk y:S\to\R$ be a functional.

Suppose, there exists $\epsilon>0$ such that for $\size{y-\hat y}_1<\epsilon$ the expression $J\sqbrk y-J\sqbrk{\hat y}$ has the same sign for all $y$.

Here $\size{~}_1 $ denotes the norm of in the space $C^1$.

Then $y=\hat y$ is a weak extremum of the functional $ J\sqbrk y$.