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Let S, H, X be groups. For two given biadditive functions A : S2 → X, B : H2 → X and for two unknown mappings T : S → H, g : S → S we will study the functional equation

We present the form of the solutions $f:S\rightarrow \mathbb{C}$ of the functional equation $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D706}\in K}f(x+\unicode[STIX]{x1D706}y)=|K|f(x)f(y)\quad \text{for }x,y\in S,\end{eqnarray}$$ where $f$ satisfies the condition $f(\sum _{\unicode[STIX]{x1D706}\in K}\unicode[STIX]{x1D706}x)\neq 0$ for all $x\in S$, $(S,+)$ is an abelian semigroup and $K$ is a subgroup of the automorphism group of $S$. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability in the sense of Hyers-Ulam for a class of the following type functional equations: ∑ λ∈Φ f(x + λy + aλ) = Nf(x) + h(y), x, y ∈ S where K is a complete valued field of characteristic zero, F is a complete normed space (Archimedean or ultrametric) over K, (S, +) is an abelian monoid, f, h: S → F, Φ is a finite automorphism group of S, N is the cardinality of Φ and aλ ∈ S. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the solution (f,g) of the equation = , x,y ∣ X, where f:X→Y , g:X→X, X,Y are inner product spaces over the same field K ∣ {R,C}. [ABSTRACT FROM AUTHOR]