F. Pittnauer’s Fixed Point Theorem. Let $(X,d)$ be a complete metric space and $T:X\rightarrow X$ continuous. Assume that there exists an integer $n$ and $0\leq k<1$ such that $$d(Tx,Ty)\leq k[d(x,T^nz)+d(y,T^nz)]$$ for all $x,y,z\in X$. Then $T$ has a unique fixed point.
I have been struggling proving this problem. I tried to apply the triangle inequality on the right hand side of the given inequality to simplify the terms, but it is getting messy by doing so. I am lost, so I literally need help.
The uniqueness is easy.
For the existence, define $x_p=T^p(x_0)$ for any $x_0$ and show that for $p \geq n$, $d(Tx_p,Tx_{p+1}) \leq kd(x_p,T^nx_{p-n})+kd(x_{p+1},T^nx_{p-n})$.
As a consequence, $\sum_p{d(x_p,x_{p+1})}$ converges, thus $(T^p(x_0))_p$ converges and the rest is standard.