Subinvariant Metric Functionals for Nonexpansive Mappings
Main results
We study common subinvariant metric functionals for commuting families of nonexpansive mappings on Banach spaces, without imposing compactness assumptions. We also investigate conditions under which subinvariant metric functionals yield fixed points and examine the geometry of metric functionals on Banach spaces.
1. Common subinvariant metric functionals for affine mappings
Theorem 3.1. Let \(E\) be a Banach space and \(\mathcal{F}\) a commuting family of affine nonexpansive mappings on \(E\). Suppose \(X\subseteq E\) is nonempty and convex, and that every \(T\in\mathcal{F}\) satisfies
\[T(X)\subseteq X,\qquad \inf_{x\in X}\|x-Tx\|=0.\]Then there exists a metric functional \(h\in X^\diamondsuit\) such that
\[h(Tx)\leq h(x)\qquad\text{for all }x\in E,\ T\in\mathcal{F}.\]Thus, the entire commuting family admits a common subinvariant metric functional, even when \(X\) is not compact.
2. Extension to general nonexpansive mappings
Theorem 3.3. We extend the existence result to commuting families of nonexpansive mappings without assuming affinity or convexity.
More precisely, let \(\mathcal{F}\) be a commuting family of nonexpansive mappings on a Banach space \(E\), and let \(X\subseteq E\) be nonempty and invariant under every mapping in \(\mathcal{F}\). If there exists \(x_0\in X\) such that, for every \(T\in\mathcal{F}\),
\[\lim_{n\to\infty}\|T^n x_0-T^{n+1}x_0\|=0,\]then there exists a metric functional \(h\in X^\diamondsuit\) satisfying
\[h(Tx)\leq h(x)\qquad\text{for all }x\in E,\ T\in\mathcal{F}.\]3. A fixed-point theorem using the Opial property
Theorem 3.6. Let \(X\) be a nonempty weakly compact subset of a Banach space having the Opial property. If a mapping \(T:X\to X\) admits a subinvariant metric functional \(h\in X^\diamondsuit\), then \(T\) has a fixed point in \(X\).
Notably, nonexpansiveness of \(T\) is not required in this theorem.
This result connects the existence of subinvariant metric functionals with fixed-point theory through the geometry of the underlying Banach space.
4. Metric functionals associated with rays
Proposition 4.12. Suppose \(X\) contains a ray \(\{tu:t\geq 0\}\), where \(\|u\|=1\). Then the limit
\[h(x)=\lim_{t\to\infty}(\|x-tu\|-t)\]defines a metric functional in \(X^\diamondsuit\), and
\[h(su)=-s\qquad\text{for every }s\geq 0.\]In particular, \(h\) is unbounded below. We also express this functional in terms of the subdifferential of the norm:
\[h(x)=\max\{-\langle x,f\rangle:f\in\partial\|\cdot\|(u)\}.\]5. Metric functionals on uniformly smooth Banach spaces
Theorem 4.16. If \(E\) is uniformly smooth, every metric functional arising from a net escaping to infinity is a continuous linear functional of norm at most one.
Conversely, every continuous linear functional of norm at most one can be obtained as a pointwise limit of metric functionals associated with points of \(E\).
This provides a characterization of the metric functionals arising at infinity in uniformly smooth Banach spaces.
6. Nontrivial metric functionals and the zero-free property
We introduce the zero-free property (ZFP): a metric space \(X\) has ZFP if none of its metric functionals vanishes identically.
We establish several results describing when this property holds:
- Proposition 4.17. The Banach space \(B(S)\) of bounded real-valued functions on any nonempty set \(S\), equipped with the supremum norm, has ZFP.
- Proposition 4.18. If \(K\) is a compact Hausdorff space, then \(C(K)\), equipped with the supremum norm, has ZFP.
- Proposition 4.20. If a Banach space \(E\) has the Radon–Riesz property and its closed unit ball \(B_E\) is weakly compact, then \(B_E\) has ZFP.
We also investigate how ZFP behaves under direct sums and changes of metric, showing that it depends on the geometry of the underlying metric space.
7. Unique minimizers of metric functionals
We introduce the unique minimizer property (UMP): a metric space \(X\) has UMP if every metric functional \(h\in X^\diamondsuit\) attains its minimum at exactly one point of \(X\).
Proposition 4.22. If \(E\) is a Banach space with the Opial property and \(X\subseteq E\) is nonempty and weakly compact, then \(X\) has UMP.
Consequently, every metric functional \(h\in X^\diamondsuit\) has a unique minimizer \(a\in X\):
\[h(a) < h(x)\qquad\text{for all }x\in X\setminus\{a\}.\]Proposition 4.25. Let \(X\) be a bounded subset of a uniformly convex Banach space \(E\). If a sequence \((a_n)\subseteq X\) generates a metric functional \(h\in X^\diamondsuit\), then \(h\) has a unique minimizer in \(E\), although this minimizer need not belong to \(X\).
These results explain how the geometry of Banach spaces can turn the existence of subinvariant metric functionals into fixed-point information. In particular, when \(X\) has UMP and \(h\in X^\diamondsuit\) is subinvariant under \(T:X\to X\), the unique minimizer of \(h\) must be a fixed point of \(T\).
8. Examples in classical sequence spaces
We investigate the affine shift
\[T(x_1,x_2,\ldots)=(1,x_1,x_2,\ldots)\]on several classical sequence spaces.
The examples reveal substantially different behavior: on \(\ell_2\), the only subinvariant metric functional for this mapping is the zero functional, whereas on \(\ell_1\) there are infinitely many nontrivial subinvariant metric functionals.
For instance, on \(\ell_1\),
\[h(x)=\sum_{k=1}^{\infty}(|x_k-1|-1)\]satisfies
\[h(Tx)=h(x)-1.\]These examples illustrate how the geometry of the ambient Banach space influences the structure and usefulness of subinvariant metric functionals.
