Solution and Stability of a Mixed Type Functional Equation

TL;DR

This paper derives the general solution and examines the stability of a new mixed additive and cubic functional equation, showing stability can be controlled by norms' powers, contributing to functional equation theory.

Contribution

It introduces the general solution and stability analysis for a novel mixed type functional equation, expanding understanding of such equations.

Findings

The general solution of the mixed functional equation is obtained.

Stability can be controlled by sums and products of powers of norms.

The results extend Hyers-Ulam-Rassias stability theory to new equations.

Abstract

In this paper, we obtain the general solution and investigate the generalized Hyers-Ulam-Rassias stability for the new mixed type additive and cubic functional equation $$3f(x+3y) - f(3x + y)=12[f(x+y)+f(x-y)]-16[f(x)+f(y)] + 12f(2y) - 4f(2x).$$ As some corollaries, we show that the stability of this equation can be controlled by the sum and product of powers of norms.