Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

Abstract

The purpose of this work is to investigate types of convergence of sequences of functions in intuitionistic fuzzy normed spaces and some properties related with these concepts.

1. Introduction and Preliminaries

The fuzzy theory has emerged as one of the most active area of research in many branches of mathematics and engineering. This new theory was introduced by Zadeh [1] in 1965 and since then a large number of research papers have appeared by using the concept of fuzzy set/numbers and fuzzification of many classical theories have also been made. It has also very useful application in various fields, for example, population dynamics [2], chaos control [3], computer programming [4], nonlinear dynamical systems [5], fuzzy physics [6], fuzzy topology [7, 8], and so forth. The notion of intuitionistic fuzzy sets, a generalization of fuzzy sets, was introduced by Atanassov [9] in 1986 and later there has been much progress in the study of intuitionistic fuzzy sets by many authors including Mursaleen et al. [10], Mursaleen et al. [11], and Yılmaz [12]. Using the idea of intuitionistic fuzzy sets, Park [13] defined the notion of intuitionistic fuzzy metric spaces with the help of the continuous t-norms and the continuous t-conorms as a generalization of fuzzy metric spaces due to George and Veeramani [14]. Samanta and Jebril [15] introduced the definitions of intuitionistic fuzzy continuity and sequential intuitionistic fuzzy continuity and proved that they are equivalent. A few of the algebraic and topological properties of intuitionistic fuzzy continuity and uniformly intuitionistic fuzzy continuity were investigated by Dinda and Samanta [16]. On the other hand, the Fast [17] introduced the concept of statistical convergence for real number sequences. Different types of statistical convergence of sequences of real functions and related notions were first studied in [18], and some important results and references on statistical convergence and function sequences can be found in [19–23].

In this paper, primarily following the line of [18], we define statistical convergence of sequences of functions in intuitionistic fuzzy normed space (IFNS for short), and we investigate some properties related with these concepts. To explain main problems of this work, we have to give some definitions and literature acknowledgment. In [24], Schweizer and Sklar introduced a continuous t-norm and a continuous t-conorm. Afterward Saadati and Park [8] introduced the following definitions by using concepts mentioned above.

We now first recall some basic notions of intuitionistic fuzzy normed spaces.

Definition 1.1 (see [8]). Let ∗ be a continuous t-norm, ⋄ a continuous t-conorm, and 𝑋 a linear space over the intuitionistic fuzzy field (ℝ or ℂ). If 𝜇 and 𝜈 are fuzzy sets on 𝑋×(0,∞) satisfying the following conditions, the five-tuple (𝑋,𝜇,𝜈,∗,⋄) is said to be an IFNS and (𝜇,𝜈) is called an intuitionistic fuzzy norm (IFN for short). For every 𝑥,𝑦∈𝑋 and 𝑠,𝑡>0,(i)𝜇(𝑥,𝑡)+𝜈(𝑥,𝑡)≤1,(ii)𝜇(𝑥,𝑡)>0,(iii)𝜇(𝑥,𝑡)=1⇔𝑥=0,(iv)𝜇(𝑎𝑥,𝑡)=𝜇(𝑥,𝑡/|𝑎|) for each 𝑎≠0,(v)𝜇(𝑥,𝑡)∗𝜇(𝑦,𝑠)≤𝜇(𝑥+𝑦,𝑡+𝑠),(vi)𝜇(𝑥,⋅)∶(0,∞)→[0,1] is continuous,(vii)lim_𝑡→∞𝜇(𝑥,𝑡)=1 and lim_𝑡→∞𝜇(𝑥,𝑡)=0,(viii)𝜈(𝑥,𝑡)<1, (ix)𝜈(𝑥,𝑡)=0⇔𝑥=0,(x)𝜈(𝑎𝑥,𝑡)=𝜈(𝑥,𝑡/|𝑎|) for each 𝑎≠0,(xi)𝜈(𝑥,𝑡)⋄𝜈(𝑦,𝑠)≥𝜈(𝑥+𝑦,𝑡+𝑠),(xii)𝜈(𝑥,⋅)∶(0,∞)→[0,1] is continuous,(xiii)lim_𝑡→∞𝜈(𝑥,𝑡)=0 and lim_𝑡→∞𝜈(𝑥,𝑡)=1.  For an IFNS, we further assume that (𝑋,𝜇,𝜈,∗,⋄) satisfy the following axiom:(see [16])(xiv)[]𝑎⋄𝑎=𝑎𝑎∗𝑎=𝑎∀𝑎∈0,1. (1.1)

Definition 1.2 (see [8]). Let (𝑋,𝜇,𝜈,∗,⋄) be a intuitionistic fuzzy normed space. A subset 𝐴 of 𝑋 is said to be IF-bounded if there exist 𝑡>0 and 0<𝑟<1 such that 𝜇(𝑥,𝑡)>1−𝑟 and 𝜈(𝑥,𝑡)<𝑟 for each 𝑥∈𝐴.

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)=inf𝜇𝑥,𝑦,𝑡∶𝑥,𝑦∈𝐴,𝜓(𝑡)=sup𝜈𝑥,𝑦,𝑡∶𝑥,𝑦∈𝐴, (1.2) (i) 𝐴 is said to be q-bounded if lim_𝑡→∞𝜙(𝑡)=1 and lim_𝑡→∞𝜓(𝑡)=0, (ii) 𝐴 is said to be semibounded if lim_𝑡→∞𝜙(𝑡)=𝑘 and lim_𝑡→∞𝜓(𝑡)=1−𝑘,0<𝑘<1 (iii) 𝐴 is said to be unbounded if lim_𝑡→∞𝜙(𝑡)=0 and lim_𝑡→∞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 (𝑥_𝑘) be a sequence in 𝑋. The sequence (𝑥_𝑘) 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(𝑥_𝑘−𝐿,𝑡)>1−𝜀 and 𝜈(𝑥_𝑘−𝐿,𝑡)<𝜀 whenever 𝑘>𝑘₀. In this case, we write (𝜇,𝜈)−lim𝑥_𝑘=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), 𝜇𝑥−𝑥₀,𝛿>1−𝛽⇒𝜇^′𝑓(𝑥)−𝑓𝑥₀𝜈,𝜀>1−𝑎,𝑥−𝑥₀,𝛿<𝛽⇒𝜈^′𝑓(𝑥)−𝑓𝑥₀,𝜀<𝑎. (1.3)

Definition 1.7 (see [16]). Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. The sequence (𝑓_𝑘) is said to be pointwise intuitionistic fuzzy convergent on 𝑋 to a function 𝑓 with respect to (𝜇^′,𝜈^′) if for each 𝑥∈𝑋, the sequence (𝑓_𝑘(𝑥))is convergent to 𝑓(𝑥) with respect to (𝜇^′,𝜈^′).

