Open Access

Statistical limit superior and limit inferior in intuitionistic fuzzy normed spaces

  • Mohammad Ali Alghamdi1,
  • Abdullah Alotaibi1,
  • Qutubuddin Mohammad Danish Lohani2 and
  • Mohammad Mursaleen3Email author
Journal of Inequalities and Applications20122012:96

DOI: 10.1186/1029-242X-2012-96

Received: 22 December 2011

Accepted: 23 April 2012

Published: 23 April 2012

Abstract

Recently, the concepts of statistical convergence, ideal convergence and lacunary statistical convergence have been studied in intuitionistic fuzzy normed spaces. In this article, we study the concepts of statistical limit superior and statistical limit inferior in intuitionistic fuzzy normed spaces. We also give an example to compute these points in intuitionistic fuzzy normed spaces.

AMS Subject Classification (2000): 40A05; 40D25; 11B05; 60H10; 60B99; 26A03.

Keywords

t-norm t-conorm fuzzy numbers intuitionistic fuzzy normed space statistical convergence statistical boundedness statistical limit point statistical cluster point statistical limit superior statistical limit inferior.

1 Introduction and preliminaries

The concept of statistical convergence was first introduced by Fast [1] which was extended for double sequences in [2, 3]. In particular, active researches on this topic were started after the study of Fridy [4]. Many of the results of the theory of ordinary convergence have been extended to the theory of statistical convergence by using the notion of density. For instance, Fridy [5] introduced the concept of statistical limit points and Fridy and Orhan [6] introduced the statistical analogs of limit superior and limit inferior of a sequence of real numbers. Recently, statistical convergence and some of its related concepts for fuzzy numbers have been studied in [79]. Quite recently, the idea of statistical convergence in intuitionistic fuzzy normed spaces for single sequences has been studied in [10, 11]; and for double sequences by Mursaleen and Mohiuddine [12, 13].

Recently, Saadati and Park [14] introduced the notion of intuitionistic fuzzy normed space and quite recently, in [15, 16] the concepts of intuitionistic fuzzy 2-normed and intuitionistic fuzzy 2-metric spaces have been introduced and studied. Certainly there are some situations where the ordinary norm does not work and the concept of intuitionistic fuzzy norm seems to be more suitable in such cases.

In this article, we study the concept of statistical limit superior and statistical limit inferior in intuitionistic fuzzy normed spaces. An example is demonstrated to determine these points in intuitionistic fuzzy normed space. We observe that our results are analogous to the results of Fridy and Orhan [6] but proofs are somewhat different when we deal with these concepts in intuitionistic fuzzy normed spaces.

We recall some basic definitions and notations.

Definition 1.1 [14]. A binary operation *: [0, 1] × [0, 1] [0, 1] is said to be a continuous t-norm if it satisfies the following conditions:
  1. (a)

    * is associative and commutative,

     
  2. (b)

    * is continuous,

     
  3. (c)

    a * 1 = a for all a [0, 1],

     
  4. (d)

    a * b ≤ c * d whenever a ≤ c and b ≤ d for each a, b, c, d [0, 1].

     

For example, a*b = max{a+b- 1, 0}, a*b = ab and a*b = min{a, b} on [0,1] are t-norms.

A binary operation : [0, 1] × [0, 1] [0, 1] is said to be a continuous t-conorm if it satisfies the conditions (a), (b), (d) as above and a0 = a for all a [0, 1].

For example, ab = min{a + b, 1} and ab = max{a, b} on [0,1] are t-conorms.

