# Hibrid Δ-Statistical Convergence for Neutrosophic Normed Space

Authors: Nazmiye Gonul Bilgin
Journal: Journal of Mathematics
 (2022)

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## Abstract

This study is about neutrosophic structures, which is one of the popular topics of recent days. In this study, different types of convergence concepts were applied to difference sequences. With the help of the properties of double type sequences, the concept of difference sequences is combined with structures that are advantageous to work like Lacunary sequences.


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## Body

## 1. Introduction

The idea of difference sequences emerged in the days when researchers focused on the idea of constructing new sequence spaces. The difference sequences were given by Kizmaz in [1]. After that, Basarir [2] applied this concept to statistical analysis. After statistical convergence was introduced in [3], different versions of statistical convergence have been defined in area of functional analysis, e.g., [4–9]. Fridy and Orhan gave this concept for Lacunary sequences in [10]. Esi and Araci applied this concept to operator theory in [11]. The properties of Lacunary A-convergence are given in [12]. Then, Hazarika applied that definition of Basarir to Lacunary sequences in normed spaces in [13]. Altundag and Kamber made an important study by evaluating Lacunary sequence and difference sequence structures together in n-dimensional intuitionistic fuzzy normed space, where Δxk=xk−xk+1, in [14]. Colak introduced Lacunary strong convergence using difference sequences for modulus function in [15]. Et et al. gave a generalization of difference sequences in [16]. Statistical convergence is defined for double sequences in [17]. Later, this concept attracted great attention of researchers. Tripathy and Sarma applied the idea of this work to difference sequences in [18]. In [19], Patterson and Savas transferred Lacunary statistical convergence to double sequences. Fuzzy-intuitionistic fuzzy sets are generalizations of classical sets established for compelling reasons in daily life. Due to the inadequacy of fuzzy set and intuitionistic set concepts, a new set concept was needed. Thus, a new concept emerged with the help of neutrosophy, which is called a subdivision of philosophy that studies the structure of neutrals: neutrosophic sets. Neutrosophic sets are a concept introduced to investigate the degrees of correctness, wrongness, and uncertainty of the elements in the set in [20]. While classical statistics uses precise data and inferences, neutrophic statistics uses methods that contain uncertain, contradictory, and partially unknown data. Kirisci and Simsek introduced classical statistical convergence in neutrosophic normed spaces in [21]. After that, Khan et al. carried this work to Lacunary sequences in [22]. Granados and Dhital adapted this worked for double sequences in [23]. In 2022, Kisi and Gurdal introduced triple difference sequences in neutrosophic normed spaces [24]. Sahin and Kargın worked neutrosophic tripled normed spaces in [25]. Apart from this, the concept of neutrosophic set has many applications in many different fields, i.e., [26, 27].Now, by evaluating all this information together, a study was prepared to fill the relevant gap in the literature. Here,Δ− statistical convergence will be applied to neutrosophic normed spaces single-double sequences, and also, Δ−Lacunary statistical convergence will be given for these two types of sequences. Many important results were given, especially the relations between these two concepts. The properties of Δ− statistical Cauchy and Δ− Lacunary statistical Cauchy sequences will be examined for these sequences.


## 2. Preliminaries

First of all, some necessary definitions will be given.Definition 1 (see [24]).
Let us considerm crisp-components: ι1,ι2,…,ιm∈0,1. If all of them are 100% independent two by two, then their sum is(1)0≤ι1+ι2+⋯∈+ιm≤m.
On the contrary, if all of them are 100% dependent, then(2)0≤ι1+ι2+⋯+ιm≤1.
As stated by Smarandache in [28], in this study, 0≤ι1+ι2+ι3≤2 will be taken when two components are dependent, while the third one is independent from them.Definition 2 (see [20]).
LetU≠∅ and μN,Ts,ϱN,ℐs, and ζN,ℱs are the degrees of correctness, uncertainty, and falsity. A neutrosophic set N is in the next form: N=s,μN,Cs,ϱN,Us,ζN,ℱs:s∈U, where, for all s in U, μN,Cs,ϱN,Us and ζN,ℱs∈0,1, 0≤μN,Cs+ζN,ℱs+ϱN,Us≤2.
It should be noted thatϱN,Us is an independent component and μN,Ts and ζN,ℱs are dependent components.Definition 3 (see [21]).
LetU be a linear spaces and ⊛ and ⊡ show the continuous t− norm and continuous t− conorm on ℝ. The notation of neutrosophic normed is s,r,μN,Cs,r,ϱN,Us,r,ζN,ℱs,r:s,r∈U×0,∞, where μN,C, ζN,ℱ, and ϱN,ℐ demonstrate the degree of correctness, uncertainty, and falsity of s,r on U×0,∞ which satisfies the following conditions, for all s1,s2∈U:(i)
For everyr∈ℝ+, μN,Cs,r+ϱN,Us,r+ζN,ℱs,r≤2.(ii)
For everyr1,r2∈ℝ+,(3)μN,Cs1,r1⊛μN,Cs2,r2≤μN,Cs1+s2,r1+r2,ϱN,Us1,r1⊡ϱN,Us2,r2≥ϱN,Us1+s2,r1+r2,ζN,ℱs1,r1⊡ζN,ℱs2,r2≥ζN,ℱs1+s2,r1+r2.(iii)
For everyr∈ℝ+, μN,Cs,r=1⇔s=0, ϱN,Us,r=0⇔s=0, and ζN,ℱs,r=0⇔s=0.(iv)
For eachK≠0, μN,CKs,r=μN,Cs,r/K, ϱN,UKs,r=ϱN,Us,r/K, and(4)ζN,ℱKs,r=ζN,ℱs,rK.(v)
μN,Cs,· is a continuous nondecreasing function; ϱN,Us,· and νPNFs,⋅ are continuous nonincreasing function,(vi)
limu⟶∞μN,Cs,r=1, limu⟶∞ϱN,Us,r=0, and limu⟶∞ζN,ℱs,r=0.(vii)
Ifr≤0, then μN,Cs,r=0, ϱN,Us,r=1, and ζN,ℱs,r=1.
In this case,U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ is called neutrosophic normed spaces. Here, μN,C and ϱN,U are interdependent and ζN,ℱ is an independent components.Definition 4 (see [3]).
LetG⊆ℕ and G denote the cardinality of G. The density of the set G is given by the following equation and is denoted by δG:(5)δG=limp⟶∞1pk≤p:k∈G.Definition 5 (see [3]).
sk is called to be statistically convergent to s, where(6)δk∈ℕ:sk−s≥ε=0,for all ε>0. Then, it will be represented by st−limsk=s. S is demonstrated, set of statistical convergence sequences.Definition 6 (see [2]).
sk is called to be Δ− statistically convergent to s, where(7)δk∈ℕ:Δsk−s≥ε=0,for all ε>0 and Δsk=sk−sk+1, i.e.,(8)limp⟶∞1pk≤p:Δsk−s≥ε=0.
Then, it is demonstratedst−limΔsk=s. SΔ is denoted, set of all Δ− statistical convergence sequences.Definition 7 (see [10]).
Letθ=kr be a sequence of increasing integers, k0=0 and also limr⟶∞hr:limr⟶∞kr−kr−1=∞. Then, θ is called to be Lacunary sequences. Let A⊂ℕ,lr=kr/kr−1, and Ir=kr−1,kr.(9)δθA=limr⟶∞1lrk∈Ir:k∈A,is said to be the θ− density of A if limit is exhibited. Let Aε=k∈Ir:sk−s≥ε; for all ε>0, if(10)δθAε=limr⟶∞1lrk∈Ir:sk−s≥ε=0,in this case, sk is called to be Lacunary statistical convergent to s. Then, it is represented as stθ−limsk=s. Sθ is a denoted set of every Lacunary statistical convergence sequences.


