Definition:Minimum Value of Functional

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.

Let $J$ have a (relative) extremum for $y=\hat y$.

Suppose, $J\sqbrk y-J \sqbrk{\hat y}\ge 0$ in the neighbourhood of $y=\hat y$.

Then this extremum is called the minimum of the functional $J$.