- 222

- 0

Let ##X## be a complete metric space and let ##T:X \to X## such that there exists ##n \in \mathbb N##: ##T^n## is a contraction. Prove that there is a unique ##x \in X## such that ##T(x)=x##.

The attempt at a solution.

Sorry but I am completely lost with this exercise and I am a little bit confused about the following: if ##T^n## is a contraction, would this mean that for any ##x,y \in X## ##d(T^n(x), T^n(y))<αd(x,y)## or ##d(T^n(x), T^n(y))<αd(T^{n-1}(x),T^{n-1}(y))##, for ##0<α<1##?. How could I begin the exercise to find the ##x## such that ##T(x)=x##?.