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 $\epsilon \in \R_{> 0}$ 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 $\size{\, \cdot \,}_1$ denotes the norm of in the space $C^1$.

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