Numerous research studies have been conducted and various classical theories have been developed to address real-life problems in science and technology. However, classical sets are inadequate in explaining uncertainties that occur in daily life situations. In modern logic, three-way decision situations, such as accepting/rejecting/pending cases, yes/no/not-applicable situations, win/lose/tie in sports, etc., cannot be adequately explained by the theory of standard analysis. To solve this problem, non-standard analysis is employed. Neutrosophic sets are a valuable advancement of classical sets, fuzzy sets and intuitionistic fuzzy sets for non-standard analysis, introduced by Smarandache [25]. Neutrosophic sets can manage inconsistent, indeterminate, and imprecise data for problems where the fundamental rules of fuzzy set theory and intuitionistic fuzzy set theory are not sufficient. They are used to investigate the degrees of correctness, wrongness, and uncertainty of the elements in the set. Each element of a neutrosophic set has a truth value, a false value and an indeterminacy value, which falls within the non-standard unit interval. Because of this nature, neutrosophic sets are more adaptable, reasonable, and efficient tools for handling not only the free components of information but also partially independent and dependent components. Neutrosophic sets are suitable for real-life situations such as databases, image processing problems, control theory, medical diagnosis problems and decision-making problems. In a neutrosophic set, elements may have inconsistent information (i.e. the sum of the components >1) or incomplete information (i.e. the sum of the components <1) or consistent information (i.e. the sum of the components =1), and other interval-valued components (i.e. without any restriction on the sum of superior or inferior components).

Definition 1.1. [25] Let U be a subset of X (which is a space of points) with a∈X. Then set U is called neutrosophic set(NS) with τ(a), υ(a) and η(a) in X and expressed as

U={<a,τ(a),υ(a),η(a)>:a∈Xandτ(a),υ(a),η(a)∈I}

where τ(a), υ(a) and η(a) denotes truth membership function, indeterminacy membership function and falsity membership function respectively, such that 0-≤τ(a)+υ(a)+η(a)≤3+. Also I=]0-,1+[ represents a non-standard unit interval.

Wang etal. [27] and Ye [28] revised the existing definitions of the neutrosophic set using the interval [0,1] by introducing the single-valued neutrosophic set and simplified neutrosophic set respectively, that can be utilized in the applications of engineering and scientific areas.

Mahapatra and Bera [5] looked at the concept of neutrosophic soft linear spaces. Kirişci and Şimşek [15] introduced the idea of neutrosophic metric spaces and established their fundamental topological and geometric properties. In [16] proposed the following notion of neutrosophic normed spaces, which is an important consideration of neutrosophic metric spaces.

Definition 1.2. [16] A neutrosophic normed space(NNS) is a 4-tuple (X,ℵ,⊛,⊙) consisting of a vector space X, a normed space ℵ={<τ(a),υ(a),η(a)>:a∈X} such that ℵ:X×R+→[0,1], a continuous t-norm ⊛ and a continuous t-conorm ⊙. For every x,y∈X and s,t>0, we have:

• (i)

  0≤τ(x,t),υ(x,t),η(x,t)≤1,

• (ii)

  τ(x,t)+υ(x,t)+η(x,t)≤3,

• (iii)

  τ(x,t)=1, υ(x,t)=0 and η(x,t)=0 for t>0 iff x=0,

• (iv)

  τ(x,t)=0, υ(x,t)=1 and η(x,t)=1 for t≤0,

• (v)

  τ(αx,t)=τx,t|α|, υ(αx,t)=υx,t|α| and η(αx,t)=ηx,t|α| for α≠0,

• (vi)

  τ(x,∘) as continuous non-decreasing function,

• (vii)

  τ(x,s)⊛τ(x,t)≤τ(x+y,s+t),

• (viii)

  υ(x,∘) as continuous non-increasing function,

• (ix)

  υ(x,s)⊙υ(y,t)≥υ(x+y,s+t),

• (x)

  η(x,∘) as continuous non-increasing function,

• (xi)

  η(x,s)⊙η(y,t)≥η(x+y,s+t),

• (xii)

  limt→∞τ(x,t)=1, limt→∞υ(x,t)=0 and limt→∞η(x,t)=0.

The tuple (τ,υ,η) is known as neutrosophic norm.

Example 1.1. [16] Let (X,|.|) be a normed space. For all t>0 and x∈X, take

• (i)

  τ(x,t)=tt+|x|,υ(x,t)=|x|t+|x| and η(x,t)=|x|t when t>|x|,

• (ii)

  τ(x,t)=0, υ(x,t)=1 and η(x,t)=1 when t≤|x|.

