General Solution and Generalized Hyers-Ulam Stability of n-dimensional Cubic Functional Equation in Various Normed Space: Direct and Fixed Point Methods

Abstract

In this paper, the authors investigate the general solution in vectors space and generalized Hyers-Ulam Stability of n-dimensional Cubic functional Equation
$$f\left(\sum^{n}_{i=1}{x_{i}}\right)+\sum^{n}_{j=1}f\left(-x_{j}+\sum_{i=1;i\neq j}{x_{i}}\right)=\left(n-5\right)\sum_{ 1\leq i<j\leq k\leq n }f\left({{x_{i}+x_{j}+x_{k}}}\right)$$
$$+\left(-n^{2}+8n-11\right)\sum_{i=1;i\neq j}{f\left(x_{i}+x_{j}\right)-\sum^{n}_{j=1}{f\left({2x_{j}}\right)}}$$ $$+\frac{1}{2}\left(n^{3}-10n^{2}+23n+2\right)\sum^{n}_{i=1}f\left({x_{i}}\right)$$
with $n>5$, and n is a positive integer using FNS,RNS,IFNS and FELBIN'S type spaces, using direct and fixed point methods.