Definition 1.2 [14]. The five-tuple (X, μ, ν, *, ) is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space,* is a continuous t-norm, is a continuous t-conorm, and μ, ν are fuzzy sets on X × (0, ∞) satisfying the following conditions. For every x, y X and s, t > 0,
  1. (a)

    μ(x, t) + ν(x, t) ≤ 1,

     
  2. (b)

    μ(x, t) > 0,

     
  3. (c)

    μ(x, t) = 1 if and only if x = 0,

     
  4. (d)

    μ ( α x , t ) = μ ( x , t α ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq1_HTML.gif for each α ≠ 0,

     
  5. (e)

    μ(x, t) * μ(y, s) ≤ μ(x + y, t + s),

     
  6. (f)

    μ(x, ·): (0, ∞) [0, 1] is continuous,

     
  7. (g)

    lim t μ ( x , t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq2_HTML.gif and lim t 0 μ ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq3_HTML.gif,

     
  8. (h)

    ν (x, t) < 1,

     
  9. (i)

    ν (x, t) = 0 if and only if x = 0,

     
  10. (j)

    ν ( α x , t ) = ν ( x , t | α | ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq4_HTML.gif for each α ≠ 0,

     
  11. (k)

    ν (x, t)ν(y, s) ≥ ν(x + y, t + s),

     
  12. (l)

    ν (x, ·): (0, ∞) [0, 1] is continuous,

     
  13. (m)

    lim t ν ( x , t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq5_HTML.gif and lim t 0 ν ( x , t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq6_HTML.gif.

     

In this case (μ, ν) is called an intuitionistic fuzzy norm.

Example. Suppose that (X, || ||) is a normed space and let a*b = ab and ab = min{a+b, 1} for all a, b [0, 1]. For all x X and every t > 0, consider
μ ( x ; t ) : = t t + x and v ( x ; t ) : = x t + x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equa_HTML.gif

Then (X, μ, ν, *, ) is an intuitionistic fuzzy normed space.

Definition 1.3 [14]. Let (X, μ, ν, *, ) be an intuitionistic fuzzy normed space. Then a sequence x = (x n ) is said to be convergent to L X with respect to the intuitionistic fuzzy norm (μ, ν) if for every ϵ > 0 and t > 0, there exists a positive integer ko such that μ(x n - L; t) > 1 - ϵ and ν(x n - L; t) < ϵ, whenever n ≥ ko. In this case, we write (μ, ν)-lim x = L or x n ( μ , ν ) L https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq7_HTML.gif as n → ∞.

Definition 1.4 [14]. Let (X, μ, ν, *, ) be an intuitionistic fuzzy normed space. Then a sequence x = (x n ) is said to be a Cauchy sequence with respect to the intuitionistic fuzzy norm (μ, ν) if for every ϵ > 0 and t > 0, there exists a positive integer ko such that μ(x n - x m ; t) > 1 - ϵ and ν(x n - x m ; t) < ϵ for all n, m ≥ ko.

Definition 1.5 [17]. If K is a subset of , then the natural density of K denoted by δ(K), is defined as
δ ( K ) : = lim n 1 n k n : k K , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equb_HTML.gif

where the vertical bars denote the cardinality of the enclosed set.

Definition 1.6 [4, 18]. A sequence x = (x n ) of numbers is said to be statistically convergent to L if
δ k n : | x k - L | ϵ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equc_HTML.gif

for every ϵ > 0. In this case we write st-lim x = L.

Definition 1.7 [5, 6]. A sequence x = (x n ) of numbers is said to be statistically bounded if there is a number B such that
δ k n : x k B = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equd_HTML.gif

Definition 1.8 [5]. If {xk(j)} is a subsequence of x = (x k ) and K: = {k(j): j }, then we abbreviate {xk(j)} by {x} K . If δ(K) = 0 then {x} K is called a subsequence of density zero or a thin subsequence. On the other hand, {x} K is a nonthin subsequence of x if K does not have density zero.

Definition 1.9 [10, 12]. Let (X, μ, ν, *, ) be an intuitionistic fuzzy normed space. We say that a sequence x = (x k ) is statistically convergent to L X with respect to the intuitionistic fuzzy normed (μ, ν) provided that for every t > 0 and b (0, 1)
δ k n : μ ( x k - L ; t ) < 1 - b or ν ( x k - L ; t ) > b = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Eque_HTML.gif

In this case we write st(μ, ν)- lim x = L.

