<p>In this paper, we study the quartic functional equation <Equation ID="Equ52"> <EquationSource Format="TEX">$ g(k\varpi +\varkappa )+g(k\varpi -\varkappa ) =k^2\bigl [g(\varpi +\varkappa )+g(\varpi -\varkappa )\bigr ] +2k^2(k^2-1)g(\varpi )-2(k^2-1)g(\varkappa ), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>ϖ</mi> <mo>+</mo> <mi>ϰ</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>ϖ</mi> <mo>-</mo> <mi>ϰ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">[</mo> </mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>ϖ</mi> <mo>+</mo> <mi>ϰ</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>ϖ</mi> <mo>-</mo> <mi>ϰ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">]</mo> </mrow> <mo>+</mo> <mn>2</mn> <msup> <mi>k</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>ϖ</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mn>2</mn> <mrow> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>ϰ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>for a fixed integer <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$k\ge 2.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The general solution of the functional equation is established, and its stability and hyperstability are proved using the fixed point method in the framework of complete quasi-2-normed spaces. Furthermore, several corollaries are established as applications of the main theorems, through which various existing results on quartic functional equations are generalized, and extended, thereby enriching the stability theory of quartic functional equations in complete quasi-2-normed spaces.</p>

Hyperstability of a quartic functional equation in complete quasi-2-normed space

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发表日期：

2026年7月18日

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Ravinder Kumar Sharma,
Meenu Prajapati,
Aasish Kumar Rawat,
Sumit Chandok

摘要

In this paper, we study the quartic functional equation $ g(k\varpi +\varkappa )+g(k\varpi -\varkappa ) =k^2\bigl [g(\varpi +\varkappa )+g(\varpi -\varkappa )\bigr ] +2k^2(k^2-1)g(\varpi )-2(k^2-1)g(\varkappa ), $ $g (k ϖ + ϰ) + g (k ϖ - ϰ) = k^{2} [g (ϖ + ϰ) + g (ϖ - ϰ)] + 2 k^{2} (k^{2} - 1) g (ϖ) - 2 (k^{2} - 1) g (ϰ),$ for a fixed integer $k\ge 2.$ $k \geq 2 .$ The general solution of the functional equation is established, and its stability and hyperstability are proved using the fixed point method in the framework of complete quasi-2-normed spaces. Furthermore, several corollaries are established as applications of the main theorems, through which various existing results on quartic functional equations are generalized, and extended, thereby enriching the stability theory of quartic functional equations in complete quasi-2-normed spaces.