A Functional Equation for the Cosine on Semigroups with Endomorphisms

Let S be a semigroup, and let IF be a quadratically closed field of characteristic not equal 2 with identity element 1. We describe, in terms of multiplicative functions of S, the solutions f : S -> F of the new functional equation f(x phi(y)) + mu(y)f (phi(y)x) = 2f(x) f(y), x, y is an element of S, where phi, phi : S -> S are two endomorphisms that need not be involutive and mu : S -> F is a multiplicative map such that mu(x phi(x)) = 1 for all x is an element of S. Significant consequences of this result are presented.