Stability of Functional Equation in Banach Space: Using Two Different Methods

Abstract

Using Direct and fixed point method, we prove the Ulam-Hyers stability of a generalized n-dimensional additive functional equation of the form \[f\left(\sum^n_{i=1}{kx_i}\right)+\sum^n_{j=1}{f\left(-kx_j+\sum^n_{\substack{i=1 \\ i\neq j}} {kx_i}\right)}=(n-1)\left[\sum^n_{i=1}{\left(2i-1\right)f\left(x_i\right)}\right]\] Where n is the positive integer with $\mathbb{N}-\{0,1,2\}$ and k is the only odd positive integers in Banach Space is discussed.