Abstract Let {(X,*)} be a hypergroup and let {w_{0}} be a fixed measure on X. In this paper we study the two functional equations \langle\delta_{x}*\delta_{y}*\omega_{0},g\rangle+\langle\delta_{x}*\delta_{% \check{y}}*\omega_{0},g\rangle=2g(x)g(y),\quad x,y\in X, and \langle\delta_{x}*\delta_{\check{y}}*\omega_{0},f\rangle-\langle\delta_{x}*% \delta_{y}*\omega_{0},f\rangle=2f(x)f(y),\quad x,y\in X, where {g,f:X\to\mathbb{C}} are continuous and bounded functions to be determined. We express the solutions of the two functional equations in terms of multiplicative maps on {(X,*)} . As an application we give the solution of the two functional equations on polynomial and Sturm–Liouville hypergroups.

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In this paper, we study the superstablity problem of the cosine and sine type functional equations: $$f(x\sigma (y)a) + f(xya) = 2f(x)f(y)$$ and $$f(x\sigma (y)a) - f(xya) = 2f(x)f(y),$$ where f : S → ℂ is a complex valued function; S is a semigroup; σ is an involution of S and a is a fixed element in the center of S.