A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations

Yang-Hi Lee,Soon-Mo Jung

doi:10.1155/2015/643969

Abstract

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while we investigate the stability of functional equations.

Highlights

The mapping f : R → R given by f(x) = ax2 + bx is a solution of the quadratic-additive type functional equation
In the study of the stability problems of quadratic-additive type functional equations, the uniqueness problem frequently occurs under various conditions
We prove a general uniqueness theorem that can be applied to the Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations

Summary

Introduction

Qf (x, y) := f (x + y) + f (x − y) − 2f (x) − 2f (y) for all x, y ∈ G1. A mapping f : G1 → G2 is called an additive mapping (or a quadratic mapping) if f satisfies the functional equation Af(x, y) = 0 (or Qf(x, y) = 0) for all x, y ∈ G1. A functional equation is called a quadratic-additive type functional equation if and only if each of its solutions is a quadratic-additive mapping. The mapping f : R → R given by f(x) = ax2 + bx is a solution of the quadratic-additive type functional equation. We prove a general uniqueness theorem that can be applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can save us much trouble in proving the uniqueness of relevant solutions repeatedly appearing in the stability problems for various quadratic-additive type functional equations

Main Result

Applications

Discussions