\text{}\text{card} \, F_T \le 1A contractive mapping is a function on a metric space that reduces the distance between any two points in the space. It is also known as a contraction or a contracting map.
A unique fixed point of a contractive mapping is a point in the metric space that remains unchanged when the mapping is applied. It is unique because there can only be one such point for a contractive mapping.
The Banach Fixed Point Theorem states that if a contractive mapping is applied on a complete metric space, then it must have a unique fixed point. Therefore, to prove the existence of a unique fixed point, one must show that the metric space is complete and the mapping is contractive.
No, a non-contractive mapping can have multiple fixed points, but it cannot have a unique fixed point. This is because a non-contractive mapping does not reduce the distance between points, so there can be more than one point that remains unchanged when the mapping is applied.
The most common notation used for contractive mappings is the "c" notation, where "c" represents the contractive constant. For example, a mapping f is said to be contractive if there exists a constant c < 1 such that for any two points x and y in the metric space, the distance between f(x) and f(y) is less than c times the distance between x and y.