On the generalized Hyers-Ulam-Rassias stability problem of radical functional equations

Seong Sik Kim,Madjid Eshaghi Gordji,Yeol Je Cho

doi:10.1186/1029-242x-2012-186

Abstract

Abstract In this paper, the generalized Hyers-Ulam-Rassias stability problem of radical quadratic and radical quartic functional equations in quasi-β-Banach spaces and then the stability by using subadditive and subquadratic functions for radical functional equations in ( β , p ) -Banach spaces are given. MSC:39B82, 39B52, 46H25.

Highlights

1 Introduction In, the stability problem of functional equations originated from the question of Ulam [, ] concerning the stability of group homomorphisms
The famous Ulam stability problem was partially solved by Hyers [ ] in Banach spaces
Rassias [, ] considered the Cauchy difference controlled by a product of different powers of norm

Summary

Introduction

In , the stability problem of functional equations originated from the question of Ulam [ , ] concerning the stability of group homomorphisms. Lee et al [ ] considered the following functional equation:. A quasi-β-norm · is a real-valued function on X satisfying the following:. F x + y + f x – y = f (x) + f (y), and discuss the generalized Hyers-Ulam-Rassias stability problem in quasi-β-normed spaces and the stability by using subadditive and subquadratic functions for the functional equations Let X be a real linear space and f : R → X be a function. Let X be a quasi-β-Banach space with the quasi-β-norm · X and K be the modulus of concavity of · X. X) exists for all x ∈ R and a function F : R → X is unique quadratic satisfying the functional equation

We can define a function

Now suppose that