## 3. Materials and Methods

In this section where important properties for difference sequences will be given, two separate parts will be given where convergence studies will be made for single and double sequences.


### 3.1.Δ−Statistical Convergence in Neutrosophic Normed Spaces

Now,Δ−convergence, Δ−statistical convergence, and Lacunary Δ−statistical convergence will be defined in neutrosophic normed spaces. The relations between these concepts will be given.Definition 8.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,∗ be neutrosophic normed spaces and Δsk=sk−sk+1. sk is called to be Δ− convergent to s according to neutrosophic normed if, for all ε∈0,1 and r>0, there exists a k˜∈ℕ such that, for every k≥k˜,(11)μN,CΔsk−s,r≤1−ε,ϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε.
This sequences is shown with−ΔNlimsk=s.Definition 9.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If there exist r>0 and 0<ε<1, for all Δsk where μN,CΔsk,r≤1−ε, ϱN,UΔsk,r≥ε, and ζN,ℱΔsk,r≥ε, then sk is called Δ−bounded sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.Definition 10.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, sk is called to be Δ−statistical convergence with respect to μN,C,ϱN,U,ζN,ℱ if, for every ε∈0,1 and r>0, there exist s such that(12)k≤n:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε,has natural density zero, i.e.,(13)limn1nk≤n:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε=0.
Therefore, it will be denoted asstΔN−limsk=s or sk⟶sSΔN, where k⟶∞. SΔN denotes set of all Δ−statistical convergence sequences.Theorem 1.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If sk is Δ−statistically convergent in this case, stΔN−limsk is unique.Lemma 1.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and sk be a Δ−statistical convergence sequences. Then, for each ε>0,r>0, the next properties are equivalent:(i)
stΔN−limsk=s(ii)
limn1/nk≤n:μN,CΔsk−s,r>1−ε,ϱN,UΔsk−s,r<ε,ζN,ℱΔsk−s,r<ε=1(iii)
limn⟶∞1/nk≤n:μN,CΔsk−s,r≤1−ε=0, limn⟶∞1/nk≤n:ϱN,UΔsk−s,r≥ε=0, and limn⟶∞1/nk≤n:ζN,ℱΔsk−s,r≥ε=0(iv)
limn⟶∞1/nk≤n:μN,CΔsk−s,r>1−ε=1, limn⟶∞1/nk≤n:ϱN,UΔsk−s,r<ε=1, and limn⟶∞1/nk≤n:ζN,ℱΔsk−s,r<ε=1(v)
st−limn⟶∞μN,CΔsk−s,r=1, st−limn⟶∞ϱN,UΔsk−s,r=0, and st−limn⟶∞ζN,ℱΔsk−s,r=0.Using Definition10, the equivalence of statements is easily demonstrated.Definition 11.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, sk is called to be Δ− Cauchy sequences if, for every ε∈0,1 and r>0, there exist a k0∈ℕ such that, for every k,p≥k0, μN,CΔsk−Δsp,r≤1−εorϱN,UΔsk−Δsp,r≥ε,ζN,ℱΔsk−Δsp,r≥ε.
Now, a new type of convergence will be defined by including the Lacunary sequence structure in the investigations for difference sequences in neutrosophic normed spaces.Definition 12.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be Lacunary sequence in this spaces. sk is named to be Δ− Lacunary statistical convergence with respect to μN,C,ϱN,U,ζN,ℱ if, for every ε∈0,1 and r>0, there exist s such that(14)k≤n:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε,has natural θ−density zero, i.e.,(15)δΔθk≤n:μN,CΔsk−,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε=0,or(16)limr⟶∞1lrk∈Ir:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε=0.
Therefore, it will denotestΔNθ−limsk=s or sk⟶sSΔN as k⟶∞. SΔNθ denotes set of all Δ−Lacunary statistical convergence sequences.Lemma 2.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, θ be a Lacunary sequence, and sk be a Δ−Lacunary statistical convergence sequences. Then, for every ε∈0,1 and r>0, the following properties are equivalent:(i)
stΔNθ−limsk=s(ii)
limn1/lrk∈Ir:μN,CΔsk−s,r>1−ε,ϱN,UΔsk−s,r<ε,ζN,ℱΔsk−s,r<ε=1(iii)
limn⟶∞1/lrk∈Ir:μN,CΔsk−s,r≤1−ε=0, limn⟶∞1/lrk∈Ir:ϱN,UΔsk−s,r≥ε=0, and limn⟶∞1/lrk∈Ir:ζN,ℱΔsk−s,r≥ε=0(iv)
limn⟶∞1/lrk∈Ir:μN,CΔsk−,r>1−ε=1, limn⟶∞1/lrk∈Ir:ϱN,UΔsk−s,r<ε=1, and limn⟶∞1/lrk∈Ir:ζN,ℱΔsk−s,r<ε=1(v)
stΔNθ−limn⟶∞μN,CΔsk−s,r=1, stΔNθ−limn⟶∞ϱN,UΔsk−s,r=0, and stΔNθ−limn⟶∞ζN,ℱΔsk−s,r=0.They are easily demonstrated using Definition12.Theorem 2.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, ε∈0,1 and r>0. If sk is Δ−Lacunary statistically convergent in this case, stΔNθ−limsk is unique.Theorem 3.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If sk is Δ− statistically convergent, then this sequences is Δ−Lacunary statistically convergent.Proof.
Letsk be Δ−statistically convergent to s and δΔθA be θ−density of A obtained with the help of the difference sequence. Then, for every ε∈0,1 and r>0, there exists a k˜∈ℕ such that, for every k≥k˜, μN,CΔsk−s,r≤1−ε or ϱN,UΔsk−s,r≥ε, and ζN,ℱΔsk−s,r<ε. So,(17)k∈ℕ:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε,has finite number of terms. Hence, θ−density of this set is zero, i.e.,(18)δΔθk∈Ir:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r=0.
Thus,sk is Δ−Lacunary statistically convergent to s.Definition 13.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be Lacunary sequence. sk is called to be Δ−Lacunary statistical Cauchy sequences when, for all ε∈0,1 and r>0, there exist a k0∈ℕ, for every k,p≥k0:(19)δΔθk∈ℕ:μN,CΔsk−Δsp,r≤1−εorϱN,UΔsk−Δsp,r≥ε,ζN,ℱΔsk−Δsp,r=0.Theorem 4.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be a Lacunary sequence. sk is Δ−Lacunary statistical convergent ⇔sk is Δ−Lacunary statistical Cauchy in neutrosophic normed spaces.Proof.
Letsk be a Δ−Lacunary statistical convergent sequence and stΔNθ−limsk=s. For a given ε∈0,1, choose ϑ>0 such that 1−ε⊛1−ε>1−ϑ and ε⊡ε<ϑ. For any r>0,(20)δΔθG≔δΔθk∈ℕ:μN,CΔsk−s,r2≤1−ε,ϱN,UΔsk−s,r2≥ε,ζN,ℱΔsk−s,r2≥ε=0,can be written so δΔθGc=1. For m∈Gc, μN,CΔsm−s,r/2>1−ε and ϱN,ℐΔsk−s,r/2<ε,ζN,ℱΔsk−s,r/2<ε Let(21)H=k∈ℕ:μN,CΔsk−Δsm,r≤1−εorϱN,UΔsk−Δsm,r≥ε,ζN,ℱΔsk−Δsm,r≥ε.
It is necessary to show thatH⊂G. So, to show this, let k∈H∩Gc. In this case, μN,CΔsk−Δsm,r≤1−ϑ and μN,CΔsk−s,r/2>1−ϑ, especially μN,CΔsm−s,r/2>1−ϑ. So,(22)1−ϑ≥μN,CΔsk−Δsm,r≥μN,CΔsk−s,r2⊛μN,CΔsm−s,r2>1−ε⊛1−ε>1−ϑ.
However, this is not possible. Moreover,ϱN,UΔsk−Δsm,r≥ϑ and ϱN,UΔsk−s,r/2<ε. Especially, ϱN,UΔsm−s,r/2<ε. Hence,(23)ϑ≤ϱN,UΔsk−Δsm,r≤ϱN,UΔsk−s,r2⊡ϱN,UΔsm−s,r2<ε⊡ε<ϑ,which is impossible. With a similar technique can be applied for ζN,ℱΔsk−Δsm,r. So, H⊂G and δΔθG=0. Then, sk is Δ−Lacunary statistical Cauchy sequences in neutrosophic normed spaces.
Otherwise, letsk be Δ− Lacunary statistical Cauchy but not Δ−Lacunary statistical convergent on U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡. For a given ε∈0,1, choose ϑ>0 such that 1−ε⊛1−ε>1−ϑ and ε⊡ε<ϑ. Because sk is not Δ−Lacunary statistical convergent,(24)μN,CΔsk−Δsm,r≥μN,CΔsk−s,r2⊛μN,CΔsm−s,r2>1−ε⊛1−ε>1−ϑ,ϱN,UΔsk−Δsm,r≤ϱN,UΔsk−s,r2⊡ϱN,UΔsm−s,r2<ε⊡ε<ϑ,ζN,ℱΔsk−Δsm,r≤ζN,ℱΔsk−s,r2⊡ζN,ℱΔsm−s,r2<ε⊡ε<ϑ.
So, for(25)I=k∈ℕ:μN,CΔsk−Δsm,r≤1−ϑ,ϱN,UΔsk−Δsm,r≥ϑ,ζN,ℱΔsk−Δsm,r≥ϑ,δΔθIc=0 and also δΔθI=1. Since sk is Δ−Lacunary statistical Cauchy, this is impossible. Thus, sk is Δ−Lacunary statistical convergent in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.


