Proving Integral Equation: f(x)= O(x)

[zetafunction]

let be the functions f(x) and g(x) defined by an integral equation

$$g(x)= \int_{0}^{\infty}dy K(yx)f(y)dy$$

then i want to prove that for example $$f(x)= O(x)$$

then using a change of varialbe yx=t i manage to put

$$g(x) \le \frac{C}{x^{2}} \int_{0}^{\infty}dtK(t)t$$

if the last integral exists , then a simple condition is that there will be a constant so $$x^{2} g(x) \le A$$ i assume K(t) is ALWAYS positive on the interval [0,oo)

 

[Eynstone]

It's difficult to decide unless one knows the order of g or K.