On two functional equations with involution on groups related to sine and cosine functions

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

“…Stetkaer [7] extends the results of Perkins and Sahoo [6] about equation (1.7) to the case where G is a semigroup and the solutions are not assumed to be abelian. The main purpose of this paper is to extend Stetkaer's results [7,8] to the following generalizations of Van Vleck's functional equation for the sine…”

“…Perkins and Sahoo [6] studied the following version of Kannappan's functional equation (1.5) f (xyz 0 ) + f (xy −1 z 0 ) = 2f (x)f (y), x, y ∈ S on groups. They found the form of any abelian solution f of (1.5).…”

“…Perkins and Sahoo [6] replaced the group inversion by an involution τ : G −→ G and they obtained the abelian, complex-valued solutions of equation…”

An extension of Van Vleck’s functional equation for the sine

“…Stetkaer [6] extends the results of Perkins and Sahoo [4] about equation (1.4) to the more general case where G is a semigroup and the solutions are not assumed to be abelian. The first purpose of this paper is to extend the results of Stetkaer [6] to the following generalization of Van Vleck's functional equation for the sine…”

“…In [5], Sahoo studied the following generalization (1.2) f (x − y + z 0 ) + g(x + y + z 0 ) = 2f (x)f (y) x, y ∈ G of the functional equations (1.1). He determined the general solutions of this equation on an abelian group G. Stetkaer [8,Exercise 9.18] found the complex-valued solution of equation (1.3) f (xy −1 z 0 ) − f (xyz 0 ) = 2f (x)f (y), x, y ∈ G, when G a group not necessarily abelian and z 0 is a fixed element in the center of G. Recently, Perkins and Sahoo [4] replaced the group inversion by the more general involution τ : G −→ G and they obtained the abelian, complex-valued solutions of equation…”