Definition 1.8 (see [16]). Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. The sequence (𝑓_𝑘) 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^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝑟,𝜈^′𝑓_𝑘(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 𝐾_𝑛={𝑘∈𝐾∶𝑘≤𝑛}. Then the asymptotic density is defined by 𝛿(𝐾)=lim_𝑛→∞(|𝐾_𝑛|/𝑛), where |𝐾_𝑛| denotes the cardinality of 𝐾_𝑛.

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).

Definition 1.11 (see [26]). A sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to the number 𝐿, or in short 𝑠𝑡−lim𝑥=𝐿, if for every 𝜀>0, the set 𝐾(𝜀) has asymptotic density zero, where 𝐾(𝜀)=𝑘∈ℕ∶𝑥_𝑘−𝐿≥𝜀, (1.5) that is, 𝑥_𝑘−𝐿<𝜀𝑎⋅𝑎⋅𝑘. (1.6)

Definition 1.12 (see [23]). Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS. Then, sequence (𝑥_𝑘) is said to be statistically convergent to 𝐿∈𝑋 with respect to IFN (𝜇,𝜈) provided that for every 𝜀>0 and 𝑡>0, 𝛿𝑘∈ℕ∶𝜇𝑥_𝑘−𝐿,𝑡≤1−𝜀or𝜈𝑥_𝑘−𝐿,𝑡≥𝜀=0, (1.7) or equivalently lim_𝑛→∞1𝑛𝑘≤𝑛∶𝜇𝑥_𝑘−𝐿,𝑡≤1−𝜀or𝜈𝑥_𝑘−𝐿,𝑡≥𝜀=0. (1.8) In this case, we write 𝑠𝑡_𝜇,𝜈−lim(𝑥_𝑘)=𝐿.

Definition 1.13 (see [23]). Let (𝑋,𝜇,𝜈,∗,⋄) be an IFNS. Sequence (𝑥_𝑘) is said to be statistically Cauchy with respect to IFN (𝜇,𝜈) provided that for every 𝜀>0 and 𝑡>0, there exists a number 𝑚∈ℕ satisfying 𝛿𝑘∈ℕ∶𝜇𝑥_𝑘−𝑥_𝑚,𝑡≤1−𝜀or𝜈𝑥_𝑘−𝑥_𝑚,𝑡≥𝜀=0. (1.9)

2. Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

In this section, we define pointwise statistically and uniformly statistically convergent sequences of functions in intuitionistic fuzzy normed spaces. Also, we give the statistical analog of the Cauchy convergence criterion for pointwise and uniformly statistical convergent in intuitionistic fuzzy normed space. Finally, we prove that uniformly statistical convergence preserves continuity.

Definition 2.1. Let (𝑋,𝜇,𝜈,∗,⋄) and (𝑌,𝜇^′,𝜈^′,∗,⋄) be two IFNS and 𝑓_𝑘:(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) a sequence of functions. We say that a sequence (𝑓_𝑘) pointwise statistically converges to 𝑓 with respect to intuitionistic fuzzy norm (𝜇^′,𝜈^′) if 𝑓_𝑘(𝑥) statistically converges to 𝑓(𝑥) for each 𝑥∈𝑋 with respect to intuitionistic fuzzy norm (𝜇^′,𝜈^′) and we write 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓.

Theorem 2.2. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. If(𝑓_𝑘) is pointwise intuitionistic fuzzy convergent on 𝑋 with respect to (𝜇,𝜈), then 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓. But the converse of this is not true.

Proof. Let (𝑓_𝑘) be pointwise intuitionistic fuzzy convergent on 𝑋. In this case the sequence (𝑓_𝑘(𝑥)) is convergent with respect to (𝜇^′,𝜈^′) for each 𝑥∈𝑋. Then for each 𝜀>0 and 𝑡>0, there is number 𝑘₀(𝜀)∈ℕ such that 𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀 (2.1) for all 𝑘≥𝑘₀ and for each 𝑥∈𝑋. Hence for each 𝑥∈𝑋 the set 𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀 (2.2) has finite numbers of terms. Since density of finite subset of ℕ is 0, hence 𝛿𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀=0. (2.3) That is, 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓.

