<p>In this research, we use a fixed point theorem to investigate the Hyers-Ulam stability in matrix fuzzy normed space (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_930_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {MFNS}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">MFNS</mi> </math></EquationSource> </InlineEquation>) connected to the generalized Jensen functional equation <Equation ID="Equ11"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_930_Article_Equ11.gif" Format="GIF" Height="55" Rendition="HTML" Resolution="72" Type="Linedraw" Width="411" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} \sum _{i=1}^mF(a_i)=\frac{1}{2m}\Bigg [\sum _{i=1}^mF\Big (ma_i+\sum _{j=1,j\ne i}^ma_j\Big )+F\Big (\sum _{i=1}^ma_i\Big )\Bigg ], \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> <mrow> <mo maxsize="2.470em" minsize="2.470em" stretchy="true">[</mo> </mrow> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mi>F</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mi>m</mi> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> <mi>m</mi> </munderover> <msub> <mi>a</mi> <mi>j</mi> </msub> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <msub> <mi>a</mi> <mi>i</mi> </msub> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mrow> <mo maxsize="2.470em" minsize="2.470em" stretchy="true">]</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <i>m</i> is a fixed positive integer with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41478_2025_930_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. This work addresses the proofs of vital theorems and presents their corresponding corollaries.</p>

Perturbation of functional equations in $\mathcal {MFN}$-spaces

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发表日期：

2025年7月1日

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Javad Shokri,
Choonkil Park

摘要

In this research, we use a fixed point theorem to investigate the Hyers-Ulam stability in matrix fuzzy normed space ( $\mathcal {MFNS}$ $MFNS$ ) connected to the generalized Jensen functional equation $\begin{aligned} \sum _{i=1}^mF(a_i)=\frac{1}{2m}\Bigg [\sum _{i=1}^mF\Big (ma_i+\sum _{j=1,j\ne i}^ma_j\Big )+F\Big (\sum _{i=1}^ma_i\Big )\Bigg ], \end{aligned}$ $\begin{matrix} \sum_{i = 1}^{m} F (a_{i}) = \frac{1}{2 m} [\sum_{i = 1}^{m} F (m a_{i} + \sum_{j = 1, j \neq i}^{m} a_{j}) + F (\sum_{i = 1}^{m} a_{i})], \end{matrix}$ where m is a fixed positive integer with $m\ge 2$ $m \geq 2$ . This work addresses the proofs of vital theorems and presents their corresponding corollaries.

Perturbation of functional equations in \(\mathcal {MFN}\)-spaces

摘要