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Let Aand Bbe unital algebras, that need not be abelian, over fields Kand K′respectively, let α,a,b,c∈K,β∈K′and λ∈C. The present work aims to determine the general solution Φ:A→Bof the functional equation Φ(x+y+αxy)=Φ(x)+Φ(y)+βΦ(x)Φ(y),x,y∈A,and to describe the solutions Φ:A→M2(C)of the functional equation Φ(ax+by+cxy)=Φ(x)+Φ(y)+λΦ(x)Φ(y),x,y∈A.We also show that, when (A,·)(as a semigroup) is commutative and regular (for instance when dimA=1), the explicit forms of the solutions of the last equation can be given.

Let Gbe a locally compact abelian Hausdorff group, let $$\sigma $$ σ be a continuous involution on G, and let $$\mu ,\nu $$ μ,ν be regular, compactly supported, complex-valued Borel measures on G. We determine the continuous solutions $$f,g:G\rightarrow {\mathbb {C}}$$ f,g:G→C of each of the two functional equations $$\begin{aligned}&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=f(x)g(y),\quad x,y\in G,\\&\int _{G}f(x+y+t)d\mu (t)+\int _{G}f(x+\sigma (y)+t)d\nu (t)=g(x)f(y),\quad x,y\in G, \end{aligned}$$ ∫Gf(x+y+t)dμ(t)+∫Gf(x+σ(y)+t)dν(t)=f(x)g(y),x,y∈G,∫Gf(x+y+t)dμ(t)+∫Gf(x+σ(y)+t)dν(t)=g(x)f(y),x,y∈G, in terms of characters and additive functions. These equations provides a common generalization of many functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s, Van Vleck’s, or Wilson’s equations. So, several functional equations will be solved.

This paper treats superstability problem of the generalized Wilson’s equation $$\begin{aligned} \underset{\varphi \in \Phi }{\sum }\int \limits _{G}\int \limits _{K}f(xtk \varphi (y)k^{-1})dw_{K}(k)d\mu (t)=\left| \Phi \right| f(x)g(y),~\ \ \ x,y\in G, \end{aligned}$$ ∑φ∈Φ∫G∫Kf(xtkφ(y)k-1)dwK(k)dμ(t)=Φf(x)g(y),x,y∈G, where Gis an arbitrary locally compact group, that need not be abelian, Kis a compact subgroup of G, $$\omega _{K}$$ ωK is the normalized Haar measure of K, $$\Phi $$ Φ is a finite group of K-invariant morphisms of G, $$\mu $$ μ is a complex measure with compact support and f, g$$ :G\longrightarrow \mathbb {C}$$ :G⟶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.

Let α∈Rand let Sbe a semigroup. The purpose of the present paper is to find the solutions f:R4→Sof the parametric (semigroup valued) functional equation f(x1,y1,z1,w1)f(x2,y2,z2,w2)=f(x1x2+αy1w2+αy2w1+αz1z2,x1y2+x2y1+αz1w2+αz2w1,x1z2+x2z1+y1y2+αw1w2,x1w2+x2w1+y1z2+y2z1),which arises from the product in a quartic number field. For α⩾0,its scalar-valued solutions are determined in [13].

The aim of this paper is to investigate the superstability problem for the variant $$\begin{aligned} f(xy)+f(\sigma (y)x)=2f(x)g(y),\ \ \ \ x,y\in G, \end{aligned}$$ f(xy)+f(σ(y)x)=2f(x)g(y),x,y∈G, of Wilson’s functional equation where $$G$$ G is an arbitrary group, $$f,g$$ f,g are complex valued functions and $$\sigma $$ σ is an involution of $$G$$ G . As a consequence, we obtain the superstability of the Pexider type functional equation $$\begin{aligned} f(xy)+f(\sigma (y)x)=2g(x)h(y),\ \ x,y\in G, \end{aligned}$$ f(xy)+f(σ(y)x)=2g(x)h(y),x,y∈G, on any group.

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