Example 2.3. Let (ℝ,|⋅|) denote the space of real numbers with the usual norm, and let 𝑎∗𝑏=𝑎⋅𝑏 and 𝑎⋄𝑏=min{𝑎+𝑏,1} for 𝑎,𝑏∈[0,1]. For all 𝑥∈ℝ and every 𝑡>0, we consider 𝜇()=𝑡𝑥,𝑡|𝑥|()=|𝑥|𝑡+,𝜈𝑥,𝑡|𝑥|.𝑡+ (2.4) In this case, (ℝ,𝜇,𝜈,∗,⋄) is an IFNS (also, ([0,1],𝜇,𝜈,∗,⋄) is intuitionistic fuzzy normed space.). Let 𝑓_𝑘∶[0,1]→ℝ be a sequence of functions whose terms are given by 𝑓_𝑘(𝑥)=𝑥^𝑘2+1if𝑘=𝑚²()1𝑚∈ℕ,𝑥∈0,2,0if𝑘≠𝑚²()1𝑚∈ℕ,𝑥∈0,2,0if𝑘=𝑚²()1𝑚∈ℕ,𝑥∈2,𝑥,1_𝑘+12if𝑘≠𝑚²()1𝑚∈ℕ,𝑥∈2,1,2if𝑥=1. (2.5) Then sequence (𝑓_𝑘) is pointwise statistically intuitionistic convergent on [0,1] with respect to (𝜇,𝜈). Indeed, for 𝑥∈⌈0,1/2), since 𝐾_𝑛(𝜀,𝑡)=𝑘≤𝑛∶𝜇𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈𝑓_𝑘(𝑥)(𝑥),𝐾−𝑓,𝑡≥𝜀_𝑛()=𝑡𝜀,𝑡𝑘≤𝑛∶𝑡+𝑓_𝑘(𝑥)𝑓−0≤1−𝜀or^𝑘(𝑥)−0𝑡+𝑓_𝑘(𝑥)=𝑓−0≥𝜀𝑘≤𝑛∶_𝑘(𝑥)≥𝜀𝑡=1−𝜀𝑘≤𝑛∶𝑓_𝑘(𝑥)=𝑥^𝑘2=+1𝑘≤𝑛∶𝑘=𝑚²,𝑚∈ℕ, (2.6) we have 𝛿𝐾_𝑛()=1𝜀,𝑡𝑛𝑘≤𝑛∶𝑘=𝑚²≤√,𝑚∈ℕ𝑛𝑛, (2.7) which yields lim_𝑛→∞𝛿(𝐾_𝑛(𝜀,𝑡))=0. Thus, for each 𝑥∈[0,1/2), sequence (𝑓_𝑘) is statistically convergent to 0 with respect to intuitionistic fuzzy norm (𝜇,𝜈).
If we take 𝑥∈[1/2,1), then we have 𝐾^′_𝑛()=𝑡𝜀,𝑡𝑘≤𝑛∶𝑡+𝑓_𝑘(𝑥)−(𝑓1/2)≤1−𝜀or^𝑘(𝑥)−(1/2)𝑡+𝑓_𝑘(𝑥)−(=𝑓1/2)≥𝜀𝑘≤𝑛∶_𝑘(𝑥)−12≥𝜀𝑡=1−𝜀𝑘≤𝑛∶𝑘=𝑚²,𝑓_𝑘(𝑥)−12≥𝜀𝑡∪1−𝜀𝑘≤𝑛∶𝑘≠𝑚²,𝑓_𝑘(𝑥)−12≥𝜀𝑡=1−𝜀𝑘≤𝑛∶𝑘=𝑚²∪𝑘≤𝑛∶𝑘≠𝑚²∶𝑥^𝑘+12−12≥𝜀𝑡=1−𝜀𝑘≤𝑛∶𝑘=𝑚²∪𝑘≤𝑛∶𝑘≠𝑚²∶𝑥^𝑘≥𝜀𝑡.1−𝜀 (2.8) Therefore, density of 𝐾^′_𝑛(𝜀,𝑡) is 0 and for each 𝑥∈[1/2,1), sequence (𝑓_𝑘) is statistically convergent to 1/2 with respect to IFN (𝜇,𝜈). If we take 𝑥=1, it can be seen easily that (𝑓_𝑘) is intuitionistic fuzzy convergent to 2. Hence (𝑓_𝑘) is intuitionistic fuzzy statistically convergent to 2. That is 𝑠𝑡_𝜇,𝜈−𝑓_𝑘(𝑥)=10,𝑥∈0,2121,𝑥∈2,12,𝑥=1. (2.9) Since 𝑓_𝑘(𝑥) is statistically convergent to different points with respect to intuitionistic fuzzy norm (𝜇,𝜈) for each 𝑥∈𝑋, it can be seen that (𝑓_𝑘) is pointwise statistically intuitionistic fuzzy convergent on [0,1].

Lemma 2.4. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequence of functions. Then the following statements are equivalent:(i)𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓,(ii)𝛿{𝑘∈ℕ∶𝜇^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)≤1−𝜀}=𝛿{𝑘∈ℕ∶𝜈^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)≥𝜀}=0 for each 𝑥∈𝑋, for each 𝜀>0 and 𝑡>0,(iii)𝛿{𝑘∈ℕ∶𝜇^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)>1−𝜀 and 𝜈^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)<𝜀}=1 for each 𝑥∈𝑋, for each 𝜀>0 and 𝑡>0,(iv)𝑠𝑡−lim𝜇^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)=1 and 𝑠𝑡−lim𝜈^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)=0 for each 𝑥∈𝑋 and 𝑡>0.

Theorem 2.5. Let (𝑓_𝑘) and (𝑔_𝑘) be two sequences of functions from (𝑋,𝜇,𝜈,∗,⋄) to (𝑌,𝜇^′,𝜈^′,∗,⋄). If 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓 and 𝑠𝑡_𝜇,𝜈−𝑔_𝑘→𝑔, then 𝑠𝑡_𝜇,𝜈−(𝛼𝑓_𝑘+𝛽𝑔_𝑘)→𝛼𝑓+𝛽𝑔 where 𝛼,𝛽∈(ℝ or ℂ).

