Abstract
In this paper, we investigated the asymptotic stability behaviour of the Pexider–Cauchy functional equation in non-Archimedean spaces. We also showed that, under some conditions, if ∥f(x+y)−g(x)−h(y)∥⩽ε, then f,g and h can be approximated by additive mapping in non-Archimedean normed spaces. Finally, we deal with a functional inequality and its asymptotic behaviour.
Highlights
The classical problem of the stability of homomorphisms was first posed by Ulam [1]
By a Banach non-Archimedean space, we mean that n =1 every Cauchy sequence is convergent
In [13,14], the authors investigated the stability of some functional equations in non-Archimedean normed spaces
Summary
The classical problem of the stability of homomorphisms was first posed by Ulam [1]. in 1940 as follows: Given a group ( G1 , ∗), a metric group ( G2 , , d) and ε > 0. Does δ > 0 exist such that if the function f : G1 → G2 satisfies the inequality: Stability of the Pexider–Cauchy d( f ( x ∗ y), f ( x ) f (y)) < δ, Functional Equation in x, y ∈ G1 , Non-Archimedean Spaces. A function k.k : X → [0, +∞) is called a non-Archimedean norm (valuation) if it satisfies the following conditions:. By a Banach non-Archimedean space, we mean that n =1 every Cauchy sequence is convergent. In [13,14], the authors investigated the stability of some functional equations in non-Archimedean normed spaces. We investigated the asymptotic stability behaviour of the Pexider–Cauchy functional equation in non-Archimedean spaces. We will use the methods and ideas presented in [15,16,17]
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