System of nonlinear integral equations

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

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