Can the Separation of Variables Technique Solve This Integral Equation?

[Karlisbad]

Let be the integral equation:

$$g(s)g(p)g(u)= \int_{0}^{\infty}dx\int_{0}^{\infty}dy\int_{0}^{\infty}dzK(sx)K(py)K(uz)f(x,y,z) <br />$$

then my question is if we could "seek" for a solution in the form:

$$f(x,y,z)=A(x)A(y)A(z)$$ where the function A satsify (for x y and z) the integral equation:

$$g(s)=\int_{0}^{\infty}dxK(xs)A(x)$$ 8and the same for the other)

¿is this approach good?

 

[HallsofIvy]

Yes, you can- though the general solution may be a linear combination of such solutions.