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On deferred f-statistical convergence for double sequences

  • Yahui Zhu , Ang Shen , Zhongzhi Wang EMAIL logo and Weicai Peng
Published/Copyright: April 26, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we first put forward the concept of deferred f-double natural density for double sequences, where f is an unbounded modulus. Then, we combine f-density with deferred statistical convergence for double sequences and investigate deferred f-statistical convergence and strongly deferred Cesàro summability with respect to modulus f. Moreover, we extend these concepts to deferred f-statistical convergence for double sequences of random variables in the Wijsman sense and prove some inclusions. Finally, we consider the concepts of deferred f-statistical convergence of order α and strongly deferred f-summability of order α for double sequences and obtain some conclusions.

Keywords: deferred f-statistical convergence; double sequences; strongly deferred Cesàro summability; in the Wijsman sense
MSC 2010: 40A35; 40G05; 40G15

1 Introduction

Fast [1] first presented statistical convergence, generalising ordinary convergence for real sequences. Buck [2] and Schoenberg [3] have proposed this for real and complex sequences. Since then, many authors have developed several extensions and further generalisations of this concept. Fridy [4] introduced the concept of a statistically Cauchy sequence and proved that it is equivalent to statistical convergence. Maddox [5] extended the idea to apply to sequences in any locally convex Hausdorff topological linear space. Gadjiev and Orhan [6] proved some Korovkin and Weierstrass type approximation theorems via statistical convergence. Miao [7] investigated statistical convergence in topological and uniform spaces and showed how this convergence can be applied to selection principles theory, function spaces and hyperspaces. Ulusu and Nuray [8] defined lacunary statistical convergence for sequences of sets and studied in detail the relationship between other convergence concepts. The notion of modulus function was first introduced by Nakano [9]. Then, many scholars applied it to the theory of statistical convergence and constructed some new sequence spaces. Connor [10] investigated strong matrix summability with respect to a modulus and the relation between A -statistical convergence and A -summability with respect to a modulus. Aizpuru et al. [11] introduced the concept of f-density and proved that the ordinary convergence is equivalent to the module statistical convergence for every unbounded modulus function. Pehlivan [12] put forward some sequence spaces that arose from the notions of strongly almost convergence and modulus function. Ghosh and Srivastava [13] defined and studied the vector valued sequence space F ( E k , f ) using modulus function f. Recently, Çolak and Altin [14] introduced statistical convergence of order α for double sequences and examined some inclusion relations. Ulusu and Nuray et al. [15] proposed the idea of Wijsman strongly lacunary summability for set sequences and discussed its relation with Wijsman strongly Cesàro summability. Nuray et al. [16] proposed the notion of statistical convergence and ℐ -statistical convergence in the Wijsman sense for double sequences of sets. Sengul et al. [17] introduced the concepts of f-lacunary statistical convergence of order α and strong f-lacunary summability of order α for double sequences and give some inclusion relations between these concepts.

A modulus f is a function from [ 0 , ∞ ) to [ 0 , ∞ ) such that

  1. f ( x ) = 0 if only if x = 0 ,

  2. f ( x + y ) ⩽ f ( x ) + f ( y ) for x , y ⩾ 0 ,

  3. f is increasing,

  4. f is continuous from the right at 0.

Obviously, f is continuous everywhere on [ 0 , ∞ ) . A modulus may be bounded or unbounded.

Aizpuru et al. [11] defined f-density of a subset K ⊂ N for any unbounded modulus f by

δ f ( K ) = lim n → ∞ f ( ∣ { k ⩽ n : k ∈ K } ∣ ) f ( n ) ,

provided the limits exists, where the vertical bars denote the cardinality of the closed set. They defined f-statistical convergence for any unbounded modulus f by

δ f ( { k ∈ N : ∣ x k − l ∣ ⩾ ε } ) = 0 ,

i.e.

lim n → ∞ f ( ∣ { k ⩽ n : ∣ x k − l ∣ ⩾ ε } ∣ ) f ( n ) = 0 ,

and denoted by S f − lim x k = l or x k → l ( S f ) .

Recall that let a subset K ⊆ N 2 be a two-dimensional set of positive integers and let K ( m , n ) be the number of ( i , j ) in K with i ⩽ m and j ⩽ n , then the two-dimensional analogue of natural density can be defined as follows.

The lower asymptotic density of a set K ⊆ N 2 is defined as follows:

(1.1) δ 2 ̲ ( K ) ≔ liminf m , n K ( m , n ) m n .

If the sequence K ( m , n ) m n has a limit in Pringshein’s [18] sense, we say that K has a double natural density and is defined as follows:

(1.2) δ 2 ( K ) ≔ P − lim m , n K ( m , n ) m n .

Dagadur and Sezgek [19] proposed the concepts of deferred Cesàro mean and deferred statistical convergence of double sequences.

Throughout this article, we denote β ( n ) = q ( n ) − p ( n ) and γ ( m ) = r ( m ) − t ( m ) by β and γ , respectively, where { p ( n ) } , { q ( n ) } , { r ( m ) } , { t ( m ) } are sequences of non-negative integers such that p ( n ) < q ( n ) , t ( m ) < r ( m ) and lim n → ∞ q ( n ) = ∞ , lim m → ∞ r ( m ) = ∞ .

Let K be a subset of N 2 and denote the set { ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , ( i , j ) ∈ K } by K β , γ ( n , m ) . The deferred double natural density of K is defined as follows:

δ D β , γ ( 2 ) ( K ) ≔ P − lim m , n → ∞ ∣ K β , γ ( n , m ) ∣ β ( n ) γ ( m )

provided the limit exists.

It is convenient to use upper deferred asymptotic density of K if δ D β , γ ( 2 ) ( K ) does not exist for all K ⊂ N 2 , and defined by

δ D β , γ * ( 2 ) ( K ) ≔ limsup m , n → ∞ ∣ K β , γ ( n , m ) ∣ β ( n ) γ ( m ) .

The following properties are obvious.

  1. If δ D β , γ * ( 2 ) ( K ) exists, then δ D β , γ ( 2 ) ( K ) = δ D β , γ * ( 2 ) ( K ) .

  2. δ D β , γ ( 2 ) ( K ) ≠ 0 if and only if δ D β , γ * ( 2 ) ( K ) > 0 .