2 Statistical limit superior and inferior in IFNS

In this section, we define limit point, statistical limit point, statistical cluster point, statistical limit superior, and statistical limit inferior in intuitionistic fuzzy normed spaces and demonstrate through an example how to compute these points in a IFN-spaces.

Definition 2.1. A sequence x in an intuitionistic fuzzy normed space (X, μ, ν, *, ) is said to be statistically bounded if there exists some to > 0 and b (0, 1) such that δ({k: μ(x k ; to) > 1 - b or ν(x k ; to) < b}) = 0.

Definition 2.2. Let (X, μ, ν,*,) be an intuitionistic fuzzy normed space. Then l X is called a limit point of the sequence x = (x k ) with respect to the intuitionistic fuzzy norm (μ, ν) provided that there is a subsequence of x that converges to l with respect to the intuitionistic fuzzy norm (μ, ν). Let L(μ, ν)(x) denotes the set of all limit points of the sequence x with respect to the intuitionistic fuzzy norm (μ, ν).

Definition 2.3. Let (X, μ, ν,*,) be an intuitionistic fuzzy normed space. Then ξ X is called a statistical limit point of the sequence x = (x k ) with respect to the intuitionistic fuzzy norm (μ, ν) provided that there is a nonthin subsequence of x that converges to ξ with respect to the intuitionistic fuzzy norm (μ, ν). In this case we say ξ is a st(μ, ν)-limit point of sequence x = (x k ). Let Λ(μ, ν)(x) denotes the set of all st(μ, ν)-limit points of the sequence x.

Definition 2.4. Let (X, μ, ν,*,) be an intuitionistic fuzzy normed space. Then η X is called a statistical cluster point of the sequence x = (x k ) with respect to the intuitionistic fuzzy norm (μ, ν) provided that for every to > 0 and a (0, 1),
δ ̄ k n : μ ( x k - η ; t o ) > 1 - a or v ( x k - η ; t o ) < a = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equf_HTML.gif

In this case we say η is a st(μ, ν)-cluster point of the sequence x. Let Γ(μ, ν)(x) denotes the set of all st(μ, ν)-cluster points of the sequence x.

Definition 2.5. For a sequence x in an intuitionistic fuzzy normed space (X, μ, ν,*,), we define the sets B x ( μ , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq8_HTML.gif and A x ( μ , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq9_HTML.gif by
B x ( μ , ν ) : = b ( 0 , 1 ) : δ k : μ ( x k ; t ) < 1 - b or ν ( x k ; t ) > b 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equg_HTML.gif
A x ( μ , ν ) : = a ( 0 , 1 ) : δ k : μ ( x k ; t ) > 1 - a or ν ( x k ; t ) < a 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equh_HTML.gif
If x is a real number sequence then the statistical limit superior of x with respect to the intuitionistic fuzzy norm (μ, ν) is defined by
s t ( μ , ν )  -  lim  sup  x : = sup B x ( μ , ν ) if B x ( μ , ν ) , 0 if B x ( μ , ν ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equi_HTML.gif
And the statistical limit inferior of x with respect to the intuitionistic fuzzy norm (μ, ν) is defined by
s t ( μ , ν ) - lim  inf  x : = inf A x ( μ , ν ) if A x ( μ , ν ) , 1 if A x ( μ , ν ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equj_HTML.gif
Example. A simple example will help to illustrate the concepts just defined. Let the sequence x = (x k ) be defined by
x k : = 2 k , if k is an odd square , - 1 , if k is an even square , 1 / 2 , if k is an odd nonsquare , 0 , if k is an even nonsquare . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equk_HTML.gif

Let μ ( x k ; t ) = t t + x k https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq10_HTML.gif and ν ( x k ; t ) = x k t + x k https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq11_HTML.gif.

