D’Alembert’s Functional Equation and Superstability Problem in Hypergroups

Abstract

Our main goal is to determine the continuous and bounded complex valued solutions of the functional equationwhere X is a hypergroup. The solutions are expressed in terms of 2 -dimensional representations of X. The papers of Davison [10] and Stetkaer [25,26] are the essential motivation for this first part of the present work and the methods used here are closely related to and inspired by those in [10,25,26]. In addition, superstability problem for this functional equation on any hypergroup and without any condition on f is considered.

Superstability problem for a large class of functional equations

“…Different generalizations of the result of Baker, Lawrence and Zorzitto was given by Badora and Ger [3,4], Székelyhidi [26,27] and Gàvruta [13]. For other superstability results, we can see for example [8,9,12,15,17,18,21,22,[30][31][32][33][34][35].…”

“…The interested reader should refer to [3,4,[6][7][8][9][10][12][13][14][15][16][17][18][19][20][21][22][26][27][28][29][30][31][32][33][34][35] for a thorough account on the subject of stability of functional equations.…”