This research examines the Hyers–Ulam stability of norm-additive functional equations expressed as <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2024_3233_Article_Equa.gif" Format="GIF" Height="46" Rendition="HTML" Resolution="72" Type="Linedraw" Width="199" /> </MediaObject> <EquationSource Format="MATHML"><math> <mtable> <mtr> <mtd columnalign="left"> <mo stretchy="false">∥</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <msup> <mi>h</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">∥</mo> <mo>=</mo> <mo stretchy="false">∥</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">∥</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mo stretchy="false">∥</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">∥</mo> <mo>=</mo> <mo stretchy="false">∥</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ξ</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">∥</mo> <mo>,</mo> </mtd> </mtr> </mtable> </math></EquationSource> <EquationSource Format="TEX"> $\begin{aligned}& \|\xi (gh^{-1})\|=\|\xi (g)-\xi (h)\|,\\& \|\xi (gh)\|=\|\xi (g)+\xi (h)\|, \end{aligned}$ </EquationSource> </Equation> through the utilization of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2024_3233_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>,</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\delta , \epsilon )$</EquationSource> </InlineEquation>-isometries. In this context, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2024_3233_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>ξ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>X</mi> </math></EquationSource> <EquationSource Format="TEX">$\xi : G \to X$</EquationSource> </InlineEquation> represents a surjective (δ-surjective) mapping, where G denotes noncommutative group (arbitrary group) and X signifies a Banach space (or a real Banach space).

Hyers–Ulam stability of norm-additive functional equations via $(\delta , \epsilon )$-isometries

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2025年1月3日

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Muhammad Sarfraz,
Jiang Zhou,
Yongjin Li

摘要

This research examines the Hyers–Ulam stability of norm-additive functional equations expressed as $\begin{matrix} ∥ ξ (g h^{- 1}) ∥ = ∥ ξ (g) - ξ (h) ∥, \\ ∥ ξ (g h) ∥ = ∥ ξ (g) + ξ (h) ∥, \end{matrix}$ $\begin{aligned}& \|\xi (gh^{-1})\|=\|\xi (g)-\xi (h)\|,\\& \|\xi (gh)\|=\|\xi (g)+\xi (h)\|, \end{aligned}$ through the utilization of $(δ, ϵ)$ $(\delta , \epsilon )$ -isometries. In this context, $ξ : G \to X$ $\xi : G \to X$ represents a surjective (δ-surjective) mapping, where G denotes noncommutative group (arbitrary group) and X signifies a Banach space (or a real Banach space).

Hyers–Ulam stability of norm-additive functional equations via \((\delta , \epsilon )\)-isometries

摘要