Proof. The proof is clear for 𝛼=0 and 𝛽=0. Now let 𝛼≠0 and 𝛽≠0. Since 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓 and 𝑠𝑡_𝜇,𝜈−𝑔_𝑘→𝑔, for each 𝑥∈𝑋, if we define 𝐴₁=𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2|𝛼|≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2|𝛼|,𝐴≥𝜀₂=𝑘∈ℕ∶𝜇^′𝑔_𝑘(𝑥)(𝑥),𝑡−𝑔2𝛽≤1−𝜀or𝜈^′𝑔_𝑘(𝑥)(𝑥),𝑡−𝑔2𝛽≥𝜀, (2.10) then 𝛿𝐴₁=0,𝛿𝐴₂=0. (2.11)
Since 𝛿(𝐴₁)=0and𝛿(𝐴₂)=0, if we state 𝐴 by (𝐴₁∪𝐴₂) then 𝛿(𝐴)=0. (2.12) Hence, 𝐴₁∪𝐴₂≠ℕ and there exists ∃𝑚∈ℕ such that 𝜇^′𝑓_𝑚(𝑥)(𝑥),𝑡−𝑓2|𝛼|>1−𝜀,𝜈^′𝑓_𝑚(𝑥)(𝑥),𝑡−𝑓2|𝛼|𝜇<𝜀,_′𝑔_𝑚(𝑥)(𝑥),𝑡−𝑔2𝛽>1−𝜀,𝜈^′𝑔_𝑚(𝑥)(𝑥),𝑡−𝑔2𝛽<𝜀. (2.13) Let 𝐵=𝑘∈ℕ∶𝜇^′𝛼𝑓_𝑘+𝛽𝑔_𝑘𝑥)−(𝑥)(𝑥)𝜈(𝛼𝑓+𝛽𝑔,𝑡>1−𝜀,_′𝛼𝑓_𝑘+𝛽𝑔_𝑘(𝑥)−𝛼𝑓(𝑥)+𝛽𝑔(𝑥),𝑡<𝜀. (2.14) We will show that for each 𝑥∈𝑋𝐴^𝑐⊂𝐵. (2.15) Let 𝑚∈𝐴^𝑐. In this case, 𝜇^′𝑓_𝑚(𝑥)(𝑥),𝑡−𝑓2|𝛼|>1−𝜀,𝜈^′𝑓_𝑚(𝑥)(𝑥),𝑡−𝑓2|𝛼|𝜇<𝜀,_′𝑔_𝑚(𝑥)(𝑥),𝑡−𝑔2𝛽>1−𝜀,𝜈^′𝑔_𝑚(𝑥)(𝑥),𝑡−𝑔2𝛽<𝜀. (2.16) Using those above, we have 𝜇^′𝛼𝑓_𝑚+𝛽𝑔_𝑚(𝑥)−𝛼𝑓+𝛽𝑔(𝑥),𝑡≥𝜇^′𝛼𝑓_𝑚(𝑥)(𝑥),𝑡−𝛼𝑓2∗𝜇^′𝛽𝑔_𝑚(𝑥)(𝑥),𝑡−𝛽𝑔2=𝜇^′𝑓_𝑚(𝑥)(𝑥),𝑡−𝑓2|𝛼|∗𝜇^′𝑔_𝑚(𝑥)(𝑥),𝑡−𝑔2𝛽>()∗()=(),𝜈1−𝜀1−𝜀1−𝜀_′𝛼𝑓_𝑚+𝛽𝑔_𝑚(𝑥)−𝛼𝑓+𝛽𝑔(𝑥),𝑡≤𝜈^′𝛼𝑓_𝑚(𝑥)(𝑥),𝑡−𝛼𝑓2∗𝜈^′𝛽𝑔_𝑚(𝑥)(𝑥),𝑡−𝛽𝑔2=𝜈^′𝑓_𝑚(𝑥)(𝑥),𝑡−𝑓2|𝛼|∗𝜈^′𝑔_𝑚(𝑥)(𝑥),𝑡−𝑔2𝛽<𝜀⋄𝜀=𝜀. (2.17)
This implies that 𝐴^𝑐⊂𝐵. (2.18) Since 𝐵^𝑐⊂𝐴 and 𝛿(𝐴)=0, hence 𝛿𝐵^𝑐=0, (2.19) that is𝛿𝑘∈ℕ∶𝜇^′𝛼𝑓_𝑘+𝛽𝑔_𝑘(𝑥)−(𝑥)𝜈𝛼𝑓+𝛽𝑔,𝑡≤1−𝜀,_′𝛼𝑓_𝑘+𝛽𝑔_𝑘(𝑥)−𝛼𝑓+𝛽𝑔(𝑥),𝑡≥𝜀=0 (2.20) which means𝑠𝑡_𝜇,𝜈−𝛼𝑓_𝑘+𝛽𝑔_𝑘→𝛼𝑓+𝛽𝑔. (2.21)

Definition 2.6. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. The sequence (𝑓_𝑘) is a pointwise statistically Cauchy sequence in IFNS provided that for each 𝜀>0 and 𝑡>0 there exists 𝑁=𝑁(𝜀,𝑡,𝑥) such that 𝛿𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝜀foreach𝑥∈𝑋=0. (2.22) That is, there exists a number 𝑁=𝑁(𝜀,𝑡,𝑥) for each 𝑥∈𝑋 such that 𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡<𝜀for𝑎⋅𝑎⋅𝑘. (2.23)

Theorem 2.7. Let 𝑓_𝑘:(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. If (𝑓_𝑘) is a pointwise statistically convergent sequence with respect to intuitionistic fuzzy norm (𝜇,𝜈), then (𝑓_𝑘) is a pointwise statistically Cauchy sequence with respect to intuitionistic fuzzy norm (𝜇,𝜈).

Proof. Suppose that 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓 and let 𝜀>0, 𝑡>0. For given each 𝜀>0, choose 𝑠>0 such that (1−𝜀)∗(1−𝜀)>1−𝑠 and 𝜀⋄𝜀<𝑠. If we state, respectively, 𝐴_𝑥(𝜀,𝑡) and 𝐴^𝑐_𝑥(𝜀,𝑡) by 𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2≥𝜀,𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2>1−𝜀,𝜈^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2<𝜀, (2.24) for each 𝑥∈𝑋. Then, we have 𝛿𝐴_𝑥(𝜀,𝑡)=0, (2.25) which implies that 𝛿𝐴^𝑐_𝑥(𝜀,𝑡)=1. (2.26) Let 𝑁∈𝐴^𝑐_𝑥(𝜀,𝑡). Then 𝜇^′𝑓_𝑁(𝑥)(𝑥),𝑡−𝑓2>1−𝜀,𝜈^′𝑓_𝑁(𝑥)(𝑥),𝑡−𝑓2<𝜀. (2.27) We want to show that there exists a number 𝑁=𝑁(𝑥,𝜀,𝑡) such that 𝛿𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝑠or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝑠foreach𝑥∈𝑋=0. (2.28) Therefore, define for each 𝑥∈𝑋, 𝐵_𝑥(𝜀,𝑡)=𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝑠or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝑠. (2.29) We have to show that 𝐵_𝑥(𝜀,𝑡)⊂𝐴_𝑥(𝜀,𝑡). (2.30) Suppose that 𝐵_𝑥(𝜀,𝑡)⊆𝐴̸_𝑥(𝜀,𝑡). (2.31) In this case, 𝐵_𝑥(𝜀,𝑡) has at least one different element which 𝐴_𝑥(𝜀,𝑡) does not has. Let 𝑘∈𝐵_𝑥(𝜀,𝑡)⧵𝐴_𝑥(𝜀,𝑡). Then we have 𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝑠,𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2>1−𝜀, (2.32) in particularly 𝜇^′(𝑓_𝑁(𝑥)−𝑓(𝑥),𝑡/2)>1−𝜀. In this case, 1−𝑠≥𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2∗𝜇^′𝑓_𝑁(𝑥)(𝑥),𝑡−𝑓2≥(1−𝜀)∗(1−𝜀)>1−𝑠, (2.33) which is not possible. On the other hand 𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝑠,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀, (2.34) in particularly 𝜈^′(𝑓_𝑁(𝑥)−𝑓(𝑥),𝑡)<𝜀. In this case, 𝑠≤𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤𝜈^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2⋄𝜈^′𝑓_𝑁(𝑥)(𝑥),𝑡−𝑓2<𝜀⋄𝜀<𝑠 (2.35) which is not possible. Hence 𝐵_𝑥(𝜀,𝑡)⊂𝐴_𝑥(𝜀,𝑡). Therefore, by 𝛿(𝐴_𝑥(𝜀,𝑡))=0,𝛿(𝐵_𝑥(𝜀,𝑡))=0. That is, (𝑓_𝑘) is a pointwise statistical Cauchy sequence with respect to intuitionistic fuzzy norm (𝜇,𝜈).

