How to solve the following equation for $F(x)$?
$$ (1-\alpha)\beta \cdot F(x+c) - \alpha \cdot F(x) = \alpha + (1-\alpha)\beta - {\alpha \cdot \gamma \over x + \phi}$$
where $\alpha, \beta, \gamma, \phi$ are all constant and $\in (0,1)$; $F(x)$ needs to be a $cdf$.
Or more general, is there a formal way to simplify and solve an equation for $f(x)$ of the following form:
$$ f(x+c) + f(x) = g(x)$$ where $g(x)$ is known?
Thank you in advance!