Abstract
In this paper, we prove the Hyers-Ulam stability of homomorphisms in fuzzy Banach algebras and of derivations on fuzzy Banach algebras associated to the Cauchy-Jensen functional equation.
Highlights
Introduction and preliminariesThe theory of fuzzy space has much progressed as developing the theory of randomness
Following Cheng and Mordeson [7], Bag and Samanta [2] gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [25] and investigated some properties of fuzzy normed spaces [3]
We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn} converging to x0 in X, the sequence {f} converges to f (x0)
Summary
The theory of fuzzy space has much progressed as developing the theory of randomness. We use the definition of fuzzy normed spaces given in [2, 29, 30] to investigate a fuzzy version of the Hyers-Ulam stability for the CauchyJensen functional equation in the fuzzy normed algebra setting. We say that a mapping f : X → Y between fuzzy normed vector spaces X and Y is continuous at a point x0 ∈ X if for each sequence {xn} converging to x0 in X, the sequence {f (xn)} converges to f (x0). The Hyers-Ulam stability of the quadratic functional equation was proved by Skof [37] for mappings f : X → Y , where X is a normed space and Y is a Banach space. We prove the Hyers-Ulam stability of homomorphisms and derivations in fuzzy Banach algebras associated with the CauchyJensen functional equation. Throughout this paper, assume that (X, NX) is a fuzzy normed algebra and that (Y, N ) is a fuzzy Banach algebra
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