  3. The function δ D β , γ * ( 2 ) ( K ) is monotone increasing.

A double sequence x = ( x ( i , j ) ) is said to be deferred statistically convergent to L ∈ R , if for every ε > 0

P − lim m , n → ∞ ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − L ∣ ⩾ ε } ∣ β ( n ) γ ( m ) = 0 ,

and denoted by ( D β , γ ) s t 2 − lim m , n → ∞ x ( i , j ) = L .

Let x = ( x ( i , j ) ) be a double sequence and L a real number. The double sequence x is said to be D β , γ -summable to L , if

P − lim m , n → ∞ 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) ∣ x ( i , j ) − L ∣ = 0

holds and denoted by ( D β , γ ) − lim m , n → ∞ x ( i , j ) = L .

We recall the concepts on Wijsman statistical convergence [20–23].

For any point y ∈ Y and any non-empty subset A of Y , we define the distance from y to A by

d ( y , A ) = inf a ∈ A ρ ( y , a ) ,

where ρ is the metric function, i.e., the function ρ : Y × Y → R + that satisfies the following conditions is called the metric function:

  1. ρ ( x , y ) = 0 ⇔ x = y ;

  2. ρ ( x , y ) = ρ ( y , x ) ;

  3. ρ ( x , y ) ⩽ ρ ( x , z ) + ρ ( z , y ) .

Let ( Y , ρ ) be a metric space and A , A k be any non-empty closed subsets of Y .

The sequence { A k } is Wijsman convergent to A if

lim k → ∞ d ( y , A k ) = d ( y , A )

for each y ∈ Y . In this case, we write W − lim A k = A .

The sequence { A k } is Wijsman statistically convergent to A if

lim n → ∞ 1 n ∣ { k ⩽ n : ∣ d ( y , A k ) − d ( y , A ) ∣ ⩾ ε } ∣ = 0 .

In this case, we write s t − lim W A k = A .

On the basis of the aforementioned work, we extend the concept of f-density to deferred f-double natural density of a subset of N 2 , where f is an unbounded modulus. We also present the notion of deferred f-statistical convergence, and strongly deferred Cesàro summability defined by modulus f for double sequences and obtain similar inclusions. Moreover, we consider deferred f-statistical convergence for double sequences of random variables in the Wijsman sense. In this article, we investigate the relations between deferred f-statistical convergence and strongly deferred Cesàro summability with respect to modulus f for double sequences and consider them in the Wijsman sense. Besides, we propose the notions of strongly deferred Cesàro summable for double sequences with respect to modulus f, deferred f-statistical convergence of order α , and strongly deferred f-summability of order α for double sequences, and we prove some inclusions between them.

Next, we will introduce some concepts of deferred f-double natural density, deferred f-statistical convergence, and strongly deferred Cesàro summability defined by modulus f for double sequences and extend them in the Wijsman sense.

Definition 1

The deferred f-double natural density of a subset K of N 2 is defined as follows:

δ β , γ f ( K ) ≔ P − lim m , n → ∞ f ( ∣ K β , γ ( n , m ) ∣ ) f ( β ( n ) γ ( m ) ) = P − lim m , n → ∞ f ( ∣ { ( i , j ) ∈ K : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) } ∣ ) f ( β ( n ) γ ( m ) )

if the limits exists. It is obvious that finite sets have zero deferred f-double natural density for any unbounded modulus f.

Remark 1

If f ( x ) = x , the deferred f-double natural density coincides with deferred double density. If p ( n ) = 0 , t ( m ) = 0 , q ( n ) = n , r ( m ) = m and f ( x ) = x , the deferred f-double natural density turns out to be double natural density.

Definition 2

Let f be an unbounded modulus. A double sequence x = ( x ( i , j ) ) is said to be deferred f-statistically convergent to L , if for each ε > 0

P − lim m , n → ∞ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − L ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) = 0 ,

and denoted by ( D β , γ ) s t 2 f − lim m , n → ∞ x ( i , j ) = L .

Definition 3

Let f be a modulus function, x = ( x ( i , j ) ) a double sequence. Then it is said to be strongly deferred Cesàro summable with respect to f, if

P − lim m , n → ∞ 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) − L ∣ ) = 0 ,

and denoted by ( D β , γ f ) − lim m , n → ∞ x ( i , j ) = L .

Definition 4

A double-indexed sequence of random variables { X ( i , j ) , ( i , j ) ∈ N 2 } is deferred Wijsman statistically convergent to X ( 0 , 0 ) , if

P − lim m , n → ∞ ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ β ( n ) γ ( m ) = 0 ,

and denoted by ( D β , γ ) s t 2 − lim W X ( i , j ) = X ( 0 , 0 ) .

Definition 5

A double-indexed sequence of random variables { X ( i , j ) , ( i , j ) ∈ N 2 } is deferred f-Wijsman statistically convergent to X ( 0 , 0 ) , if

P − lim m , n → ∞ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) = 0 ,

and denoted by ( D β , γ ) s t 2 f − lim W X ( i , j ) = X ( 0 , 0 ) .

Definition 6

A double-indexed sequence of random variables { X ( i , j ) , ( i , j ) ∈ N 2 } is strongly deferred f-Wijsman Cesàro summable to X ( 0 , 0 ) , if

P − lim m , n → ∞ 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ) = 0 ,

and denoted by ( D β , γ f ) − lim W X ( i , j ) = X ( 0 , 0 ) .

2 Main results and proofs

In this section, we mainly discuss the relations between deferred f-statistical convergence and strongly deferred Cesàro summability with respect to modulus f for double sequences and the relations in the Wijsman sense.

Theorem 1

Let f be an unbounded modulus, and there exists a positive constant c such that f ( x y ) > c f ( x ) f ( y ) for all x ⩾ 0 , y ⩾ 0 , and liminf t → ∞ f ( t ) t > 0 . If a double sequence x = ( x ( i , j ) ) is strongly deferred Cesàro summable to L with respect to f, then it is deferred f-statistically convergent to L.

