𝜆-Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

Abstract

We study 𝜆-statistically convergent sequences of functions in intuitionistic fuzzy normed spaces. We define concept of 𝜆-statistical pointwise convergence and 𝜆-statistical uniform convergence in intuitionistic fuzzy normed spaces and we give some basic properties of these concepts.

1. Introduction and Some Definitions

Fuzzy logic was introduced by Zadeh [1]. Since then, the importance of fuzzy logic has come increasingly to the present. There are many applications of fuzzy logic in the field of science and engineering, for example, population dynamics [2], chaos control [3, 4], computer programming [5], nonlinear dynamical systems [6], fuzzy topology [7], and so forth. The concept of intuitionistic fuzzy set, as a generalization of fuzzy logic, was introduced by Atanassov [8] in 1983.

In the literature, 𝑡-norm and 𝑡-conorm were defined by Schweizer and Sklar [9]. The norms on intuitionistic fuzzy sets are introduced firstly in [10]. Recently Park [11] has introduced the concept of intuitionistic fuzzy metric space and in [12], Saadati and Park introduced intuitionistic fuzzy normed spaces and concept of convergence of a sequence in intuitionistic fuzzy normed spaces. In light of these developments, intuitionistic fuzzy analogues of many concepts in classical analysis were studied by many authors [13–17], and so forth.

The concept of statistical convergence was introduced by Fast [18] and Steinhaus [19] independently. Mursaleen defined 𝜆-statistical convergence in [20]. Also the concept of statistical convergence was studied in intuitionistic fuzzy normed space in [21]. Quite recently, Karakaya et al. [22] defined and studied statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces. Mohiuddine and Lohani [23] defined and studied 𝜆-statistical convergence in intuitionistic fuzzy normed spaces.

In this paper, we will study concept 𝜆-statistical convergence for sequences of functions and investigate some basic properties related to the concept in intuitionistic fuzzy normed space.

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

Definition 1.1 (see [12]). Let ∗ be a continuous 𝑡-norm, let ⋄ be a continuous 𝑡-conorm, and 𝑋 be a linear space over the field IF (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 intuitionistic fuzzy normed space and (𝜇,𝜈) is called an intuitionistic fuzzy norm. 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→0𝜇(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→0𝜈(𝑥,𝑡)=1.

Definition 1.2 (see [19, 24]). Let 𝐾⊂ℕ and 𝐾_𝑛={𝑘∈𝐾∶𝑘≤𝑛}. Then, the natural density is defined by 𝛿(𝐾)=lim_𝑛→∞(|𝐾_𝑛|/𝑛), where |𝐾_𝑛| denotes the cardinality of 𝐾_𝑛. A sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to the number 𝐿 if for every 𝜀>0, the set 𝑁(𝜀) has asymptotic density zero, where 𝑁(𝜀)=𝑘∈ℕ∶𝑥_𝑘−𝐿≥𝜀. (1.1) This case is stated by 𝑠𝑡−lim𝑥=𝐿.

Definition 1.3 (see [22]). Let (𝑋,𝜇,𝜈,∗,⋄) and (𝑌,𝜇^′,𝜈^′,∗,⋄) be two intuitionistic fuzzy normed linear spaces and let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequences of functions. If for each 𝑥∈𝑋 and for all 𝜀>0,𝑡>0, 𝛿𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀=0, (1.2) then we say that the sequence (𝑓_𝑘) is pointwise statistically convergent to 𝑓 with respect to intuitionistic fuzzy norm (𝜇,𝜈) and we write it 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓.

