Good day!
The sense of the problem: there is the following functional
J=\int_{x_1 }^{x_2 } {F\left( {\frac{b(x)}{\int {b(x)dx} }} \right)dx}
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, {x_1 } , {x_2} – some numbers, a(..), F(..) - some known functions.
We need to find such function q(x,y) that the functional J will possess it’s minimum value.
Thanks in anticipation for Your answers and advices.