let be the integral equation for two functions $ f(x) $ and $ g(x) $
in the form $$ g(s)= \int_{0}^{s}\sqrt{s-f(x)}dx $$
is valid to accept that in the sense of fractional calculus, the ONLY solution to this equation will be
$$ f^{-1}(x)= a \frac{d^{1/2}}{dx^{1/2}}g(x) $$
here of course $ f^{-1}(x) $ means the inverse function of $ f(x) $ so $ f(f^{-1}(x))=x $
for some real constant 'a'
EDIT: sorry i forgot in order to obtain the solution i make the change of variables $ x=f^{-1}(t) $ inside the first nonlinear equation.
my interest is due to the fact that the two integral equations
$ g(s)= \int_{0}^{s}dt \frac{f(t)}{\sqrt{s-t}}$ and $ g(s)= \int_{0}^{s}\sqrt{s-f(t)}dt $ appear in several branches of physics dealing with WKB approximation.
i say so because this is important for my theoretical model implying the RIemann zeros :) thanks in advance