### 3.2.Δ−Lacunary Statistical Convergence with Double Sequences

Now,Δ−lacunary statistical convergence will be applied to neutrosophic normed spaces using double sequences. First of all, let us remind a few definitions necessary for the section.Letδn,n˜ be numbers of k,l∈D, where k≤nandl≤n˜. D⊂ℕ×ℕ with double density is defined [17] as(26)limn,n˜⟶∞δn,n˜nn˜=δn,n˜D.In [17], statistical convergence is defined using double sequences: skl is called statistical convergence if, for all ε>0, δn,n˜k,l,k≤nandl≤n˜:skl−s≥ε=0.Then, in [19], double Lacunary sequences are defined as follows. Let there exist two increasing integers sequences, where k0,k˜0=0, limr⟶∞hr≔limr⟶∞kr−kr−1=∞, and limr˜⟶∞hr˜≔limr˜⟶∞k˜r˜−k˜r˜−1=∞; then, θ=kr,k˜r˜ is called double Lacunary sequences. Here, lr=kr/kr−1, l˜r˜=k˜r˜/k˜r˜−1, and lr,r˜=lrl˜r˜; also, Ir=kr−1,kr, Ir˜=k˜r˜−1,k˜r˜, and Ir,r˜=k,k˜:k∈Irandk˜∈Ir˜. Also, statistical convergence for double Lacunary sequences is given in [19]. Then, for neutrosophic normed spaces, statistical convergence is defined using double sequences in [23].Double difference sequences is given in [29] as Δsk,l=sk+1,l+1−sk,l+1+sk,l−sk+1,l.Definition 14.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and Δsk,l=sk+1,l+1−sk,l+1+sk,l−sk+1,l. Double sequences skl is called to be Δ− convergent to s according to neutrosophic normed if, for all ε∈0,1 and r>0, there exists k˜,l˜∈ℕ such that, for every k˜≤kandl˜≤l,(27)μN,CΔsk,l−s,r≤1−ε,ϱN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε.
This sequences is shown with−ΔNlimk,lskl=s.Definition 15.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If there exist r>0 and 0<ε<1, for each Δsk,l, where μN,CΔsk,l,r≤1−ε, ϱN,UΔsk,l,r≥ε, and ζN,ℱΔsk,l,r≥ε, then skl is called bounded sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.Lemma 3.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces.(1)
If a sequenceskl is Δ− convergent to s, then −ΔNlimk,lskl is unique(2)
Ifskl is Δ− convergent to s and tkl is Δ− convergent to t, then skl+tkl is Δ− convergent to s+t(3)
Ifskl is Δ− convergent to s and c≠0, then cskl is Δ− convergent to csDefinition 16.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces; skl is called to be Δkl− Cauchy sequences if, for every ε∈0,1 and r>0, there exist a k0∈ℕ such that, for every k,l,j,m≥k0,(28)μN,CΔsk,l−Δsj,m,r≤1−εorϱN,UΔsk,l−Δsj,m,r≥ε,ζN,ℱΔsk,l−Δsj,m,r≥ε.Lemma 4.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and skl be a Δkl− Cauchy sequences. Then,(1)
U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ is a complete if all Δkl− Cauchy sequences have a Δ− convergent subsequence in this spaces(2)
For theΔkl− Cauchy skl and tkl sequences, skl+tkl is Δkl− Cauchy sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡(3)
Letckl be scalar and skl be a Δkl− Cauchy sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡. Then, cklskl is a Δkl− Cauchy sequences in this spaces.Definition 17.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces; ckl is called to be Δkl− statistical convergence with respect to μN,C,ϱN,U,ζN,ℱ, if, for every ε∈0,1 and r>0, there exist s such that(29)k≤nandl≤n˜:μN,CΔsk,l−s,r≤1−εorϱN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε,has natural density zero, i.e.,(30)limn,n˜1nn˜k≤nandl≤n˜:μN,CΔsk,l−s,r≤1−εorϱN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε=0.
Therefore, it will be denotestΔklN−limskl=s or skl⟶sSΔklN as k⟶∞. SΔklN denote set of each Δkl− statistical convergence sequences.Lemma 5.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, and skl is Δkl− statistical convergence; then, stΔklN−limskl=s is unique.Definition 18.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces; skl is called to be Δkl− statistical Cauchy sequences if, for every ε∈0,1 and r>0, there exist j,m∈ℕ such that(31)δn,n˜k≤nandl≤n˜≤l:μN,CΔskl−Δsjm,r≤1−εorϱN,UΔskl−Δsjm,r≥ε,ζN,ℱΔskl−Δsjm,r≥ε=0.Lemma 6.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. Every Δkl− statistical convergence sequences is Δkl− statistical Cauchy sequences.Proof.
