The notion of statistical convergence was introduced by Fast [1] and also independently by Buck [2] and Schoenberg [3] for real and complex sequences.Over the years and under different names statistical convergence has been discussed in the theory of Fourier analysis, Ergodic theory and Number theory. Later on it was further investigated from the sequence spaces point of view and linked with summability theory by Altinok and Et [4], Connor [5], Et et al. ( [6] [7] [8] ), Fridy [9], Fridy and Orhan [10], Mursaleen [11] and many others.

Matloka [12] defined the notion of fuzzy sequence and introduced bounded and convergent sequences of fuzzy real numbers and studied their some properties. After then, Nuray and Savas [13] defined the notion of statistical convergence for sequences of fuzzy numbers. Since then, there has been increasing interest in the study of statistical convergence of fuzzy sequences (see [14] - [19] ).

Lindesstrauss and Tzafriri [20] used the idea of Orlicz sequence space,

l M : = { x ∈ ω : ∑ k = 1 ∞     M ( | x | ρ ) < ∞ ,   for   some   ρ > 0 } , which is Banachi space with the norm: ‖ x ‖ M = inf { ρ > 0 : ∑ k = 1 ∞     M ( | x | ρ ) ≤ 1 } . The space l M is closely related to

the space l p , which is an Orlicz sequence space with M ( x ) = x p for 1 ≤ p < ∞ .

Connor, Fridy and Kline [21] proved that statistical convergent sequences are statistically pre-Cauchy and any bounded statistically pre-Cauchy sequence with nowhere dense set of limit points is statistically convergent. They also gave an example showing statistically pre-Cauchy sequences are not necessarily statistically convergent.

In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. Also, some properties of these new sequence spaces are investigated. It popularized the work of predecessors.

In this section, we give some basic notions which will be used throughout the paper.

Let A ˜ ∈ F ˜ ( R ) be a fuzzy subset on R. If A ˜ is convex, normal, upper semi-continuous and has compact support, we say that A ˜ is a fuzzy number. Let R ˜ c denote the set of all fuzzy numbers.

For A ˜ ∈ R ˜ c , we write the level set of A ˜ as A λ = { x : A ( x ) ≥ λ } and A λ = [ A λ − , A λ + ] . Let A ˜ , B ˜ ∈ R ˜ c , we define A ˜ + B ˜ = C ˜ iff A λ + B λ = C λ , λ ∈ [ 0 , 1 ] iff A λ − + B λ − = C λ − and A λ + + B λ + = C λ + for any λ ∈ [ 0 , 1 ] . A λ ⋅ B λ = C λ , where

C λ − = min { A λ − ⋅ B λ − , A λ − ⋅ B λ + , A λ + ⋅ B λ − , A λ + ⋅ B λ + } ,

C λ + = max { A λ − ⋅ B λ − , A λ − ⋅ B λ + , A λ + ⋅ B λ − , A λ + ⋅ B λ + } .

Define

D ( A ˜ , B ˜ ) = sup λ ∈ [ 0 , 1 ] d ( A λ , B λ ) = sup λ ∈ [ 0 , 1 ] max { | A λ − − B λ − | , | A λ + − B λ + | } ,

where d is the Hausdorff metric. D ( A ˜ , B ˜ ) is called the distance between A ˜ and B ˜ .

Using the results of [22] [23], we see that

1) ( R ˜ c , D ) is a complete metric space,

2) D ( u + w , v + w ) = D ( u , v ) ,

3) D ( k u , k v ) = | k | D ( u , v ) ,   k ∈ R ,

4) D ( u + v , w + e ) ≤ D ( u , w ) + D ( v , e ) ,

5) D ( u + v , 0 ¯ ) ≤ D ( u , 0 ¯ ) + D ( v , 0 ¯ ) ,

6) D ( u + v , w ) ≤ D ( u , w ) + D ( v + 0 ¯ ) ,

Where u , v , w , e ∈ R ˜ c , 0 ˜ represents zero fuzzy number.

A sequence { x n } of fuzzy numbers is said to be statistically convergent to a fuzzy number x 0 if for each ε > 0 the set A ( ε ) = { n ∈ N : D ( x n , x 0 ) ≥ ε } has natural density zero. The fuzzy number x 0 is called the statistical limit of the sequence { x n } and we write st- l i m n → ∞ x n = x 0 [24].

The concept of Orlicz function was introduced by Parashar and Choudhary [25], A mapping M : [ 0, ∞ ) → [ 0, ∞ ) is said to be an Orlicz funtion [26]

1) M ( 0 ) = 0 iff x = 0 ,

2) M ( x ) > 0 for x > 0 ,

3) M ( x ) → ∞ as x → ∞ ,

4) M is continuous, nondecreasing and convex.

An Orlicz function may be bounded or unbounded. For example, M ( x ) = x p ( 0 < p ≤ 1 ) is bounded.

A triple sequence can be defined as a function X : N × N × N → R ( C ) where N, R nad C denote the set of natural numbers, real numbers and complec numbers respectively. A triple sequence { x i j k } is said to be Cauchy sequence if for every ε > 0 , there exist N ( ε ) ∈ N such that | x i j k − x p q r | < ε whenever i , p ≥ N , j , q ≥ N , k , r ≥ N [27].

