Integral Van Vleck’s and Kannappan’s functional equations on semigroups

Abstract

Abstract. In this paper we study the solutions of the integral Van Vleck's functional equation for the sineand the integral Kannappan's functional equationwhere S is a semigroup, τ is an involution of S and µ is a measure that is linear combinations of point measures (δz i ) i∈I , such that for all i ∈ I, z i is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of the classic d'Alembert's functional equation with involution.

A generalization of the cosine addition law on semigroups

“…He proved that the solutions are all abelian even in the setting of semigroups. Elqorachi [7] described the solutions of the functional equation…”