Abstract
In this paper, we first introduce the notions of $(n,\beta)$ -normed space and non-Archimedean $(n,\beta)$ -normed space, then we study the Hyers-Ulam stability of the Cauchy functional equation and the Jensen functional equation in non-Archimedean $(n,\beta)$ -normed spaces and that of the pexiderized Cauchy functional equation in $(n,\beta)$ -normed spaces.
Highlights
1 Introduction The stability problem of functional equations originated from a question of Ulam [ ] in concerning the stability of group homomorphisms
The case of approximately additive functions was solved by Hyers [ ] under the assumption that G and G are Banach spaces
The result of Rassias has provided a lot of influence during the past years in the development of a generalization of the Hyers-Ulam stability concept. This new concept is known as the Hyers-Ulam-Rassias stability of functional equation
Summary
The stability problem of functional equations originated from a question of Ulam [ ] in concerning the stability of group homomorphisms. In Section , we investigate the Hyers-Ulam stability of the pexiderized Cauchy functional equation in (n, β)-normed spaces. A sequence {xm} in a linear (n, β)-normed space X is called a convergent sequence if there is x ∈ X such that lim m→∞
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