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<p>In this paper, we study the Hyers–Ulam stability of the Euler–Lagrange cubic functional equation <Equation ID="Equ68"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1272_Article_Equ68.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="525" /> </MediaObject> <EquationSource Format="TEX">\( f(ax+y) + f(x+ay) = (a+1)(a-1)^{2}[f(x)+f(y)] + a(a+1) f(x+y), \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">[</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>+</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>within certain restricted domains in normed spaces, where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2025_1272_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(a\ne 0,\pm 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is a fixed real constant. Additionally, we investigate the hyperstability of this functional equation and analyze its asymptotic behavior under various conditions.</p>

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Hyers–Ulam stability and hyperstability of a cubic functional equation of Euler–Lagrange type on restricted domains

  • 期刊论文

  • 发表日期：

    2025年7月11日

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  • Abbas Najati,
  • Mohammad Amin Tareeghee

摘要

In this paper, we study the Hyers–Ulam stability of the Euler–Lagrange cubic functional equation \( f(ax+y) + f(x+ay) = (a+1)(a-1)^{2}[f(x)+f(y)] + a(a+1) f(x+y), \) f ( a x + y ) + f ( x + a y ) = ( a + 1 ) ( a - 1 ) 2 [ f ( x ) + f ( y ) ] + a ( a + 1 ) f ( x + y ) , within certain restricted domains in normed spaces, where \(a\ne 0,\pm 1\) a ≠ 0 , ± 1 is a fixed real constant. Additionally, we investigate the hyperstability of this functional equation and analyze its asymptotic behavior under various conditions.