Nearly Quadratic Mappings over p -Adic Fields

Abstract

We establish some stability results over $p$-adic fields for the generalized quadratic functional equation ${\sum }_{k=2}^n{\sum }_{i_1=2}^k{\sum }_{i_2=i_1+1}^{k+1}\cdots {\sum }_{i_{n-k+1}=i_{n-k}+1}^{n}f({\sum }_{i=1,i\ne i_1,\dots ,i_{n-k+1}}^n{x}_i-{\sum }_{r=1}^{n-k+1}{x}_{i_r})+f\left({\sum }_{i=1}^n{x}_i\right)=2^{n-1}{\sum }_{i=1}^{n}f(x_i),$ where $n\in \mathbb{N}$ and $n\ge 2$.