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On the quadratic functional equation on groups
Publicationes Mathematicae Debrecen, 2006Given two groups \(G,H\), the functional equation \[ f(xy)+f(xy^{-1})=2f(x)+2f(y),\qquad x,y\in G \tag{(1)} \] is called the quadratic functional equation, where \(f:G\to H\) is considered as an unknown function. Assuming that \(G\) and \(H\) are abelian and \(H\) is uniquely 2-divisible, \textit{J. Aczél} [Period. Math.-Phys. Astron., II. Ser. 20, 65--
Friis, P.d.P., Stetkær, H.
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Alienation of the Quadratic and Additive Functional Equations
Analysis Mathematica, 2019Let \(G\) and \(H\) be uniquely \(2\)-divisible abelian groups. The Pexider-type functional equation \(f(x+y) + f(x-y) + g(x+y) = 2f(x) + 2f(y) + g(x) + g(y)\) is constructed by summing up the quadratic functional equation and additive Cauchy functional equation side by side. Here \(f, g : G \to H\).
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Quadratic Functional Equations
2011So far, we have discussed the stability problems of functional equations in connection with additive or linear functions. In this chapter, the Hyers–Ulam–Rassias stability of quadratic functional equations will be proved. Most mathematicians may be interested in the study of the quadratic functional equation since the quadratic functions are applied to
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The quadratic functional equation on groups
Publicationes Mathematicae Debrecen, 2005The quadratic functional equation \[ f(xy)+f(xy^{-1})=2f(x)+2f(y) \] is considered on free groups. The author presents the result on a general solution of the above equation defined on a free group with values in an abelian group. In the proof some results concerning the Jensen functional equation are utilized.
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Stability of the Quadratic Functional Equation
1998The quadratic functional equation $$ f\left( {x + y} \right) + f\left( {x - y} \right) - 2f\left( x \right) - 2f\left( y \right) = 0$$ (3.1) clearly has f(x) = cx2 as a solution with c an arbitrary constant when f is a real function of a real variable.
Donald H. Hyers +2 more
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Stability of a generalization of Cauchy’s and the quadratic functional equations
Journal of Fixed Point Theory and Applications, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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The quadratic function and the quadratic equation
1983B.D. Bunday, H. Mulholland
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THE QUADRATIC FUNCTION AND THE QUADRATIC EQUATION
1968C. PLUMPTON, W.A. TOMKYS
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Stability of a Quadratic Functional Equation
Advances in Dynamical Systems and Applications, 2021S. Jaikumar +4 more
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