Also let g⊛h=gh and g⊙h=g+h-gh for g,h∈[0,1]. The 4-tuple (X,ℵ,⊛,⊙) is a NNS which satisfies above mentioned conditions.

A generalized version of intuitionistic fuzzy norms has been considered in neutrosophic normed spaces that helped to investigate fundamental properties such as convergence and completeness in these spaces. Omran and Elrawy [23] discussed the relationship between continuous operators with bounded operators in the structure of neutrosophic normed spaces. Khan and Khan [13] also contributed to this topic by studying various topological properties and characterizations of these spaces. Further, Kirişci and Şimşek [16]established the concept of convergence for sequences on neutrosophic normed spaces.

Definition 1.3. [16] Let (X,ℵ,⊛,⊙) be a NNS with neutrosophic norm (τ,υ,η). A sequence x={xk} from X is called convergent to x0∈X with respect to neutrosophic norm (τ,υ,η) if for every ε>0 and t>0 we can find k0∈N provided τ(xk-x0,t)>1-ε, υ(xk-x0,t)<ε and η(xk-x0,t)<ε for k≥k0. It is represented symbolically by (τ,υ,η)-limk→∞xk=x0 or xk→(τ,υ,η)x0.

Remark 1.1. In the previous example of the NNS (X,ℵ,⊛,⊙), we have xk→(τ,υ,η)x0 if and only if xk→|.|x0.

Kirişci and Şimşek[16] established the statistical convergence for sequences in the neutrosophic normed spaces using natural density. Although, natural density of set A(A⊆N)has given by δ(A)=limn→∞1n∣{a≤n:a∈A}∣, provided limit exists and ∣.∣ designates the order of the enclosed set. Further, sequence x={xk} is statistically convergent to x0, if A(ε)={k∈N:|xk-x0|>ε} has zero natural density (see [11]).

Definition 1.4.[16] Let (X,ℵ,⊛,⊙) be a NNS with neutrosophic norm (τ,υ,η). A sequence x={xk} from X is called statistically convergent to x0∈X with respect to neutrosophic norm (τ,υ,η) if for every ε>0 and t>0, we have

δ({k∈N:τ(xk-x0,t)≤1-ε or υ(xk-x0,t)≥ε,η(xk-x0,t)≥ε})=0.

It is represented symbolically by St(τ,υ,η)-limk→∞xk=x0 or xk→St(τ,υ,η)x0 .

Some noteworthy results related to statistical convergence on neutrosophic normed spaces in different directions have been studied (c.f. [13, 17, 18]). In this paper, we have associated the theory of neutrosophic normed spaces with λ-statistical convergence of sequences. The λ-statistical convergence is a generalized form of sequence convergence presented by Mursaleen [22], using a non-decreasing sequence λ={λn} which tends to ∞ such that λn+1≤λn+1 and λ1=1. Also, the generalized de la Vallée-Poussin mean has been given by tn(x)=1λn∑k∈Inxk, where In=[1+n-λn,n].

Throughout the paper we use In for [1+n-λn,n].

Definition 1.5.[22] A sequence x={xk} is called λ-statistically convergent to x0 provided for every ε>0 satisfies

limn→∞1λn|{k∈In:|xk-x0|≥1-ε}|=0,

or

δλ({k∈In:|xk-x0|≥1-ε})=0.

It is represented symbolically by Sλ-limn→∞xk=x0.

Kizmaz[19] discovered the difference sequence spaces conception by considering Z(△)={x={xk}:{△xk}∈Z} with the spaces Z=l∞(spaces of all the bounded sequences), c0(spaces of all the convergent sequences) and c0(spaces of all the null sequences), where △x={△xk}={xk-xk+1}, and x={xk} is a real sequence for all k∈N. The spaces l∞(△),c(△) and c0(△) are Banach spaces, due to the norm endowed by |x|△=|x1|+supk|△xk|. Moreover, the generalized difference sequence spaces were defined by Et and Çolak[8] considering Z(△m)={x={xk}:{△mxk}∈Z}, where m be any fixed positive integer, for Z=l∞,c,c0 and △mx={△mxk}={△m-1xk-△m-1xk+1} so that △mxk=∑r=0m(-1)rmrxk+r. The △m-statistical convergence concept studied and established by Mikail and Nuray[9] with the help of statistical convergence.

Definition 1.6.[9] A sequence x={xk} is called Δm- statistically convergent to x0 provided with every ε>0, we have

δ({k≤n:|Δmxk-x0|≥ε}=0.

