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Cleon S. Barroso

Publications and source records attributed to Cleon S. Barroso.

At least 19 recordsLinked to original sources
arXiv2026

Fixed point results for mappings of asymptotically Hölder nonexpansive type

We introduce asymptotically Hölder nonexpansive mappings and mappings of asymptotically Hölder nonexpansive type, in which the Hölder exponents converge to one along the iterates. We prove that if \(K\) is a nonempty closed bounded convex subset of a Banach space \(X\) with characteristic of convexity \(ε_0(X)<1\), then every self-mapping of \(K\) of asymptotically Hölder nonexpansive type admits a point \(x\in K\) such that \(T^n x\to x\). Consequently, a fixed point exists whenever some positive iterate of \(T\) is continuous; in particular, this applies to asymptotically Hölder nonexpansive mappings. We also obtain fixed-point-free constructions in spaces containing copies of \(c_0\), and a related construction on a bounded convex, not necessarily closed, subset of every Banach space containing an isomorphic copy of \(\ell_1\). Further examples show that the new classes properly extend their classical counterparts and may contain mappings with no continuous positive iterate. Finally, we examine Lin's renorming of \(\ell_1\), identify an obstruction related to shift-type constructions, and establish a shrinking-diagonal criterion that reduces the remaining closed-set problem to the construction of a suitably controlled uniformly Lipschitzian mapping.

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arXiv2026

Transfinite contractive propagation and the ball fixed point property in \texorpdfstring{$C(K)$}{C(K)}

We prove that, for a compact Hausdorff space $K$, the real Banach space $C(K)$ has the ball fixed point property if and only if $K$ is extremally disconnected. The new implication is obtained by constructing a fixed-point-free nonexpansive self-map of the whole closed unit ball whenever $K$ is a compact $F$-space which is not extremally disconnected. The construction uses a transfinite contractive propagation system with two increasing profiles. A successor--limit delay preserves their tail constraint and has no subsolution in the profile domain. Analysis records boundary deficits; positive operators on ordinal $C_0$-spaces and a common center determined by the two tails yield a nonexpansive synthesis satisfying an order domination inequality. The required topological families are obtained from a gap in a maximal Boolean chain in the zero-dimensional case, and from a transfinite extension of signs otherwise. The argument works in ZFC and resolves the question posed by Avilés, Japón, Lennard, Martínez-Cervantes, and Stawski.

math.FA↗
arXiv2025

On $L$-$ω$-nonexpansive maps

We consider $L$-$ω$-nonexpansive maps $T\colon K\to K$ on a convex subset $K$ of a Banach space $X$, i.e., maps in which $ω_T(δ)\leq Lδ+ω(δ)$ with $L\in [0,1]$, $ω$ being a modulus of continuity and $ω_T$ is the minimal modulus of continuity of $T$. Both AFPP and FPP are studied. For moduli $ω$ with $ω'(0)=\infty$, we show that if $X$ contains an isomorphic copy of $\co$ then it fails the FPP for $0$-$ω$-nonexpansive maps with minimal displacement zero. In the affirmative direction, we prove for certain class of moduli $ω$ that $0$-$ω$-nonexpansive maps are constant on certain domains. Also, when $ω'(0)\leq 1-L$ we show that AFPP works and FPP also works under a monotonicity condition on $ω$. Further related results and examples are given.

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arXiv2023

Remarks on the FPP in Banach spaces with unconditional Schauder basis

This paper brings new results on the FPP in Banach spaces $X$ with a Schauder basis. We first deal with the problem of whether there is a Banach space isomorphic to $\co$ having the FPP. We show that the answer is negative if $X$ contains a pre-monotone basic sequence equivalent to the unit basis of $\co$. We then study sufficient conditions to ensure the existence of such sequences. Interesting results are obtained, including the case when $X$ has a $1$-suppression unconditional basis and its unit ball fails the PCP. With regarding the weak-FPP, we establish two fixed-point results. First, we show that under certain conditions this property is invariant under Banach-Mazur distance one. Next, it is shown that when the basis is either $1$-suppression unconditional or $1$-spreading then $X$ has the weak-FPP provided that a Rosenthal's type property on block basis is verified.

math.FA↗
arXiv2023

$\ell_1$ spreading models and FPP in Banach spaces with monotone Schauder basis

The main result of this paper is a fixed point result relating the spreading model structure of Banach spaces and Schauder basis with not too large basis constant. As a striking consequence, we deduce that every super-reflexive space has the fixed point property thus solving a long-standing open question in metric fixed point theory.

