Suppose we are given the integral equation $$ u(x;a) =v(x)+\int_0^a K(x,y)\,u(y;a)\,dy, $$ where $K(x,y)$ and $v(x)$ are known functions, and $a>0$ is a constant. What I am interested in is the asymptotic behavior of the solution $u(x;a)$ as $a\to\infty$. Is there a standard method of some kind to approach this problem?
To be more specific, suppose $v(x)=1$ for all $x$ and $$ K(x,y)= \frac{\partial}{\partial y} P\left(\frac{y}{x}\right), $$ where $P(t)$ is the cdf of a nonnegative random variable, so $\int_0^\infty dP(t)=1$. Many thanks.