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.