Superstability problem for a large class of functional equations

This paper treats superstability problem of the generalized Wilson’s equation $$\begin{aligned} \underset{\varphi \in \Phi }{\sum }\int \limits _{G}\int \limits _{K}f(xtk \varphi (y)k^{-1})dw_{K}(k)d\mu (t)=\left| \Phi \right| f(x)g(y),~\ \ \ x,y\in G, \end{aligned}$$ where G is an arbitrary locally compact group, that need not be 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, \(\mu \) is a complex measure with compact support and f, g \( :G\longrightarrow \mathbb {C}\) are continuous complex-valued functions. We dont impose any condition on the continuous function f. In addition, superstability problem for a large class of related functional equations are considered.