Definition:Extremum/Functional

Definition
Let $S$ be a set of mappings.

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

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

Suppose for $y = \hat y \paren x$ there exists a neighbourhood of the curve $y = \hat y \paren x$ such that the difference $J \sqbrk y - J \sqbrk {\hat y}$ does not change its sign in this neighbourhood.

Then $y = \hat y$ is called a (relative) extremum of the functional $J$.