<p>We introduce the finite variable cubic functional equation of the form <Equation ID="Equ38"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1350_Article_Equ38.gif" Format="GIF" Height="162" Rendition="HTML" Resolution="72" Type="Linedraw" Width="525" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \sum _{a=1}^{m}\phi \left( -t_{a}+\sum _{b=1;a \ne b}^{m}t_{b}\right) -\sum _{a=1}^{m}\phi \left( 2t_{a}\right) =\left( m-6\right) \sum _{1 \le a&lt; b&lt; c \le m}\phi \left( t_{a}+t_{b}+t_{c}\right) \\ +\left( -m^{2}+9m-14\right) \sum _{1\le a&lt;b\le m}\phi \left( t_{a}+t_{b}\right) \\ +\left( \frac{m^{3}-11 m^{2}+28 m-36}{2}\right) \sum _{a=1}^{m}\phi \left( t_{a}\right) \end{aligned}\)</EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1350_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">\(m \ge 4\)</EquationSource> </InlineEquation> is a fixed integer, and we establish the Hyers–Ulam–Rassias stability results in paranormed spaces and matrix paranormed spaces.</p>

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Hyers–Ulam–Rassias stability of a finite variable cubic functional equation in matrix paranormed spaces

  • Kandhasamy Tamilvanan,
  • G. Balasubramanian,
  • Choonkil Park,
  • Jung Rye Lee

摘要

We introduce the finite variable cubic functional equation of the form \(\begin{aligned} \sum _{a=1}^{m}\phi \left( -t_{a}+\sum _{b=1;a \ne b}^{m}t_{b}\right) -\sum _{a=1}^{m}\phi \left( 2t_{a}\right) =\left( m-6\right) \sum _{1 \le a< b< c \le m}\phi \left( t_{a}+t_{b}+t_{c}\right) \\ +\left( -m^{2}+9m-14\right) \sum _{1\le a<b\le m}\phi \left( t_{a}+t_{b}\right) \\ +\left( \frac{m^{3}-11 m^{2}+28 m-36}{2}\right) \sum _{a=1}^{m}\phi \left( t_{a}\right) \end{aligned}\) where \(m \ge 4\) is a fixed integer, and we establish the Hyers–Ulam–Rassias stability results in paranormed spaces and matrix paranormed spaces.