It is represented symbolically by St-limΔmxk=x0.

A lot of work related to convergence of difference sequences as fusion with different structures, has been done by various researchers [1, 10, 26, 4, 24, 21, 12, 2, 7, 3, 6, 14, 20] which leads us to investigate and explore λ-△m-statistical convergence with the theory of neutrosophic normed spaces.

We first mention the conception of λ-△m-statistical convergence of the sequences on neutrosophic normed spaces (NNS) that will be helpful in studying the major results of work.

Definition 2.1. Let (X,ℵ,⊛,⊙) be a NNS with neutrosophic norm (τ,υ,η). A sequence x={xk} from X is said to be λ-△m-statistically convergent to x0∈X with respect to neutrosophic norm (τ,υ,η) if for every t>0 and ε∈(0,1), satisfies

δλ({k∈In:τ(△mxk-x0 t)≤1-ε or υ(△mxk-x0,t)≥ε,η(△mxk-x0,t)≥ε})=0.

It is represented symbolically by Sλ(τ,υ,η)-lim△mx=x0 or △mxk→λ-St(τ,υ,η)x0.

Example 2.1. Let (R,|.|) be any normed space. For every x∈X, we take τ(x,t)=tt+|x|,υ(x,t)=|x|t+|x| and η(x,t)=p|x|t+p|x| when p∈R. Also, g⊛h=min{g,h} and g⊙h=max{g,h} for g,h∈[0,1]. Then, a 4-tuple (X,ℵ,⊛,⊙) is a NNS. Consider a sequence x={xk} such that

△mxk=2n-λn+1≤k≤ni.e.k∈In0otherwise

For t>0 and ε>0, we have

A(ε,t)={k∈​In:τ(△mxk-0,t)≤1-ε or υ(△mxk-0,t)≥ε,η(△mxk-0,t)≥ε} ={k∈In:tt+|△mxk|≤1-εor|△mxk|t+|△mxk|≥ε,p|△mxk|t+p|△mxk|≥ε} ={k∈In:|△mxk|>0} ={k∈In:|△mxk|=2} ={k∈In:k∈[n-λn+1≤k≤n]}.

Now,

1λn|A(ε,t)|≤λnλn→0 as n→∞.

⇒limn→∞1λn|A(ε,t|=0. Thus, Sλ(τ,υ,η)-lim△mx=0, i.e. x={xk} is λ-△m-statistical convergent on (X,ℵ,⊛,⊙).

Definition 2.2. Let (X,ℵ,⊛,⊙) be a NNS with neutrosophic norm (τ,υ,η). A sequence x={xk} from X is said to be λ-△m-statistically Cauchy with respect to neutrosophic norm (τ,υ,η) for some non-negative number r if for every t>0 and ε∈(0,1), we can find k0∈In such that

δλ({k∈In:τ(△mxk-△mxk0 t)≤1-ε or υ(△mxk-△mxk0,t)≥ε,η(△mxk-△mxk0,t)≥ε})=0.

Example 2.2. Consider a real normed space (X,|.|). For every t>0 and all x∈X, we take(i) τ(x,t)=tt+|x|,υ(x,t)=|x|t+|x| and η(x,t)=|x|t when t>|x|, (ii) τ(x,t)=0, υ(x,t)=1 and η(x,t)=1 when t≤|x|. Also, g⊛h=gh and g⊙h=g+h-gh for g,h∈[0,1]. Then, 4-tuple (X,ℵ,⊛,⊙) is a NNS. Consider a sequence x={xk} such that

△mxk=12k1+n-λn≤k≤n0otherwise

For t>0 and ε>0, choose k0 with 2-k0<ε we have

A(ε,t)={k∈In:τ(△mxk-△mxk0,t)≤1-ε  or υ(△mxk-△mxk0,t)≥ε, η(△mxk-△mxk0,t)≥ε}={k∈In:tt+|△mxk-△mxk0|≤1-ε  or |△mxk-△mxk0|t+|△mxk-△mxk0|≥ε, |△mxk-△mxk0|t≥ε}={k∈In:|△mxk-△mxk0|≥ε}={k∈In:k∈[1+n-λn≤k≤n]}

Now,

1λn|A(ε,t)|≤λnλn→0 as n→∞.

⇒limn→∞1λn|A(ε,t|=0. Thus, x={xk} is λ-△m-statistical Cauchy sequence on (X,ℵ,⊛,⊙).

The next result can be obtained using above Definition 2.1.