Proof

From the definition of modulus function (ii) and (iii), for any ε > 0 , we have

∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) − L ∣ ) ⩾ f ( ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) ∣ x ( i , j ) − L ∣ ) ⩾ f ( ε ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − L ∣ ⩾ ε } ∣ ) ⩾ c f ( ε ) f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − L ∣ ⩾ ε } ∣ ) .

Note that ( D β , γ f ) − lim m , n → ∞ x ( i , j ) = L and liminf t → ∞ f ( t ) t > 0 , we have ( D β , γ ) s t 2 f − lim m , n → ∞ x ( i , j ) = L .□

Theorem 2

Let x = ( x ( i , j ) ) ∈ l ∞ 2 , where l ∞ 2 is the set of all bounded double sequences, liminf t → ∞ f ( t ) t > 0 and limsup t → ∞ f ( t ) t < ∞ . If x is deferred f-statistically convergent to l, then x is strongly deferred Cesàro summable to l with respect to f.

Proof

From the assumption, there exists a positive real number M such that ∣ x ( i , j ) − l ∣ ⩽ M for all ( i , j ) ∈ N 2 and a number c > 0 such that f ( t ) > c t for t > 0 .

Note that

1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) − l ∣ ) = 1 β ( n ) γ ( m ) ( ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) − l ∣ ⩾ ε q ( n ) , r ( m ) + ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) − l ∣ < ε q ( n ) , r ( m ) ) f ( ∣ x ( i , j ) − l ∣ ) = 1 β ( n ) γ ( m ) f ( M ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) − l ∣ ⩾ ε q ( n ) , r ( m ) + f ( ε ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) − l ∣ < ε q ( n ) , r ( m ) ⩽ f ( M ) β ( n ) γ ( m ) ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ + f ( ε ) β ( n ) γ ( m ) ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ < ε } ∣ ⩽ f ( M ) f ( β ( n ) γ ( m ) ) c β ( n ) γ ( m ) f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) + f ( ε ) f ( β ( n ) γ ( m ) ) c β ( n ) γ ( m ) ∣ f ( { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ < ε } ∣ ) f ( β ( n ) γ ( m ) )

and ( D β , γ ) s t 2 f − lim m , n → ∞ x ( i , j ) = l , we have ( D β , γ f ) − lim m , n → ∞ x ( i , j ) = l .□

Remark 2

By Theorem 2, we are easily seen that the proof on Proposition 3.11 in [24] is not valid.

Proposition 1

Let f be an unbounded modulus. If a double sequence x = ( x ( i , j ) ) is deferred f-statistically convergent to l, then it is deferred statistically convergent to l, but the converse need not be true.

Proof

Since P − lim m , n → ∞ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) = 0 , then for ∀ p ∈ N , ∃ n 0 ∈ N such that for any m , n ⩾ n 0 , we have

f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ) ⩽ 1 p f ( β ( n ) γ ( m ) ) ⩽ 1 p f p ⋅ β ( n ) γ ( m ) p ⩽ f β ( n ) γ ( m ) p .

Noticing that f is increasing, we have ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ⩽ β ( n ) γ ( m ) p , then P − lim m , n → ∞ ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } β ( n ) γ ( m ) ∣ ⩽ 1 p , which follows that x = ( x ( i , j ) ) is deferred statistically convergent to l .

Define the double sequence x = ( x ( i , j ) ) by

x ( i , j ) = i j , if ( i , j ) ∈ [ 0 , n ] × [ 0 , m ] , 0 , otherwise .

And, let p ( n ) = 0 , t ( m ) = 0 , q ( n ) = n , r ( m ) = m , f ( x ) = log x , l = 0 , we have

P − lim m , n → ∞ ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ β ( n ) γ ( m ) = P − lim m , n → ∞ ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) ∣ ⩾ ε } ∣ m n = P − lim m , n → ∞ m n m n → 0 ,

but

P − lim m , n → ∞ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) = P − lim m , n → ∞ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) ∣ ⩾ ε } ∣ ) f ( m n ) = P − lim m , n → ∞ log m n log m n → 1 2 .

Therefore, we obtain that x = ( x ( i , j ) ) is deferred statistically convergent to l , but it is not deferred f-statistically convergent to l .□

Theorem 3

Let f be an unbounded modulus, and let x = ( x ( i , j ) ) be a double sequence, q ( n ) = n , r ( m ) = m for all n , m ∈ N and let { p ( n ) } , { t ( m ) } be arbitrary sequences. If the double sequence x = ( x ( i , j ) ) is deferred f-statistically convergent to l, then it is statistically convergent to l, but the converse need not be true.

Proof

From [19], we have ( D β , γ ) s t 2 − lim m , n → ∞ x ( i , j ) = l ⇒ s t 2 − lim m , n → ∞ x ( i , j ) = l and Proposition 1, so ( D β , γ ) s t 2 f − lim m , n → ∞ x ( i , j ) = l ⇒ s t 2 − lim m , n → ∞ x ( i , j ) = l .

Define a double sequence x = ( x ( i , j ) ) by

x ( i , j ) = i j , if ( i , j ) ∈ [ 0 , n ] × [ 0 , m ] , 0 , otherwise .

And, let p ( n ) = 0 , t ( m ) = 0 , q ( n ) = n , r ( m ) = m , f ( x ) = log x , l = 0 , we have

P − lim m , n → ∞ 1 m n ∣ { ( i , j ) : 0 ⩽ i ⩽ n , 0 ⩽ j ⩽ m , ∣ x ( i , j ) ∣ ⩾ ε } ∣ = P − lim m , n → ∞ m n m n → 0 ,

but

P − lim m , n → ∞ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) = P − lim m , n → ∞ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) ∣ ⩾ ε } ∣ ) f ( m n ) = P − lim m , n → ∞ log m n log m n → 1 2 .

Therefore, we obtain that x = ( x ( i , j ) ) is statistically convergent to l , but not deferred f-statistically convergent to l .□

Theorem 4

Assume that x = ( x ( i , j ) ) is a deferred f-statistically convergent double sequence, then there exists a convergent double sequence y = ( y ( i , j ) ) and a deferred f-statistically null double sequence z = ( z ( i , j ) ) such that x = y + z .

