Firm non-expansive mappings in weak metric spaces
Main results
We introduce firm non-expansive mappings in arbitrary weak metric spaces, extending the classical notion beyond Banach spaces and geodesic metric spaces.
1. Equality of asymptotic quantities
Our main theorem establishes that, for every firm non-expansive mapping \(T:X\to X\) on a weak metric space \((X,\delta)\), every \(x\in X\), and every integer \(k\geq 1\),
\[\begin{aligned} \inf_{y\in X}\delta(y,Ty) &= \lim_{n\to\infty}\frac{\delta(x,T^n x)}{n} \\ &= \lim_{n\to\infty}\delta(T^n x,T^{n+1}x) \\ &= \frac{1}{k}\lim_{n\to\infty}\delta(T^n x,T^{n+k}x). \end{aligned}\]Thus, the minimal displacement, linear rate of escape, and asymptotic step size coincide. This generalizes the Reich–Shafrir theorem without requiring a linear or geodesic structure.
2. Metric functionals and zero escape rate
If \(T\) is a firm non-expansive mapping of a metric space with zero linear escape rate, then there exists a metric functional \(h\) such that
\[h(Tx)\leq h(x)\qquad\text{for every }x\in X.\]Consequently, \(h\) is non-increasing along every orbit of \(T\).
3. Monotone escape in finite-dimensional normed spaces
If \(T\) is a firm non-expansive mapping of a finite-dimensional normed space with zero linear escape rate, then either \(T\) has bounded orbits or there exists a metric functional \(h\) such that
\[h(T^n x)\longrightarrow -\infty\qquad\text{monotonically, for every }x.\]Moreover, \(h\) can be chosen as a limit point of the orbit \((T^n0)_{n\geq 0}\).
4. A broader class of mappings
We show that this notion includes classical firmly non-expansive mappings in Banach spaces and \(W\)-hyperbolic spaces, while also admitting mappings that are not firmly non-expansive in the classical sense. We additionally characterize firmness for non-expansive mappings on one-dimensional asymmetric normed spaces.
