Approximate Generalized Jensen Mappings in 2-Banach Spaces

Our aim is to investigate the generalized Hyers-Ulam-Rassias stability for the following general Jensen functional equation: ∑ k = 0 n − 1 f ( x + b k y ) = n f ( x ) , $$\displaystyle \sum _{k=0}^{n-1} f(x+ b_{k}y)=nf(x), $$ where n ∈ ℕ 2 $$n \in \mathbb {N}_{2}$$ , b k = exp ( 2 i π k n ) $$b_{k}=\exp (\frac {2i\pi k}{n})$$ for 0 ≤ k ≤ n − 1, in 2-Banach spaces by using a new version of Brzdȩk’s fixed point theorem. In addition, we prove some hyperstability results for the considered equation and the general inhomogeneous Jensen equation ∑ k = 0 n − 1 f ( x + b k y ) = n f ( x ) + G ( x , y ) . $$\displaystyle \sum _{k=0}^{n-1} f(x+ b_{k}y)=nf(x)+G(x,y). $$