Stability of an Alternative Functional Equation Related to the Quadratic Equation
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Abstract
Given a rational number \(\alpha \ne \pm 2\), a criterion is established for the existence of the general solution to the alternative quadratic functional equation of the form
\(f(xy) + f(xy^{-1}) = 2(f(x) + f(y))\) or \(f(xy) + f(xy^{-1}) = \alpha(f(x) + f(y)),\)
where \(f\) is a mapping from an abelian group \((G,\cdot)\) to a uniquely divisible abelian group \((H,+)\). Subsequently, the Hyers-Ulam stability of this equation is proved for mappings from an abelian group to a Banach space, provided that \(\alpha \notin \{0, \pm\frac{1}{2}, \pm 1, \pm 2\}\) is a rational number.
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References
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