The above sequence is clearly unbounded with respect to (μ, ν). On the other hand, it is statistically bounded with respect to (μ, ν). For this,
δ k n : μ ( x k ; t o ) < 1 - b or ν ( x k ; t o ) > b = δ k n : t o t o + x k < 1 - b or x k t o + x k > b , = δ k n : x k > b t o 1 - b . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equl_HTML.gif
Since 0 < b < 1 , 1 b - 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq12_HTML.gif. Choose t o = 1 - b 3 b https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq13_HTML.gif. Then to > 0 and
δ k n : μ ( x k ; t o ) < 1 - b or ν x k ; t o > b = δ k n : x k > b 1 - b × 1 - b 3 b = 1 3 = δ k n : x k > 1 3 = lim n 1 n × n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equm_HTML.gif

Hence it is statistically bounded with respect to (μ, ν).

To find B x ( μ , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq8_HTML.gif, we have to find those b (0, 1) such that
δ k n : μ x k ; t < 1 - b or ν x k ; t > b 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equn_HTML.gif
Now,
δ k n : μ ( x k ; t ) < 1 - b or ν ( x k ; t ) > b = δ k n : t t + x k < 1 - b or x k t + x k > b , = δ k n : x k > b t 1 - b . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equo_HTML.gif
We can easily choose any t > 0 as t < 1 3 ( 1 b - 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq14_HTML.gif for 0 < b < 1, so that
0 < b t 1 - b < b 1 - b × 1 - b 3 b = 1 3 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equp_HTML.gif
Therefore
δ k n : x k > b t 1 - b = δ k n : x k > r = b t 1 - b , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equq_HTML.gif
and by the above condition r (0, 1). Now the number of members of the sequence which satisfy the above condition is always greater than n - n 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq15_HTML.gif or n - n - 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq16_HTML.gif for the case n is even or odd, respectively. Therefore
δ k n : x k > r = b t 1 - b > lim n 1 n × n 2 = 1 2 o r lim n 1 n × n + 1 2 = 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equr_HTML.gif
Thus
δ k n : x k > r = b t 1 - b 0 for all b ( 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equs_HTML.gif
Hence
B x ( μ , ν ) = ( 0 , 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equt_HTML.gif
and
s t ( μ , ν ) - lim sup  x = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equu_HTML.gif
The above sequence has two subsequences
x = ( x n i ) where x n i = 1 for each n i 3 , 5 , 7 , 11 , 13 , . . . , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equv_HTML.gif
and
x = ( x n j ) where x n j = 0 for each n j 2 , 6 , 8 , 10 , 12 , . . . , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equw_HTML.gif
i, j ; which are of positive density and clearly convergent to 1 and 0, respectively. Therefore, x is not statistically convergent. Similarly, we have
A x ( μ , ν ) = ( 0 , 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equx_HTML.gif
and
s t ( μ , ν ) - lim inf  x = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equy_HTML.gif

Hence the set of statistical cluster points of x is {0, 1}, where st(μ,ν)- lim inf x = least element and st(μ,ν)- lim sup x = greatest element of the above set.

This observation suggests the main idea of our first theorem of the following section.

3 Main results

The following results are analogs of the results due to Fridy and Orhan [6], while the proofs are different which show the technique to work with IFN-spaces. We observe that in contrast to the real case here from the definition limit sup cannot be infinite, as it can be at most 1.

Theorem 3.1. Let b = st(μ,ν)- lim sup x. Then for every positive numbers t and γ
δ k : μ x k ; t < 1 - b + γ or v x k ; t > b - γ 0 , and δ k : μ x k ; t < 1 - b + γ or v x k ; t > b + γ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ1_HTML.gif
(1)

Conversely, if (1) holds for every positive t and γ then b = st(μ,ν)- lim sup x.

Proof. Let b = st(μ,ν)-lim sup x, where b be finite. Then
δ ( { k : μ ( x k ; t ) < 1 - b or ν ( x k ; t ) > b } ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ2_HTML.gif
(2)
Since μ(x k ; t) < 1 - b + γ or ν(x k ; t) > b - γ for every k and for any t, γ > 0,
δ k n : μ ( x k ; t ) < 1 - b + γ or ν ( x k ; t ) > b - γ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equz_HTML.gif

Now applying the definition of st(μ,ν)- lim sup x we have 1 - b as the least value and b as the greatest value satisfying (2).

