Abstract
We prove the stability of some functional equations in the random normed spaces under arbitraryt-norms.
Highlights
Introduction and PreliminariesThe 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]
We prove the stability of some functional equations in the random normed spaces under arbitrary t-norms
The result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference
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]. Every solution of the cubic functional equation is said to be a cubic mapping. The stability problem for the cubic functional equation was solved by Skof [23] for mappings f : X → Y, where X is a normed space and Y is a Banach space. Every solution of the quadratic functional equation is said to be a quadratic mapping (see [8, 9]). A random normed space (briefly, RNspace) is a triple (X, μ, T), where X is a vector space, T is a continuous t-norm, and μ is a mapping from X into D+ such that, the following conditions hold:. (3) An RN-space (X, μ, T) is said to be complete if every Cauchy sequence in X is convergent to a point in X.
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