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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 $0<L<1$ and the series $$ \sum_{n=1}^{\infty} w^n(t)$$ is uniformly convergent in $(0,1)$.

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

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For example each function $f: (0,1)\rightarrow \mathbb{R}$ of the form: $$f(t)=t-bt^{\alpha+1},$$ where $\alpha \in (0,1)$ and $0<b\leq\frac{1}{1+\alpha}$ are arbitrary constants, satisfies requirments. For details see J.W Thron estimation for iterations of function of the form $v(t)=t-bt^{\alpha+1}$. (answered by Robert Israel).

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