Extensions of Kannappan’s and Van Vleck’s Functional Equations on Semigroups

This paper treats two functional equations, the Kannappan-Van Vleck functional equation μ ( y ) f ( x τ ( y ) z 0 ) ± f ( x y z 0 ) = 2 f ( x ) f ( y ) , x , y ∈ S $$\displaystyle \mu (y)f(x\tau (y)z_0)\pm f(xyz_0) =2f(x)f(y), \;x,y\in S $$ and the following variant of it μ ( y ) f ( τ ( y ) x z 0 ) ± f ( x y z 0 ) = 2 f ( x ) f ( y ) , x , y ∈ S , $$\displaystyle \mu (y)f(\tau (y)xz_0)\pm f(xyz_0) = 2f(x)f(y), \;x,y\in S, $$ in the setting of semigroups S that need not be abelian or unital, τ is an involutive morphism of S, μ : S→C is a multiplicative function such that μ(xτ(x)) = 1 for all x ∈ S and z 0 is a fixed element in the center of S. We find the complex-valued solutions of these equations in terms of multiplicative functions and solutions of d’Alembert’s functional equation.