# Definition:Extremum/Functional

## Definition

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 $J$.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 1.3$: The Variation of a Functional. A Necessary Condition for an Extremum