Proof

Note that x = ( x ( i , j ) ) is a deferred f-statistically convergent sequence, there exists l such that for each ε > 0 , δ β , γ f ( A ) = 0 , where A = { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ > ε } . Define double sequences y = ( y ( i , j ) ) and z = ( z ( i , j ) ) as follows:

y ( i , j ) ( ω ) = x ( i , j ) , if ( i , j ) ∈ N 2 − A , l , if ( i , j ) ∈ A .

z ( i , j ) ( ω ) = 0 , if ( i , j ) ∈ N 2 − A , x ( i , j ) − l , if ( i , j ) ∈ A .

We have x = y + z , where y is a convergent double sequence and z is a deferred f-statistically null double sequence. S β , γ f ⊂ c ∪ S β , γ ; 0 f , where S β , γ f denotes the space of deferred f-statistically convergent double sequences, c and S β , γ ; 0 f denote the spaces of convergent and deferred f-statistically null double sequences, respectively. Since c , S β , γ ; 0 f ⊂ S β , γ f , then c ∪ S β , γ ; 0 f ⊂ S β , γ f . Therefore, S β , γ f = c ∪ S β , γ ; 0 f .□

Theorem 5

Let f be an unbounded modulus and there exists a positive constant c such that f ( x y ) > c f ( x ) f ( y ) for all x ⩾ 0 , y ⩾ 0 , and liminf t → ∞ f ( t ) t > 0 . If a double-indexed sequence of random variables { X ( i , j ) , ( i , j ) ∈ N 2 } is strongly deferred f-Wijsman Cesàro summable to X ( 0 , 0 ) with respect to f, then it is deferred f-Wijsman statistically convergent to X ( 0 , 0 ) .

Proof

From the definition of modulus function (ii) and (iii), for any ε > 0 , we have

∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ) ⩾ f ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ f ( ε ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) ⩾ c f ( ε ) f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) .

Note that ( D β , γ f ) − lim W X ( i , j ) = X ( 0 , 0 ) and liminf t → ∞ f ( t ) t > 0 , we have ( D β , γ ) s t 2 f − lim W X ( i , j ) = X ( 0 , 0 ) .□

A double-indexed sequence of random variables { X ( i , j ) , ( i , j ) ∈ N 2 } is bounded that satisfies sup d ( y , X ( i , j ) ) < ∞ , i.e., there exists a positive real number M such that ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩽ M for each y ∈ Y and all ( i , j ) ∈ N 2 .

Theorem 6

If a double-indexed sequence of random variables { X ( i , j ) , ( i , j ) ∈ N 2 } ∈ l ∞ 2 , where l ∞ 2 is the set of all bounded double sequences, liminf t → ∞ f ( t ) t > 0 and limsup t → ∞ f ( t ) t < ∞ . If { X ( i , j ) , ( i , j ) ∈ N 2 } is deferred f-Wijsman statistically convergent to X ( 0 , 0 ) , then it is strongly deferred f-Wijsman Cesàro summable to X ( 0 , 0 ) .

Proof

From the assumption, there exists a positive real number M such that ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩽ M for all ( i , j ) ∈ N 2 and a number c > 0 such that f ( t ) > c t for t > 0 . Note that

1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ) = 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε q ( n ) , r ( m ) + ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ < ε q ( n ) , r ( m ) f ( ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ) ⩽ 1 β ( n ) γ ( m ) f ( M ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε q ( n ) , r ( m ) + f ( ε ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ < ε q ( n ) , r ( m ) ⩽ f ( M ) β ( n ) γ ( m ) ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ + f ( ε ) β ( n ) γ ( m ) ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ < ε } ∣ ⩽ f ( M ) f ( β ( n ) γ ( m ) ) c β ( n ) γ ( m ) f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) + f ( ε ) f ( β ( n ) γ ( m ) ) c β ( n ) γ ( m ) f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ < ε } ∣ ) f ( β ( n ) γ ( m ) ) .

Thus, ( D β , γ ) s t 2 f − lim W X ( i , j ) = X ( 0 , 0 ) , we have ( D β , γ f ) − lim W X ( i , j ) = X ( 0 , 0 ) .□

Theorem 7

Let f be an unbounded modulus and there exists a positive constant c such that f ( x y ) > c f ( x ) f ( y ) for all x ⩾ 0 , y ⩾ 0 , and ( β 1 ( n ) γ 1 ( m ) β 2 ( n ) γ 2 ( m ) ) is bounded, where p 1 ( n ) ⩽ p 2 ( n ) < q 2 ( n ) ⩽ q 1 ( n ) , t 1 ( m ) ⩽ t 2 ( m ) < r 2 ( m ) ⩽ r 1 ( m ) , β 1 = q 1 ( n ) − p 1 ( n ) , β 2 = q 2 ( n ) − p 2 ( n ) , and γ 1 = r 1 ( m ) − t 1 ( m ) , γ 2 = r 2 ( m ) − t 2 ( m ) . If ( D β 1 , γ 1 ) s t 2 − lim W X ( i , j ) = X ( 0 , 0 ) , then ( D β 2 , γ 2 ) s t 2 − lim W X ( i , j ) = X ( 0 , 0 ) .

Proof

For every ε > 0 , we have

{ ( i , j ) : p 2 ( n ) < i ⩽ q 2 ( n ) , t 2 ( m ) < j ⩽ r 2 ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ⊂ { ( i , j ) : p 1 ( n ) < i ⩽ q 1 ( n ) , t 1 ( m ) < j ⩽ r 1 ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε }

for each y ∈ Y , we can obtain

f ( ∣ { ( i , j ) : p 2 ( n ) < i ⩽ q 2 ( n ) , t 2 ( m ) < j ⩽ r 2 ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) f ( β 2 ( n ) γ 2 ( m ) ) ⩽ f ( β 1 ( n ) γ 1 ( m ) ) f ( β 2 ( n ) γ 2 ( m ) ) f ( ∣ { ( i , j ) : p 1 ( n ) < i ⩽ q 1 ( n ) , t 1 ( m ) < j ⩽ r 1 ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) f ( β 1 ( n ) γ 1 ( m ) ) ⩽ 1 c f β 1 ( n ) γ 1 ( m ) β 2 ( n ) γ 2 ( m ) f ( ∣ { ( i , j ) : p 1 ( n ) < i ⩽ q 1 ( n ) , t 1 ( m ) < j ⩽ r 1 ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) f ( β 1 ( n ) γ 1 ( m ) ) .

