Definition:Extremum/Functional

Definition
Let $ S $ be a set of mappings.

Let $ J \left [ { y } \right ] : S \to \R $ be a functional.

Suppose for $ y = \hat { y } \left ( { x } \right ) $ there exists a neighbourhood of the curve $ y = \hat { y } \left ( { x } \right ) $ such that the difference $ J \left [ { y } \right ] - J \left [ { \hat { y } } \right ] $ does not change its sign in this neighbourhood.

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