Now if possible,
μ ( x k ; t ) < 1 - b - γ or ν ( x k ; t ) > b + γ for some γ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equaa_HTML.gif

Then 1 - b - γ and b + γ are another values with 1 - b - γ < 1 - b and b + γ > b which satisfies (2). This observation contradicts the fact that 1 - b and b are least and greatest values, respectively, which satisfies the above condition.

Hence,
δ k n : μ ( x k ; t ) < 1 - b - γ or ν ( x k ; t ) > b + γ = 0 for every γ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equab_HTML.gif
Conversely, if (1) holds for every positive t and γ, then
δ k n : μ ( x k ; t ) < 1 - b + γ or ν ( x k ; t ) > b - γ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equac_HTML.gif
and
δ { k n : μ x k ; t < 1 - b - γ  or  ν x k ; t > b + γ } = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equad_HTML.gif
Therefore
δ { k n : μ x k ; t 1 - b  or  ν x k ; t b } 0 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equae_HTML.gif
and
δ { k n : μ x k ; t = 1 - b  or  ν x k ; t = b } = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equaf_HTML.gif
That is
δ { k n : μ x k ; t < 1 - b  or  ν x k ; t > b } 0 for every t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equag_HTML.gif

Hence b = st(μ,ν)- lim sup x.

This completes the proof of the theorem.

The dual statement for st(μ,ν)- lim inf x can also be proved similarly.

Theorem 3.1 '. Let a = st(μ,ν)- lim inf x. Then for every positive number t and γ
δ ( { k n : μ x k ; t > 1 - a - γ  or  ν x k ; t < a + γ } ) 0 ,  and δ ( k n : μ x k ; t > 1 - a + γ  or  ν x k ; t < a - γ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ3_HTML.gif
(1A)

Conversely, if (1') holds for every positive t and γ then a = st(μ,ν)- lim inf x.

Remark. From the definition of statistical cluster points we see that Theorems 3.1 and 3.1' can be interpreted as saying that st(μ,ν)- lim sup x and st(μ,ν)- lim inf x are the greatest and the least statistical cluster points of x, respectively.

Theorem 3.2. For any sequence x, st(μ,ν)- lim inf x ≤ st(μ,ν)- lim sup x.

Proof. First consider the case in which st(μ,ν)- lim sup x = 0, which implies that
B x ( μ , ν ) = . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equah_HTML.gif
Then for every b (0, 1),
B x ( μ , ν ) = δ { k : μ x k ; t < 1 - b  or  ν x k ; t > b } = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equai_HTML.gif
that is
δ { k : μ x k ; t 1 - b  or  ν x k ; t b } = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equaj_HTML.gif
Also for every a (0, 1), we have
δ { k : μ x k ; t > 1 - a  or  ν x k ; t < a } 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equak_HTML.gif

Hence, st(μ,ν)- lim inf x = 0.

The case in which st(μ,ν)- lim sup x = 1, is trivial.

Suppose that b = st(μ,ν)- lim sup x, and a = st(μ,ν)- lim inf x; where a and b are finite.

Now for given any γ, we show that 1 - b - γ A x ( μ , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq17_HTML.gif.

By Theorem 3.1,
δ k : μ x k ; t < 1 - b - γ 2  or  ν x k ; t > b + γ 2 = 0 , where 1 - b = least upper bound of  B x ( μ , ν ) . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equal_HTML.gif
Therefore
δ k : μ x k ; t 1 - b - γ 2  or  ν x k ; t b + γ 2 = 1 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equam_HTML.gif
which in turn gives
δ { k : μ x k ; t > 1 - b - γ  or  ν x k ; t < b + γ } = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equan_HTML.gif

Hence, 1 - b - γ A x ( μ , ν ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq17_HTML.gif.

By definition
a = inf  A x ( μ , ν ) , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equao_HTML.gif
so we conclude that
1 - b - γ 1 - a , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equap_HTML.gif
and since γ is arbitrary,
1 - b 1 - a , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equaq_HTML.gif
that is
- b - a , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equar_HTML.gif
a b . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equas_HTML.gif

This completes the proof of the theorem.

