@Simple in the given inequality $$\lVert f(x)-x-(f(y)-y)\rVert\leq c\lVert x-y\rVert$$ just note $$ f(x)-x-(f(y)-y) = g_z(y)-g_z(x)$$

oh

Let $(M, d)$ be a complete metric space, let $T:M\to M$ be acontinuous map and let $\varphi:M\to\mathbb{R}$ be a function which is bounded below. Assume that together they satisfy $$d(x,Tx)\leq\varphi(x)-\varphi(Tx)$$ Prove that for every $x\in M$ the sequence $\{T^nx\}$ converges to a fixed point of $T$

I think we will use the banach fixed point theorem. We need to show $T$ is a contraction map