Afterward this step, we introduce a uniformly statistical convergence of sequences of function in an IFNS. To do this, we need the following definition

Definition 2.8. Let (𝑋,𝜇,𝜈,∗,⋄) and (𝑌,𝜇^′,𝜈^′,∗,⋄) be two intuitionistic fuzzy normed linear space over the same field IF and 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. (𝑓_𝑘) converges uniform statistically to f with respect to (𝜇,𝜈)⇔∀𝜀>0,∃𝑀⊂ℕ, 𝛿(𝑀)=1 and ∃𝑘₀=𝑘₀(𝜀,𝑡)∈𝑀϶   ∀𝑘>𝑘₀ and 𝑘∈𝑀 and ∀𝑥∈𝑋, 𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀. (2.36)

Lemma 2.9. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. Then the following statements are equivalent:(i)𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓.(ii)𝛿{𝑘∈ℕ∶𝜇^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)≤1−𝜀}=𝛿{𝑘∈ℕ∶𝜈^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)≥𝜀}=0 for all 𝑥∈𝑋, for every 𝜀>0 and 𝑡>0.(iii)𝛿{𝑘∈ℕ∶𝜇^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)>1−𝜀 and 𝜈^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)<𝜀}=1 for all 𝑥∈𝑋, for every 𝜀>0 and 𝑡>0.(iv)𝑠𝑡−lim𝜇^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)=1 and 𝑠𝑡−lim𝜈^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)=0 for all 𝑥∈𝑋 and 𝑡>0.

Proposition 2.10. Let the sequence (𝑓_𝑘) and 𝑓 be bounded functions from (𝑋,𝜇,𝜈,∗,⋄) to (𝑌,𝜇^′,𝜈^′,∗,⋄). 𝑓_𝑘 is intuitionistic fuzzy uniformly statistically convergent to 𝑓 if and only if 𝑠𝑡−liminf𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=1,𝑠𝑡−limsup𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=0, (2.37) where the supremum and infimum are taken over all 𝑥∈𝑋.

Proof. Suppose that 𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓 on 𝑋. Since (𝑓_𝑘) and 𝑓 are bounded in (𝑋,𝜇,𝜈,∗,⋄) for each 𝑘∈ℕ, by using Definition 1.3 and Theorem 1.4, we have inf𝜇^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)=𝑟_𝑘 and sup𝜈^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)=1−𝑟_𝑘 for each 𝑘∈ℕ and for each 𝑡>0,(0<𝑟_𝑘<1). By using (𝑖𝑣) of Lemma 2.9, we get 𝑠𝑡−liminf𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=1,𝑠𝑡−limsup𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=0, (2.38) where the supremum and infimum are taken over all 𝑥∈𝑋.
Conversely, suppose that 𝑠𝑡−liminf𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=1,𝑠𝑡−limsup𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=0, (2.39) where the supremum and infimum are taken over all 𝑥∈𝑋. Since inf𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡∈(0,1],sup𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡∈[0,1), (2.40) from definition statistical convergence, for 𝑎⋅𝑎⋅𝑘, for every 𝜀>0 and 𝑡>0inf𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡−1<𝜀,sup𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡−0<𝜀. (2.41) For all 𝑥∈𝑋𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥inf𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡⇒−𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<−inf𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡⇒1−𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<1−inf𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡since𝜇^′𝑓_𝑘(𝑥)(𝑥)∈(]⇒−𝑓,𝑡0,1∀𝑥∈𝑋1−𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<1−inf𝜇^′𝑓_𝑘(𝑥)(𝑥)⇒−𝑓,𝑡1−𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<1−inf𝜇^′𝑓_𝑘(𝑥)(𝑥)⇒−𝑓,𝑡<𝜀1−𝜇^′𝑓_𝑘(𝑥)(𝑥)𝜈−𝑓,𝑡<𝜀𝑎⋅𝑎⋅𝑘,_′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤sup𝜈^′𝑓_𝑘(𝑥)(𝑥)⇒𝜈−𝑓,𝑡_′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡−0≤sup𝜈^′𝑓_𝑘(𝑥)(𝑥)⇒𝜈−𝑓,𝑡−0<𝜀_′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡−0<𝜀𝑎⋅𝑎⋅𝑘. (2.42) Therefore, 𝑠𝑡−lim𝜇^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)=1 and 𝑠𝑡−lim𝜈^′(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)=0 for all 𝑥∈𝑋 and 𝑡>0. From Lemma 2.9, we get 𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓.

