| Summary: | We discuss the solution to the minimum functional equation
$ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $
for a real-valued function $ \eta: G \to \mathbb{R} $ defined on arbitrary group $ G $. In addition, we examine the Pexider-type functional equation
$ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \chi(x)\eta(y)+\psi(x), \qquad x, y \in G, \end{align} $
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
$ \begin{align} \max \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $
and
$ \begin{align} \min \{\, \eta(xy^{-1}), \eta(xy) \, \} = \eta(x)\eta(y), \qquad x, y \in G, \end{align} $
with the restriction that the function $ \eta $ satisfies the Kannappan condition.
|
|---|