Letskl be a Δkl− statistical convergence sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ and stΔklN−limskl=s. For a given ε∈0,1, choose ϑ>0 such that 1−ε⊛1−ε>1−ϑ and ε⊡ε<ϑ. For any r>0,(32)δn,n˜Gn,n˜≔δn,n˜k≤nandl≤n˜:μN,CΔskl−s,r2≤1−ε,ϱN,UΔskl−s,r2≥ε,ζN,ℱΔskl−s,r2≥ε=0,can be written, so δn,n˜Gn,n˜c=1. For j,m∈Gn,n˜c, μN,CΔsjm−s,r>1−ε and ϱN,UΔsjm−s,r<ε,ζN,ℱΔsjm−s,r<ε. Let(33)Hn,n˜=k≤nandl≤n˜:μN,CΔskl−Δsjm,r≤1−ϑorϱN,UΔskl−Δsjm,r≥ϑ,ζN,ℱΔskl−Δsjm,r≥ϑ.
It is necessary to show thatHn,n˜⊂Gn,n˜. So, to show this, let u,w∈H∩Gc. In this case, μN,CΔsuw−Δsjm,r≤1−ϑ and μN,CΔsuw−s,r/2>1−ϑ, especially μN,CΔsjm−s,r>1−ϑ. So,(34)1−ϑ≥μN,CΔsuw−Δsjm,r≥μN,CΔsuw−s,r2⊛μN,CΔsjm−s,r2>1−ε⊛1−ε>1−ϑ.
However, this is not possible. Moreover,ϱN,UΔsuw−Δsjm,r≥ϑ and ϱN,UΔsuw−s,r/2<ϑ. Especially, ϱN,UΔsjm−s,r/2<ε. Hence,(35)ϑ≤ϱN,UΔsuw−Δsjm,r≤ϱN,UΔsuw−s,r2⊡ϱN,UΔsjm−s,r2<ε⊡ε<ϑ,which is impossible. With a similar technique, we can apply for ζN,ℱΔsuw−Δsjm,r. So, Hn,n˜⊂Gn,n˜ and δn,n˜Gn,n˜=0. Then, sk is Δ− statistical Cauchy sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.Theorem 5.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. Then, every Δkl− statistical Cauchy sequences is Δkl− statistical convergence in this spaces.Proof.
Letskl be Δ− statistical Cauchy but not Δ− statistical convergent on U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡. For a given ε∈0,1, choose ϑ>0 such that 1−ε⊛1−ε>1−ϑ and ε⊡ε<ϑ. Then,(36)μN,CΔsk,l−Δsj,m,r≥μN,CΔsk,l−s,r2⊛μN,CΔsj,m−s,r2>1−ε⊛1−ε>1−ϑ,ϱN,UΔsk,l−Δsj,m,r≤ϱN,UΔsk,l−s,r2⊡ϱN,UΔsj,m−s,r2<ε⊡ε<ϑ,ζN,ℱΔsk,l−Δsj,m,r≤ζN,ℱΔsk,l−s,r2⊡ζN,ℱΔsj,m−s,r2<ε⊡ε<ϑ.
So, for(37)In,n˜=k≤nandl≤n˜:μN,CΔsk−Δsm,r≤1−ϑ,ϱN,UΔsk−Δsm,r≥ϑ,ζN,ℱΔsk−Δsm,r≥ϑ,and δn,n˜In,n˜c=0; also, δn,n˜In,n˜=1. Since skl is Δ− statistical Cauchy, this is impossible. Thus, sk is Δ− statistical convergent in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.Definition 19.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If every Δkl− statistical Cauchy sequences is Δkl− statistical convergence in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡, then this spaces is called complete.Result 1.
IfU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ is neutrosophic normed spaces, then this spaces is complete.Result 2.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and skl be a double sequences in this spaces. Then, skl is a Δkl− statistical convergence sequences, sskl is a Δkl− statistical Cauchy sequences, and ⇔U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ is complete neutrosophic normed spaces.Lemma 7.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and skl be a double sequences in this spaces. For all ε∈0,1 and r>0,(38)stΔklN−limskl=s,limn,n˜1nn˜k≤nandl≤n˜:μN,CΔsk,l−s,r>1−ε,ϱN,UΔsk,l−s,r<ε,ζN,ℱΔsk,l−s,r<ε=1,limn,n˜1nn˜k≤nandl≤n˜:μN,CΔsk,l−s,r≤1−ε=0,limn,n˜1nn˜k≤nandl≤n˜:ϱN,UΔsk,l−s,r≥ε=0,limn,n˜1nn˜k≤nandl≤n˜:ζN,ℱΔsk,l−s,r≥ε=0,limn,n˜1nn˜k≤nandl≤n˜:μN,CΔsk,l−s,r>1−ε=1,limn,n˜1nn˜k≤nandl≤n˜:ϱN,UΔsk,l−s,r<ε=1,limn,n˜1nn˜k≤nandl≤n˜:ζN,ℱΔsk,l−s,r<ε=1,stΔN−limn⟶∞μN,CΔsk,l−s,r=1,stΔN−limn⟶∞ϱN,UΔsk−s,r=0,stΔN−limn⟶∞ζN,ℱΔsk−s,r=0.Now, with the help of double Lacunary sequences, the definition of convergence and important properties for difference sequences will be given.Definition 20.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be a double Lacunary sequences. skl is called to be Δkl−Lacunary statistical convergence in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ if, for every ε∈0,1 and r>0, there exist a s such that(39)k≤nandl≤n˜:μN,CΔsk,l−s,r≤1−εorϱN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε,has natural θ− density zero, i.e.,(40)limr,r˜⟶∞1lr,r˜k,k˜∈Ir,r˜:μN,CΔsk,l−s,r≤1−εorN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε=0.
Therefore, it will be denotestΔklNθ−limskl=s or skl⟶sSΔklN as k⟶∞. SΔklNθ denote set of all Δkl−Lacunary statistical convergence sequences.Definition 21.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be a double Lacunary sequence. In this case, skl is a called to be Δkl− Lacunary statistical Cauchy sequences if, for all ε∈0,1 and r>0, there exist a k0∈ℕ such that, for every k,l,j,m≥k0,(41)δn,n˜θ,Δk≤nandl≤n˜≤l:μN,CΔsk,l−Δsj,m,r≤1−εorϱN,UΔsk,l−Δsj,m,r≥ε,ζN,ℱΔsk,l−Δsj,m,r≥ε=0.Theorem 6.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, θ be Lacunary sequence. skl is Δ−Lacunary statistical convergent ⇔skl is Δ− Lacunary statistical Cauchy in neutrosophic normed spaces.