Example 2.11. Let (ℝ,𝜇,𝜈,∗,⋄) be as Example 2.3. Consider 𝑓_𝑘∶[0,1)→ℝ be sequence of functions whose terms are given by 𝑓_𝑘=𝑥^𝑘+1,if𝑘≠𝑚²(𝑚∈ℕ)2,otherwise. (2.43) Then, for every 0<𝜀<1 and for every 𝑡>0, we define 𝐾_𝑛=𝑘≤𝑛∶𝜇𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀. (2.44) For all 𝑥∈𝑋, we have 𝛿𝐾_𝑛=𝛿𝑘≤𝑛∶𝜇𝑓_𝑘(𝑥)−1,𝑡≤1−𝜀or𝜈𝑓_𝑘(𝑥)−1,𝑡≥𝜀=0. (2.45) Since 𝑠𝑡_𝜇,𝜈−𝑓_𝑘(𝑥)=1, (2.46) for all 𝑥∈𝑋, the sequence (𝑓_𝑘) is uniformly statistically intuitionistic fuzzy convergent to 1 on [0,1).

Example 2.12. Let (ℝ,𝜇,𝜈,∗,⋄) be as Example 2.3. Consider 𝑓_𝑘∶[−1,1]→ℝ be sequence of functions whose terms are given by 𝑓_𝑘=1𝑥−𝑘²,if𝑘≠𝑚²(𝑚∈ℕ)2,otherwise. (2.47) Since 𝑠𝑡_𝜇,𝜈−𝑓_𝑘(𝑥)=𝑥² for all 𝑥∈[−1,1],𝑓_𝑘 is uniformly statistically intuitionistic fuzzy convergent to 𝑥² on [−1,1]. We can show this using Proposition 2.10 as the following, since 𝑓_𝑘 is bounded functions sequence on [−1,1].
We want to find lim_𝑘→∞inf_{[]𝑥∈−1,1}𝜇𝑓_𝑘(𝑥)(𝑥)=𝑡−𝑓,𝑡𝑡+−(2𝑥/𝑘)+1/𝑘²,lim_𝑘→∞sup_{𝑥∈[−1,1]}𝜈𝑓_𝑘(𝑥)(𝑥)=−()−𝑓,𝑡2𝑥/𝑘+1/𝑘²𝑡+−(2𝑥/𝑘)+1/𝑘²⋅ (2.48) Firstly, we need to find lim_𝑘→∞inf𝜇(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡) and lim_𝑘→∞sup𝜈(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡) for over all 𝑥∈[−1,1/2𝑘] and for over all 𝑥∈(1/2𝑘,1], respectively. In case of 𝑥∈[−1,1/2𝑘], we have −(2𝑥/𝑘)+(1/𝑘²)≥0. Since 0≤−2𝑥𝑘+1𝑘²≤2𝑘+1𝑘²⇒𝑡≤𝑡−2𝑥𝑘+1𝑘²2≤𝑡+𝑘+1𝑘²⇒1𝑡+(2/𝑘)+1/𝑘²≤1𝑡−(2𝑥/𝑘)+1/𝑘²≤1𝑡⇒𝑡𝑡+(2/𝑘)+1/𝑘²≤𝑡𝑡−(2𝑥/𝑘)+1/𝑘²≤1, (2.49) we get lim_𝑘→∞inf_{𝑥∈[−1,1/2𝑘]}𝜇𝑓_𝑘(𝑥)−𝑥²,𝑡=lim_𝑘→∞inf_{𝑥∈[−1,1/2𝑘]}𝑡𝑡−(2𝑥/𝑘)+1/𝑘²=lim_𝑘→∞𝑡𝑡+(2/𝑘)+1/𝑘²=1. (2.50) On the other hand, we have −(2𝑥/𝑘)+1/𝑘²𝑡−(2𝑥/𝑘)+1/𝑘²≤(2/𝑘)+1/𝑘²𝑡, (2.51) and so lim_𝑘→∞sup_{𝑥∈[−1,1/2𝑘]}𝜈𝑓_𝑘(𝑥)−𝑥²,𝑡=lim_𝑘→∞sup_{𝑥∈[−1,1/2𝑘]}−(2𝑥/𝑘)+1/𝑘²𝑡−(2𝑥/𝑘)+1/𝑘²=lim_𝑘→∞(2/𝑘)+1/𝑘²𝑡=0. (2.52) In case of 𝑥∈(1/2𝑘,1],we have −(2𝑥/𝑘)+(1/𝑘²)<0. Since −2𝑘+1𝑘²<−2𝑥𝑘+1𝑘²≤0⇒0≤2𝑥𝑘−1𝑘²<2𝑘−1𝑘²⇒𝑡<𝑡+2𝑥𝑘−1𝑘²2≤𝑡+𝑘−1𝑘²⇒1𝑡+(2/𝑘)−1/𝑘²<1𝑡+(2𝑥/𝑘)−1/𝑘²≤1𝑡⇒𝑡𝑡+(2/𝑘)−1/𝑘²≤𝑡𝑡+(2𝑥/𝑘)−1/𝑘²<1, (2.53) we get lim_𝑘→∞inf_{(]𝑥∈1/2𝑘,1}𝜇𝑓_𝑘(𝑥)−𝑥²,𝑡=lim_𝑘→∞inf_{(]𝑥∈1/2𝑘,1}𝑡𝑡+(2𝑥/𝑘)−1/𝑘²=lim_𝑘→∞𝑡𝑡+(2/𝑘)−1/𝑘²=1. (2.54) On the other hand, we have (2𝑥/𝑘)−1/𝑘²𝑡+(2𝑥/𝑘)−1/𝑘²≤(2/𝑘)−1/𝑘²𝑡, (2.55) and so lim_𝑘→∞sup_{(]𝑥∈1/2𝑘,1}𝜈𝑓_𝑘(𝑥)−𝑥²,𝑡=lim_𝑘→∞sup_{(]𝑥∈1/2𝑘,1}(2𝑥/𝑘)−1/𝑘²𝑡+(2𝑥/𝑘)−1/𝑘²=lim_𝑘→∞(2/𝑘)−1/𝑘²𝑡=0. (2.56) That is, the sequence (𝑓_𝑘) is uniformly statistically intuitionistic fuzzy convergent to 𝑥² on [−1,1].

