Fuzzy Stability of the Cauchy Additive and Quadratic Type Functional Equation

Abstract

In this paper, we investigate a fuzzy version of stability for the functional equation 2f (x + y) + f (x − y) + f (y − x) − 3f (x) − f (−x) − 3f (y) − f (−y) = 0 in the sense of M. Mirmostafaee and M. S. Moslehian.

Stability of an n-dimensional functional equation related to quadratic-additive mappings in fuzzy normed spaces

“…Since q > 1, together with (N5), we can deduce that the last term of (10) also tends to 1 as m → ∞. It follows from (10) that…”

“…Therefore, we can say that DF ≡ 0. Moreover, using the similar argument after (10) in Case 1, we get the inequality (6) from (12) in this case. To prove the uniqueness of F , let F : X → Y be another quadratic additive function satisfying (6).…”

A General Uniqueness Theorem concerning the Stability of Additive and Quadratic Functional Equations

“…for all ∈ \ {0}. Thus, it follows that ( ) = lim → ∞ ( / ) for all ∈ \ {0} provided Φ satisfies (4). Similarly, if Φ satisfies condition (4), then we have…”

“…In this paper, we prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. In Section 4, we apply our uniqueness theorem to complement stability theorems of the papers [4,5] where the uniqueness has not been proved. Indeed, this uniqueness theorem can save us much trouble in proving the uniqueness of relevant solutions repeatedly appearing in the stability problems for various quadratic-additive type functional equations.…”