Let X be a normed space and Y be an ulrametric n-Banach space. In this paper, we investigate some hyperstability results for the following quadratic functional equation <Equation ID="Equ14"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_916_Article_Equ14.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="321" /> </MediaObject> <EquationSource Format="TEX">$f\big (x+\sigma (y)\big )+f\big (x+\tau (y)\big )=2f(x)+2f(y),$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_916_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$f:X\rightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> is a mapping and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_916_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 involutions. We also examine the hyperstability of the given equation in its inhomogeneous version <Equation ID="Equ15"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_916_Article_Equ15.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="390" /> </MediaObject> <EquationSource Format="TEX">$f\big (x+\sigma (y)\big )+f\big (x+\tau (y)\big )=2f(x)+2f(y)+\chi (x,y),$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_916_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$\chi :X\times X\rightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we elucidate the hyperstability of various special cases of our main results.

Hyperstability of a quadratic functional equation with involutions in ultrametric n-Banach spaces via fixed point approach

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发表日期：

2025年7月21日

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Mohamed El Fatini,
Muaadh Almahalebi,
Choonkil Park,
Ahmad M. Alghamdii

摘要

Let X be a normed space and Y be an ulrametric n-Banach space. In this paper, we investigate some hyperstability results for the following quadratic functional equation $f\big (x+\sigma (y)\big )+f\big (x+\tau (y)\big )=2f(x)+2f(y),$ $f (x + σ (y)) + f (x + τ (y)) = 2 f (x) + 2 f (y),$ where $f:X\rightarrow Y$ $f : X \to Y$ is a mapping and $\sigma ,\tau :X\rightarrow X$ $σ, τ : X \to X$ are involutions. We also examine the hyperstability of the given equation in its inhomogeneous version $f\big (x+\sigma (y)\big )+f\big (x+\tau (y)\big )=2f(x)+2f(y)+\chi (x,y),$ $f (x + σ (y)) + f (x + τ (y)) = 2 f (x) + 2 f (y) + χ (x, y),$ where $\chi :X\times X\rightarrow Y$ $χ : X \times X \to Y$ . Additionally, we elucidate the hyperstability of various special cases of our main results.