<p>We consider the conditional Jensen functional equation<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1527_Article_Equa.gif" Format="GIF" Height="35" Rendition="HTML" Resolution="72" Type="Linedraw" Width="403" /> </MediaObject> <EquationSource Format="TEX">$f(x+y)\ne 0 \implies 2f\big(\frac{x+y}{2}\big)=f(x)+f(y), \quad x,y\in \mathcal{G}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mn>2</mn> <mi>f</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mfrac> <mrow> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mrow> <mn>2</mn> </mfrac> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi mathvariant="script">G</mi> </mrow> </math></EquationSource> </Equation>for functions <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1527_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$f \colon \mathcal{G} \to\mathcal{V}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo lspace="0pt">:</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">V</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1527_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$(\mathcal{G},+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1527_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$(\mathcal{V},+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">V</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are uniquely 2-divisible groups,with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1527_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$(\mathcal{V},+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">V</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> being abelian. Additionally, we investigate the hyperstability of thisfunctional equation.</p>

On Jensen-like form of Mikusiński's functional equation

期刊论文
发表日期：

2025年4月1日

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E. Imani,
A. Najati,
M. A. Tareeghee

摘要

We consider the conditional Jensen functional equation $f(x+y)\ne 0 \implies 2f\big(\frac{x+y}{2}\big)=f(x)+f(y), \quad x,y\in \mathcal{G}$ $f (x + y) \neq 0 \Rightarrow 2 f (\frac{x + y}{2}) = f (x) + f (y), x, y \in G$ for functions $f \colon \mathcal{G} \to\mathcal{V}$ $f : G \to V$ , where $(\mathcal{G},+)$ $(G, +)$ and $(\mathcal{V},+)$ $(V, +)$ are uniquely 2-divisible groups,with $(\mathcal{V},+)$ $(V, +)$ being abelian. Additionally, we investigate the hyperstability of thisfunctional equation.