Remark 2.13. If 𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓, then 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓. But the converse of this is not true.

We prove this with the following example.

Example 2.14. Let us define the sequence of functions 𝑓_𝑘(𝑥)=0𝑘=𝑛²𝑘²𝑥1+𝑘³𝑥²otherwise, (2.57) on [0,1]. This sequence of functions is pointwise statistically intuitionistic fuzzy convergent to 0 (indeed, for 𝑥=1/𝑘,𝑠𝑡_𝜇,𝜈−𝑓_𝑘(1/𝑘)=1 and for 𝑥=0,𝑠𝑡_𝜇,𝜈−𝑓_𝑘(0)=0). But, it is not uniformly statistical intuitionistic fuzzy convergent. Since the sequence of functions is bounded on [0,1], we can use Proposition 2.10 to prove our claim. Let us take infumum for 𝜇(𝑓_𝑘(𝑥)−0,𝑡)=𝜇(𝑘²𝑥/(1+𝑘³𝑥²),𝑡)=𝑡/(𝑡+(𝑘²𝑥)/(1+𝑘³𝑥²)) and supremum for 𝜈(𝑓_𝑘(𝑥)−0,𝑡)=(𝜈(𝑘²𝑥/(1+𝑘³𝑥²),𝑡))=(𝑘²𝑥/(1+𝑘³𝑥²)/(𝑡+(𝑘²𝑥)/(1+𝑘³𝑥²)), over all 𝑥∈[0,1]. Firstly, we try to find sup_𝑥∈[0,1](𝑘²𝑥/(1+𝑘³𝑥²)) and inf_𝑥∈[0,1](𝑘²𝑥/(1+𝑘³𝑥²)). For this, we have 𝑘²𝑥1+𝑘³𝑥^2′=𝑘²1−𝑘³𝑥²1+𝑛³𝑥²²=0⇒𝑥=𝑘^−3/2. (2.58) Since 𝑓_𝑘(0)=0,𝑓_𝑘(1)=𝑘²/(1+𝑘³) and 𝑓_𝑘(𝑘^−3/2)=√𝑘/2, we get sup_𝑥∈[0,1](𝑘²𝑥/(1+𝑘³𝑥²))=√𝑘/2 and inf_𝑥∈[0,1](𝑘²𝑥/(1+𝑘³𝑥²))=0. Then, 𝑘²𝑥1+𝑘³𝑥²≤√𝑘2𝑘⇒𝑡+²𝑥1+𝑘³𝑥²√≤𝑡+𝑘2⇒1𝑡+√≤1𝑘/2𝑡+𝑘²𝑥/1+𝑘³𝑥²⇒𝑡𝑡+√≤𝑡𝑘/2𝑡+𝑘²𝑥/1+𝑘³𝑥²,𝑘0≤²𝑥1+𝑘³𝑥²𝑘⇒𝑡+0≤𝑡+²𝑥1+𝑘³𝑥²⇒1𝑡+𝑘²𝑥/1+𝑘³𝑥²≤1𝑡⇒𝑘²𝑥/1+𝑘³𝑥²𝑡+𝑘²𝑥/1+𝑘³𝑥²≤𝑘²𝑥/1+𝑘³𝑥²𝑡≤√𝑘,2𝑡 (2.59) so, we get lim_𝑘→∞𝑘inf𝜇²𝑥1+𝑘³𝑥²,𝑡=lim_𝑘→∞𝑡inf𝑡+𝑘²𝑥/1+𝑘³𝑥²=lim_𝑘→∞𝑡𝑡+√𝑘/2=lim_𝑘→∞2𝑡2𝑡+√𝑘=0≠1,lim_𝑘→∞𝑘sup𝑣²𝑥1+𝑘³𝑥²,𝑡=lim_𝑘→∞𝑘sup²𝑥/1+𝑘³𝑥²𝑡+𝑘²𝑥/1+𝑘³𝑥²=lim_𝑘→∞√𝑘2𝑡=∞≠0. (2.60) Therefore, we conclude that (𝑓_𝑘) does not intuitionistic fuzzy uniformly statistical convergent to 0.

Theorem 2.15. Let 𝑓_𝑘 :(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. If (𝑓_𝑘) is uniformly intuitionistic fuzzy convergent on 𝑋 to a function 𝑓 with respect to (𝜇,𝜈), then 𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓. But the converse of this is not true.

Proof. Let 𝑓_𝑘 be uniformly intuitionistic fuzzy convergent on 𝑋 to a function 𝑓. In this case, given 0<𝜀<1,𝑡>0, there exist a positive integer 𝑘₀=𝑘₀(𝑟,𝑡) such that ∀𝑥∈𝑋 and ∀𝑘>𝑘₀, 𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀. (2.61) That is, for 𝑘≤𝑘₀𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀, (2.62) is satisfied and these 𝑘’s are finite. Since finite set has 0-density, density of complement of finite set is 1. If complement of this finite set is stated by 𝑀, for every 𝜀>0, there exist 𝑀⊂ℕ, 𝛿(𝑀)=1 and ∃𝑘₀=𝑘₀(𝜀,𝑡)∈𝑀 such that ∀𝑘>𝑘₀ and 𝑘∈𝑀 and ∀𝑥∈𝑋, 𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀. (2.63) This shows that 𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓.

Definition 2.16. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions.The sequence (𝑓_𝑘) is a uniformly statistically Cauchy sequence in intuitionistic fuzzy normed space provided that for every 𝜀>0 and 𝑡>0, there exists a number 𝑁=𝑁(𝜀,𝑡) such that 𝛿𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝜀∀𝑥∈𝑋=0. (2.64)

Theorem 2.17. Let 𝑓_𝑘 :(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequence of functions. If (𝑓_𝑘) is a uniformly statistically convergent sequence with respect to intuitionistic fuzzy norm (𝜇,𝜈), then (𝑓_𝑘) is uniformly statistically Cauchy sequence with respect to intuitionistic fuzzy norm (𝜇,𝜈).