Thus, ( D β 2 , γ 2 ) s t 2 f − lim W X ( i , j ) = X ( 0 , 0 ) .□

Theorem 8

Let { X ( i , j ) , ( i , j ) ∈ N 2 } , { Y ( i , j ) , ( i , j ) ∈ N 2 } , { Z ( i , j ) , ( i , j ) ∈ N 2 } be double-indexed sequence of random variables and satisfy that { Y ( i , j ) , ( i , j ) ∈ N 2 } ⊂ { X ( i , j ) , ( i , j ) ∈ N 2 } ⊂ { Z ( i , j ) , ( i , j ) ∈ N 2 } . Then, ( D β , γ ) s t 2 f − lim W Y ( i , j ) = X ( 0 , 0 ) and ( D β , γ ) s t 2 f − lim W Z ( i , j ) = X ( 0 , 0 ) ⇒ ( D β , γ ) s t 2 f − lim W X ( i , j ) = X ( 0 , 0 ) .

Proof

From the assumption, for each y ∈ Y , d ( y , Z ( i , j ) ) ⩽ d ( y , X ( i , j ) ) ⩽ d ( y , Y ( i , j ) ) . Hence, for every ε > 0 , we have

{ ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } = { ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , d ( y , X ( i , j ) ) ⩾ d ( y , X ( 0 , 0 ) ) + ε } ∪ { ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , d ( y , X ( i , j ) ) ⩽ d ( y , X ( 0 , 0 ) ) − ε } ⊂ { ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , d ( y , Y ( i , j ) ) ⩾ d ( y , X ( 0 , 0 ) ) + ε } ∪ { ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , d ( y , Z ( i , j ) ) ⩽ d ( y , X ( 0 , 0 ) ) − ε } ,

and so

f ( ∣ { ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , ∣ d ( y , X ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) ⩽ f ( ∣ { ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , ∣ d ( y , Y ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) + f ( ∣ { ( i , j ) : p ( n ) < i ⩽ q ( n ) , t ( m ) < j ⩽ r ( m ) , ∣ d ( y , Z ( i , j ) ) − d ( y , X ( 0 , 0 ) ) ∣ ⩾ ε } ∣ ) f ( β ( n ) γ ( m ) ) .

Thus, ( D β , γ ) s t 2 f − lim W X ( i , j ) = X ( 0 , 0 ) .□

3 Strongly deferred Cesàro summability with respect to modulus f

In this section, we begin by introducing the notion of strongly deferred Cesàro summable for double sequences with respect to modulus f, deferred f-statistical convergence of order α and strongly deferred f-summability of order α for double sequences, then we discuss some inclusions between them.

Definition 7

Let f be a modulus. Define

ω β , γ ; 0 f = x : P − lim m , n → ∞ 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) ∣ ) = 0 ,

ω β , γ f = x : P − lim m , n → ∞ 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) − l ∣ ) = 0 for some number l ,

and

ω β , γ ; ∞ f = x : sup m , n 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) ∣ ) < ∞ .

Definition 8

Let f be an unbounded modulus, x = ( x ( i , j ) ) be a double sequence and α a real number such that 0 < α ⩽ 1 . The double sequence x = ( x ( i , j ) ) is f-statistically convergent of order α , if there exists a real number l satisfies that

P − lim m , n → ∞ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ) [ f ( β ( n ) γ ( m ) ) ] α = 0 ,

and denoted by S β , γ f , α − lim x ( i , j ) = l or x ( i , j ) → l ( S β , γ f , α ) and the set of all double sequences which are f-statistically convergent of order α is denoted by S β , γ f , α .

Definition 9

Let f be a modulus function, x = ( x ( i , j ) ) be a double sequence, α a real number such that 0 < α ⩽ 1 , and p = ( p k ) a sequence of strictly positive real numbers. The double sequence x = ( x ( i , j ) ) is strongly ω α [ β , γ , f , p ] -summable to l (a real number), if there exists a real number l such that

P − lim m , n → ∞ 1 [ β ( n ) γ ( m ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ f ( ∣ x ( i , j ) − l ∣ ) ] p k = 0 ,

and denoted by ω α [ β , γ , f , p ] − lim x ( i , j ) = l and the set of all ω α [ β , γ , f , p ] -summable double sequences is denoted by ω α [ β , γ , f , p ] .

Definition 10

Let f be an unbounded modulus, x = ( x ( i , j ) ) be a double sequence , α a real number such that 0 < α ⩽ 1 and p = ( p k ) a sequence of strictly positive real numbers. The double sequence x = ( x ( i , j ) ) is strongly ω β , γ f , α ( p ) -summable to l (a real number), if there is a real number l such that

P − lim m , n → ∞ 1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ f ( ∣ x ( i , j ) − l ∣ ) ] p k = 0 ,

and denoted by ω β , γ f , α ( p ) − lim x ( i , j ) = l and the set of all ω β , γ f , α ( p ) -summable double sequences is denoted by ω β , γ f , α ( p ) . In case of p k = p for all k ∈ N , we write ω β , γ f , α [ p ] instead of ω β , γ f , α ( p ) .

Definition 11

Let f be an unbounded modulus, x = ( x ( i , j ) ) be a double sequence , α a real number such that 0 < α ⩽ 1 , and p = ( p k ) a sequence of strictly positive real numbers. The double sequence x = ( x ( i , j ) ) is strongly ω β , γ , f α ( p ) -summable to l (a real number), if there exists a real number l such that

P − lim m , n → ∞ 1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ ∣ x ( i , j ) − l ∣ ] p k = 0 ,

and denoted by ω β , γ , f α ( p ) − lim x ( i , j ) = l and the set of all ω β , γ , f α ( p ) -summable double sequences is denoted by ω β , γ , f α ( p ) . In case of p k = p for all k ∈ N , we write ω β , γ , f α [ p ] instead of ω β , γ , f α ( p ) .

Proposition 2

For any modulus f, we have ω β , γ ; 0 ⊂ ω β , γ f ⊂ ω β , γ ; ∞ f .

Proof

ω β , γ ; 0 ⊂ ω β , γ f is obvious.

