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Abstract
We establish some stability results for the cubic functional equations 3 f x + 3 y + f 3 x - y = 15 f x + y + 15 f x - y + 8 0 f y , f 2 x + y + f 2 x - y = 2 f x + y + 2 f x - y + 12 f x and f 3 x + y + f 3 x - y = 3 f x + y + 3 f x - y + 48 f x in the setting of various -fuzzy normed spaces that in turn generalize a Hyers-Ulam stability result in the framework of classical normed spaces. First, we shall prove the stability of cubic functional equations in the -fuzzy normed space under arbitrary t-norm which generalizes previous studies. Then, we prove the stability of cubic functional equations in the non-Archimedean -fuzzy normed space. We therefore provide a link among different disciplines: fuzzy set theory, lattice theory, non-Archimedean spaces, and mathematical analysis.Mathematics Subject Classification (2000): Primary 54E40; Secondary 39B82, 46S50, 46S40.
Highlights
The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and it was affirmatively answered for Banach spaces by Hyers [2]
The article [4] of Rassias has provided a lot of influence in the development of what we call Hyers-Ulam-Rassias stability of functional equations
The stability problem for the cubic functional equations was studied by Jun and Kim [16] for mappings f : X ® Y, where X is a real normed space and Y is a Banach space
Summary
The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and it was affirmatively answered for Banach spaces by Hyers [2]. The stability problem for the cubic functional equations was studied by Jun and Kim [16] for mappings f : X ® Y, where X is a real normed space and Y is a Banach space. Let (X, || · ||) be a Banach space, (X, Pμ,ν , TM) be an intuitionistic fuzzy normed space in which TM(a, b) = (min{a1, b1}, max{a2, b2}) and
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