Rough variation on lacunary quasi Cauchy triple difference sequences

Nagarajan Subramanian; Ayhan Esi

doi:10.1515/anly-2020-0030

Publicly Available Published by Oldenbourg Wissenschaftsverlag October 9, 2020

Rough variation on lacunary quasi Cauchy triple difference sequences

Nagarajan Subramanian and Ayhan Esi

From the journal Analysis

https://doi.org/10.1515/anly-2020-0030

Abstract

In the present paper, we extend the notion of rough ideal lacunary statistical quasi Cauchy triple difference sequence of order α using the concept of ideals, which automatically extends the earlier notions of rough convergence and rough statistical convergence. We prove several results associated with this notion.

Keywords: Rough variation; lacunary sequence; triple sequence; ideal convergence; compactness; continuity

MSC 2010: 40A35; 40A05

1 Introduction and background

The idea of statistical convergence was introduced by Steinhaus [16] and also independently by Fast [10] for real or complex sequences. Statistical convergence is a generalization of the usual notion of convergence, which parallels the theory of ordinary convergence. The theory of statistical convergence has been discussed in trigonometric series, summability theory, measure theory, turnpike theory, approximation theory, fuzzy set theory and so on.

Let K be a subset of the set of positive integers ℕ×ℕ×ℕ=ℕ3, and let us denote the set

{(m,n,k)∈K:m≤u,n≤v,k≤w}

by Ku⁢v⁢w. Then the natural density of K is given by

δ⁢(K)=limu,v,w→∞⁡|Ku⁢v⁢w|u⁢v⁢w,

where |Ku⁢v⁢w| denotes the number of elements in Ku⁢v⁢w. Clearly, a finite subset has natural density zero, and we have δ⁢(Kc)=1-δ⁢(K), where Kc=ℕ∖K is the complement of K. If K1⊆K2, then δ⁢(K1)≤δ⁢(K2).

A triple sequence (real or complex) can be defined as a function x:ℕ3→ℝ⁢(ℂ), where ℕ,ℝ and ℂ denote the set of natural numbers, real numbers and complex numbers, respectively. The different types of notions of triple sequence were introduced and investigated initially by Sahiner et al. [14, 15], Esi et al. [5, 6, 8], Dutta, Esi and Tripathy [4], Debnath, Sarma and Das [3], Aiyub, Esi and Subramanian [1], Subramanian et al. [17, 18, 19, 9, 20, 23, 11, 7, 22, 21] and many others.

Throughout the paper, we consider a triple sequence x=(xm⁢n⁢k) such that xm⁢n⁢k∈ℝ,m,n,k∈ℕ.

A triple sequence x=(xm⁢n⁢k) is said to be statistically convergent to 0∈ℝ, written as st-lim⁡x=0, provided that the set

{(m,n,k)∈ℕ3:|xm⁢n⁢k|≥ϵ}

has natural density zero for any ϵ>0. In this case, 0 is called the statistical limit of the triple sequence x.

If a triple sequence is statistically convergent, then for every ϵ>0 infinitely many terms of the sequence may remain outside the ϵ-neighborhood of the statistical limit, provided that the natural density of the set consisting of the indices of these terms is zero. This is an important property that distinguishes statistical convergence from ordinary convergence. Because the natural density of a finite set is zero, we can say that every ordinary convergent sequence is statistically convergent.

The idea of rough convergence was introduced by Phu [13], who also introduced the concepts of rough limit points and roughness degree. The idea of rough convergence occurs very naturally in numerical analysis and has interesting applications. Aytar [2] extended the idea of rough convergence into rough statistical convergence using the notion of natural density just as usual convergence was extended to statistical convergence. Pal, Chandra and Dutta [12] extended the notion of rough convergence using the concept of ideals, which automatically extends the earlier notions of rough convergence and rough statistical convergence.

The difference triple sequence space was introduced by Debnath et al. [3] and is defined by

Δ⁢xm⁢n⁢k=xm⁢n⁢k-xm,n+1,k-xm,n,k+1+xm,n+1,k+1-xm+1,n,k+xm+1,n+1,k+xm+1,n,k+1-xm+1,n+1,k+1,

Δ0⁢xm⁢n⁢k=〈xm⁢n⁢k〉.

Throughout the paper, let β be a nonnegative real number.

The main purpose of this paper is to investigate rough ideal lacunary statistical ward continuity of order α, where a function f is called I-lacunary statistically ward continuous of order α if it preserves I-lacunary statistically quasi Cauchy triple difference sequences of order α, i.e. (f⁢(xr⁢s⁢t)) is an Sθα⁢(I)-quasi Cauchy triple difference sequence whenever (xr⁢s⁢t).

2 Definitions and preliminaries

Throughout the paper, we assume that β>0.

Definition 2.1.

The triple sequence θi,ℓ,j={(mi,nℓ,kj)} is called triple lacunary if there exist three increasing sequences of integers such that

m0=0, ⁢hi=mi-mr-1→∞ ⁢as ⁢i→∞,

n0=0, ⁢hℓ¯=nℓ-nℓ-1→∞ ⁢as ⁢ℓ→∞,

k0=0, ⁢hj¯=kj-kj-1→∞ ⁢as ⁢j→∞.