## 3.1.Δ−Statistical Convergence in Neutrosophic Normed Spaces

Now,Δ−convergence, Δ−statistical convergence, and Lacunary Δ−statistical convergence will be defined in neutrosophic normed spaces. The relations between these concepts will be given.Definition 8.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,∗ be neutrosophic normed spaces and Δsk=sk−sk+1. sk is called to be Δ− convergent to s according to neutrosophic normed if, for all ε∈0,1 and r>0, there exists a k˜∈ℕ such that, for every k≥k˜,(11)μN,CΔsk−s,r≤1−ε,ϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε.
This sequences is shown with−ΔNlimsk=s.Definition 9.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If there exist r>0 and 0<ε<1, for all Δsk where μN,CΔsk,r≤1−ε, ϱN,UΔsk,r≥ε, and ζN,ℱΔsk,r≥ε, then sk is called Δ−bounded sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.Definition 10.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, sk is called to be Δ−statistical convergence with respect to μN,C,ϱN,U,ζN,ℱ if, for every ε∈0,1 and r>0, there exist s such that(12)k≤n:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε,has natural density zero, i.e.,(13)limn1nk≤n:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε=0.
Therefore, it will be denoted asstΔN−limsk=s or sk⟶sSΔN, where k⟶∞. SΔN denotes set of all Δ−statistical convergence sequences.Theorem 1.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If sk is Δ−statistically convergent in this case, stΔN−limsk is unique.Lemma 1.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and sk be a Δ−statistical convergence sequences. Then, for each ε>0,r>0, the next properties are equivalent:(i)
stΔN−limsk=s(ii)
limn1/nk≤n:μN,CΔsk−s,r>1−ε,ϱN,UΔsk−s,r<ε,ζN,ℱΔsk−s,r<ε=1(iii)
limn⟶∞1/nk≤n:μN,CΔsk−s,r≤1−ε=0, limn⟶∞1/nk≤n:ϱN,UΔsk−s,r≥ε=0, and limn⟶∞1/nk≤n:ζN,ℱΔsk−s,r≥ε=0(iv)
limn⟶∞1/nk≤n:μN,CΔsk−s,r>1−ε=1, limn⟶∞1/nk≤n:ϱN,UΔsk−s,r<ε=1, and limn⟶∞1/nk≤n:ζN,ℱΔsk−s,r<ε=1(v)
st−limn⟶∞μN,CΔsk−s,r=1, st−limn⟶∞ϱN,UΔsk−s,r=0, and st−limn⟶∞ζN,ℱΔsk−s,r=0.Using Definition10, the equivalence of statements is easily demonstrated.Definition 11.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, sk is called to be Δ− Cauchy sequences if, for every ε∈0,1 and r>0, there exist a k0∈ℕ such that, for every k,p≥k0, μN,CΔsk−Δsp,r≤1−εorϱN,UΔsk−Δsp,r≥ε,ζN,ℱΔsk−Δsp,r≥ε.
Now, a new type of convergence will be defined by including the Lacunary sequence structure in the investigations for difference sequences in neutrosophic normed spaces.Definition 12.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be Lacunary sequence in this spaces. sk is named to be Δ− Lacunary statistical convergence with respect to μN,C,ϱN,U,ζN,ℱ if, for every ε∈0,1 and r>0, there exist s such that(14)k≤n:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε,has natural θ−density zero, i.e.,(15)δΔθk≤n:μN,CΔsk−,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε=0,or(16)limr⟶∞1lrk∈Ir:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε=0.
Therefore, it will denotestΔNθ−limsk=s or sk⟶sSΔN as k⟶∞. SΔNθ denotes set of all Δ−Lacunary statistical convergence sequences.Lemma 2.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, θ be a Lacunary sequence, and sk be a Δ−Lacunary statistical convergence sequences. Then, for every ε∈0,1 and r>0, the following properties are equivalent:(i)
stΔNθ−limsk=s(ii)
limn1/lrk∈Ir:μN,CΔsk−s,r>1−ε,ϱN,UΔsk−s,r<ε,ζN,ℱΔsk−s,r<ε=1(iii)
limn⟶∞1/lrk∈Ir:μN,CΔsk−s,r≤1−ε=0, limn⟶∞1/lrk∈Ir:ϱN,UΔsk−s,r≥ε=0, and limn⟶∞1/lrk∈Ir:ζN,ℱΔsk−s,r≥ε=0(iv)
limn⟶∞1/lrk∈Ir:μN,CΔsk−,r>1−ε=1, limn⟶∞1/lrk∈Ir:ϱN,UΔsk−s,r<ε=1, and limn⟶∞1/lrk∈Ir:ζN,ℱΔsk−s,r<ε=1(v)
stΔNθ−limn⟶∞μN,CΔsk−s,r=1, stΔNθ−limn⟶∞ϱN,UΔsk−s,r=0, and stΔNθ−limn⟶∞ζN,ℱΔsk−s,r=0.They are easily demonstrated using Definition12.Theorem 2.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, ε∈0,1 and r>0. If sk is Δ−Lacunary statistically convergent in this case, stΔNθ−limsk is unique.Theorem 3.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If sk is Δ− statistically convergent, then this sequences is Δ−Lacunary statistically convergent.Proof.
Letsk be Δ−statistically convergent to s and δΔθA be θ−density of A obtained with the help of the difference sequence. Then, for every ε∈0,1 and r>0, there exists a k˜∈ℕ such that, for every k≥k˜, μN,CΔsk−s,r≤1−ε or ϱN,UΔsk−s,r≥ε, and ζN,ℱΔsk−s,r<ε. So,(17)k∈ℕ:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r≥ε,has finite number of terms. Hence, θ−density of this set is zero, i.e.,(18)δΔθk∈Ir:μN,CΔsk−s,r≤1−εorϱN,UΔsk−s,r≥ε,ζN,ℱΔsk−s,r=0.
Thus,sk is Δ−Lacunary statistically convergent to s.Definition 13.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be Lacunary sequence. sk is called to be Δ−Lacunary statistical Cauchy sequences when, for all ε∈0,1 and r>0, there exist a k0∈ℕ, for every k,p≥k0:(19)δΔθk∈ℕ:μN,CΔsk−Δsp,r≤1−εorϱN,UΔsk−Δsp,r≥ε,ζN,ℱΔsk−Δsp,r=0.Theorem 4.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be a Lacunary sequence. sk is Δ−Lacunary statistical convergent ⇔sk is Δ−Lacunary statistical Cauchy in neutrosophic normed spaces.Proof.
Letsk be a Δ−Lacunary statistical convergent sequence and stΔNθ−limsk=s. For a given ε∈0,1, choose ϑ>0 such that 1−ε⊛1−ε>1−ϑ and ε⊡ε<ϑ. For any r>0,(20)δΔθG≔δΔθk∈ℕ:μN,CΔsk−s,r2≤1−ε,ϱN,UΔsk−s,r2≥ε,ζN,ℱΔsk−s,r2≥ε=0,can be written so δΔθGc=1. For m∈Gc, μN,CΔsm−s,r/2>1−ε and ϱN,ℐΔsk−s,r/2<ε,ζN,ℱΔsk−s,r/2<ε Let(21)H=k∈ℕ:μN,CΔsk−Δsm,r≤1−εorϱN,UΔsk−Δsm,r≥ε,ζN,ℱΔsk−Δsm,r≥ε.
It is necessary to show thatH⊂G. So, to show this, let k∈H∩Gc. In this case, μN,CΔsk−Δsm,r≤1−ϑ and μN,CΔsk−s,r/2>1−ϑ, especially μN,CΔsm−s,r/2>1−ϑ. So,(22)1−ϑ≥μN,CΔsk−Δsm,r≥μN,CΔsk−s,r2⊛μN,CΔsm−s,r2>1−ε⊛1−ε>1−ϑ.
However, this is not possible. Moreover,ϱN,UΔsk−Δsm,r≥ϑ and ϱN,UΔsk−s,r/2<ε. Especially, ϱN,UΔsm−s,r/2<ε. Hence,(23)ϑ≤ϱN,UΔsk−Δsm,r≤ϱN,UΔsk−s,r2⊡ϱN,UΔsm−s,r2<ε⊡ε<ϑ,which is impossible. With a similar technique can be applied for ζN,ℱΔsk−Δsm,r. So, H⊂G and δΔθG=0. Then, sk is Δ−Lacunary statistical Cauchy sequences in neutrosophic normed spaces.
Otherwise, letsk be Δ− Lacunary statistical Cauchy but not Δ−Lacunary statistical convergent on U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡. For a given ε∈0,1, choose ϑ>0 such that 1−ε⊛1−ε>1−ϑ and ε⊡ε<ϑ. Because sk is not Δ−Lacunary statistical convergent,(24)μN,CΔsk−Δsm,r≥μN,CΔsk−s,r2⊛μN,CΔsm−s,r2>1−ε⊛1−ε>1−ϑ,ϱN,UΔsk−Δsm,r≤ϱN,UΔsk−s,r2⊡ϱN,UΔsm−s,r2<ε⊡ε<ϑ,ζN,ℱΔsk−Δsm,r≤ζN,ℱΔsk−s,r2⊡ζN,ℱΔsm−s,r2<ε⊡ε<ϑ.
So, for(25)I=k∈ℕ:μN,CΔsk−Δsm,r≤1−ϑ,ϱN,UΔsk−Δsm,r≥ϑ,ζN,ℱΔsk−Δsm,r≥ϑ,δΔθIc=0 and also δΔθI=1. Since sk is Δ−Lacunary statistical Cauchy, this is impossible. Thus, sk is Δ−Lacunary statistical convergent in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.