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arXiv2022

Hölder-contractive mappings, nonlinear extension problem and fixed point free results

For a bounded closed convex set $K$, in this note, we study the FPP for $α$-Hölder nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^α$ for all $x, y\in K$, $α\in (0,1)$. First, we note that only finite-dimensional spaces have the Hölder-FPP. Moreover, the unit ball $B_X$ of any infinite-dimensional space fails the FPP for Hölder maps with $\mathrm{d}(T, B_X)>0$, where $\mathrm{d}(T, K)$ denotes the minimal displacement of $T$. We further show that reflexivity and weak sequential continuity are sufficient conditions to capture fixed points of Hölder-Lipschitz maps with bounded orbits. Next we focus on the existence of fixed point free $α$-Hölder maps $T\colon K\to K$ with $\mathrm{d}(T, K)\leq φ(α)$ where either $φ(α)=0$ or $φ(α)\to 0$ as $α\to 1$. Interesting results are obtained for the spaces $\mathrm{c}$, $\co$, $\ell_1$ and $\ell_2$, and also for $L_p$-spaces with $p\in[ 1, \infty]$. We also study the problem in spaces containing copies of $\co$ and $\ell_1$. Some questions are left open.

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arXiv2021

A proof of the asymptotic conjecture

In this paper we prove that if $f$ is a self-mapping of a nonempty subset $K$ of a normed space $X$ that satisfies some mild conditions, then the minimal displacement of large iterations $f^n$ always dominates that of $f$ along certain $f^n$-invariant regions. As a consequence, we deduce that when $X$ is a Banach space, $K$ is closed convex and $f$ is continuous with $f^n$ being compact for some $n\geq 1$, then $f$ has at least one fixed point. This offers a new approach resulting in a streamlined proof of the long-standing asymptotic conjecture.

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arXiv2021

Isometric embeddings of Banach spaces under optimal projection constants

Let $X$ be a Banach space with separable dual. It is proved that for every $\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$ has further an FDD $(E_n)$ whose strong bimonotonicity projection constant is not larger than $\mathcal{D}$, then $(w_n)$ has strong bimonotonicity projection constant not exceeding $\mathcal{D}(1 +\varepsilon)$. Further, if $(E_n)$ is $\mathcal{C}$-unconditional then $(w_n)$ is $\mathcal{C}(1 + \varepsilon)$-unconditional. The proof uses renorming and skipped blocking decomposition techniques. As an application, we prove that every Banach space having a shrinking $\mathcal{D}$-unconditional basis with $\mathcal{D}<\sqrt{6}-1$, has the weak fixed point property.

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arXiv2020

Weak compactness and fixed point property for affine bi-Lipschitz maps

Let $X$ be a Banach space and let $C$ be a closed convex bounded subset of $X$. It is proved that $C$ is weakly compact if, and only if, $C$ has the {it generic} fixed point property ($\mathcal{G}$-FPP) for the class of $L$-bi-Lipschitz affine mappings for every $L>1$. It is also proved that if $X$ has Pełczyński's property $(u)$, then either $C$ is weakly compact, contains an $\ell_1$-sequence or a $\mathrm{c}_0$-summing basic sequence. In this case, weak compactness of $C$ is equivalent to the $\mathcal{G}$-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We also introduce a generalized form of property $(u)$, called {it property $(\mathfrak{su})$}, and use it to prove that if $X$ has property $(\mathfrak{su})$ then either $C$ is weakly compact or contains a wide-$(s)$ sequence which is uniformly shift equivalent. In this case, weak compactness in such spaces can also be characterized in terms of the $\mathcal{G}$-FPP for affine uniformly bi-Lipschitz mappings. It is also proved that every Banach space with a spreading basis has property $(\mathfrak{su})$, thus property $(\mathfrak{su})$ is stronger than property $(u)$. These results yield a significant strengthening of an important theorem of Benavides, Japón-Pineda and Prus published in 2004.

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arXiv2019

A note on asymptotically monotone basic sequences and well-separated sets

We remark that if $X$ is an infinite dimensional Banach space then every seminormalized weakly null sequence in $X$ has an asymptotic monotone basic subsequence. We also observe that if $X$ contains an isomorphic copy of $\ell_1$, then for every $\varepsilon>0$ there exist a $(1 +\varepsilon)$-equivalent norm $\vertiii{\cdot}$ on $X$ such that the unit sphere $(S_{(X, \vertiii{\cdot})})$ contains a normalized bimonotone basic sequences which is symmetrically $2$-separated.

math.FA↗
arXiv2019

Selection type results and fixed point property for affine bi-Lipschitz maps

We obtain a refinement of a selection principle for $(\mathcal{K}, λ)$-wide-$(s)$ sequences in Banach spaces due to Rosenthal. This result is then used to show that if $C$ is a bounded, non-weakly compact, closed convex subset of a Banach space $X$, then there exists a Hausdorff vector topology $τ$ on $X$ which is weaker than the weak topology, a closed, convex $τ$-compact subset $K$ of $C$ and an affine bi-Lipschitz map $T: K\to K$ without fixed points.