Definition 1.4 (see [20]). Let 𝜆=(𝜆_𝑛) be a nondecreasing sequence of positive numbers tending to ∞ such that 𝜆_𝑛+1≤𝜆_𝑛,𝜆₁=0. (1.3) Let 𝐾⊂ℕ. The number 𝛿_𝜆(𝐾)=lim_𝑛→∞1𝜆_𝑛𝑘∈𝐼_𝑛∶𝑘∈𝐾 (1.4) is said to be 𝜆-density of 𝐾, where 𝐼_𝑛=[𝑛−𝜆_𝑛+1,𝑛].
If 𝜆_𝑛=𝑛, then 𝜆-density is reduced to asymptotic density. A sequence 𝑥=(𝑥_𝑘) is said to be 𝜆-statistically convergent to the number 𝐿 if for every 𝜀>0, the set 𝑁(𝜀) has 𝜆-density zero, where 𝑁(𝜀)=𝑘∈ℕ∶𝑥_𝑘−𝐿≥𝜀. (1.5) This case is stated by 𝑠𝑡_𝜆−lim𝑥=𝐿.

Definition 1.5 (see [23]). Let (𝑋,𝜇,𝜈,∗,⋄) be an intuitionistic fuzzy normed space. Then, a sequence (𝑥_𝑘) is said to be 𝜆-statistically convergent to 𝐿∈𝑋 with respect to intuitionistic fuzzy norm (𝜇,𝜈) provided that for every 𝜀>0 and 𝑡>0, 𝛿_𝜆𝑘∈ℕ∶𝜇𝑥_𝑘−𝐿,𝑡≤1−𝜀or𝜈𝑥_𝑘−𝐿,𝑡≥𝜀=0, (1.6) or equivalently 𝛿_𝜆𝑘∈ℕ∶𝜇𝑥_𝑘−𝐿,𝑡>1−𝜀,𝜈𝑥_𝑘−𝐿,𝑡<𝜀=1. (1.7) This case is stated by 𝑠𝑡^𝜆_𝜇,𝜈−lim𝑥=𝐿.

2. 𝜆-Statistical Convergence of Sequence of Functions in Intuitionistic Fuzzy Normed Spaces

In this section, we define pointwise 𝜆-statistical and uniformly 𝜆-statistical convergent sequences of functions in intuitionistic fuzzy normed spaces. Also, we give the 𝜆-statistical analog of the Cauchy convergence criterion for pointwise and uniformly 𝜆-statistical convergent in intuitionistic fuzzy normed space. We investigate relations of these concepts with continuity. Let us start definition of pointwise 𝜆-statistical convergence in intuitionistic fuzzy normed spaces.

Definition 2.1. Let (𝑋,𝜇,𝜈,∗,⋄) and (𝑌,𝜇^′,𝜈^′,∗,⋄) be two intuitionistic fuzzy normed linear spaces over the same field IF and let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequences of functions. If for each 𝑥∈𝑋 and for all 𝜀>0,𝑡>0, 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀=0, (2.1) or equivalently 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀=1, (2.2) then we say that the sequence (𝑓_𝑘) is pointwise 𝜆-statistically convergent with respect to intuitionistic fuzzy norm (𝜇,𝜈) and we write it 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓.

This means that for every 𝛿>0, there is integer 𝑁 such that, for all 𝑛>𝑁(𝜀,𝛿,𝑡,𝑥) and for every 𝜀>0, 1𝜆_𝑛𝑘∈𝐼_𝑛∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀foreach𝑥∈𝑋<𝛿. (2.3)

Remark 2.2. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequences of functions. If 𝜆_𝑛=𝑛, since 𝜆-density is reduced to asymptotic density, then(𝑓_𝑘) is pointwise statistically convergent on 𝑋 with respect to (𝜇,𝜈), that is, 𝑠𝑡_𝜇,𝜈−𝑓_𝑘→𝑓.

Lemma 2.3. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequences of functions. Then for every 𝜀>0 and 𝑡>0, the following statements are equivalent. (i)Consider 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓.(ii)For each 𝑥∈𝑋𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀=𝛿_𝜆𝑘∈ℕ∶𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀=0. (2.4)(iii)For each 𝑥∈𝑋𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀=1. (2.5)(iv)For each 𝑥∈𝑋𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀=𝛿_𝜆𝑘∈ℕ∶𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀=1. (2.6)(v)For each 𝑥∈𝑋𝑠𝑡_𝜆−lim𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=1,𝑠𝑡_𝜆−lim𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=0. (2.7)

