I have an equation for which I want to show existence of at least one equilibrium. The equation, call it $h(x)$, is:
$$h(x)=[a-q'(y)]f(x-y) - q''(y)[1-F(x-y)],$$
where $a$ is a positive constant, $f$ is a probability density function, $F$ is its CDF, $q$ is a function of $y$ (which is assumed to be constant at a predefined level) such that $q'>0$ and $q''<0$. The PDF is general (not defined specifically).
I denoted the function by $h(x)$ and attempted to use the Extreme value theorem to show that there exists at least an $x$ such that $h'(x)=0$. But, I got it difficult to find the limit of $f(x)$ as $x$ approaches $\pm \infty$.
Can you please offer some help?