A triple sequence { x i j k } is called statistically pre-Cauchy if for every ε > 0 there exist p = p ( ε ) , q ( ε ) and r ( ε ) such that

l i m m , n , t → ∞ | 1 m 2 n 2 t 2 | x i j k − x p q r | ≥ ε , i ≤ m , j ≤ n , t ≤ k | = 0.

where the vertical bars indicate the number of elements in the set [28].

Definition 3.1. A triple sequence of fuzzy numbers is said to be ideal statistically pre-Cauchy if for every ε > 0 , δ > 0 there exist p = p ( ε ) , q ( ε ) and r ( ε ) such that

{ ( m , n , t ) ∈ N × N × N : 1 m 2 n 2 t 2 | { ( i , j , k ) : D ( x i j k , x p q r ) ≥ ε , i ≤ m , j ≤ n , t ≤ k } | ≥ δ } ∈ I .

where the I denote the nontrivival ideal of N.

Theorem 3.1. Let x = { x i j k } be a triple sequence of fuzzy number and let M be a bounded Oricz function. Then x is ideal statistically pre-Cauchy if and only if

{ ( m , n , t ) ∈ N × N × N : 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t     M ( D ( x i j k , x p q r ) ρ ) ≥ δ } ∈ I .

Proof. Suppose that

{ ( m , n , t ) ∈ N × N × N : 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t     M ( D ( x i j k , x p q r ) ρ ) ≥ δ } ∈ I .

For each ε > 0 , δ > 0 and ρ > 0 , m , n , t ∈ N , we have

1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t     M ( D ( x i j k , x p q r ) ρ ) = 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t D ( x i j k , x p q r ) ≤ ε     M ( D ( x i j k , x p q r ) ρ )       + 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t D ( x i j k , x p q r ) ≥ ε     M ( D ( x i j k , x p q r ) ρ ) ≥ 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t D ( x i j k , x p q r ) ≤ ε     M ( D ( x i j k , x p q r ) ρ ) ≥ M ( ε ) { 1 m 2 n 2 t 2 | { ( i , j , k ) : D ( x i j k , x p q r ) ≥ ε , i ≤ m , j ≤ n , k ≤ t } | ≥ δ } ∈ I .

Now suppose that x is ideal statistically pre-Cauchy and that ε has been given.

Let ε > 0 , δ > 0 be such that M ( ξ ) < ε 2 .

Since M is bounded Orlicz function, there exist an integer G such that M ( x ) < G 2 for all x ≥ 0 .

Not that, for each m , n , t ∈ N

1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t     M ( D ( x i j k , x p q r ) ρ ) = 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t D ( x i j k , x p q r ) < ξ     M ( D ( x i j k , x p q r ) ρ )       + 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t D ( x i j k , x p q r ) ≥ ξ     M ( D ( x i j k , x p q r ) ρ ) ≤ M ( ξ ) + 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t D ( x i j k , x p q r ) ≥ ξ     M ( D ( x i j k , x p q r ) ρ ) ≤ ε 2 + G 2 { 1 m 2 n 2 t 2 | { ( i , j , k ) : D ( x i j k , x p q r ) ≥ ξ , i ≤ m , j ≤ n , k ≤ t } | ≥ δ } ≤ ε + G { 1 m 2 n 2 t 2 | { ( i , j , k ) : D ( x i j k , x p q r ) ≥ ξ , i ≤ m , j ≤ n , k ≤ t } | ≥ δ } ∈ I .

Hence

{ ( m , n , t ) ∈ N × N × N : 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , q ≤ n   ∑ k , r ≤ t     M ( D ( x i j k , x p q r ) ρ ) ≥ δ } ∈ I .

Theorem 3.2. Let x = { x i j k } be a triple sequence of fuzzy numbers and let M be a bounded Orlicz function. Then x is ideal statistically convergent to x 0 if and only if

{ ( m , n , t ) ∈ N × N × N : 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     M ( D ( x i j k , x 0 ) ρ ) ≥ δ } ∈ I .

Proof. Suppose that

{ ( m , n , t ) ∈ N × N × N : 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     M ( D ( x i j k , x 0 ) ρ ) ≥ δ } ∈ I .

For each ε > 0 , δ > 0 and ρ > 0 , m , n , t ∈ N , we have

1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     M ( D ( x i j k , x 0 ) ρ ) = 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 , D ( x i j k , x 0 ) ≥ ε t     M ( D ( x i j k , x 0 ) ρ )     + 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 , D ( x i j k , x 0 ) < ε t     M ( D ( x i j k , x 0 ) ρ ) ≥ 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 , D ( x i j k , x 0 ) ≥ ε t     M ( D ( x i j k , x 0 ) ρ ) ≥ M ( ε ) { 1 m n t | { ( i , j , k ) : D ( x i j k , x 0 ) ≥ ε , i ≤ m , j ≤ n , k ≤ t } | ≥ δ } ∈ I .

