The following is from O. Zeitouni's "Lecture Notes on Random Walks in Random Environments":
Suppose I have a function $V_{a,b,c}:\mathbb{Z}\longrightarrow\mathbb{R}$ (for $a,b\in\mathbb{Z}$ and $c$ a probability environment) that satisfies \begin{equation*} \begin{matrix}(w^{+}_{z}+w^{-}_{z})V_{a,b,w} & = & w ^{-}_{z}V_{a,b,w}(z)+w^{-}_{z}V_{a,b,w} (z+1)\\ V_{a,b,w}(-a) & = & 1\\ V_{a,b,w}(b)& = & 0\text{.} \end{matrix} \end{equation*} Zeitouni claims the function \begin{equation} V_{a,b,w}(z)=\frac{\sum_{i=z+1}^{b}\prod_{j=z+1}^{i-1}\rho_{j}}{\sum_{i=z+1}^{b}\prod_{j=z+1}^{i-1}\rho_{j}+\sum_{i=-a+1}^{z}\prod_{j=i}^{z}\rho_{j}^{-1}}\text{.} \end{equation} where $\rho_{i}=\frac{w^{+}_{i}}{w^{-}_{i}}$ and $w^{+}_{i},w^{-}_{i},w^{0}_{i}$ are the probabilities to walk right, left or to stay at $i$ respectively.
It is easy to see that this does indeed solve the equation. Zeitouni now makes the claim that this solution is unique "due to the maximum principle".
Why? What maximum principle applies here?