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arXiv2018

On the fixed point property in Banach spaces isomorphic to $c_0$

We prove that every Banach space containing a subspace isomorphic to $\co$ fails the fixed point property. The proof is based on an amalgamation approach involving a suitable combination of known results and techniques, including James's distortion theorem, Ramsey's combinatorial theorem, Brunel-Sucheston spreading model techniques and Dowling, Lennard and Turett's fixed point methodology employed in their characterization of weak compactness in $\co$.

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arXiv2015

On topological groups with an approximate fixed point property

A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $(x_i)$, $x_i\in C$, such that $x_{i}-gx_{i}\longrightarrow 0$ for all $g\in G$. We study the relationship of this property with amenability.

math.GR↗
arXiv2014

On the minimal space problem and a new result on existence of basic sequences in quasi-Banach spaces

We prove that if $X$ is a quasi-normed space which possesses an infinite countable dimensional subspace with a separating dual, then it admits a strictly weaker Hausdorff vector topology. Such a topology is constructed explicitly. As an immediate consequence, we obtain an improvement of a well-known result of Kalton-Shapiro and Drewnowski by showing that a quasi-Banach space contains a basic sequence if and only if it contains an infinite countable dimensional subspace whose dual is separating. We also use this result to highlight a new feature of the minimal quasi-Banach space constructed by Kalton. Namely, which all of its $\aleph_0$-dimensional subspaces fail to have a separating family of continuous linear functionals.

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arXiv2012

An interplay between the weak form of Peano's theorem and structural aspects of Banach spaces

In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space $E$ admits a fundamental biorthogonal system, then there exists a continuous vector field $f\colon E\to E$ such that the autonomous differential equation $u'=f(u)$ has no solutions at any time. The proof relies on a key result asserting that every infinite-dimensional Fréchet space with a fundamental biorthogonal system possesses a nontrivial separable quotient. The later, is the byproduct of a mixture of known results on barrelledness and two fundamental results of Banach space theory (namely, a result of Pełczyński on Banach spaces containing $L_1(μ)$ and the $\ell_1$-theorem of Rosenthal). Next, we introduce a natural notion of weak-approximate solutions for the non-autonomous Cauchy-Peano problem in Banach spaces, and prove that a necessary and sufficient condition for the existence of such an approximation is the absence of $\ell_1$-isomorphs inside the underline space. We also study a kind of algebraic genericity for the Cauchy-Peano problem in spaces $E$ having complemented subspaces with unconditional Schauder basis. It is proved that if $\mathscr{K}(E)$ denotes the family of all continuous vector fields $f\colon E\to E$ for which $u'=f(u)$ has no solutions at any time, then $\mathscr{K}(E)\bigcup \{0\}$ is spaceable in sense that it contains a closed infinite dimensional subspace of $C(E)$, the locally convex space of all continuous vector fields on $E$ with the linear topology of uniform convergence on bounded sets.

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arXiv2012

Optimal approximate fixed point results in locally convex spaces

Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate fixed point net. Next, it is shown that if $C$ is bounded but not totally bounded, then there is a uniformly continuous map $f\colon C\to C$ without approximate fixed point nets. We also exhibit an example of a sequentially continuous map defined on a compact convex set with no approximate fixed point sequence. In contrast, it is observed that every affine (not-necessarily continuous) self-mapping a bounded convex subset of a topological vector space has an approximate fixed point sequence. Moreover, it is constructed a affine sequentially continuous map from a compact convex set into itself without fixed points.

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arXiv2011

On the approximate fixed point property in abstract spaces

Let $X$ be a Hausdorff topological vector space, $X^*$ its topological dual and $Z$ a subset of $X^*$. In this paper, we establish some results concerning the $σ(X,Z)$-approximate fixed point property for bounded, closed convex subsets $C$ of $X$. Three major situations are studied. First when $Z$ is separable in the strong topology. Second when $X$ is a metrizable locally convex space and $Z=X^*$, and third when $X$ is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fréchet-Urysohn property for certain sets with regarding the $σ(X,Z)$-topology. The support tools include the Brouwer's fixed point theorem and an analogous version of the classical Rosenthal's $\ell_1$-theorem for $\ell_1$-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces.

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