For questions involving the Banach Fixed Point Theorem and its applications
Let $(X,d)$ be a metric space and $T$ a map from $X$ to itself. The map $T$ is called a contraction if there is some $k\in[0,1)$ such that$$(\forall x,y\in X):d\bigl(T(x),T(y)\bigr)\leqslant kd(x,y).$$The Banach fixed point theorem states that if $(X,d)$ is complete and $T$ is a contraction, then $T$ has one and only one fixed point (that is, a point $x\in X$ such that $T(x)=x$).
The fact that the fixed point is unique is very easy to establish, even without the assumption that $X$ is complete. In fact, if $x$ and $y$ are fixed points, then$$d(x,y)=d\bigl(T(x),T(y)\bigr)\leqslant kd(x,y)$$and therefore, if $d(x,y)\neq0$, we would have $1\leqslant k$, which is false. So, $d(x,y)=0$, which means that $x=y$. On the other hand, the completeness of $X$ is essential for the existence of a fixed point.
