On the Stability of an 𝑚-Variables Functional Equation in Random Normed Spaces via Fixed Point Method

Abstract

At first we find the solution of the functional equation 𝐷𝑓(𝑥1,…,𝑥𝑚)∶=∑𝑚𝑘=2(∑𝑘𝑖1=2∑𝑘+1𝑖2=𝑖1+1⋯∑𝑚𝑖𝑚−𝑘+1=𝑖𝑚−𝑘+1)𝑓(∑𝑚𝑖=1,𝑖≠𝑖1,…,𝑖𝑚−𝑘+1𝑥𝑖−∑𝑚−𝑘+1𝑟=1𝑥𝑖𝑟)+𝑓(∑𝑚𝑖=1𝑥𝑖)−2𝑚−1𝑓(𝑥1)=0, where 𝑚≥2 is an integer number. Then, we obtain the generalized Hyers-Ulam-Rassias stability in random normed spaces via the fixed point method for the above functional equation.