In this paper, we prove the generalized Hyers-Ulam-Rassias stability for the Drygas functional equation$$f(x+y)+f(x-y)=2f(x)+f(y)+f(-y)$$ in Banach spaces by using the Brzc{d}ek's fixed point theorem. Moreover, we give a general result on the hyperstability of this equation. Our results are improvements and generalizations of the main result of M. Piszczek and J. Szczawi'{n}ska [21].

Let $$(S,+)$$ be an abelian semigroup, let $$(H,+)$$ a uniquely 2-divisible, group and $$\sigma ,\tau $$ two endomorphisms of S. In this paper, we find the solutions $$f:S\rightarrow H$$ following Drygas type functional Eq. 1 $$\begin{aligned} f(x+\sigma (y))+f(x+\tau (y))=2f(x)+f(\sigma (y))+f(\tau (y)), \quad x,y\in S, \end{aligned}$$ in terms additive bi-additive maps. Further, solve partly ...

Abstract Let ( S, +) be an abelian semigroup, let H, group which is uniquely 2-divisible, and ϕ endomorphism of S . We find the solutions f, h : → H each functional equations f ( x + y )...