how to prove $ F(\bar{x})=\bar{x} $ I need to have a hint. I even don't know how to start. Please leave your comments in the bellow.

Let 
  $(X,d)$ 
  be a complete metric space and let 
  $F:X\to X.$ 
  Define the 
  $ n-th $ 
  iteration of 
  $F$ 
  as follows: 
  $$ F_n(x) :=\underbrace{(F\circ F\circ\cdots \circ F)}_{n-times}(x), $$ 
  and assume for each 
  $ n \in \mathbb{N} $ 
  the existence of 
  $ \alpha_{n} \geq 0 $ 
  such that
  $$ d(F_{n}(x),F_{n}(y))\leq \alpha_{n}d(x,y),\;\; \forall x,y\in X. $$
  Prove that, if
  $ \sum_{n}\alpha_{n}<+\infty, $
  then there exists a unique
  $\bar{x}\in X$
  such that
  $ F(\bar{x})=\bar{x}. $

 A: First show that for some $k\in\Bbb N$, $\alpha_k\lt1$, which is obvious from $\sum_{n\in\Bbb N}\alpha_n\lt\infty$. So we have that $F_k$ is a contraction mapping, and hence has a unique fixed point $\bar x$ by Banach fixed point theorem.
To prove uniqueness of fixed point of $F$, let $y\in X$ be a fixed point of $F$ (if such a point $y$ exists) and show that $y=\bar x$, which is obvious because $y$ is a fixed point of $F_k$ as well and hence equal to $\bar x$, i.e. if $F$ has a fixed point, it would be $\bar x$.
To prove $\bar x$ is really a fixed point of $F$, we pay attention to the two points $F(\bar x),F_k(F(\bar x))$. Note that $F_k(F(\bar x))=F_{k+1}(\bar x)=F(F_k(\bar x))=F(\bar x)$, i.e. $F(\bar x)$ is also a fixed point of $F_k$, which means $F(\bar x)=\bar x$ by uniqueness of fixed point of $F_k$.
A: This is a very weak generalization of the Banach fixed point theorem and is easily google-able. If you're interested in tackling it on your own, here's an outline:


*

*establish uniqueness by assuming $F(x) = x$ and $F(y) = y$ and proving that $x = y$ (hint: you need an implication of $\sum a_n < \infty$ and $a_n \geq 0$

*establish the continuity of $F$ (hint: this is effectively given to you in the problem)

*establish existence of $\bar x$ by taking an arbitrary $x \in X$ and considering the sequence $F(x), F_2(x), F_3(x),\dots$. Show this sequence converges by showing that it is Cauchy (hint: you need that the partial sums of $\sum a_n$ are Cauchy)

*Call the limit $\bar x$. Show that $F(\bar x) = \bar x$ by considering $\lim F_n(x)$ and $\lim F_{n+1}(x)$ and using part 2.

