Let $(X, \rho)$ be a metric space and suppose $f: X\rightarrow X$ to be a continuous function. We say that the function $f$ admits $w: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ as a modulus of continuity if $\rho(f(x),f(y))\leqslant w(\rho(x,y))\;\;\; (x,y\in X).$ Clearly, when $f$ is a contraction then $w(t)=Lt$ ($t\in\mathbb{R}^+$), where $0
I am looking for an example of a function $f$ on noncompact (but at least separable and complete) metric space which is not contraction but admits the modulus of continuity $w$ such that the series $ \sum_{n=1}^{\infty} \varphi(w^n(t))$ is uniformly convergent in some neighborhood of zero, where $\varphi:\mathbb{R}^+\rightarrow\mathbb{R}^+$ is an arbitrary continuous and nondecreasing function.
I'll be very grateful for every hint