This research delves into the stability and general solution of two m-functional equalities: <Equation ID="Equ1"> <EquationNumber>0.1</EquationNumber> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3295_Article_Equ1.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="395" /> </MediaObject> <EquationSource Format="MATHML"><math> <mtable columnalign="right left" columnspacing="0.2em"> <mtr> <mtd /> <mtd> <mi>m</mi> <mo stretchy="false">{</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>c</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>c</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>m</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>m</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mspace width="1em" /> <mo>=</mo> <mn>2</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo stretchy="false">{</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mn>4</mn> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </math></EquationSource> <EquationSource Format="TEX"> $\begin{aligned} &m\{I(mc + u) + I(mc - u)\} + I(c+mu)+I(c-mu) \\ &\quad =2m^{2}\{I(c+u)+I(c-u)\}+2(m^{4}-2m^{2}+1)I(c), \end{aligned}$ </EquationSource> </Equation> and <Equation ID="Equ2"> <EquationNumber>0.2</EquationNumber> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3295_Article_Equ2.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="463" /> </MediaObject> <EquationSource Format="MATHML"><math> <mtable columnalign="right left" columnspacing="0.2em"> <mtr> <mtd /> <mtd> <mi>m</mi> <mo stretchy="false">{</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>c</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mi>c</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>m</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>m</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mspace width="1em" /> <mo>=</mo> <mn>2</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo stretchy="false">{</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mn>4</mn> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>m</mi> <mn>2</mn> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">{</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> <mo>,</mo> </mtd> </mtr> </mtable> </math></EquationSource> <EquationSource Format="TEX"> $\begin{aligned} &m\{I(mc + u) + I(mc - u)\} + I(c+mu)+I(c-mu) \\ &\quad =2m^{2}\{I(c+u)+I(c-u)\}+2(m^{4}-2m^{2}+1)\{I(c)+I(u)\}, \end{aligned}$ </EquationSource> </Equation> for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3295_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>m</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </math></EquationSource> <EquationSource Format="TEX">$m\in \mathbb{R}$</EquationSource> </InlineEquation> not 0 and ±1 in the setting of Matrix Banach spaces.These equations have far-reaching implications in various fields, including image processing and security. Using a direct approach, we employ a combination of analytical and numerical techniques to analyze the stability and general solution of the given equations. Our analysis reveals that the equations exhibit stability for specific ranges of the parameter m, and the general solution of the equations provides valuable insight into the behavior of the solutions. As an application of our results, we demonstrate how the stability and general solution of the m-functional equalities can be used to enhance image security systems, including image encryption and decryption techniques. Our research has significant implications for image security and Matrix Banach spaces. Future work will focus on extending the results to other functional equations, developing more efficient image encryption and decryption techniques, and exploring applications in other fields.

Generalized m-functional equations in matrix Banach space with an application

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2025年9月30日

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Sundas Nawaz,
Ali Akgül,
Murad Khan Hassani,
Mohammad A. Tashtoush

摘要

This research delves into the stability and general solution of two m-functional equalities: 0.1 $\begin{aligned} m {I (m c + u) + I (m c - u)} + I (c + m u) + I (c - m u) \\ = 2 m^{2} {I (c + u) + I (c - u)} + 2 (m^{4} - 2 m^{2} + 1) I (c), \end{aligned}$ $\begin{aligned} &m\{I(mc + u) + I(mc - u)\} + I(c+mu)+I(c-mu) \\ &\quad =2m^{2}\{I(c+u)+I(c-u)\}+2(m^{4}-2m^{2}+1)I(c), \end{aligned}$ and 0.2 $\begin{aligned} m {I (m c + u) + I (m c - u)} + I (c + m u) + I (c - m u) \\ = 2 m^{2} {I (c + u) + I (c - u)} + 2 (m^{4} - 2 m^{2} + 1) {I (c) + I (u)}, \end{aligned}$ $\begin{aligned} &m\{I(mc + u) + I(mc - u)\} + I(c+mu)+I(c-mu) \\ &\quad =2m^{2}\{I(c+u)+I(c-u)\}+2(m^{4}-2m^{2}+1)\{I(c)+I(u)\}, \end{aligned}$ for $m \in R$ $m\in \mathbb{R}$ not 0 and ±1 in the setting of Matrix Banach spaces.

These equations have far-reaching implications in various fields, including image processing and security. Using a direct approach, we employ a combination of analytical and numerical techniques to analyze the stability and general solution of the given equations. Our analysis reveals that the equations exhibit stability for specific ranges of the parameter m, and the general solution of the equations provides valuable insight into the behavior of the solutions. As an application of our results, we demonstrate how the stability and general solution of the m-functional equalities can be used to enhance image security systems, including image encryption and decryption techniques. Our research has significant implications for image security and Matrix Banach spaces. Future work will focus on extending the results to other functional equations, developing more efficient image encryption and decryption techniques, and exploring applications in other fields.