It seems like you don't need to find a "c" to show the existence of a fixed point of x such that

.

First, choose a point

in M and define

for

.

You might need to mark some points in M (to figure out the points indeed converge) and make sure each

where

.

Since is a contractive function, is a Cauchy sequence. Since M is a complete metric space, the sequence has a limit in M and we call it x. A contractive function in a metric space is a continuous, so

converges. We know that

is simply

whose limit is x. Thus

.

To show the uniqueness, suppose on the contrary that you have another point

. Now you can draw a contradiction if you check your distance function formula