System of nonlinear integral equations

[jannyhuggy]

Hello everybody!

While solving some physical problem I got stuck with some system of integral equations.
The problem is formulated in .pdf file below.

I will be over-satisfied with the following
1) to know whether and why this system has/doesn't have a solution
2) how it could be produced approximately
3) if there are any methods for exact solution of equations of such a type, perhaps with other V(k)

View attachment 11111.pdf

ANY comments are highly welcome!

 

[jannyhuggy]

I tried two iteration methods but perturbations seem to be divergent. Perhaps any regularization ideas?