Hyers–Ulam stability of generalized Wilson’s and d’Alembert’s functional equations

We study the Hyers–Ulam stability problem for the generalized Wilson’s equation $$\begin{aligned} \underset{\varphi \in \Phi }{\sum }\int \limits _{K}f(xk\varphi (y)k^{-1})dw_{K}(k)=\left| \Phi \right| f(x)g(y),~\ \ \ x,y\in G, \end{aligned}$$ where \(G\) is an arbitrary locally compact group, not necessarily abelian, \(K\) is a compact subgroup of \(G\), \(\omega _{K}\) is the normalized Haar measure of \(K\), \(\Phi \) is a finite group of \(K\)-invariant morphisms of \(G\) and \(f,g\) \(:G\longrightarrow \mathbb C \) are continuous complex-valued functions. We dont impose that \(f\) satisfies the Kannappan type condition. In addition, superstability problem for some related functional equations are considered.