Is this recursion well-defined?

Question

I have a recursion defined by $$ f(n)=\max\{0,-c+pf(n-1)+(1-p)f(n+1)\} $$

with $0.5<p \leq 1$ and $f(0)=R>0$ and $f(m)=0$ for some $m>0$. I am trying to show that $f(n)$ is decreasing in $n$ but I am not even sure if it is well-defined. I'd appreciate any help.

Answer 1

Suppose $m$ is the first positive integer where $f$ is $0$. Then for $0 < n < m$ you have $f(n) = -c + p f(n-1) + (1-p) f(n+1)$. That equation has solutions of the form $f(n) = - c n/(2p-1) + A + B (p/(1-p))^n$. That function is concave or convex, depending on the sign of $B$...