On the Stability of Cubic Mappings and Quadratic Mappings in Random Normed Spaces

R Saadati,Sm Vaezpour,E Baktash,Yj Cho,M Jalili

doi:10.1155/2008/902187

Abstract

Recently, the stability of the cubic functional equation in fuzzy normed spaces was proved in earlier work; and the stability of the additive functional equations in random normed spaces was proved as well. In this paper, we prove the stability of the cubic functional equation in random normed spaces by an alternative proof which provides a better estimation. Finally, we prove the stability of the quartic functional equation in random normed spaces.

Highlights

Introduction and preliminariesThe study of stability problems for functional equations is related to a question of Ulam 1 concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers 2
We prove the stability of the cubic functional equation f 2x y f 2x − y 2f x y 2f x − y 12f x in random normed spaces by an alternative proof which provides a better estimation
The stability problem for the cubic functional equation was proved by Jun and Kim 10 for mappings f : X → Y, where X is a real normed space and Y is a Banach space

Summary

Introduction and preliminaries

The study of stability problems for functional equations is related to a question of Ulam 1 concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers 2. Every solution of the cubic functional equation is said to be a cubic mapping. The stability problem for the cubic functional equation was proved by Jun and Kim 10 for mappings f : X → Y , where X is a real normed space and Y is a Banach space. Every solution of the quadratic functional equation is said to be a quadratic mapping. The stability problem for the quadratic functional equation first was proved by J. We have μxn−x η > 1 − λ ⇔ Eλ,μ xn − x < η for every η > 0 We establish the stability of the cubic and quadratic functional equations in the setting of random normed spaces

On the stability of cubic mappings in RN-spaces

On the stability of quadratic mappings in RN-spaces