Let X, Y be real vector metric spaces and K be a closed convex cone in Y. We prove that every K-continuous K-sublinear set-valued map <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">$F:X\rightarrow 2^Y\setminus \{\emptyset \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <msup> <mn>2</mn> <mi>Y</mi> </msup> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mrow> <mo stretchy="false">{</mo> <mi mathvariant="normal">∅</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which has compact convex values, satisfies <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$F(tx)=_KtF(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>=</mo> <mi>K</mi> </msub> <mi>t</mi> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$t\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we show that in the case <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$X={\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, F is K-continuous and K-sublinear if and only if there are convex compact sets <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$A,B\subset Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>⊂</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq7.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$A-B\subset K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>-</mo> <mi>B</mi> <mo>⊂</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$F(t)=_KtA$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>=</mo> <mi>K</mi> </msub> <mi>t</mi> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$t\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="88" /> </InlineMediaObject> <EquationSource Format="TEX">$F(t)=_KtB$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>=</mo> <mi>K</mi> </msub> <mi>t</mi> <mi>B</mi> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_664_Article_IEq11.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$t<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Our results refer mainly to the papers [<CitationRef CitationID="CR7">7</CitationRef>] and [<CitationRef CitationID="CR9">9</CitationRef>].

On K-sublinear set-valued maps

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2025年8月1日

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Eliza Jabłońska

摘要

Let X, Y be real vector metric spaces and K be a closed convex cone in Y. We prove that every K-continuous K-sublinear set-valued map $F:X\rightarrow 2^Y\setminus \{\emptyset \}$ $F : X \to 2^{Y} \ {\emptyset}$ , which has compact convex values, satisfies $F(tx)=_KtF(x)$ $F (t x) =_{K} t F (x)$ for $x\in X$ $x \in X$ and $t\ge 0$ $t \geq 0$ . Moreover, we show that in the case $X={\mathbb {R}}$ $X = R$ , F is K-continuous and K-sublinear if and only if there are convex compact sets $A,B\subset Y$ $A, B \subset Y$ such that $A-B\subset K$ $A - B \subset K$ and $F(t)=_KtA$ $F (t) =_{K} t A$ for $t\ge 0$ $t \geq 0$ and $F(t)=_KtB$ $F (t) =_{K} t B$ for $t<0$ $t < 0$ . Our results refer mainly to the papers [7] and [9].