Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

and Applied Analysis 3 Definition 1.3 see 25 . Let X, μ, ν, ∗, be a intuitionistic fuzzy metric space. Let A be any subset of X. Define φ t inf { μ ( x, y, t ) : x, y ∈ A}, ψ t sup{ν(x, y, t) : x, y ∈ A}, 1.2 i A is said to be q-bounded if limt→∞φ t 1 and limt→∞ψ t 0, ii A is said to be semibounded if limt→∞φ t k and limt→∞ψ t 1 − k, 0 < k < 1 iii A is said to be unbounded if limt→∞φ t 0 and limt→∞ψ t 1. Theorem 1.4 see 25 . Let X, μ, ν, ∗, be a intuitionistic fuzzy metric space. A subset of X is IF bounded if and only if A is q-bounded or semibounded. Definition 1.5 see 8 . Let X, μ, ν, ∗, be an IFNS and xk be a sequence inX. The sequence xk is said to be convergent to L ∈ X with respect to IFN μ, ν if for every ε > 0 and t > 0, there exists a positive integer k0 ε such that μ xk −L, t > 1− ε and ν xk −L, t < ε whenever k > k0. In this case, we write μ, ν − lim xk L as k → ∞. Definition 1.6 see 16 . Let X, μ, ν, ∗, and Y, μ′, ν′, ∗, be two IFNS. A mapping f from X, μ, ν, ∗, to Y, μ′, ν′, ∗, is said to be intuitionistic fuzzy continuous at x0 ∈ X if for any given ε > 0, there exist δ δ a, ε , β β a, ε ∈ 0, 1 such that for all x ∈ X and for all a ∈ 0, 1 , μ x − x0, δ > 1 − β ⇒ μ′ ( f x − f x0 , ε ) > 1 − a, ν x − x0, δ < β ⇒ ν′ ( f x − f x0 , ε ) < a. 1.3 Definition 1.7 see 16 . Let fk : X, μ, ν, ∗, → Y, μ′, ν′, ∗, be a sequence of functions. The sequence fk is said to be pointwise intuitionistic fuzzy convergent onX to a function f with respect to μ′, ν′ if for each x ∈ X, the sequence fk x is convergent to f x with respect to μ′, ν′ . Definition 1.8 see 16 . Let fk : X, μ, ν, ∗, → Y, μ′, ν′, ∗, be a sequence of functions. The sequence fk is said to be uniformly intuitionistic fuzzy convergent on X to a function f with respect to μ, ν , if given 0 < r < 1, t > 0, there exist a positive integer k0 k0 r, t such that ∀x ∈ X and ∀k > k0, μ′ ( fk x − f x , t ) > 1 − r, ν′(fk x − f x , t) < r. 1.4 Now, we recall the notion of the statistical convergence of sequences in intuitionistic fuzzy normed spaces. Definition 1.9 see 26 . Let K ⊂ N and Kn {k ∈ K : k ≤ n}. Then the asymptotic density is defined by δ K limn→∞ |Kn|/n , where |Kn| denotes the cardinality of Kn. Definition 1.10. Let A be subset of N. If a property P k holds for all k ∈ A with δ A 1, we say that P holds for almost all k a · a · k . 4 Abstract and Applied Analysis Definition 1.11 see 26 . A sequence x xk is said to be statistically convergent to the number L, or in short st − lim x L, if for every ε > 0, the set K ε has asymptotic density zero, where K ε {k ∈ N : |xk − L| ≥ ε}, 1.5