Lemma 2.1. Consider (X,ℵ,⊛,⊙) as a NNS with neutrosophic norm (τ,υ,η). Then following statements are equivalent for the sequence x={xk} from X for ε>0 and t>0,

• (i)

  Sλ(τ,υ,η)-lim△mx=x0,

• (ii)

  δλ({k∈In:τ(△mxk-x0 t)≤1-ε or υ(△mxk-x0,t)≥ε,η(△mxk-x0,t)≥ε})=0,

• (iii)

  δλ({k∈In:τ(△mxk-x0 t)>1-εandυ(△mxk-x0,t)<ε,η(△mxk-x0,t)<ε})=1,

• (iv)

  Sλ(τ,υ,η)-limτ(△mxk-x0,t)=1 and Sλ(τ,υ,η)-limυ(△mxk-x0,t)=0.

Using above lemma and definitions we obtain our results on λ-△m-statistical convergence on NNS:

Theorem 2.1. Let x={xk} be any sequence from a NNS (X,ℵ,⊛,⊙). If Sλ(τ,υ,η)-lim△mx=x0, then limit x0 is unique.

Proof. Assume, Sλ(τ,υ,η)-lim△mx=x0 and Sλ(τ,υ,η)-lim△mx=x1 and x0≠x1. For t>0 and ε>0, take κ>0 with (1-κ)⊛(1-κ)>1-ε and κ⊙κ<ε. Define

A1,τ(κ,t)={k∈In:τ(△mxk-x0,t/3)≤1-κ},
A2,τ(κ,t)={k∈In:τ(△mxk-x1,t/3)≤1-κ},
A3,υ(κ,t)={k∈In:υ(△mxk-x0,t/3)≥κ},
A4,υ(κ,t)={k∈In:υ(△mxk-x1,t/3)≥κ},
A5,η(κ,t)={k∈In:η△mxk-x0,t/3)≥κ},
A6,η(κ,t)={k∈In:η(△mxk-x1,t/3)≥κ}.

Since Sλ(τ,υ,η)-lim△mx=x0, then due to Definition 2.1 we get

δλ(A1,τ(κ,t))=δλ(A3,υ(κ,t))=δλ(A5,η(κ,t))=0.

Further Sλ(τ,υ,η)-lim△mx=x1, due to Definition 2.1 we get

δλ(A2,τ(κ,t))=δλ(A4,υ(κ,t))=δλ(A6,η(κ,t))=0.

Consider

Aτ,υ,η(κ,t)​=​(A1,τ(κ,t)∪A2,τ(κ,t))∩(A3,υ(κ,t)∪A4,υ(κ,t))∩(A5,η(κ,t)∪A6,η(κ,t)).

Clearly,

δλ(Aτ,υ,η(κ,t))=0⇔δλ(In-Aτ,υ,η(κ,t))=1.

If k∈In-Aτ,υ,η(κ,t) then either k∈In-(A1,τ(κ,t)∪A2,τ(κ,t)) or k∈In-(A3,υ(κ,t)∪A4,υ(κ,t)) or k∈In-(A5,η(κ,t)∪A6,η(κ,t)). If k∈In-(A1,τ(κ,t)∪A2,τ(κ,t)) , then

τ(x0-x1,t)≥τ(△mxk-x0,t/2)⊛τ(△mxk-x1,t/2)>(1-κ)⊛(1-κ)>1-ε.

As ε>0, we get τ(x0-x1,t)=1 for all t>0, then x0=x1. Also if k∈In-(A3,υ(κ,t)∪A4,υ(κ,t)), then

υ(x0-x1,t)≤υ(△mxk-x0,t/2)⊙υ(△mxk-x1,t/2)<κ⊙κ<ε.

As ε>0, we get υ(x0-x1,t)=0 for all t>0, then x0=x1. Further if k∈In-(A5,η(κ,t)∪A6,η(κ,t)), then

η(x0-x1,t)≤η(△mxk-x0,t/2)⊙η(△mxk-x1,t/2)<κ⊙κ<ε.

As ε>0, we get η(x0-x1,t)=0 for all t>0, then x0=x1. Hence, limit is unique.

Theorem 2.2. Consider (X,ℵ,⊛,⊙) as a NNS with neutrosophic norm (τ,υ,η). If (τ,υ,η)-lim△mx=x0, then Sλ(τ,υ,η)-lim△mx=x0. But counter part does not hold.

