Good day!
The sense of the problem: there is the following functional
$\displaystyle J=\int_{x_1 }^{x_2 } {F\left( {\frac{b(x)}{\int {b(x)dx} }} \right)dx} $
$\displaystyle b(x)=\int {a\left( {x,y,q(x,y),\int {q(x,y)dx} )} \right)dy} $

*All " indefinite" integrals are in the infinite limits, $\displaystyle {x_1 } $, $\displaystyle {x_2} $ – some numbers, $\displaystyle a(..), F(..) $ - some known functions.
We need to find such function $\displaystyle q(x,y) $ that the functional $\displaystyle J $ will possess it’s minimum value.
Thanks in anticipation for Your answers and advices.