Intuitionistic Fuzzy Stability of Functional Equations Associated with Inner Product Spaces

and Applied Analysis 3 x ν αx, t ν x, t/|α| for each α/ 0, xi ν x, t ν y, s ≥ ν x y, t s , xii ν x, · : 0,∞ → 0, 1 is continuous, xiii limt→∞ν x, t 0 and limt→ 0ν x, t 1. In this case μ, ν is called an intuitionistic fuzzy norm. Example 1.4 cf. 37 . Let X, ‖·‖ be a normed space, a∗b ab, and a b min a b, 1 for all a, b ∈ 0, 1 . For all x ∈ X and every t > 0 and k 1, 2, consider μk x, t ⎧ ⎨ ⎩ t t k‖x‖ , if t > 0, 0, if t ≤ 0, νk x, t ⎧ ⎪⎨ ⎪⎩ k‖x‖ t k‖x‖ , if t > 0, 0, if t ≤ 0. 1.3 Then X, μ, ν, ∗, is an IFNS. The concepts of convergence and Cauchy sequences in an intuitionistic fuzzy normed space are studied in 27 . Let X, μ, ν, ∗, be an IFNS. Then, a sequence {xk} is said to be intuitionistic fuzzy convergent to x ∈ X if, for every ε > 0 and t > 0, there exists k0 ∈ N such that μ xk−x, t > 1−ε and ν xk − x, t < ε for all k ≥ k0. In this case we write μ, ν − limxk x. The sequence {xk} is said to be intuitionistic fuzzy Cauchy sequence if, for every ε > 0 and t > 0, there exists k0 ∈ N such that μ xk − x , t > 1 − ε and ν xk − x , t < ε for all k, ≥ k0. X, μ, ν, ∗, is said to be complete if every intuitionistic fuzzy Cauchy sequence in X, μ, ν, ∗, is intuitionistic fuzzy convergent in X, μ, ν, ∗, . 2. Intuitionistic Fuzzy Stability Throughout this section, assume that X, Z, μ′, ν′ , and Y, μ, ν are linear space, IFNS, and intuitionistic fuzzy Banach space, respectively. For convenience, we use the following abbreviation for a given function f : X → Y : Δf x1, . . . , xn n ∑ i 1 f ⎛ ⎝xi − 1 n n ∑