I'm currently working on a problem about differential equations and I came across the following problem.

Let $f$ be a continuous function defined on $[0,\infty)$ such that $f(x)\ge 0$, $f(0)\ne 0$. Let $g$ be a integrable function defined on $[0,\infty)$ such that $\int_0^\infty g(x)\mathrm{d}x<\infty$. Let $0<p<1$, suppose $$ f(x)\le f(0)+\int_0^x (x-y)^{p-1}f(y)g(y)\mathrm{d}y, $$ does it follow $f(x)\le Kf(0)$ for some constant $K$?

I was unable to prove this or find a counterexample. However I personally believe that there may exist a counterexample. Further I would like to ask

Under what condition of $g$ does it follow such constant $K$? The best result I obtain for now is $\int_0^\infty e^{2x}g^2(x)\mathrm{d}x<\infty$. However such a condition is so strong that I'm looking for a improvement. For instance, I would like to get rid of $e^{2x}$.