<p>Using the Brzdȩk fixed point theorem, we establish the Hyers–Ulam stability of the following generalized Drygas functional equation <Equation ID="Equ25"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1246_Article_Equ25.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="415" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \mathcal {F}(x+\sigma (y))+\mathcal {F}(x+\tau (y))=2\mathcal {F}(x)+\mathcal {F}(\sigma (y))+\mathcal {F}(\tau (y)) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for all <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1246_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$x,y\in X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1246_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma ,\tau : X\rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>,</mo> <mi>τ</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> are additive isometries.</p>

Stability of a generalized Drygas functional equation via Brzdȩk’s fixed point method

期刊论文
发表日期：

2025年1月28日

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Mehdi Dehghanian,
Choonkil Park,
Yamin Sayyari

摘要

Using the Brzdȩk fixed point theorem, we establish the Hyers–Ulam stability of the following generalized Drygas functional equation $\begin{aligned} \mathcal {F}(x+\sigma (y))+\mathcal {F}(x+\tau (y))=2\mathcal {F}(x)+\mathcal {F}(\sigma (y))+\mathcal {F}(\tau (y)) \end{aligned}$ $\begin{matrix} F (x + σ (y)) + F (x + τ (y)) = 2 F (x) + F (σ (y)) + F (τ (y)) \end{matrix}$ for all $x,y\in X$ $x, y \in X$ , where $\sigma ,\tau : X\rightarrow X$ $σ, τ : X \to X$ are additive isometries.