I have on domain $[0,\infty)$ a known and positive function $f(x)$ and two unknown functions $g(x), h(x)$ that start positive when $x=0$.


I also know that if $h(x)$ is positive, then $g(x)$ is also positive.


I define the induction


$$h(x)=\int\_0^x f(t)g(t)h(t) dt$$


I imagine that is possible to say that $h(x)$ (and so $g(x)$) remains positive, but I could not find a formal argument.


Thank you in advance.



Seek help to formalize an argument to positiveness of function defined inductively by integral
Taking the derivative of your "induction" w.r.t. $x$, one gets
$$
h'(x) = f(x)g(x)h(x)
$$
and thus $h'(x)>0$ if $h(x)>0$, according to your conditions. Since $h(0)>0$, one concludes that $h(x)>0$ for $x\in[0;\infty)$.