Let x = ( x ( i , j ) ) ∈ ω β , γ f . From the definition of modulus function (ii) and (iii), we have

1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) ∣ ) ⩽ 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) − l ∣ ) + 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ l ∣ ) .

Hence, ω β , γ f ⊂ ω β , γ ; ∞ f .□

Proposition 3

Let f be a modulus and 0 < δ < 1 . Then for each ‖ x ‖ ⩾ δ , we have f ( ‖ x ‖ ) ⩽ 2 f ( 1 ) δ − 1 ‖ x ‖ , where ‖ x ‖ = sup k ‖ x k ‖ < ∞ .

Proof

Note that

f ( ‖ x ‖ ) ⩽ f ( 1 + [ δ − 1 ‖ x ‖ ] ) ⩽ f ( 1 ) + f ( [ δ − 1 ‖ x ‖ ] ) ⩽ f ( 1 ) + ( δ − 1 ‖ x ‖ ) f ( 1 ) = f ( 1 ) ( 1 + δ − 1 ‖ x ‖ ) ⩽ 2 f ( 1 ) δ − 1 ‖ x ‖ .

It follows.□

Proposition 4

For any modulus f, we have ω β , γ ; 0 ⊂ ω β , γ ; 0 f , ω β , γ ⊂ ω β , γ f , and ω β , γ ; ∞ ⊂ ω β , γ ; ∞ f .

Proof

The first two inclusions are easily proved, so we consider only the last inclusion. Let x ∈ ω β , γ ; ∞ so that sup m , n 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) ∣ x ( i , j ) ∣ < ∞ . Let ε > 0 and choose δ with 0 < δ < 1 such that f ( t ) < ε for 0 < t ⩽ δ .

Note that

1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) ∣ ) = 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) ∣ ⩽ δ q ( n ) , r ( m ) f ( ∣ x ( i , j ) ∣ ) + 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) ∣ > δ q ( n ) , r ( m ) f ( ∣ x ( i , j ) ∣ ) ⩽ 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) ∣ ⩽ δ q ( n ) , r ( m ) ε + 2 f ( 1 ) δ 1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) ∣ > δ q ( n ) , r ( m ) ∣ x ( i , j ) ∣ .

Thus, x ∈ ω β , γ ; ∞ f and the proof is complete.□

Proposition 5

Let f be a modulus. If liminf t → ∞ f ( t ) t > 0 , then

  1. ω β , γ ; 0 f ⊂ ω β , γ ; 0 ,

  2. ω β , γ f ⊂ ω β , γ ,

  3. ω β , γ ; ∞ f ⊂ ω β , γ ; ∞ .

Proof

(i) Let x ∈ ω β , γ ; 0 f . If liminf t → ∞ f ( t ) t > 0 , there exists a number c > 0 such that f ( t ) > c t for t > 0 .

Note that

1 β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) ∣ ) ⩾ 1 β ( n ) γ ( m ) f ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) ∣ x ( i , j ) ∣ ⩾ c β ( n ) γ ( m ) ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) ∣ x ( i , j ) ∣ .

Thus, x ∈ ω β , γ ; 0 .

The proofs for (ii) and (iii) are quite similar to that of (i) and so are omitted.□

It follows immediately from Propositions 4 and 5.

Proposition 6

Let f be any modulus with liminf t → ∞ f ( t ) t > 0 , then

  1. ω β , γ ; 0 f = ω β , γ ; 0 ,

  2. ω β , γ f = ω β , γ ,

  3. ω β , γ ; ∞ f = ω β , γ ; ∞ .

Theorem 9

Let f be an unbounded modulus, α a real number with 0 < α ⩽ 1 and p > 1 . If liminf t → ∞ f ( t ) t > 0 , then ω β , γ f , α [ p ] = ω β , γ , f α [ p ] .

Proof

(i) Let p > 1 be a positive real number and x ∈ ω β , γ f , α [ p ] . If liminf t → ∞ f ( t ) t > 0 , then there exists a number c > 0 such that f ( t ) > c t for t > 0 . Note that

1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ f ( ∣ x ( i , j ) − l ∣ ) ] p ⩾ 1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ c ∣ x ( i , j ) − l ∣ ] p = c p [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ ∣ x ( i , j ) − l ∣ ] p .

Thus x ∈ ω β , γ , f α [ p ] .

(ii) Since x ∈ ω β , γ , f α [ p ] , then P − lim m , n → ∞ 1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ ∣ x ( i , j ) − l ∣ ] p = 0 .

Let 0 < δ < 1 ,

1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ ∣ x ( i , j ) − l ∣ ] p ⩾ 1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) − l ∣ ⩾ δ q ( n ) , r ( m ) [ ∣ x ( i , j ) − l ∣ ] p ⩾ 1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) − l ∣ ⩾ δ q ( n ) , r ( m ) δ f ( ∣ x ( i , j ) − l ∣ ) 2 f ( 1 ) p ⩾ 1 [ f ( β ( n ) γ ( m ) ) ] α δ p 2 p [ f ( 1 ) ] p ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) − l ∣ ⩾ δ q ( n ) , r ( m ) [ f ( ∣ x ( i , j ) − l ∣ ) ] p .

Thus, x ∈ ω β , γ f , α [ p ] . □

Remark 3

If liminf t → ∞ f ( t ) t = 0 , the equality ω β , γ f , α [ p ] = ω β , γ , f α [ p ] cannot be hold as shown in the following example.

Let f ( x ) = x and define a double sequence x = ( x ( i , j ) ) by

x ( i , j ) = β ( n ) , γ ( m ) 3 , if ( i , j ) ∈ [ p ( n ) + 1 , q ( n ) ] × [ t ( m ) + 1 , r ( m ) ] , 0 , otherwise .

For l = 0 , α = 3 4 , p = 6 5 , we have

1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ f ( ∣ x ( i , j ) ∣ ) ] p = [ ( β ( n ) γ ( m ) ) 1 6 ] 6 5 [ ( β ( n ) γ ( m ) ) 1 2 ] 3 4 → 0 , m , n → ∞ .

Thus, x ∈ ω β , γ f , α [ p ] .