We have x is ideal statistically convergent to x 0 .

Now suppose that x is ideal statistically convergent to x 0 , let ε > 0 , δ > 0 be such that M ( ξ ) < ε 2 .

Since M is bounded Orlicz function, there exist an integer G such that M ( X ) < G 2 for all x ≥ 0 .

Note that, for each m , n , t ∈ N

1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     M ( D ( x i j k , x 0 ) ρ ) = 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 , D ( x i j k , x 0 ) ≥ ε t     M ( D ( x i j k , x 0 ) ρ )         + 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 , D ( x i j k , x 0 ) < ε t     M ( D ( x i j k , x 0 ) ρ ) ≤ M ( ξ ) + 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 , D ( x i j k , x 0 ) ≥ ε t     M ( D ( x i j k , x 0 ) ρ ) ≤ ε 2 + G 2 { 1 m n t | { ( i , j , k ) : D ( x i j k , x p q r ) ≥ ξ , i ≤ m , j ≤ n , k ≤ t } | ≥ δ } ≤ ε + G { 1 m n t | { ( i , j , k ) : D ( x i j k , x p q r ) ≥ ξ , i ≤ m , j ≤ n , k ≤ t } | ≥ δ } ∈ I .

Hence

{ ( m , n , t ) ∈ N × N × N : 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     M ( D ( x i j k , x 0 ) ρ ) ≥ δ } ∈ I .

Corollary 3.3. Let x = { x i j k } be a bound triple sequence of fuzzy number. Then x is ideal statistically pre-Cauchy if and only if

{ ( m , n , t ) ∈ N × N × N : 1 m 2 n 2 t 2 ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     D ( x i j k , x p q r ) ≥ δ } ∈ I .

Proof. Let K = sup i , j , k D ( x i j k , 0 ¯ ) and defined M ( x ) = ( 1 + 2 K ) x 1 + x , then M ( D ( x i j k , x p q r ) ρ ) ≤ ( 1 + 2 K ) D ( x i j k , x p q r ) , and

M ( D ( x i j k , x p q r ) ρ ) = ( 1 + 2 K ) D ( x i j k , x p q r ) 1 + D ( x i j k , x p q r ) ≥ ( 1 + 2 K ) D ( x i j k , x p q r ) ρ 1 + D ( x i j k , x p q r ) ≥ ( 1 + 2 K ) D ( x i j k , x p q r ) 1 + 2 A = D ( x i j k , x p q r ) .

Hence { ( m , n , t ) ∈ N × N × N : 1 m 2 n 2 t 2 ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     D ( x i j k , x p q r ) ≥ δ } ∈ I , if and only if { ( m , n , t ) ∈ N × N × N : 1 m 2 n 2 t 2 ∑ i , p ≤ m   ∑ j , k ≤ n   ∑ t , r ≤ t     M ( D ( x i j k , x p q r ) ρ ) ≥ δ } ∈ I , and an immediate application of Theorem 3.1 completes the proof.

Corollary 3.4. Let x = { x i j k } be a bound triple sequence of fuzzy number. Then x is ideal statistically convergent x 0 if and only if

{ ( m , n , t ) ∈ N × N × N : 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     D ( x i j k , x 0 ) ≥ δ } ∈ I .

Proof. Let K = sup i , j , k D ( x i j k , 0 ¯ ) and defined M ( x ) = ( 1 + K + x 0 ) x 1 + x , then M ( D ( x i j k , x 0 ) ρ ) ≤ ( 1 + K + x 0 ) D ( x i j k , x 0 ) , and

M ( D ( x i j k , x 0 ) ρ ) = ( 1 + K + x 0 ) D ( x i j k , x 0 ) 1 + D ( x i j k , x 0 ) ≥ ( 1 + K + x 0 ) D ( x i j k , x 0 ) ρ 1 + D ( x i j k , x 0 ) ≥ ( 1 + K + x 0 ) D ( x i j k , x 0 ) 1 + K + x 0 = D ( x i j k , x 0 ) .

Hence { ( m , n , t ) ∈ N × N × N : 1 m n t ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t   D ( x i j k , x 0 ) ≥ δ } ∈ I , if and only if { ( m , n , t ) ∈ N × N × N : 1 m 2 n 2 t 2 ∑ i = 1 m   ∑ j = 1 n   ∑ k = 1 t     M ( D ( x i j k , x 0 ) ρ ) ≥ δ } ∈ I , and an immediate application of Theorem 3.1 completes the proof.

In this article, we introduced ideal statistically pre-Cauchy triple sequences of fuzzy numbers about Orlicz function. At the same time, we have proved some properties and relationships.

This work is supported by National Natural Science Fund of China (11761056); the Natural Science Foundation of Qinghai Province (2020-ZJ-920); University level planning project of Qinghai Minzu University (2021XJGH24).

The author declares no conflicts of interest regarding the publication of this paper.

Feng, X. (2021) Ideal Statistically Pre-Cauchy Triple Sequences of Fuzzy Number and Orlicz Functions. Applied Mathematics, 12, 767-774. https://doi.org/10.4236/am.2021.129051