A New Approach to Hyers-Ulam Stability of r -Variable Quadratic Functional Equations

Abstract

In this paper, we investigate the general solution of a new quadratic functional equation of the form ${\sum }_{1\le i<j<k\le r}\varphi \left(l_i+l_j+l_k\right)=\left(r-2\right){\sum }_{i=1,i\ne j}^r\varphi \left(l_i+l_j\right)+\left(\left(-r^2+3r-2\right)/2\right){\sum }_{i=1}^r\varphi \left(l_i\right).$ We prove that a function admits, in appropriate conditions, a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.