μ-Trigonometric Functional Equations and Hyers–Ulam Stability Problem in Hypergroups

Let (X, ∗ ) be a hypergroup and μ be a complex bounded measure on X. We determine the continuous and bounded solutions of each of the following three functional equations $$\begin{array}{rcl} \left \langle {\delta }_{x} {_\ast} \mu {_\ast} {\delta }_{y},f\right \rangle & =& f(x)g(y) \pm g(x)f(y),\;x,y \in X, \\ \left \langle {\delta }_{x} {_\ast} \mu {_\ast} {\delta }_{y},g\right \rangle & =& g(x)g(y) + f(x)f(y),\;x,y \in X.\end{array}$$ In addition, when μ = δ e , Hyers–Ulam stability problems for these functional equations on hypergroups are considered. The results obtained in this paper are natural extensions of previous works done in groups especially by Stetkaer, Elqorachi, Redouani, and Szekelyhidi.