Let mi,ℓ,j=mi⁢nℓ⁢kj, hi,ℓ,j=hi⁢hℓ¯⁢hj¯ and θi,ℓ,j be determined by

Ii,ℓ,j={(m,n,k):mi-1<m<mi⁢ and ⁢nℓ-1<n≤nℓ⁢ and ⁢kj-1<k≤kj},

qk=mkmk-1,

qℓ¯=nℓnℓ-1,

qj¯=kjkj-1.

Definition 2.2.

A rough difference triple sequence (xm⁢n⁢k) of points in ℝ is called Sθα⁢(I)-convergent to a real number ℓ (or I-lacunary statistically convergent of order α to ℓ) if

{(r,s,t)∈ℕ3:1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|xm⁢n⁢k-ℓ|≥β+ϵ}|≥δ}∈I

for each ϵ,δ>0. Moreover, Sθα⁢(I) is a sequential method.

Definition 2.3.

A subset A of ℝ is called Sθα⁢(I)-sequentially compact if any rough difference triple sequence of points in A has an Sθα⁢(I)-convergent subsequence with Sθα⁢(I)-limit in A.

Definition 2.4.

A function f defined on a subset A of ℝ is Sθα⁢(I)-sequentially continuous at a point x0 if given a rough difference triple sequence x=(xr⁢s⁢t) of points in A, Sθα⁢(I)-lim⁡(xr⁢s⁢t)=x0 implies that Sθα⁢(I)-lim⁡f⁢(xr⁢s⁢t)=f⁢(x0).

Definition 2.5.

A rough difference triple sequence (xm⁢n⁢k) of points in ℝ is called Sθα⁢(I)-quasi Cauchy (or I-lacunary statistically quasi Cauchy of order α ) if Sθα⁢(I)-limm,n,k→∞⁡Δ⁢xm⁢n⁢k=0, i.e.

{(r,s,t)∈ℕ3:1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|Δ⁢xm⁢n⁢k|≥β+ϵ}|≥δ}∈I

for each ϵ,δ>0.

Definition 2.6.

A subset A of ℝ is called Sθα⁢(I)-ward compact (or I-lacunary statistically ward compact of order α) if any rough difference triple sequence of points in A has an Sθα⁢(I)-quasi Cauchy subsequence.

Definition 2.7.

A function defined on a subset A of ℝ is called Sθα⁢(I)-ward continuous (or I-lacunary statistically ward continuous of order α) if it preserves I-lacunary statistically quasi Cauchy sequences of order α, i.e. (f⁢(xr⁢s⁢t)) is an Sθα⁢(I)-quasi Cauchy sequence whenever (xr⁢s⁢t).

3 Main results

Theorem 3.1.

Let {fr⁢s⁢t} be a rough difference triple sequence of Sθα⁢(I)-ward continuous functions on a subset A of R and let {fr⁢s⁢t} be uniformly convergent to a function f. Then f is Sθα⁢(I)-ward continuous on A.

Proof.

Let ϵ be a positive real number and let (xm⁢n⁢k) be any Sθα⁢(I)-quasi Cauchy sequence of points in A. By uniform convergence of {fr⁢s⁢t}, there exists a positive integer N such that

|fr⁢s⁢t⁢(xm⁢n⁢k)-f⁢(xm⁢n⁢k)|<β+ϵ3

for all x∈A whenever r,s,t≥N. As fN is Sθα⁢(I)-ward continuous on A, we have

{(r,s,t)∈ℕ3:1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|fN⁢(xm+1⁢n+1⁢k+1)-fN⁢(xm⁢n⁢k)|≥β+ϵ3}|≥δ3}∈I.

On the other hand, we have

{(m,n,k)∈Ir⁢s⁢t:|f⁢(xm+1⁢n+1⁢k+1)-f⁢(xm⁢n⁢k)|≥β+ϵ}

⊂{(m,n,k)∈Ir⁢s⁢t:|f⁢(xm+1⁢n+1⁢k+1)-fN⁢(xm+1⁢n+1⁢k+1)|≥β+ϵ3}

∪{(m,n,k)∈Ir⁢s⁢t:|fN⁢(xm+1⁢n+1⁢k+1)-fN⁢(xm⁢n⁢k)|≥β+ϵ3}

∪{(m,n,k)∈Ir⁢s⁢t:|fN⁢(xm⁢n⁢k)-f⁢(xm⁢n⁢k)|≥β+ϵ3}.

Now it follows from this inclusion that

1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|f⁢(xm+1⁢n+1⁢k+1)-f⁢(xm⁢n⁢k)|≥β+ϵ}|

≤1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|f⁢(xm+1⁢n+1⁢k+1)-fN⁢(xm+1⁢n+1⁢k+1)|≥β+ϵ3}|

+1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|fN⁢(xm+1⁢n+1⁢k+1)-fN⁢(xm⁢n⁢k)|≥β+ϵ3}|

+1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|fN⁢(xm⁢n⁢k)-f⁢(xm⁢n⁢k)|≥β+ϵ3}|.