Theorem 3.3. In an intuitionistic fuzzy normed space (X, μ, ν, *, ), the statistically bounded sequence x is statistically convergent if and only if
s t ( μ , ν ) - lim inf  x = s t ( μ , ν ) - lim sup  x . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equat_HTML.gif
Proof. Let α, β be st(μ, ν)- lim inf x and st(μ, ν)- lim sup x, respectively. Now we assume that st(μ,ν)- lim x = L. Then for every ϵ > 0 and b (0, 1),
δ { k : μ x k ; t 1 - b  or  ν x k ; t b } = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equau_HTML.gif
so that
δ k : μ x k ; t 2 * μ L ; t 2 1 - b  or  ν x k ; t 2 ν L ; t 2 b = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equav_HTML.gif
Let for every t > 0,
sup t μ x k ; t 2 = 1 - b 1  and  sup t μ L ; t 2 = 1 - b 2 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equaw_HTML.gif
or
inf t ν x k ; t 2 = b 1  and  inf t ν L ; t 2 = b 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equax_HTML.gif
such that
1 - b 1 * 1 - b 2 1 - b  or  b 1 b 2 b . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ4_HTML.gif
(3.1)
Then
δ k : μ x k ; t 2 1 - b 1  or  ν x k ; t 2 b 1 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ5_HTML.gif
(3.2)
and therefore
δ k : μ x k ; t 2 < 1 - b 1 - γ  or  ν x k ; t 2 > b 1 + γ = 0  for every  γ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ6_HTML.gif
(3.3)
Now applying Theorem 3.1 and the definition of st(μ,ν)- lim sup x, we get
δ k : μ x k ; t 2 < 1 - β - γ  or  ν x k ; t 2 > β + γ = 0  for every  γ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ7_HTML.gif
(3.4)
From (3.3) and (3.4) and by the definition of st(μ,ν)- lim sup x, we get
1 - b 1 - γ 1 - β - γ  or  b 1 + γ β + γ , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equay_HTML.gif
that is,
β b 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ8_HTML.gif
(3.5)
Now we find those k such that
μ x k ; t 2 > 1 - b 1  +  γ  or  ν x k ; t 2 < b 1 - γ . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equaz_HTML.gif

We can easily observe that no such k exists which satisfy (3.1) and above condition together.

Therefore this implies that
δ k : μ x k ; t 2 > 1 - b 1  +  γ  or  ν x k ; t 2 < b 1 - γ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equba_HTML.gif
Since α = st(μ,ν)- lim inf x, by Theorem 3.1', we get
δ k : μ x k ; t 2 > 1 - α  +  γ  or  ν x k ; t 2 < α - γ = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbb_HTML.gif
By the definition of st(μ, ν)- lim inf x, we have
1 - α + γ 1 - b 1 + γ  or  α - γ b 1 - γ , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbc_HTML.gif
that is,
b 1 α . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ9_HTML.gif
(3.6)

From (3.4) and(3.5), we get β ≤ α. Now combining Theorem 3.2 and the above inequality, we conclude α = β.

Conversely, suppose that α = β and let sup t μ(L, t) = 1 - α or inf t ν(L, t) = α. Then for any γ > 0, Theorems 3.1 and 3.1 ' will together imply that
δ k : μ x k ; t 2 < 1 - α  +  γ 2  or  ν x k ; t 2 > α - γ 2 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ10_HTML.gif
(3.7)
and
δ k : μ x k ; t 2 > 1 - α  +  γ 2  or  ν x k ; t 2 < α - γ 2 = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ11_HTML.gif
(3.8)
Now
1 - α μ L ; t = μ x k - x k - L ; t μ x k ; t 2 * μ x k - L ; t 2 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbd_HTML.gif
and
α ν L ; t = ν x k - x k - L ; t ν x k ; t 2 ν x k - L ; t 2 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Eqube_HTML.gif
Therefore
μ x k ; t 2 * μ x k - L ; t 2 1 - α  or  ν x k ; t 2 ν x k - L ; t 2 α . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ12_HTML.gif
(3.9)
Let sup t { μ ( x k - L ; t 2 ) } = 1 - a 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq18_HTML.gif or inf t { ν ( x k - L ; t 2 ) } = a 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq19_HTML.gif, where a1 (0, 1) and (3.7) and (3.9) hold. Then
δ k : μ x k - L ; t 2 < 1 - α 1 - γ 2  or  ν x k - L ; t 2 > α 1 + γ 2 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbf_HTML.gif
which is true for all γ > 0. Hence
δ k : μ x k - L ; t 2 1 - α 1  or  ν x k - L ; t 2 α 1 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbg_HTML.gif

