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<p>Let <i>X</i> be a unital algebra with unit <i>e</i> and <i>Y</i> a real Hausdorff topological vector space. In this article, we obtain the general set-valued solution of the Davison functional equation <Equation ID="Equ31"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10_2025_1157_Article_Equ31.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="279" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} F(xy+x)+F(y)=F(xy)+F(x+y) \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for functions <i>F</i> defined on <i>X</i> with values in the set of nonempty compact and convex subsets of <i>Y</i>. Additionally, we investigate various extensions of the Davison set-valued functional equation.</p>

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A study on Davison’s functional equation for set-valued functions

  • 期刊论文

  • 发表日期：

    2025年4月16日

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  • Elham Mohammadi,
  • Abbas Najati,
  • Iz-iddine EL-Fassi

摘要

Let X be a unital algebra with unit e and Y a real Hausdorff topological vector space. In this article, we obtain the general set-valued solution of the Davison functional equation \(\begin{aligned} F(xy+x)+F(y)=F(xy)+F(x+y) \end{aligned}\) F ( x y + x ) + F ( y ) = F ( x y ) + F ( x + y ) for functions F defined on X with values in the set of nonempty compact and convex subsets of Y. Additionally, we investigate various extensions of the Davison set-valued functional equation.