Sufficient Conditions for Weak Extremum

Theorem
Let $ J $ be a functional such that:


 * $ \displaystyle J \left [ { y } \right ] = \int_a^b F \left ( { x, y, y' } \right ) \mathrm d x, \quad y \left ( { a } \right ) = A, \quad y \left ( { b } \right ) = B $

Let $ y = y \left ( { x } \right ) $ be an extremum.

Let the strengthened Legendre's condition hold.

Let the strengthened Jacobi's necessary condition hold.

Then the functional $ J $ has a weak extremum for $ y = y \left ( { x } \right ) $.

Proof
By the continuity of function $ P \left ( { x } \right ) $ and the solution of Jacobi's equation:


 * $ \displaystyle \exists \epsilon > 0 : \left ( { \forall x \in \left [ { a \,.\,.\, b + \epsilon } \right ] \quad P \left ( { x } \right ) > 0 } \right ) \land \left ( { \left [ { a \,.\,.\, b + \epsilon } \right ] \text{ does not contain a point conjugate to a} } \right )   $

Consider the quadratic functional


 * $ \int_a^b \left ( { Ph'^2 + Qh^2 } \right ) \mathrm d x - \alpha^2 \int_a^b h'^2 \mathrm d x$

together with Euler's equation


 * $ - \frac{ \mathrm d }{ \mathrm d x } \left [ { (P - \alpha^2 )h' } \right ] + Qh = 0 $