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

摘要

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 o 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.