## 3.2.Δ−Lacunary Statistical Convergence with Double Sequences

Now,Δ−lacunary statistical convergence will be applied to neutrosophic normed spaces using double sequences. First of all, let us remind a few definitions necessary for the section.Letδn,n˜ be numbers of k,l∈D, where k≤nandl≤n˜. D⊂ℕ×ℕ with double density is defined [17] as(26)limn,n˜⟶∞δn,n˜nn˜=δn,n˜D.In [17], statistical convergence is defined using double sequences: skl is called statistical convergence if, for all ε>0, δn,n˜k,l,k≤nandl≤n˜:skl−s≥ε=0.Then, in [19], double Lacunary sequences are defined as follows. Let there exist two increasing integers sequences, where k0,k˜0=0, limr⟶∞hr≔limr⟶∞kr−kr−1=∞, and limr˜⟶∞hr˜≔limr˜⟶∞k˜r˜−k˜r˜−1=∞; then, θ=kr,k˜r˜ is called double Lacunary sequences. Here, lr=kr/kr−1, l˜r˜=k˜r˜/k˜r˜−1, and lr,r˜=lrl˜r˜; also, Ir=kr−1,kr, Ir˜=k˜r˜−1,k˜r˜, and Ir,r˜=k,k˜:k∈Irandk˜∈Ir˜. Also, statistical convergence for double Lacunary sequences is given in [19]. Then, for neutrosophic normed spaces, statistical convergence is defined using double sequences in [23].Double difference sequences is given in [29] as Δsk,l=sk+1,l+1−sk,l+1+sk,l−sk+1,l.Definition 14.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and Δsk,l=sk+1,l+1−sk,l+1+sk,l−sk+1,l. Double sequences skl is called to be Δ− convergent to s according to neutrosophic normed if, for all ε∈0,1 and r>0, there exists k˜,l˜∈ℕ such that, for every k˜≤kandl˜≤l,(27)μN,CΔsk,l−s,r≤1−ε,ϱN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε.
This sequences is shown with−ΔNlimk,lskl=s.Definition 15.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If there exist r>0 and 0<ε<1, for each Δsk,l, where μN,CΔsk,l,r≤1−ε, ϱN,UΔsk,l,r≥ε, and ζN,ℱΔsk,l,r≥ε, then skl is called bounded sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.Lemma 3.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces.(1)
If a sequenceskl is Δ− convergent to s, then −ΔNlimk,lskl is unique(2)
Ifskl is Δ− convergent to s and tkl is Δ− convergent to t, then skl+tkl is Δ− convergent to s+t(3)
Ifskl is Δ− convergent to s and c≠0, then cskl is Δ− convergent to csDefinition 16.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces; skl is called to be Δkl− Cauchy sequences if, for every ε∈0,1 and r>0, there exist a k0∈ℕ such that, for every k,l,j,m≥k0,(28)μN,CΔsk,l−Δsj,m,r≤1−εorϱN,UΔsk,l−Δsj,m,r≥ε,ζN,ℱΔsk,l−Δsj,m,r≥ε.Lemma 4.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and skl be a Δkl− Cauchy sequences. Then,(1)
U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ is a complete if all Δkl− Cauchy sequences have a Δ− convergent subsequence in this spaces(2)
For theΔkl− Cauchy skl and tkl sequences, skl+tkl is Δkl− Cauchy sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡(3)
Letckl be scalar and skl be a Δkl− Cauchy sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡. Then, cklskl is a Δkl− Cauchy sequences in this spaces.Definition 17.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces; ckl is called to be Δkl− statistical convergence with respect to μN,C,ϱN,U,ζN,ℱ, if, for every ε∈0,1 and r>0, there exist s such that(29)k≤nandl≤n˜:μN,CΔsk,l−s,r≤1−εorϱN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε,has natural density zero, i.e.,(30)limn,n˜1nn˜k≤nandl≤n˜:μN,CΔsk,l−s,r≤1−εorϱN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε=0.
Therefore, it will be denotestΔklN−limskl=s or skl⟶sSΔklN as k⟶∞. SΔklN denote set of each Δkl− statistical convergence sequences.Lemma 5.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, and skl is Δkl− statistical convergence; then, stΔklN−limskl=s is unique.Definition 18.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces; skl is called to be Δkl− statistical Cauchy sequences if, for every ε∈0,1 and r>0, there exist j,m∈ℕ such that(31)δn,n˜k≤nandl≤n˜≤l:μN,CΔskl−Δsjm,r≤1−εorϱN,UΔskl−Δsjm,r≥ε,ζN,ℱΔskl−Δsjm,r≥ε=0.Lemma 6.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. Every Δkl− statistical convergence sequences is Δkl− statistical Cauchy sequences.Proof.
