asymptotic decreasing function

Question

I'm currently working on my university thesis, I'm trying to model a particular behavior but I can't think of the right function that works for me.

I'm looking for:

1. Decreasing function
2. $f(0) = 1$
3. $\lim_\limits{x\to\infty} f(x) = 0.1$
4. $a\cdot f(a) = b\cdot f(b) - c\cdot f(c)$ s.t. $a=b-c$ for every $a,b,c$

I can easily go through features 1, 2 and 3 but I can't find anything suitable with number 4. Any help?

edit: I can also solve property number 4 with integrals, i.e. $\int f(6) = \int f(4) + \int f(2)$