But

1 [ f ( β ( n ) γ ( m ) ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) [ ∣ x ( i , j ) ∣ ] p = [ ( β ( n ) γ ( m ) ) 1 3 ] 6 5 [ ( β ( n ) γ ( m ) ) 1 2 ] 3 4 → ∞ , m , n → ∞ ,

and so x ∉ ω β , γ , f α [ p ] .

Theorem 10

Let f be an unbounded modulus, α a real number, p k = 1 for all k ∈ N , and there exists a positive constant c such that f ( x y ) ⩾ c f ( x ) f ( y ) . If liminf t → ∞ [ f ( t ) ] α t α > 0 , then ω α [ β , γ , f , p ] ⊂ S β , γ f , α .

Proof

Let x ∈ ω α [ β , γ , f , p ] and liminf t → ∞ [ f ( t ) ] α t α > 0 . For ε > 0 , we have

1 [ β ( n ) γ ( m ) ] α ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) f ( ∣ x ( i , j ) − l ∣ ) ⩾ 1 [ β ( n ) γ ( m ) ] α f ( ∑ i = p ( n ) + 1 , j = t ( m ) + 1 q ( n ) , r ( m ) ∣ x ( i , j ) − l ∣ ) ⩾ 1 [ β ( n ) γ ( m ) ] α f ( ∑ i = p ( n ) + 1 , j = t ( m ) + 1 , ∣ x ( i , j ) − l ∣ ⩾ ε q ( n ) , r ( m ) ∣ x ( i , j ) − l ∣ ) ⩾ f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ε ) [ β ( n ) γ ( m ) ] α ⩾ c f ( ε ) f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ) [ β ( n ) γ ( m ) ] α = [ f ( β ( n ) γ ( m ) ) ] α [ β ( n ) γ ( m ) ] α c f ( ε ) f ( ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ ) [ f ( β ( n ) γ ( m ) ) ] α .

Thus, ω α [ β , γ , f , p ] − lim x ( i , j ) = l implies that S β , γ f , α − lim x ( i , j ) = l .□

Theorem 11

Let f be an unbounded modulus and there exists a positive constant c such that f ( x y ) ⩾ c f ( x ) f ( y ) , x = ( x ( i , j ) ) be a double sequence and α a real number such that 0 < α ⩽ 1 . If S f , β α − lim x i = l and S f , γ α − lim x j = l , then S f , β , γ α − lim x ( i , j ) = l .

Proof

As S f , β α − lim x i = l and S f , γ α − lim x j = l , then for any ε > 0 ,

P − lim n → ∞ 1 [ f ( β ( n ) ) ] α ∣ { p ( n ) + 1 ⩽ i ⩽ q ( n ) : ∣ x i − l ∣ ⩾ ε } ∣ = 0 ,

P − lim m → ∞ 1 [ f ( γ ( m ) ) ] α ∣ { t ( m ) + 1 ⩽ j ⩽ r ( m ) : ∣ x j − l ∣ ⩾ ε } ∣ = 0 .

So we have

∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ [ f ( β ( n ) γ ( m ) ) ] α ⩽ ∣ { ( i , j ) : p ( n ) + 1 ⩽ i ⩽ q ( n ) , t ( m ) + 1 ⩽ j ⩽ r ( m ) , ∣ x ( i , j ) − l ∣ ⩾ ε } ∣ c α [ f ( β ( n ) ) ] α [ f ( γ ( m ) ) ] α ⩽ ∣ { p ( n ) + 1 ⩽ i ⩽ q ( n ) : ∣ x i − l ∣ ⩾ ε } ∣ [ f ( β ( n ) ) ] α ⋅ ∣ { t ( m ) + 1 ⩽ j ⩽ r ( m ) : ∣ x j − l ∣ ⩾ ε } ∣ [ f ( γ ( m ) ) ] α .

Hence, S f , β , γ α − lim x ( i , j ) = l .□

Theorem 12

Let f be an unbounded modulus and α 1 and α 2 two real numbers such that 0 < α 1 ⩽ α 2 ⩽ 1 . If

liminf m , n → ∞ [ f ( β 1 ( n ) γ 1 ( m ) ) ] α 1 [ f ( β 2 ( n ) γ 2 ( m ) ) ] α 2 > 0 ,

then, ω β 2 , γ 2 f , α 2 ( p ) ⊂ ω β 1 , γ 1 f , α 1 ( p ) , where β 1 = q 1 ( n ) − p 1 ( n ) , β 2 = q 2 ( n ) − p 2 ( n ) , γ 1 = r 1 ( m ) − t 1 ( m ) , γ 2 = r 2 ( m ) − t 2 ( m ) and [ p 1 ( n ) , q 1 ( n ) ] × [ t 1 ( m ) , r 1 ( m ) ] ⊂ [ p 2 ( n ) , q 2 ( n ) ] × [ t 2 ( m ) , r 2 ( m ) ] .

Proof

Let x ∈ ω β 2 , γ 2 f , α 2 ( p ) , we have

1 [ f ( β 2 ( n ) γ 2 ( m ) ) ] α 2 ∑ i = p 2 ( n ) + 1 , j = t 2 ( m ) + 1 q 2 ( n ) , r 2 ( m ) [ f ( ∣ x ( i , j ) − l ∣ ) ] p = 1 [ f ( β 2 ( n ) γ 2 ( m ) ) ] α 2 ∑ ( i , j ) ∈ [ p 2 ( n ) , q 2 ( n ) ] × [ t 2 ( m ) , r 2 ( m ) ] \ [ p 1 ( n ) , q 1 ( n ) ] × [ t 1 ( m ) , r 1 ( m ) ] [ f ( ∣ x ( i , j ) − l ∣ ) ] p + 1 [ f ( β 2 ( n ) γ 2 ( m ) ) ] α 2 ∑ i = p 1 ( n ) + 1 , j = t 1 ( m ) + 1 q 1 ( n ) , r 1 ( m ) [ f ( ∣ x ( i , j ) − l ∣ ) ] p ⩾ 1 [ f ( β 2 ( n ) γ 2 ( m ) ) ] α 2 ∑ i = p 1 ( n ) + 1 , j = t 1 ( m ) + 1 q 1 ( n ) , r 1 ( m ) [ f ( ∣ x ( i , j ) − l ∣ ) ] p = [ f ( β 1 ( n ) γ 1 ( m ) ) ] α 1 [ f ( β 2 ( n ) γ 2 ( m ) ) ] α 2 1 [ f ( β 1 ( n ) γ 1 ( m ) ) ] α 1 ∑ i = p 1 ( n ) + 1 , j = t 1 ( m ) + 1 q 1 ( n ) , r 1 ( m ) [ f ( ∣ x ( i , j ) − l ∣ ) ] p .

