In this paper, we consider the class of Lipschitz maps on the unit ball <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B_X$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation> of a Banach space X, and the question we deal with is whether for any <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> there exists a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-Lipschitz fixed-point free mapping <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T:B_X\rightarrow B_X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <msub> <mi>B</mi> <mi>X</mi> </msub> <mo stretchy="false">→</mo> <msub> <mi>B</mi> <mi>X</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{d}(T,B_X)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>d</mtext> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>B</mi> <mi>X</mi> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We also consider its Hölder version. New related results are obtained. We show that if X has a spreading Schauder basis then such mappings can always be built, answering a question posed by the first author in [<CitationRef CitationID="CR7">7</CitationRef>]. In the general case, using a recent approach of Medina [<CitationRef CitationID="CR33">33</CitationRef>] concerning Hölder retractions of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(r_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-flat closed convex sets, we show that for any decreasing null sequence <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(r_n)\subset \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, there exists a fixed-point free mapping T on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$B_X$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation> so that <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Vert T^nx - T^n y\Vert \le r_n(\Vert x - y\Vert ^\alpha +1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msup> <mi>T</mi> <mi>n</mi> </msup> <mi>x</mi> <mo>-</mo> <msup> <mi>T</mi> <mi>n</mi> </msup> <mrow> <mi>y</mi> <mo stretchy="false">‖</mo> <mo>≤</mo> </mrow> <msub> <mi>r</mi> <mi>n</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">‖</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">‖</mo> </mrow> <mi>α</mi> </msup> <mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$x, y\in B_X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>B</mi> <mi>X</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$n\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>.

Retraction methods and fixed point free maps on unit balls

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2026年5月28日

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Cleon S. Barroso,
Valdir Ferreira

摘要

In this paper, we consider the class of Lipschitz maps on the unit ball $B_X$ $B_{X}$ of a Banach space X, and the question we deal with is whether for any $\lambda >1$ $λ > 1$ there exists a $\lambda $ $λ$ -Lipschitz fixed-point free mapping $T:B_X\rightarrow B_X$ $T : B_{X} \to B_{X}$ with $\textrm{d}(T,B_X)=0$ $d (T, B_{X}) = 0$ . We also consider its Hölder version. New related results are obtained. We show that if X has a spreading Schauder basis then such mappings can always be built, answering a question posed by the first author in [7]. In the general case, using a recent approach of Medina [33] concerning Hölder retractions of $(r_n)$ $(r_{n})$ -flat closed convex sets, we show that for any decreasing null sequence $(r_n)\subset \mathbb {R}$ $(r_{n}) \subset R$ and $\alpha \in (0,1)$ $α \in (0, 1)$ , there exists a fixed-point free mapping T on $B_X$ $B_{X}$ so that $\Vert T^nx - T^n y\Vert \le r_n(\Vert x - y\Vert ^\alpha +1)$ $‖ T^{n} x - T^{n} y ‖ \leq r_{n} {(‖ x - y ‖}^{α} + 1)$ for all $x, y\in B_X$ $x, y \in B_{X}$ and $n\in \mathbb {N}$ $n \in N$ .