# proving existence of an equilibrium in an equation involving PDF and CDF

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?

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 I took the lim h(x) as x approaches - or + infinity, the exact problem I faced is calculating (or knowing) the lim f(x) as x approaches + or - infinity – Daniel Lårs Jul 9 '12 at 15:56