Letskl be a Δkl− statistical convergence sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ and stΔklN−limskl=s. For a given ε∈0,1, choose ϑ>0 such that 1−ε⊛1−ε>1−ϑ and ε⊡ε<ϑ. For any r>0,(32)δn,n˜Gn,n˜≔δn,n˜k≤nandl≤n˜:μN,CΔskl−s,r2≤1−ε,ϱN,UΔskl−s,r2≥ε,ζN,ℱΔskl−s,r2≥ε=0,can be written, so δn,n˜Gn,n˜c=1. For j,m∈Gn,n˜c, μN,CΔsjm−s,r>1−ε and ϱN,UΔsjm−s,r<ε,ζN,ℱΔsjm−s,r<ε. Let(33)Hn,n˜=k≤nandl≤n˜:μN,CΔskl−Δsjm,r≤1−ϑorϱN,UΔskl−Δsjm,r≥ϑ,ζN,ℱΔskl−Δsjm,r≥ϑ.
It is necessary to show thatHn,n˜⊂Gn,n˜. So, to show this, let u,w∈H∩Gc. In this case, μN,CΔsuw−Δsjm,r≤1−ϑ and μN,CΔsuw−s,r/2>1−ϑ, especially μN,CΔsjm−s,r>1−ϑ. So,(34)1−ϑ≥μN,CΔsuw−Δsjm,r≥μN,CΔsuw−s,r2⊛μN,CΔsjm−s,r2>1−ε⊛1−ε>1−ϑ.
However, this is not possible. Moreover,ϱN,UΔsuw−Δsjm,r≥ϑ and ϱN,UΔsuw−s,r/2<ϑ. Especially, ϱN,UΔsjm−s,r/2<ε. Hence,(35)ϑ≤ϱN,UΔsuw−Δsjm,r≤ϱN,UΔsuw−s,r2⊡ϱN,UΔsjm−s,r2<ε⊡ε<ϑ,which is impossible. With a similar technique, we can apply for ζN,ℱΔsuw−Δsjm,r. So, Hn,n˜⊂Gn,n˜ and δn,n˜Gn,n˜=0. Then, sk is Δ− statistical Cauchy sequences in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.Theorem 5.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. Then, every Δkl− statistical Cauchy sequences is Δkl− statistical convergence in this spaces.Proof.
Letskl be Δ− statistical Cauchy but not Δ− statistical convergent on U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡. For a given ε∈0,1, choose ϑ>0 such that 1−ε⊛1−ε>1−ϑ and ε⊡ε<ϑ. Then,(36)μN,CΔsk,l−Δsj,m,r≥μN,CΔsk,l−s,r2⊛μN,CΔsj,m−s,r2>1−ε⊛1−ε>1−ϑ,ϱN,UΔsk,l−Δsj,m,r≤ϱN,UΔsk,l−s,r2⊡ϱN,UΔsj,m−s,r2<ε⊡ε<ϑ,ζN,ℱΔsk,l−Δsj,m,r≤ζN,ℱΔsk,l−s,r2⊡ζN,ℱΔsj,m−s,r2<ε⊡ε<ϑ.
So, for(37)In,n˜=k≤nandl≤n˜:μN,CΔsk−Δsm,r≤1−ϑ,ϱN,UΔsk−Δsm,r≥ϑ,ζN,ℱΔsk−Δsm,r≥ϑ,and δn,n˜In,n˜c=0; also, δn,n˜In,n˜=1. Since skl is Δ− statistical Cauchy, this is impossible. Thus, sk is Δ− statistical convergent in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡.Definition 19.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces. If every Δkl− statistical Cauchy sequences is Δkl− statistical convergence in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡, then this spaces is called complete.Result 1.
IfU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ is neutrosophic normed spaces, then this spaces is complete.Result 2.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and skl be a double sequences in this spaces. Then, skl is a Δkl− statistical convergence sequences, sskl is a Δkl− statistical Cauchy sequences, and ⇔U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ is complete neutrosophic normed spaces.Lemma 7.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and skl be a double sequences in this spaces. For all ε∈0,1 and r>0,(38)stΔklN−limskl=s,limn,n˜1nn˜k≤nandl≤n˜:μN,CΔsk,l−s,r>1−ε,ϱN,UΔsk,l−s,r<ε,ζN,ℱΔsk,l−s,r<ε=1,limn,n˜1nn˜k≤nandl≤n˜:μN,CΔsk,l−s,r≤1−ε=0,limn,n˜1nn˜k≤nandl≤n˜:ϱN,UΔsk,l−s,r≥ε=0,limn,n˜1nn˜k≤nandl≤n˜:ζN,ℱΔsk,l−s,r≥ε=0,limn,n˜1nn˜k≤nandl≤n˜:μN,CΔsk,l−s,r>1−ε=1,limn,n˜1nn˜k≤nandl≤n˜:ϱN,UΔsk,l−s,r<ε=1,limn,n˜1nn˜k≤nandl≤n˜:ζN,ℱΔsk,l−s,r<ε=1,stΔN−limn⟶∞μN,CΔsk,l−s,r=1,stΔN−limn⟶∞ϱN,UΔsk−s,r=0,stΔN−limn⟶∞ζN,ℱΔsk−s,r=0.Now, with the help of double Lacunary sequences, the definition of convergence and important properties for difference sequences will be given.Definition 20.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be a double Lacunary sequences. skl is called to be Δkl−Lacunary statistical convergence in U,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ if, for every ε∈0,1 and r>0, there exist a s such that(39)k≤nandl≤n˜:μN,CΔsk,l−s,r≤1−εorϱN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε,has natural θ− density zero, i.e.,(40)limr,r˜⟶∞1lr,r˜k,k˜∈Ir,r˜:μN,CΔsk,l−s,r≤1−εorN,UΔsk,l−s,r≥ε,ζN,ℱΔsk,l−s,r≥ε=0.
Therefore, it will be denotestΔklNθ−limskl=s or skl⟶sSΔklN as k⟶∞. SΔklNθ denote set of all Δkl−Lacunary statistical convergence sequences.Definition 21.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces and θ be a double Lacunary sequence. In this case, skl is a called to be Δkl− Lacunary statistical Cauchy sequences if, for all ε∈0,1 and r>0, there exist a k0∈ℕ such that, for every k,l,j,m≥k0,(41)δn,n˜θ,Δk≤nandl≤n˜≤l:μN,CΔsk,l−Δsj,m,r≤1−εorϱN,UΔsk,l−Δsj,m,r≥ε,ζN,ℱΔsk,l−Δsj,m,r≥ε=0.Theorem 6.
LetU,μN,C,ϱN,U,ζN,ℱ,⊛,⊡ be neutrosophic normed spaces, θ be Lacunary sequence. skl is Δ−Lacunary statistical convergent ⇔skl is Δ− Lacunary statistical Cauchy in neutrosophic normed spaces.


