Minimum functional equation and some Pexider-type functional equation on any group

Abstract

<abstract><p>We discuss the solution to the minimum functional equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $\end{document} </tex-math></disp-formula></p> <p>for a real-valued function $ \eta: G \to \mathbb{R} $ defined on arbitrary group $ G $. In addition, we examine the Pexider-type functional equation</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \chi(x)\eta(y)+\psi(x), \qquad x, y \in G, \end{align} $\end{document} </tex-math></disp-formula></p> <p>where $ \eta $, $ \chi $ and $ \psi $ are real mappings acting on arbitrary group $ G $. We also investigate this Pexiderized functional equation that generalizes two functional equations</p> <p><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE4"> \begin{document}$ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $\end{document} </tex-math></disp-formula></p> <p>with the restriction that the function $ \eta $ satisfies the Kannappan condition.</p></abstract>