Example 2.4. 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, consider 𝜇()=𝑡𝑥,𝑡|𝑥|()=|𝑥|𝑡+,𝜈𝑥,𝑡|𝑥|.𝑡+ (2.8) In this case (ℝ,𝜇,𝜈,∗,⋄) is intuitionistic fuzzy normed space. (Also, ([0,1],𝜇,𝜈,∗,⋄) is intuitionistic fuzzy normed space.) Let 𝑓_𝑘∶[0,1]→ℝ be sequences of functions whose terms are given by 𝑓_𝑘(𝑥)=𝑥^𝑘+1if𝑘∈𝑛−𝜆_𝑛1+1,𝑛,𝑥∈0,20if𝑘∉𝑛−𝜆_𝑛1+1,𝑛,𝑥∈0,2𝑥^𝑘+12if𝑘∈𝑛−𝜆_𝑛1+1,𝑛,𝑥∈21,1if𝑘∉𝑛−𝜆_𝑛1+1,𝑛,𝑥∈22,1if𝑥=1. (2.9)(𝑓_𝑘) is pointwise 𝜆-statistically convergent on [0,1] with respect to intuitionistic fuzzy norm (𝜇,𝜈). It is fact that for each 𝑥∈[0,1/2), and since 𝐾(𝜀,𝑡)=𝑘∈𝐼_𝑛∶𝜇𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀, (2.10) hence we have 𝐾(𝜀,𝑡)=𝑘∈𝐼_𝑛∶𝑡𝑡+𝑓_𝑘(𝑥)−0≤1−𝜀or𝑓_𝑘(𝑥)−0𝑡+𝑓_𝑘(𝑥)=−0≥𝜀𝑘∈𝐼_𝑛∶𝑓_𝑘(𝑥)≥𝜀𝑡=1−𝜀𝑘∈𝐼_𝑛∶𝑓_𝑘(𝑥)=𝑥^𝑘,|𝐾(≤+1𝜀,𝑡)|𝜆_𝑛. (2.11) Thus, for each 𝑥∈[0,1/2), since 𝛿_𝜆(𝐾(𝜀,𝑡))=lim_𝑛→∞|𝐾(𝜀,𝑡)|𝜆_𝑛=lim_𝑛→∞𝜆_𝑛𝜆_𝑛=0, (2.12)𝑓_𝑘 is 𝜆-statistically convergent to 0 with respect to intuitionistic fuzzy norm (𝜇,𝜈).

If we take 𝑥∈[1/2,1), then we have 𝐾^′(𝜀,𝑡)=𝑘∈𝐼_𝑛∶𝑡𝑡+𝑓_𝑘(𝑥)−1≤1−𝜀or𝑓_𝑘(𝑥)−1𝑡+𝑓_𝑘(𝑥)=−1≥𝜀𝑘∈𝐼_𝑛∶𝑓_𝑘(𝑥)−1≥𝜀𝑡=1−𝜀𝑘∈𝐼_𝑛∶𝑓_𝑘(𝑥)=𝑥^𝑘+12,𝐾^′(𝜀,𝑡)≤𝜆_𝑛. (2.13) Thus 𝑥∈[1/2,1), then 𝑓_𝑘 is 𝜆-statistically convergent to 1 with respect to intuitionistic fuzzy norm (𝜇,𝜈).

If we take 𝑥=1, it can be seen easily that 𝐾^′′(𝜀,𝑡)=𝑘∈𝐼_𝑛∶𝑡𝑡+𝑓_𝑘(𝑥)−2≤1−𝜀or𝑓_𝑘(𝑥)−2𝑡+𝑓_𝑘(𝑥)=−2≥𝜀𝑘∈𝐼_𝑛∶0≥𝜀𝑡𝐾1−𝜀=⊘,_′′(𝛿𝜀,𝑡)=0,_𝜆𝐾^′′(𝜀,𝑡)=lim_𝑛→∞𝐾^′′(𝜀,𝑡)𝜆_𝑛=lim_𝑛→∞0𝜆_𝑛=0. (2.14) Thus, at 𝑥=1, 𝑓_𝑘 is 𝜆-statistically convergent to 2 with respect to intuitionistic fuzzy norm (𝜇,𝜈).