Proof. Assume (τ,υ,η)-lim△mx=x0. For given ε>0 and t>0 we get k0∈N satisfying

τ(△mxk-x0,t)>1-ε,υ(△mxk-x0,t)<ε and η(△mxk-x0,t)<ε

for all k≥k0. This provides the set

{k∈In:τ(△mxk-x0,t)≤1-ε or υ(△mxk-x0,t)≥ε,η(△mxk-x0,t)≥ε},

have finite members. As λ-density of every finite set is zero. Then,

δλ{k∈In:τ(△mxk-x0,t)≤1-ε or υ(△mxk-x0,t)≥ε,η(△mxk-x0,t)≥ε}=0.

i.e,

Sλ(τ,υ,η)-lim△mx=x0.

However, counter part of the above mentioned result fails to exist. That can be explained from the following example:

Example 2.3. Consider any real normed space (X,|.|). For every t>0 and all x∈X, we take(i) τ(x,t)=tt+|x|,υ(x,t)=|x|t+|x| and υ(x,t)=|x|t when t>|x|, (ii) τ(x,t)=0, υ(x,t)=1 and υ(x,t)=1 when t≤|x|. Also, g⊛h=gh and g⊙h=g+h-gh for g,h∈[0,1]. Then, 4-tuple (X,ℵ,⊛,⊙) is a NNS. Consider a sequence x={xk} such that

△mxk=1n-λn+1≤k≤n0otherwise

For t>0 and ε>0, we have

A(ε,t)={k∈​In:τ(△mxk-0,t)≤​1-ε or υ(△mxk-0,t)≥ε,η(△mxk-0,t)≥ε} ={k∈In:tt+|△mxk|≤1-εor|△mxk|t+|△mxk|≥ε,|△mxk|t≥ε} ={k∈In:|△mxk|>0} ={k∈In:|△mxk|=1} ={k∈In:k∈[n-λn+1≤k≤n]}

Now,

1λn|A(ε,t)|≤λnλn→0 as n→∞.

⇒limn→∞1λn|A(ε,t|=0. Thus, Sλ(τ,υ,η)-lim△mx=0, i.e. x={xk} is λ-△m-statistical convergent on (X,ℵ,⊛,⊙). Using above defined sequence, we get

τ(△mxk,t)=tt+1n-λn+1≤k≤n1otherwise
i.eτ(△mxk,t)≤1,∀k,

and

υ(△mxk,t)=1t+1n-λn+1≤k≤n0otherwise
i.eυ(△mxk,t)≥0,∀k,

and

η(△mxk,t)=1tn-λn+1≤k≤n0otherwise
i.eη(△mxk,t)≥0,∀k.

This implies (τ,υ,η)-lim△mx≠0.

Next, we will discuss some algebraic properties of λ-△m-statistical sequences in NNS as follows:

Theorem 2.3. Let (X,ℵ,⊛,⊙) be a NNS. Let x={xk} and y={yk} be any sequences from X. Then (i) If Sλ(τ,υ,η)-lim△mx=x0 then Sλ(τ,υ,η)-lim△max=ax0 a∈R, (ii) If Sλ(τ,υ,η)-lim△mx=x0 and Sλ(τ,υ,η)-lim△my=y0 then Sλ(τ,υ,η)-lim△m(x+y)=x0+y0.

Proof. (i) Assume Sλ(τ,υ,η)-lim△mx=x0 .Then, for the fixed ε>0 and any t>0, we can take

A(ε,t)={k∈In:τ(△mxk-x0,t)≤1-ε or υ(△mxk-x0,t)≥ε,η(△mxk-x0,t)≥ε}.

Which provides

δλ(A(ε,t))=0 so that δλ([A(ε,t)]c)=1.

Let k∈[A(ε,t)]c and a≠0, then

τ(△m(axk)-ax0,t)=τ(a(△mxk-x0),t)=τ△mxk-x0,t|a| ≥τ(△mxk-x0,t)⊛τ0,t|a|-t =τ(△mxk-x0,t)⊛1 >1-ε,

and

υ(△m(axk)-ax0,t)=υ(a(△mxk-x0),t)=υ△mxk-x0,t|a| ≤υ(△mxk-x0,t)⊙υ0,t|a|-t ≤υ(△mxk-x0,t)⊙0 <ε,

and

η(△m(axk)-ax0,t)=η(a(△mxk-x0),t)=η△mxk-x0,t|a| ≤η(△mxk-x0,t)⊙η0,t|a|-t ≤η(△mxk-x0,t)⊙0 <ε.

Therefore, δλ([A(ε,t)]c)=1. Hence, Sλ(τ,υ,η)-limax=ax0,a≠0. When a=0, we get

τ(0△mxk,t)>1-ε,υ(0△mxk,t)<εandη(0△mxk,t)<ε.