## 4. Results and Discussion

Since the concept of set with membership degrees can explain the situations in daily life more realistically, spaces with neutrosophic norms have been chosen as the space to be studied as it also includes uncertainty situations. By using difference sequences in neutrosophic normed spaces, the characteristic structure of some convergence types for single-double sequences is established.It is an important study for researchers working in engineering, where convergence situations are important and for approximation theory researchers who want to approach with the help of useful sequences. In this study, the work of Granados and Dhital was transferred to difference sequences, and a more comprehensive study was created since it was studied for single-double sequences. When this study is evaluated together with the study examining the statistical convergence properties for neutrosophic normed spaces using triple difference sequences, in [24], this study can be considered as a study containing more recent information, since it also includes statistical convergence and Lacunary convergence studies using single and double difference sequences.The concept of Turiyam feature, first defined by Singh, was examined in [30] with the structure of Turiyam R-Modules and Turiyam modulo integers. Since it is considered that the 4th component, which is described as the Turiyam feature, is included in the 3rd component where the uncertainty situations are given, the study was built with 3 components.


## 5. Conclusions

In this study, which is based on ∆-statistical convergence in neutrosophic normed spaces, the concepts such as ∆-statistical bounded, Lacunary ∆-statistical convergence, related to this concept and important theorems including their transitions to each other are given. Researching these concepts in double sequences is one of the advantages of this study. By making use of the relationship between the concepts ofΔkl−Lacunary statistical Cauchy sequences with Δkl−lacunary statistical convergence, results related to the completeness of the space were obtained. This study was created due to the need in the literature. It is a guiding study for researchers who will study convergence types in different senses.



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