which is true for all a ≤ a1 (0, 1), because 1 - a1 is the least upper bound or a1 is the greatest lower bound.

Now repeat the process by taking (3.8) and (3.9) instead of (3.7) and (3.9). If (3.8) and (3.9) are satisfied, then inf t μ ( x k - L ; t 2 ) = 1 - a 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq20_HTML.gif or sup t ν ( x k - L ; t 2 ) = a 1 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq21_HTML.gif,

On contrary suppose that 1 - a 1 inf t μ ( x k - L ; t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq22_HTML.gif or a 1 sup t ν ( x k - L ; t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq23_HTML.gif and conditions (3.8) and (3.9) be satisfied. This implies that there exists some r (0, 1) such that either 1 - r = μ x k - L ; t 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq24_HTML.gif or r = ν x k - L ; t 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq25_HTML.gif for some t > 0 where 1 - a1 > 1 - r or a1 < r.

As (3.8) and (3.9) are satisfied, and let us suppose that inf t μ ( x k - L ; t 2 ) = 1 - a 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq26_HTML.gif or sup t ν ( x k - L ; t 2 ) = a 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq27_HTML.gif.

Then
1 - a 1 > 1 - a 2  or  a 1 < a 2 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ13_HTML.gif
(3.10)
and from (3.9), we get
μ x k - L ; t 2 * ( 1 - a 2 ) 1 - α  or  ν x k - L ; t 2 α 2 α . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbh_HTML.gif
Using (3.8), we get
1 - α + γ 2 * ( 1 - a 2 ) 1 - α  or  ν α - γ 2 ( α 2 ) α  for all  γ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbi_HTML.gif
Clearly,
1 - α + γ 2 * ( 1 - a 2 ) 1 - α  or  α + γ 2 ( α 2 ) α  for all  γ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equ14_HTML.gif
(3.11)
Now
1 - a 1 = sup t μ x k - L ; t 2  or  α 1 = inf t ν x k - L ; t 2 https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbj_HTML.gif

where a1 (0, 1) and which satisfy (3.7) and (3.9).

From (3.11) we conclude that 1 - a2 is another value satisfying (3.7) and (3.9).

Hence
1 - a 1 < 1 - a 2  or  a 2 < a 1 . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbk_HTML.gif

This contradicts (3.10). Hence 1 - a 1 = inf t μ ( x k - L ; t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq28_HTML.gif or a 1 = sup t ν ( x k - L ; t 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_IEq29_HTML.gif satisfying conditions (3.8) and (3.9).

Therefore the inequality becomes true for all a ≥ a1 (0, 1), because 1 - a1 is the greatest lower bound, and hence
δ k : μ x k - L ; t 2 1 - α  or  ν x k - L ; t 2 α = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbl_HTML.gif
for each t > 0 and a (0, 1). Therefore
s t ( μ , ν )  - lim  x = L . https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2012-96/MediaObjects/13660_2011_Article_220_Equbm_HTML.gif

This completes the proof of the theorem.

Declarations

Acknowledgements

The authors would like to thank the Research Deanship at King Abdulaziz University for its financial support under grant number 147/130/1431.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University
(2)
Department of Computer Science, South Asian University
(3)
Department of Mathematics, Aligarh Muslim University

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© Alghamdi et al; licensee Springer. 2012

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