Hyers-Ulam-Rassias Stability of a General Septic Functional Equation

Abstract

In this paper, we investigate the stability of the following general septic functional equation: \(\sum_{i=0}^8{ }_8 C_i(-1)^{8-i} f(x+(i-4) y)=0\)which is a generalization of many functional equations such as the additive functional equation, the quadratic functional equation, the cubic functional equation, the quartic functional equation, the quintic functional equation, and the sextic functional equation. The equation is analysed from the perspective of Hyers-Ulam-Rassias stability.

Representation and Stability of General Nonic Functional Equation

“…In the papers [16,[25][26][27][28][29][30][31][32][33], one can find hyperstability results for various types of functional equation. Recent results regarding the stability of the functional equation described in Equation (1) with n < 9 can be found in [34][35][36][37][38][39], and hyperstability results for this functional equation with n < 9 can be found in [38,40]. Note that the superstability of a functional equation requires that any mapping satisfying the equation approximately (in some sense) must be either a real solution to it or a bounded mapping, while hyperstability requires that any mapping satisfying the equation approximately (in some sense) must be a real solution to it.…”