Consequently, since 𝑓_𝑘(𝑥) is 𝜆-statistically convergent to different points with respect to intuitionistic fuzzy norm (𝜇,𝜈) for each 𝑥∈𝑋, (𝑓_𝑘) is pointwise 𝜆-statistically intuitionistic fuzzy convergent on [0,1].

Theorem 2.5. Let (𝑋,𝜇,𝜈,∗,⋄) be an intuitionistic fuzzy normed space and let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequences of functions. If sequence (𝑓_𝑘) is pointwise intuitionistic fuzzy convergent on 𝑋 to a function 𝑓 with respect to (𝜇,𝜈), then (𝑓_𝑘) is pointwise 𝜆-statistically convergent with respect to intuitionistic fuzzy norm (𝜇,𝜈).

Proof. Let (𝑓_𝑘) be pointwise intuitionistic fuzzy convergent in 𝑋. In this case, (𝑓_𝑘(𝑥)) is convergent with respect to (𝜇^′,𝜈^′) for each 𝑥∈𝑋. Then for every 𝜀>0 and 𝑡>0, there is number 𝑘₀∈ℕ such that 𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀 (2.15) 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 (2.16) has finite numbers of terms. Since finite subset of ℕ has 𝜆-density 0, hence 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀=0. (2.17) That is, 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓.

Theorem 2.6. Let (𝑓_𝑘) and (𝑔_𝑘) be two sequences of functions from intuitionistic fuzzy normed space (𝑋,𝜇,𝜈,∗,⋄) to (𝑌,𝜇^′,𝜈^′,∗,⋄). If 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓 and 𝑠𝑡^𝜆_𝜇,𝜈−𝑔_𝑘→𝑔, then 𝑠𝑡^𝜆_𝜇,𝜈−(𝛼𝑓_𝑘+𝛽𝑔_𝑘)→𝛼𝑓+𝛽𝑔 where 𝛼,𝛽∈ IF (ℝorℂ).

