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

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.

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

“…The stability of a general quadratic function equation was obtained by Y. H. Lee [11], Y. H. Lee et al [12], and S. S. Jin et al [13]. On the other hand, the stability of a general cubic function equation was studied by Y. H. Lee [14,15], S. M. Jun et al [16], and Y. H. Lee et al [17,18], and the stability of the general quartic function equation are discussed in Y. H. Lee [20] and Y. H. Lee et al [?,18,21,22,23]. Moreover, the stability of a general quintic functional equation has been studied by S. S. Jin et al [24], and the stability of the general sextic function equation has been obtained by Y. H. Lee [25], I.…”

A General Theorem on the Stability of a Class of Functional Equations including Quartic-Cubic-Quadratic-Additive Equations

“…Till now, we have followed out a routine and monotonous procedure for studying the stability problems of the quartic-cubic-quadratic-additive functional equations. However, the stability theorems of this paper can save us much trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations (see [8,9,10,12,13]).…”

A General Theorem on the Stability of a Class of Functional Equations Including Quartic-Cubic-Quadratic-Additive Equations

“…Until now, we have followed out a routine and monotonous procedure for studying the stability problems of the quartic-cubic-quadratic-additive functional equations. However, the stability theorems of this paper can save us the trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations (see [16][17][18][19][20]).…”