Notion of fixed point for the sequence $x_{n+1}=f_n(x_n)$ Let $(X,d)$ be a complete metric space and let $f_n:X\to X,\ n\in \mathbb{N}$ be a sequence of contractions. 

Is there any notion of fixed point for the following sequence? $$x_{n+1}=f_n(x_n),\ n\in \mathbb{N},\ x_1\in X$$ 

I know that when $f_n$ are all the same function $f$ $\ \forall n$, and $f$ is a contraction then by Banach fixed point theorem the sequence $\{x_n\}$ converges to the unique fixed point of $f$. But I was unable to find a similar result for the present case. The idea that I have is that maybe we have to have some kind of mutual contraction relations between the different functions, i.e. we have to have some $L(m,n)\in (0,1),\ \forall n,m\in \mathbb{N}$ such that $d(f_m(x),f_n(y))\le L(m,n) d(x,y)\ \forall x,y\in X,\ \forall m,n\in \mathbb{N}$.
Can anybody kindly provide any related literature or maybe some idea regarding analysis of this kind of sequence? Thanks in advance.
 A: Okay, I come up with a little bit technical condition. Basically, I specify the rate of convergence for $f_n\to f$.
I am lazy and assume $X$ is a closed subset of some Banach space (just for notation) and write the norm  as $|\cdot|$. Supremum norm for bounded functions from $X$ to $X$ is denoted by $\|\cdot\|$.
For $n\in\mathbb N$ let $e_{n} = \| f_{n+1} - f_{n} \|$.
Assume $\sum_{n=0}^\infty e_n < \infty$ and there is some $c < 1$ such $|f_n(y) - f_n(x)| \le c |y-x|$ for every $n\in\mathbb N$ and $x,y\in X$.
I will modify the usual proof of Banach fix point theorem and show that $(x_n)_{n\in\mathbb N}$ is a Cauchy sequence.
For large $n$ notice that
\begin{align} 
|x_{n+1} - x_n| 
&= |f_n(x_n) - f_{n-1}(x_{n-1})| \\
&= |f_n(x_n) - f_{n-1}(x_{n}) + f_{n-1}(x_{n}) - f_{n-1}(x_{n-1})| \\
&\le e_{n-1} + c |x_n - x_{n-1}| \\
&\le e_{n-1} + c e_{n-2} + c^2 |x_{n-1} - x_{n-2}| \\
&\le \underbrace{\sum_{k=0}^{n-1} c^k e_{n-1-k}}_{=:\eta_{n-1}} + c^n |x_{1} - x_{0}| \\
\end{align}
Thus, for $m\ge 1$ we have
\begin{align} 
|x_{n+m} - x_n| 
&\le |x_{n+m} - x_{n+m-1}| + \dotsb + |x_{n+1} - x_n| \\
&\le \eta_{n+m-2} + c^{n+m-1} |x_1 - x_0| + \dotsb + \eta_{n-1} + c^n |x_1 - x_0| \\
&= \underbrace{\sum_{k=n-1}^{n+m-2} \eta_{k}}_{(*)} + |x_1 - x_0| \underbrace{\sum_{k=n}^{n+m-1} c^{k}}_{(**)}.
\end{align}
As $\eta_{n-1}$ is a summand of a Cauchy product, the sum $(*)$ converges to $0$ for $n\to\infty$. The sum $(**)$ also converges to $0$, as geometric series converges.
Thus, $(x_n)_{n\in\mathbb N}$ is a Cauchy sequence and has a limit, which is a fix point of $f$.