Hence, Sλ(τ,υ,η)-lim△max=ax0,a∈R. (ii) As Sλ(τ,υ,η)-lim△mx=x0 and Sλ(τ,υ,η)-lim△my=y0. Then, for t>0 and ε>0, take κ>0 with (1-κ)⊛(1-κ)>1-ε and κ⊙κ<ε. Define sets for the given sequences x={xk} and y={yk} sets

Ax(κ,t)={k∈​In:​τ(△mxk-x0,t2)​≤​1-κorυ(△mxk-x0,t2)​≥​κ,η(△mxk-x0,t2)​≥​κ},

and

Ay(κ,t)={k∈​In:τ(△myk-y0,t2)≤​1-κorυ(△myk-y0,t2)≥​κ,η(△myk-y0,t2)≥​κ}.

We have, δλ(Ax(κ,t))=δλ(Ay(κ,t))=0. Consider A(κ,t)=Ax(κ,t)∩Ay(κ,t), then δλ(A(κ,t))=0i.eδ([A(κ,t)]c)=1. For all k∈[A(κ,t)]c,

τ(△m(xk+yk)-(x0+y0),t)=τ(△mxk-x0+△mykr-y0,t) ≥τ(△mxk-x0,t/2)⊛τ(△myk-y0,t/2) ≥(1-κ)⊛(1-κ) >1-ε,

and

υ(△m(xk+yk)-(x0+y0),t)=υ(△mxk-x0+Λyk-y0,t) ≤υ(△mxk-x0,t/2)⊙υ(△myk-y0,t/2) ≤κ⊙κ <ε,

and

η(△m(xk+yk)-(x0+y0),t)=η(△mxk-x0+Λyk-y0,t) ≤η(△mxk-x0,t/2)⊙η(△myk-y0,t/2) ≤κ⊙κ <ε.
⇒Sλ(τ,υ,η)-lim△m(x+y)=x0+y0.

Theorem 2.4. Consider (X,ℵ,⊛,⊙) as a NNS. A sequence x={xk} from X is Sλ(τ,υ,η)-lim△mx=x0 if and only if set J={j1<j2<j3.....}⊆In exists with δλ(J)=1 and (τ,υ,η)λ-lim△mxjn=x0.

Proof. Necessary part:

Consider Sλ(τ,υ,η)-lim△mx=x0 . For t>0 and κ∈N, we consider

A(κ,t)​=​{k∈​In​:τ(△mxk-x0,t)​>​1-1κ and υ(△mxk-x0,t)​<​1κ,η(△mxk-x0,t)​<​1κ},

and

K(κ,t)​=​{k∈​In​:τ(△mxk-x0,t)​≤​1-1κ or υ(△mxk-x0,t)​≥​1κ,η(△mxk-x0,t)​≥​1κ}.

Since Sλ(τ,υ,η)-lim△mx=x0, then δλ(K(κ,t))=0.

Moreover, A(κ,t)⊃A(κ+1,t), and

(2.1)δλ(A(κ,t))=1.

Next, for any k∈A(κ,t), we have (τ,υ,η)λ-lim△mx=x0. We prove this part by contradiction. If for any k∈A(κ,t) we have μ>0 and k0∈N satisfying

τ(△mxk-x0,t)≤1-μ or υ(△mxk-x0,t)≥μ,η(△mxk-x0,t)≥μ, for allk≥k0,

This implies that

τ(△mxk-x0,t)>1-μ and υ(△mxk-x0,t)<μ,η(△mxk-x0,t)<μ, for allk<k0.

Therefore,

δλ{k∈In:τ(△mxk-x0,t)>1-μ and υ(△mxk-x0,t)<μ,η(△mxk-x0,t)<μ}=0.

As α>1κ, we've δλ(A(κ,t))=0, which leads contradiction to (2.1). Then, we get set A(κ,t) with δλ(A(κ,t))=1. Hence x={xk} is λ-△m-statistical convergent to x0.

Sufficient Part:

Suppose there exists a subset J={j1<j2<j3<…}⊆N such that δλ(J)=1 and (τ,υ,η)λ-lim△myjn=x0. i.e.∃ N0∈N for every ε>0 and any t>0 satisfying

τ(△mxk-x0,t)>1-ε,υ(△mxk-x0,t)<ε and η(△mxk-x0,t)<ε k≥N0.

