Definition:Weak Extremum

Definition
Let $ S $ be a set of mappings.

Let $ y \in S $.

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

Suppose, there exists $ \epsilon > 0 $ such that for $ \left \Vert y - \hat { y } \right \Vert_1 < \epsilon $ the expression $ J \left [ { y } \right ] - J \left [ { \hat { y } }\right ] $ has the same sign for all $ y $.

Here $ \left \Vert ~ \right \Vert_1 $ denotes the norm of in the space $ C^1 $.

Then $ y = \hat { y } $ is a weak extremum of the functional $ J \left [ { y } \right ] $.