On the Weierstrass type condition for nonlocal functionals

G.A.Kamenskii, Ju.P.Zabrodina

e-mail: malina@rfbr.ru

There is considered the problem of extremum of the nonlocal functional

where ,, are some integers. Here denotes the partial derivative of with respect to . Denote

It was proved that if the function furnishes the functional with an extremum, then satisfies the analog of the Euler equation for the considered problem

,

where

The generalized function of Weierstrass will be called the function

Theorem 1. If the functional attains on a strong minimum, then satisfies the generalized Weierstrass condition for any .

Theorem 2. The function satisfies the minimum principle if it satisfies the generalized Euler equation and the Weierstrass inequality.

Theorem 3. If the functional attains on a strong minimum, then satisfies the minimum principle.

Theorem 4. Suppose that functional attains on an extremum,

and is a corner point of . Then at this point the generalized Weierstrass-Erdman conditions are fulfilled.