Thus, x ∈ ω β 1 , γ 1 f , α 1 ( p ) . □

Acknowledgements

The authors thank the referee(s) for reading the manuscript very carefully and making several valuable and kind comments which improved the presentation.

  1. Funding information: This research was supported by University Key Project of the Natural Science Foundation of Anhui Province (KJ2021A1031, KJ2021A1032, KJ2021A0386, and KJ2020A0679), Key Project of the Natural Science Foundation of Chaohu University (XLZ-202201), Key Construction Discipline of Chaohu University (kj22zdjsxk01, kj22yjzx05, and kj22xjzz01), University Outstanding Young Talents Project of Anhui Province (gxyq2021018), and Anhui Province Graduate Student Academic Innovation Project (2022xscx071).

  2. Author contributions: Conceptualization: Zhongzhi Wang; writing–original draft preparation: Yahui Zhu and Zhongzhi Wang; writing–review and editing: Yahui zhu, Weicai Peng, and Ang Shen; funding acquisition: Weicai Peng.

  3. Conflict of interest: The authors state no conflicts of interest.

  4. Data availability statement: No data were used to support this study.

References

[1] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. 10.4064/cm-2-3-4-241-244Search in Google Scholar

[2] R. C. Buck, Generalized asymptotic density, Amer. J. Math. 75 (1953), 335–346. 10.2307/2372456Search in Google Scholar

[3] I. J. Schoenberg, The inequality of certain functions and related summability method, Amer. Math. Monthly 66 (1959), 361–375. 10.2307/2308747Search in Google Scholar

[4] J. A. Fridy, On statistical convergence, Analysis 5 (1985), 301–313. 10.1524/anly.1985.5.4.301Search in Google Scholar

[5] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1988), 141–145. 10.1017/S0305004100065312Search in Google Scholar

[6] A. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32 (2002), no. 1, 129–138. 10.1216/rmjm/1030539612Search in Google Scholar

[7] G. D. Miao, Statistical convergence in topology, Topology Appl. 156 (2008), 28–45. 10.1016/j.topol.2008.01.015Search in Google Scholar

[8] U. Ulusu and F. Nuray, Lacunary statistical convergence of sequences of sets, Prog. Appl. Math. 4 (2012), no. 2, 99–109. Search in Google Scholar

[9] H. Nakano, Concave modulars, J. Math. Soc. Japan 5 (1953), 29–49. 10.2969/jmsj/00510029Search in Google Scholar

[10] J. Connor, On strong matrix summability with respect to a moludus and statistical convergence, Canad. Math. Bull. 32 (1989), 194–198. 10.4153/CMB-1989-029-3Search in Google Scholar

[11] A. Aizpuru, M. C. Listan-Garcia, and F. Rambla-Barreno, Density by moduli and statistical convergence, Quaest. Math. 37 (2014), 525–530. 10.2989/16073606.2014.981683Search in Google Scholar

[12] S. Pehlivan, Strongly almost convergent sequences defined by a modulus and uniformly statistical convergence, Soochow J. Math. 20 (1994), 205–211. Search in Google Scholar

[13] D. Ghosh and P. D. Srivastava, On some vector valued sequence spaces defined using a modulus function, Indian J. Pure Appl. Math. 30 (1999), 819–826. Search in Google Scholar

[14] R. Çolak and Y. Altin, Statistical convergence of double sequences of order ã, J. Funct. Spaces Appl. 2013 (2013), 682823. 10.1155/2013/682823Search in Google Scholar

[15] U. Ulusu and F. Nuray, On strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinform. 3 (2013), 75–88. 10.1155/2013/310438Search in Google Scholar

[16] F. Nuray, U. Ulusu, and D. Erdinc, Cesàro summability of double sequences of sets, Gen. Math. Notes 25 (2014), 8–18. Search in Google Scholar

[17] H. Sengul, M. Et, and Y. Altin, f-lacunary statistical convergence and strong f-lacunary summability of order α of double sequences, F. U. Math. Inform. 35 (2020), no. 2, 495–506. 10.22190/FUMI2002495SSearch in Google Scholar

[18] A. Pringshein, Zur theorie der zweifach unendlichen Zablenfolgen, Math. Ann. 53 (1900), 289–321. 10.1007/BF01448977Search in Google Scholar

[19] I. Dagadur and S. Sezgek, Deferred Cesàro mean and deferred statistical convergence of double sequences, J. Inequal. Spec. Funct. 4 (2016), 118–136. 10.13164/ma.2015.06Search in Google Scholar

[20] J. P. Aubin and H. Frankowska, Set-valued Analysis, Birkhauser, Boston, 1990. Search in Google Scholar

[21] G. Beer, On convergence of closed sets in a metric space and distance functions, Bull. Aust. Math. Soc. 31 (1985), 421–432. 10.1017/S0004972700009370Search in Google Scholar

[22] M. Baronti and P. Papini, Convergence of sequences of sets, in: C. A. Micchelli, D.V. Pai, and B. V. Limaye (Eds.), Methods of Functional Analysis in Approximation Theory: Proceedings of the International Conference Held at the Indian Institute of Technology, Bomb, ISNM 76, Birkhauser, Basel, 1986, pp. 133–155. Search in Google Scholar

[23] G. Beer, Wijsman convergence: A survey, Set-Valued Anal. 2 (1994), 77–94. 10.1007/BF01027094Search in Google Scholar

[24] S. Gupta and V. K. Bhardwaj, On deferred f-statistical convergence, Kyungpook Math. J. 58 (2018), 91–103. Search in Google Scholar
Received: 2023-10-04
Revised: 2023-12-04
Accepted: 2023-12-25
Published Online: 2024-04-26

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2023-0174
Keywords for this article
deferred f-statistical convergence; double sequences; strongly deferred Cesàro summability; in the Wijsman sense
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