Abstract
Abstract In this article, we prove the nonlinear stability of the quartic functional equation 1 6 f ( x + 4 y ) + f ( 4 x - y ) = 3 0 6 9 f x + y 3 + f ( x + 2 y ) (1) + 1 3 6 f ( x - y ) - 1 3 9 4 f ( x + y ) + 4 2 5 f ( y ) - 1 5 3 0 f ( x ) (2) (3) in the setting of random normed spaces Furthermore, the interdisciplinary relation among the theory of random spaces, the theory of non-Archimedean space, the theory of fixed point theory, the theory of intuitionistic spaces and the theory of functional equations are also presented in the article.
Highlights
The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [2]
This result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference
The article of Rassias [4] has provided a lot of influence in the development of what we call generalized Ulam-Hyers stability of functional equations
Summary
The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [2]. A function || · || : X → [0, ∞] is called a non-Archimedean norm if it satisfies the following conditions:. (X , || · ||) is called a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent. Let (X , ||.||) be is a non-Archimedean normed linear space. If each Cauchy sequence is convergent, the random norm is said to be complete and the non-Archimedean RN-space is called a non-Archimedean random Banach space. 4. Generalized Ulam-Hyers stability for a quartic functional equation in nonArchimedean RN-spaces Let K be a non-Archimedean field, X a vector space over K and let (Y, μ, T) be a nonArchimedean random Banach space over K
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