Proof. Suppose that 𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓. In this case, ∀𝜀>0, there exists 𝑀⊂ℕ, 𝛿(𝑀)=1 and 𝑘₀=𝑘₀(𝜀,𝑡)∈𝑀 such that ∀𝑘>𝑘₀,𝑘∈𝑀 and ∀𝑥∈𝑋, 𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2>1−𝜀,𝜈^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2<𝜀. (2.65) Choose 𝑁=𝑁(𝜀,𝑡)∈𝑀,𝑁>𝑘₀. So, 𝜇^′(𝑓_𝑁(𝑥)−𝑓(𝑥),𝑡/2)>1−𝜀 and 𝜈^′(𝑓_𝑁(𝑥)−𝑓(𝑥),𝑡/2)<𝜀. We investigate 𝑁=𝑁(𝜀,𝑡) such that 𝛿𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝜀∀𝑥∈𝑋=0, (2.66) or 𝛿𝐾^′=𝛿𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡>1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡<𝜀∀𝑥∈𝑋=1. (2.67) For every 𝑘∈𝑀, we have 𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡=𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥)+𝑓(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2∗𝜇^′𝑓(𝑥)−𝑓_𝑁(𝑥),𝑡2>()∗()=(),𝜈1−𝜀1−𝜀1−𝜀_′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡=𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥)+𝑓(𝑥)−𝑓_𝑁(𝑥),𝑡≤𝜈^′′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓2∗𝜈^′𝑓(𝑥)−𝑓_𝑁(𝑥),𝑡2>𝜀⋄𝜀=𝜀. (2.68) Since 𝛿(𝑀)=1,(𝑓_𝑘) is a uniformly statistically Cauchy sequence in intuitionistic fuzzy normed space.

Theorem 2.18. Let (𝑋,𝜇,𝜈,∗,⋄) and (𝑌,𝜇^′,𝜈^′,∗,⋄) be two IFNS and the mapping 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be the intuitionistic fuzzy continuous on 𝑋 of sequence of functions. If 𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓, the mapping 𝑓∶𝑋→𝑌 is the intuitionistic fuzzy continuous on 𝑋.

Proof. Let 𝑥₀∈𝑋 be an arbitrary point. By the intuitionistic fuzzy continuity of 𝑓_𝑘’s, for every 𝜀>0 and 𝑡>0 there exists 𝛿=𝛿(𝑥₀,𝜀,𝑡/3)>0 such that 𝜇^′𝑓_𝑘𝑥₀−𝑓_𝑘(𝑥),𝑡3>1−𝜀,𝜈^′𝑓_𝑘𝑥₀−𝑓_𝑘(𝑥),𝑡3<𝜀, (2.69) for every 𝑘∈ℕ and all 𝑥 such that 𝜇(𝑥₀−𝑥,𝑡)>1−𝛿 and 𝜈(𝑥₀−𝑥,𝑡)<𝛿. Let 𝑥∈𝐵(𝑥₀,𝛿,𝑡) be fixed (𝐵(𝑥₀,𝛿,𝑡) stands for an open ball in (𝑋,𝜇,𝜈,∗,⋄) with center 𝑥₀ and radius 𝛿). Since 𝑠𝑡_𝜇,𝜈−𝑓_𝑘⇉𝑓 on 𝑋, for all 𝑥∈𝑋, if we state, respectively, 𝐴 and 𝐵 by these sets 𝐴=𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓3≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓3≥𝜀∀𝑥∈𝑋,𝐵=𝑘∈ℕ∶𝜇^′𝑓_𝑘𝑥₀−𝑓𝑥₀,𝑡3≤1−𝜀or𝜈^′𝑓_𝑘𝑥₀−𝑓𝑥₀,𝑡3≥𝜀∀𝑥∈𝑋 (2.70) then, 𝛿(𝐴)=0 and 𝛿(𝐵)=0, hence 𝛿(𝐴∪𝐵)=0 and 𝐴∪𝐵 is different from ℕ. Thus, there exists 𝑚∈ℕ such that 𝜇^′𝑓_𝑚(𝑥)(𝑥),𝑡−𝑓3>1−𝜀,𝜈^′𝑓_𝑚(𝑥)(𝑥),𝑡−𝑓3𝜇<𝜀,_′𝑓_𝑚𝑥₀−𝑓𝑥₀,𝑡3>1−𝜀,𝜈^′𝑓_𝑚𝑥₀−𝑓𝑥₀,𝑡3<𝜀. (2.71) Now, we will show that 𝑓 is intuitionistic fuzzy contunious at 𝑥₀. Using Definition 1.1, we have 𝜇^′𝑓(𝑥)−𝑓𝑥₀,𝑡=𝜇^′𝑓(𝑥)−𝑓_𝑚(𝑥)+𝑓_𝑚(𝑥)−𝑓_𝑚𝑥₀+𝑓_𝑚𝑥₀−𝑓𝑥₀,𝑡≥𝜇^′𝑓(𝑥)−𝑓_𝑚(𝑥),𝑡3∗𝜇^′𝑓_𝑚(𝑥)−𝑓_𝑚𝑥₀,𝑡3∗𝜇^′𝑓_𝑚𝑥₀−𝑓𝑥₀,𝑡3>()∗()∗()𝜈1−𝜀1−𝜀1−𝜀=1−𝜀,_′𝑓(𝑥)−𝑓𝑥₀,𝑡=𝜈^′𝑓(𝑥)−𝑓_𝑚(𝑥)+𝑓_𝑚(𝑥)−𝑓_𝑚𝑥₀+𝑓_𝑚𝑥₀−𝑓𝑥₀,𝑡≤𝜈^′𝑓(𝑥)−𝑓_𝑚(𝑥),𝑡3∗𝜈^′𝑓_𝑚(𝑥)−𝑓_𝑚𝑥₀,𝑡3∗𝜈^′𝑓_𝑚𝑥₀−𝑓𝑥₀,𝑡3<𝜀⋄𝜀⋄𝜀=𝜀. (2.72) Thus, the proof is completed.

Acknowledgments

This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK) under the Project number 110T699. We have benefited a lot from the report of the anonymous referee. So, the authors are thankful for his/her valuable comments and careful corrections on the first draft of this paper which improved the presentation and readability.