I would like to get the solution of $\int_{x}^{y}f(t,y)f(x,t)dt=af(x,y)$ for all $\{y>x>0\}$, satisfying $\int _{0}^{x}f(t,x)dt=1$.
Or a more simple question may be if we assume $f(x,y)=g(x/y)/y$ where $g$ is defined on $(0,1)$ satisfying $\int_{0}^{1}g(t)dt=1$, and the equation turns out to be $\int_{x}^{y}g(t/y)g(x/t)/tdt=ag(x/y)$.
Can we get a solution or approximation of the above? Thanks