Proof. The proof is clear for 𝛼=0 and 𝛽=0. Now let 𝛼≠0 and 𝛽≠0. Since 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓 and 𝑠𝑡^𝜆_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, (2.18) then 𝛿_𝜆(𝐴₁)=0 and 𝛿_𝜆(𝐴₂)=0. Since 𝛿_𝜆(𝐴₁)=0 and 𝛿_𝜆(𝐴₂)=0, if we state 𝐴 by (𝐴₁∪𝐴₂), then 𝛿_𝜆(𝐴)=0. (2.19) Hence 𝐴₁∪𝐴₂≠ℕ and there exists 𝑘₀∈ℕ such that 𝜇^′𝑓_𝑘0(𝑥)(𝑥),𝑡−𝑓2|𝛼|>1−𝜀,𝜈^′𝑓_𝑘0(𝑥)(𝑥),𝑡−𝑓2|𝛼|𝜇<𝜀,_′𝑔_𝑘0(𝑥)(𝑥),𝑡−𝑔2𝛽>1−𝜀,𝜈^′𝑔_𝑘0(𝑥)(𝑥),𝑡−𝑔2𝛽<𝜀. (2.20) 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. (2.21) We will show that for each 𝑥∈𝑋𝐴^𝑐⊂𝐵. (2.22) Let 𝑘₀∈𝐴^𝑐. In this case 𝜇^′𝑓_𝑘0(𝑥)(𝑥),𝑡−𝑓2|𝛼|>1−𝜀,𝜈^′𝑓_𝑘0(𝑥)(𝑥),𝑡−𝑓2|𝛼|𝜇<𝜀,_′𝑔_𝑘0(𝑥)(𝑥),𝑡−𝑔2𝛽>1−𝜀,𝜈^′𝑔_𝑘0(𝑥)(𝑥),𝑡−𝑔2𝛽<𝜀. (2.23) Using those mentioned previously, we have 𝜇^′𝛼𝑓_𝑘0+𝛽𝑔_𝑘0(𝑥)−𝛼𝑓(𝑥)+𝛽𝑔(𝑥),𝑡≥𝜇^′𝛼𝑓_𝑘0(𝑥)(𝑥),𝑡−𝛼𝑓2∗𝜇^′𝛽𝑔_𝑘0(𝑥)(𝑥),𝑡−𝛽𝑔2=𝜇^′𝑓_𝑘0(𝑥)(𝑥),𝑡−𝑓2|𝛼|∗𝜇^′𝑔_𝑘0(𝑥)(𝑥),𝑡−𝑔2𝛽>()∗()=(),𝜈1−𝜀1−𝜀1−𝜀_′𝛼𝑓_𝑘0+𝛽𝑔_𝑘0(𝑥)−𝛼𝑓(𝑥)+𝛽𝑔(𝑥),𝑡≤𝜈^′𝛼𝑓_𝑘0(𝑥)(𝑥),𝑡−𝛼𝑓2⋄𝜈^′𝛽𝑔_𝑘0(𝑥)(𝑥),𝑡−𝛽𝑔2=𝜈^′𝑓_𝑘0(𝑥)(𝑥),𝑡−𝑓2|𝛼|⋄𝜈^′𝑔_𝑘0(𝑥)(𝑥),𝑡−𝑔2𝛽<𝜀⋄𝜀=𝜀. (2.24) This implies that 𝐴^𝑐⊂𝐵. (2.25) Since 𝐵^𝑐⊂𝐴 and 𝛿_𝜆(𝐴)=0, hence 𝛿_𝜆𝐵^𝑐=0. (2.26) That is 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝛼𝑓_𝑘+𝛽𝑔_𝑘(𝑥)−(𝑥)(𝑥)𝜈𝛼𝑓+𝛽𝑔,𝑡≤1−𝜀,_′𝛼𝑓_𝑘+𝛽𝑔_𝑘(𝑥)−𝛼𝑓(𝑥)+𝛽𝑔(𝑥),𝑡≥𝜀=0,𝑠𝑡^𝜆_𝜇,𝜈−𝛼𝑓_𝑘+𝛽𝑔_𝑘→𝛼𝑓+𝛽𝑔. (2.27)

Definition 2.7. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequences of functions. (𝑓_𝑘) is a pointwise 𝜆-statistical Cauchy sequence in intuitionistic fuzzy normed space provided that for every 𝜀>0 and 𝑡>0 there exists a number 𝑁=𝑁(𝑥,𝜀,𝑡) such that 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝜀foreach𝑥∈𝑋=0. (2.28)

Theorem 2.8. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. If (𝑓_𝑘) is a pointwise 𝜆-statistical convergent sequence with respect to intuitionistic fuzzy norm (𝜇,𝜈), then (𝑓_𝑘) is a pointwise 𝜆-statistical Cauchy sequence with respect to intuitionistic fuzzy norm (𝜇,𝜈).

Proof. Suppose that 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓 and let 𝜀>0,𝑡>0. For a given 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 (2.29) for each 𝑥∈𝑋. Then, we have 𝛿_𝜆𝐴_𝑥(𝜀,𝑡)=0, (2.30) which implies that 𝛿_𝜆𝐴^𝑐_𝑥(𝜀,𝑡)=1. (2.31) 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. (2.32) We want to show that there exists a number 𝑁=𝑁(𝑥,𝜀,𝑡) such that 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝑠or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝑠foreach𝑥∈𝑋=0. (2.33) 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. (2.34) We have to show that 𝐵_𝑥(𝜀,𝑡)⊂𝐴_𝑥(𝜀,𝑡). (2.35) Suppose that 𝐵_𝑥(𝜀,𝑡)⊆𝐴̸_𝑥(𝜀,𝑡). (2.36) In this case 𝐵_𝑥(𝜀,𝑡) has at least one different element which 𝐴_𝑥(𝜀,𝑡) does not have. 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, (2.37) in particular 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, (2.38) 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, (2.39) in particular 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, (2.40) which is not possible. Hence 𝐵_𝑥(𝜀,𝑡)⊂𝐴_𝑥(𝜀,𝑡). Therefore, by 𝛿_𝜆(𝐴_𝑥(𝜀,𝑡))=0,𝛿_𝜆(𝐵_𝑥(𝜀,𝑡))=0. That is, (𝑓_𝑘) is a pointwise 𝜆-statistical Cauchy sequence with respect to intuitionistic fuzzy norm (𝜇,𝜈).