Take

K(ε,t)={k∈​In:τ(△mxk-x0,t)≤1-ε or υ(△mxk-x0,t)≥​ε,η(△mxk-x0,t)≥​ε}.

Then,

K(ε,t)⊆In-{jN0+1,jN0+2,....}.

Since δλ(J)=1 then we get δλ(K(ε,t))≤0. Therefore, Sλ(τ,υ,η)-lim△mx=x0.

Theorem 2.5. Consider (X,ℵ,⊛,⊙) as a NNS. Then Sλ(τ,υ,η)-lim△mx=x0 if and only if there exists a sequence y={yk} with (τ,υ,η)λ-lim△my=x0 and δλ({k∈In:△mx=△my})=1.

Proof. Necessary part:

Consider Sλ(τ,υ,η)-lim△mx=x0. By Theorem 2.4, we get a set J⊆In with δλ(J)=1 and (τ,υ,η)λ-lim△mxjn=x0. Consider a sequence y={yk} such that

△myk=△mxkk∈Jx0otherwise

Then y={yk} serve our purpose.

Sufficient part:

Consider x={xk} and y={yk} from X with (τ,υ,η)λ-lim△my=x0 and δλ({k∈In:△mx=△my})=1. Then for any t>0 and every ε>0, we've

{k∈In:τ(△myk-x0,t)≤1-ε or υ(△myk-x0,t)≥ε,η(△myk-x0,t)≥ε}⊆A∪B

where A={k∈In:τ(△mxk-x0,t)≤1-ε or υ(△mxk-x0,t)≥ε,η(△mxk-x0,t)≥ε},

B={k∈In:△myk≠△mxk}.

Since (τ,υ,η)λ-lim△mx=x0 then above defined set A has at most finitely many elements. Also δλ(B)=0 as δλ(Bc)=1 where Bc={k∈In:△myk=△mxk}. Therefore

δλ({k∈In:τ(△mxk-x0,t)≤1-εorυ(△mxk-x0,t)≥ε,η(△mxk-x0,t)≥ε})=0.

Hence Sλ(τ,υ,η)-lim△mx=x0.

Theorem 2.6. Let x={xk} be a sequence from a NNS (X,ℵ,⊛,⊙). Then Sλ(τ,υ,η)-lim△mx=x0 if and only if there are sequences y={yk} and z={zk} from X with △mxk=△myk+△mzk for all k∈In where (τ,υ,η)λ-lim△my=x0 and Sλ(τ,υ,η)-lim△mz=x0.

Proof. Necessary part:

Let Sλ(τ,υ,η)-lim△mx=x0. By Theorem 2.4 we get a set J={kq:q=1,2,3,....}⊆N with δλ(J)=1 and (τ,υ,η)λ-limkq→∞△mykq=x0. Consider the sequences y={yk} and z={zk}

△myk=△mzkk∈Jx0otherwise

and

△mxk=0k∈J△myjk-x0otherwise

which gives the required result.

Sufficient Part:

If two such sequences y={yk} and z={zk} exists in X with the required properties, then the result follows using Theorem 2.2 and Theorem 2.3.

Theorem 2.7. Consider (X,ℵ,⊛,⊙) as a NNS with norm (τ,υ,η). Then S(τ,υ,η)(△m)⊆Sλ(τ,υ,η)(△m) if and only if limk→∞infλnn>0.

Proof. For given ε>0 and t>0 we have

{k≤n:τ(△mxk-x0 t)≤1-ε or υ(△mxk-x0 t)≥ε,η(△mxk-x0 t)≥ε} ⊇{k∈In:τ(△mxk-x0 t)≤1-ε or υ(△mxk-x0 t)≥ε,η(△mxk-x0 t)≥ε}.

Therefore,

1n|{k≤n:τ(△mxk-x0 t)≤1-εorυ(△mxk-x0 t)≥ε,η(△mxk-x0 t)≥ε}| ≥1λn|{k∈​In:τ(△mxk-x0 t)≤1-εorυ(△mxk-x0 t)≥ε,η(△mxk-x0 t)≥ε}| ≥λnn1λn|{k∈​In:τ(△mxk-x0 t)≤​1-εorυ(△mxk-x0 t)≥​ε,η(△mxk-x0 t)≥​ε}|

Take limit as n→∞ then we get S(τ,υ,η)-lim△mx=x0 (As limk→∞infλnn>0). Hence Sλ(τ,υ,η)-lim△mx=x0. Conversely, Suppose that limk→∞infλnn=0. We can take a sub-sequence {nj} such that λnjnj<1j. Consider a sequence x={xk} such that

△myk=1k∈Inj0otherwise.

