Valerii A. Faiziev, Tver State Agricultural Academy, Tver Sakharovo, Russia, e-mail: valeriy.faiz@mail.ru; Prasanna K. Sahoo, Department of Mathematics, University of Louisville, Louisville, Kentucky 40292 USA, e-mail: sahoo@louisville.edu
Abstract: Let $G$ be a group and $H$ an abelian group. Let $J^* (G, H)$ be the set of solutions $f G \to H$ of the Jensen functional equation $f(xy)+f(xy^{-1}) = 2f(x)$ satisfying the condition $f(xyz) - f(xzy) = f(yz)-f(zy)$ for all $x, y , z \in G$. Let $Q^* (G, H)$ be the set of solutions $f G \to H$ of the quadratic equation $f(xy)+f(xy^{-1}) = 2f(x) + 2f(y)$ satisfying the Kannappan condition $f(xyz) = f(xzy)$ for all $x, y, z \in G$. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f G \to H$ of the Whitehead equation is of the form $4f = 2 \varphi+ 2 \psi$, where $2\varphi\in J^* (G, H)$ and $2\psi\in Q^* (G, H)$. Moreover, if $H$ has the additional property that $2h = 0$ implies $h = 0$ for all $h \in H$, then every solution $f G \to H$ of the Whitehead equation is of the form $2f = \varphi+ \psi$, where $\varphi\in J^*(G,H)$ and $2\psi(x) = B(x, x)$ for some symmetric bihomomorphism $B G \times G \to H$.
Keywords: homomorphism, Frechet functional equation, Jensen functional equation, symmetric bihomomorphism, Whitehead functional equation
Classification (MSC 2010): 39B52
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