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

Abstract

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.

Keywords

• minimum functional equation
• pexider functional equation
• kannappan condition
• strictly positive solution