<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_894_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation> be a real normed linear space and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_894_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {Y}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">Y</mi> </math></EquationSource> </InlineEquation> be a Banach space. In this work, we demonstrate the Hyers-Ulam stability theorem and the associated hyperstability results for the generalized cubic functional equation <Equation ID="Equ20"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_894_Article_Equ20.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="465" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned}\varphi (cx + y) + \varphi (cx-y)= c\varphi (x+y) +c\varphi (x- y) +2c(c^2-1)\varphi (x)\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>c</mi> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>-</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>c</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>within some restricted domains, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_894_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$\varphi :\mathcal {X}\rightarrow \mathcal {Y}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>:</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">Y</mi> </mrow> </math></EquationSource> </InlineEquation> is an unknown mapping and the parameter <i>c</i> denotes a fixed integer, excluding 0, 1 and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_894_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The asymptotic properties of this functional equation are discussed as an application.</p>

A cubic functional equation of the Euler–Lagrange type on restricted domains

期刊论文
发表日期：

2025年2月24日

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Saeid Ostadbashi,
Abbas Najati,
Iz-iddine EL-Fassi

摘要

Let $\mathcal {X}$ $X$ be a real normed linear space and $\mathcal {Y}$ $Y$ be a Banach space. In this work, we demonstrate the Hyers-Ulam stability theorem and the associated hyperstability results for the generalized cubic functional equation $\begin{aligned}\varphi (cx + y) + \varphi (cx-y)= c\varphi (x+y) +c\varphi (x- y) +2c(c^2-1)\varphi (x)\end{aligned}$ $\begin{matrix} φ (c x + y) + φ (c x - y) = c φ (x + y) + c φ (x - y) + 2 c (c^{2} - 1) φ (x) \end{matrix}$ within some restricted domains, where $\varphi :\mathcal {X}\rightarrow \mathcal {Y}$ $φ : X \to Y$ is an unknown mapping and the parameter c denotes a fixed integer, excluding 0, 1 and $-1$ $- 1$ . The asymptotic properties of this functional equation are discussed as an application.