Then take t>0 and ε∈(0,1) such that 1∉B(0,ε,t). Also, to each n∈N we get nj∈N such that nj≤n for j>0.

1n|{k≤n:τ(△mxk t)≤1-ε or υ(△mxk t)≥ε,η(△mxk t)≥ε}|<1j.

Then Sλ(τ,υ,η)-lim△mx=0. For k∉Inj we get

limj→∞1λnj|{k∈​Inj:τ(△mxk t)≤1-ε or υ(△mxk t)≥ε,η(△mxk t)≥ε}|=1.
limn→∞1λn|{k∈​In:τ(△mxk-1 t)≤1-εorυ(△mxk-1 t)≥ε,η(△mxk-1 t)≥ε}|​=​1.

This implies that x∉Sλ(τ,υ,η)(△m).

Next we establish the result related to Cauchy criterion for λ-△m-statistical convergent sequences in NNS.

Theorem 2.8. A sequence x={xk} from a NNS (X,ℵ,⊛,⊙) is λ-△m-statistical convergent corresponding to (τ,υ,η) if and only if it is λ-△m-statistical Cauchy corresponding to (τ,υ,η).

Proof. Necessary part:

Consider Sλ(τ,υ,η)-lim△mx=x0. Then, for any t>0 and ε>0, take κ>0 with (1-κ)⊛(1-κ)>1-ε and κ⊙κ<ε. Consider A(κ,t)={k∈In:τ(△mxk-x0,t/2)≤1-κ or υ(△mxk-x0,t/2)≥κ,η(△mxk-x0,t/2)≥κ}. ∴δλ(A(κ,t))=0 and δλ([A(κ,t)]c)=1. Let B(ε,t)={k∈​In:τ(△mxk-Λxs,t)≤1-εorυ(△mxk-△mxs,t)≥ε,η(△mxk-△mxs,t)≥ε}. Here, for the result we show that B(ε,t)⊂A(κ,t). As k∈B(ε,t)-A(κ,t)⇒τ(△mxk-x0,t/2)≤1-κ or υ(△mxk-x0,t/2)≥κ,η(△mxk-x0,t/2)≥κ.

1-ε≥τ(△mxk-△mxs,t)≥τ(△mxk-x0,t/2)⊛τ(△mxs-x0,t/2) >(1-κ)⊛(1-κ) >1-ε,
ε≤υ(△mxk-△mxs,t)≤υ(△mxk-x0,t/2)⊙υ(△mxs-x0,t/2) <κ⊙κ <ε,

and

ε≤η(△mxk-△mxs,t)≤η(△mxk-x0,t/2)⊙η(△mxs-x0,t/2) <κ⊙κ <ε,

which is not possible. This implies that B(ε,t)⊂A(κ,t) and δλ(B(ε,t))=0 i.e. λ-△m-statistical Cauchy corresponding to (τ,υ,η).

Sufficient part:

Let x={xk} be λ-△m-statistical Cauchy corresponding to (τ,υ,η) but not λ-△m-statistical convergent corresponding to (τ,υ,η). Then, for any t>0 and ε>0, we have δλ(C(ε,t))=0 where

C(ε,t)={k∈In:τ(△mxk-△mxk0,t)≤1-εorυ(△mxk-△mxk0,t)≥ε,η(△mxk-△mxk0,t)≥ε}.

Take κ>0 with (1-κ)⊛(1-κ)>1-ε and κ⊙κ<ε.

Let D(κ,t)={k∈In:τ(△mxk-x0,t/2)>1-κ or υ(△mxk-x0,t/2)<κ}. Now for k∈D(ε,t) we get

τ(△mxk-△mxk0,t)≥τ(△mxk-x0,t/2)⊛τ(△mxk0-ξ,t/2) >(1-κ)⊛(1-κ) >1-ε,
υ(△mxk-△mxk0,t)≤υ(△mxk-ξ,t/2)⊙υ(△mxk0-ξ,t/2) <κ⊙κ <ε,

and

η(△mxk-△mxk0,t)≤η(△mxk-ξ,t/2)⊙η(△mxk0-ξ,t/2) <κ⊙κ <ε.

Since x={xk} is not λ-△m-statistical convergent corresponding to (τ,υ,η). Therefore, δλ([C(ε,t)]c)=0 i.e. δλ(C(ε,t))=1 , which results contradiction for x={xk}, assumed to be λ-△m-statistical Cauchy. Thus, x={xk} converges λ-△m-statistically corresponding to (τ,υ,η).