In the following, we introduce uniformly 𝜆-statistical convergence in intuitionistic fuzzy normed spaces.

Definition 2.9. Let (𝑋,𝜇,𝜈,∗,⋄) and (𝑌,𝜇^′,𝜈^′,∗,⋄) be two intuitionistic fuzzy normed linear spaces over the same field IF and let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. If for all 𝑥∈𝑋 and for all 𝜀>0,𝑡>0, 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀=0, (2.41) or equivalently 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀=1, (2.42) then we say that the sequence 𝑓_𝑘 is uniformly 𝜆-statistically convergent with respect to intuitionistic fuzzy norm (𝜇,𝜈) and we write it 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘⇉𝑓.

Remark 2.10. If 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘⇉𝑓, then 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓.

Lemma 2.11. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be a sequence of functions. Then for every 𝜀>0 and 𝑡>0, the following statements are equivalent:(i)Consider 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘⇉𝑓.(ii)For all 𝑥∈𝑋𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≤1−𝜀=𝛿_𝜆𝑘∈ℕ∶𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡≥𝜀=0. (2.43)(iii)For all 𝑥∈𝑋𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀,𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀=1. (2.44)(iv)For all 𝑥∈𝑋𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡>1−𝜀=𝛿_𝜆𝑘∈ℕ∶𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡<𝜀=1. (2.45)(v)For all 𝑥∈𝑋𝑠𝑡_𝜆−lim𝜇^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=1,𝑠𝑡_𝜆−lim𝜈^′𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡=0. (2.46)

Example 2.12. Let (ℝ,𝜇,𝜈,∗,⋄) be as Example 2.4. Let 𝑓_𝑘∶[0,1]→ℝ be a sequence of functions whose terms are given by 𝑓_𝑘(𝑥)=𝑥^𝑘+1,if𝑛−𝜆_𝑛+1≤𝑘≤𝑛0,otherwise. (2.47) Since 𝐾(𝜀,𝑡)={𝑘∈𝐼_𝑛∶𝜇(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)≤1−𝜀or𝜈(𝑓_𝑘(𝑥)−𝑓(𝑥),𝑡)≥𝜀}, hence we have 𝐾(𝜀,𝑡)=𝑘∈𝐼_𝑛∶𝑡𝑡+𝑓_𝑘(𝑥)−0≤1−𝜀or𝑓_𝑘(𝑥)−0𝑡+𝑓_𝑘(𝑥)=−0≥𝜀𝑘∈𝐼_𝑛∶𝑓_𝑘(𝑥)≥𝜀𝑡=1−𝜀𝑘∈𝐼_𝑛∶𝑓_𝑘(𝑥)=𝑥^𝑘,|𝐾(≤+1𝜀,𝑡)|𝜆_𝑛. (2.48) Thus, for all 𝑥∈[0,1], since 𝛿_𝜆(𝐾(𝜀,𝑡))=lim_𝑛→∞|𝐾(𝜀,𝑡)|𝜆_𝑛=lim_𝑛→∞𝜆_𝑛𝜆_𝑛=0, (2.49)𝑓_𝑘 is uniformly 𝜆-statistically convergent to 0 with respect to intuitionistic fuzzy norm (𝜇,𝜈).

