Abstract
In this article, we propose to determine some stability results for the functional equation of cubic in random 2-normed spaces which seems to be a quite new and interesting idea. Also, we define the notion of continuity, approximately and conditional cubic mapping in random 2-normed spaces and prove some interesting results.
Highlights
1 Introduction and preliminaries In 1940, Ulam [1] proposed the following question concerning the stability of group homomorphisms: Let G1 be a group and let G2 be a metric group with the metric d(., .)
Hyers [2] answers the problem of Ulam under the assumption that the groups are Banach spaces and generalized by Aoki [3] and Rassias [4] for additive mappings and linear mappings, respectively
The stability problem for the cubic functional equation was proved by Jun and Kim [5] for mappings f: X ® Y, where X is a real normed space and Y is a Banach space
Summary
RTN-space (X, F , ∗) is said to be complete if every F-Cauchy is F-convergent. In this case (X, F , ∗) is called random 2-Banach space. Suppose that a function : X × X ® Z satisfies (2x, 2y) = a(x, y) for all x, y Î X and a ≠ 0. Let f : X ® Y be a -approximately cubic function. For all x Î X, t > 0 and nonzero zÎX f. ≥ Fφ(2nx,0),z(t) ≥ Fφ(x,0),z(t/αn), for all x Î X, t > 0 and nonzero z Î X; and for all n ≥ 0.
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