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

Our main goal is to determine the continuous and bounded complex valued solutions of the functional equation $$ \langle \delta_{x}\ast \delta_{y},g\rangle +\langle \delta_{x}\ast \delta_{\check{y}},g\rangle =2~g(x)g(y),\;x,y\in X,$$ where 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.