Definition 2.13. Let 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄) be sequences of functions. The sequence (𝑓_𝑘) is a uniformly 𝜆-statistical 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 𝛿_𝜆𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)−𝑓_𝑁(𝑥),𝑡≥𝜀∀𝑥∈𝑋=0. (2.50)

Definition 2.14. Let (𝑋,𝜇,𝜈,∗,⋄) and (𝑌,𝜇^′,𝜈^′,∗,⋄) be two intuitionistic fuzzy normed spaces, and let 𝐹 be a family of functions from 𝑋 to 𝑌. The family 𝐹 is intuitionistic fuzzy equicontinuous at a point 𝑥₀∈𝑋 if for every 𝜀>0 and 𝑡>0, there exists a 𝛿>0 such that 𝜇^′(𝑓(𝑥₀)−𝑓(𝑥),𝑡)>1−𝜀 and 𝜈^′(𝑓(𝑥₀)−𝑓(𝑥),𝑡)<𝜀 for all 𝑓∈𝐹 and all 𝑥 such that 𝜇^′(𝑥₀−𝑥,𝑡)>1−𝛿 and 𝜈^′(𝑥₀−𝑥,𝑡)<𝛿. The family is intuitionistic fuzzy equicontinuous if it is equicontinuous at each point of 𝑋. (For continuity, 𝛿 may depend on 𝜀, 𝑥₀ and 𝑓; for equicontinuity, 𝛿 must be independent of 𝑓.)

Theorem 2.15. Let (𝑋,𝜇,𝜈,∗,⋄),(𝑌,𝜇^′,𝜈^′,∗,⋄) be intuitionistic fuzzy normed space. Assume that 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓 on 𝑋 where functions 𝑓_𝑘∶(𝑋,𝜇,𝜈,∗,⋄)→(𝑌,𝜇^′,𝜈^′,∗,⋄),𝑘∈ℕ are intuitionistic fuzzy equicontinuous on 𝑋 and 𝑓∶𝑋→𝑌. Then 𝑓 is continuous on 𝑋.

Proof. Let 𝑥₀∈𝑋 be an arbitrary point. By the intuitionistic fuzzy equicontinuity of 𝑓_𝑘’s, for every 𝜀>0 and 𝑡>0 there exist 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 (2.51) 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) (𝐵(𝑥₀,𝛿,𝑡) stands for an open ball in (𝑋,𝜇,𝑣,∗,⋄) with center 𝑥₀ and radius 𝛿) be fixed. Since 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓 on 𝑋, for each 𝑥∈𝑋, if we state, respectively, 𝐴_𝑥(𝜀,𝑡) and 𝐵_𝑥(𝜀,𝑡) by the sets 𝐴_𝑥(𝜀,𝑡)=𝑘∈ℕ∶𝜇^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓3≤1−𝜀or𝜈^′𝑓_𝑘(𝑥)(𝑥),𝑡−𝑓3≥𝜀foreach,𝐵𝑥∈𝑋_𝑥(𝜀,𝑡)=𝑘∈ℕ∶𝜇^′𝑓_𝑘𝑥₀−𝑓𝑥₀,𝑡3≤1−𝜀or𝜈^′𝑓_𝑘𝑥₀−𝑓𝑥₀,𝑡3≥𝜀foreach,𝑥∈𝑋 (2.52) then 𝛿_𝜆(𝐴_𝑥(𝜀,𝑡))=0 and 𝛿_𝜆(𝐵_𝑥(𝜀,𝑡))=0; hence 𝛿_𝜆(𝐴_𝑥(𝜀,𝑡)∪𝐵_𝑥(𝜀,𝑡))=0 and 𝐴_𝑥(𝜀,𝑡)∪𝐵_𝑥(𝜀,𝑡) 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. (2.53) Now, we will show that 𝑓 is intuitionistic fuzzy continuous at 𝑥₀. Since 𝑠𝑡^𝜆_𝜇,𝜈−𝑓_𝑘→𝑓 and for every 𝑘∈ℕ𝑓_𝑘’s are continuous, 𝑓_𝑚 is also continuous for 𝑚∈ℕ, 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_𝑘𝑥₀,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. (2.54) Thus, the proof is completed.

Acknowledgment

This work is supported by The Scientific and Technological Research Council of Turkey (TUBITAK) under the project no. 110T699.