A joint generalization of Van Vleck's and Kannappan's equations on groups

Abstract

<jats:title>Abstract</jats:title><jats:p>Let</jats:p>

An integral functional equation on groups under two measures

“…The proof follows on putting z 1 = z 0 in Corollary 3.3. 2With z 1 = z 0 in Corollary 3.4 we obtain: Corollary 4.3 of[2]). The non-zero solutions f : G → C of the functional equationf (σ(y)xz 0 ) − f (xyz 0 ) = 2f (x)f (y), x,y ∈ G,are the functions of the formf = − χ(z 0 ) 2 (χ − χ • σ), where χ is a character of G such that χ • σ(z 0 ) = −χ(z 0 ).We complete the paper with an important result concerning Eq.…”

“…by expressing its continuous solutions in terms of continuous characters. The following theorem is proved in [2]. For the notation N (G, σ) see the section Introduction.…”

“…Let G be a group, z 0 ∈ G be a fixed element, and σ : G → G be an involutive automorphism. In [2], the authors determined the general solutions f, g : which contains the solutions of a functional equation due to Van Vleck [10]. In 1910, he studied the continuous solutions f : R → R, f 6 = 0, of the functional equation…”

The generalized Van Vleck's equation on locally compact groups

“…Results of [1,2,3,5,10] have been an inspiration by their treatments of similar functional equations on groups.…”

“…Van Vleck's functional equation (1.1) was generalized in another direction by the authors in [3], viz. to the functional equation f (σ(y)xz 0 ) − f (xyz 0 ) = 2f (x)f (y), x,y ∈ G, (1.3) where z 0 ∈ G is a fixed element that need not belongs to Z(G) and σ : G → G is an involutive automorphism (that is involutive means that σ(σ(x)) = x for all x ∈ G).…”