The setup of my problem is as follows. I have a mapping $f:X \rightarrow X$ where $X$ is a general space that may be changed through adjusting a set of parameters that define it. $f$ is always a contraction mapping over $X$, regardless of the parameters we use to define $X$.
For almost all parameter choices that define $X$, sequential composition of $f$ will converge to a fixed point in the interior of $X$ for any initial point. However, in cases of ambiguity, for a subset of parameter choices, iteration of $f$ over any initial point in $X$ will converge to the boundary of $X$, which is always a fixed point.
I am mostly concerned with the first case where application of $f$ converges to a fixed point in the interior of $X$, so our entire space is just the interior of $X$ without the boundary. Is there some specific way I should go about studying such a system? How can I deal with the edge cases when the the parameters defining $X$ are such that the map converges to the boundary of the closure of $X$?
Example: Consider the real interval $(0, a)$ where $a \in \mathbb{R}^+$, and let $f$ be a contraction mapping over this interval. Then, if $a$ is a positive integer, for any point $x \in (0, a)$, $f^k(x) \rightarrow a$ as $k \rightarrow \infty$. However if $a \in \mathbb{R}^+ \setminus \mathbb{N}^+$, then for any point $x \in (0, a)$, $f^k(x) \rightarrow b$ as $k \rightarrow \infty$, where $b \in (0, a)$. As has been pointed out in the comments, such an interval $(0, a)$ is incomplete, and thus the Banach fixed point theorem cannot be used. So then how can I prove that such a map will converge to a fixed point? What if instead we took $[0, a]$ which is complete, and let $f(a), f(0)$ be undefined?