Therefore, we have

{(r,s,t)∈ℕ3:1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|f⁢(xm+1⁢n+1⁢k+1)-f⁢(xm⁢n⁢k)|≥β+ϵ3}|≥δ}∈I.∎

Theorem 3.2.

The rough difference triple sequence set of Sθα⁢(I)-ward continuous functions on a subset A of R is a closed subset of the set of continuous functions on A.

Proof.

Let ϵ be a positive real number and let (xm⁢n⁢k) be any Sθα⁢(I)-quasi Cauchy sequence of points in A. There exists a positive integer N such that

|xm+1⁢n+1⁢k+1-xm⁢n⁢k|<β+ϵ3

for all x∈A whenever r,s,t≥N. As N is Sθα⁢(I)-ward continuous on A, we have

{(r,s,t)∈ℕ3:1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|xm+1⁢n+1⁢k+1-xm⁢n⁢k|≥β+ϵ3}|≥δ3}∈I.

On the other hand, we have

{(m,n,k)∈Ir⁢s⁢t:|xm+1⁢n+1⁢k+1-xm⁢n⁢k|≥β+ϵ}

⊂{(m,n,k)∈Ir⁢s⁢t:|xm+1⁢n+1⁢k+1-xm+1⁢n+1⁢k+1|≥β+ϵ3}

∪{(m,n,k)∈Ir⁢s⁢t:|xm+1⁢n+1⁢k+1-xm⁢n⁢k|≥β+ϵ3}

∪{(m,n,k)∈Ir⁢s⁢t:|xm⁢n⁢k-xm⁢n⁢k|≥β+ϵ3}.

Now it follows from this inclusion that

1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|xm+1⁢n+1⁢k+1-xm⁢n⁢k|≥β+ϵ}|

≤1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|xm+1⁢n+1⁢k+1-xm+1⁢n+1⁢k+1|≥β+ϵ3}|

+1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|xm+1⁢n+1⁢k+1-xm⁢n⁢k|≥β+ϵ3}|

+1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|xm⁢n⁢k-xm⁢n⁢k|≥β+ϵ3}|.

Therefore, we have

{(r,s,t)∈ℕ3:1hr⁢s⁢tα⁢|{(m,n,k)∈Ir⁢s⁢t:|xm+1⁢n+1⁢k+1-xm⁢n⁢k|≥β+ϵ3}|≥δ}∈I.∎

Corollary 3.3.

The rough difference triple sequence set of Sθα⁢(I)-ward continuous functions on a subset A of R is a complete subspace of the set of continuous functions on A.

Theorem 3.4.

If the rough difference triple sequence of f is Sθα⁢(I)-ward continuous on a subset A of R, then it is Sθα⁢(I)-continuous on A.

Proof.

Assume that the rough difference triple sequence of f is Sθα⁢(I)-ward continuous on a subset A of ℝ. Let (xr⁢s⁢t) be any Sθα⁢(I)-convergent rough triple sequence with Sθα⁢(I)-limm,n,k→∞⁡xm⁢n⁢k=x000. Then the triple sequence

(x111x000x222x000…xr-1⁢s-1⁢t-1x000xr⁢s⁢tx000…x111x000x222x000…xr-1⁢s-1⁢t-1x000xr⁢s⁢tx000…x111x000x222x000…xr-1⁢s-1⁢t-1x000xr⁢s⁢tx000…⋮x111x000x222x000…xr-1⁢s-1⁢t-1x000xr⁢s⁢tx000…⋮)

is Sθα⁢(I)-convergent to x000. Hence it is Sθα⁢(I)-quasi Cauchy. As f is Sθα⁢(I)-ward continuous, the rough triple sequence

(f⁢(x111)f⁢(x000)f⁢(x222)f⁢(x000)…f⁢(xr-1⁢s-1⁢t-1)f⁢(x000)f⁢(xr⁢s⁢t)f⁢(x000)…f⁢(x111)f⁢(x000)f⁢(x222)f⁢(x000)…f⁢(xr-1⁢s-1⁢t-1)f⁢(x000)f⁢(xr⁢s⁢t)f⁢(x000)…f⁢(x111)f⁢(x000)f⁢(x222)f⁢(x000)…f⁢(xr-1⁢s-1⁢t-1)f⁢(x0)f⁢(xn)f⁢(x0)…⋮f⁢(x111)f⁢(x000)f⁢(x222)f⁢(x000)…f⁢(xr-1⁢s-1⁢t-1)f⁢(x000)f⁢(xr⁢s⁢t)f⁢(x000)…⋮)

is Sθα⁢(I)-quasi Cauchy. Therefore, we obtain for the rough difference triple sequence (f⁢(xr⁢s⁢t)) that it Sθα⁢(I)-converges to f⁢(x000). ∎

Lemma 3.5.

Any rough difference triple sequence of an Sθα-convergent sequence of points in R with an Sθα-limit ℓ has a convergent subsequence with the same limit ℓ in the ordinary sense.

Theorem 3.6.

A rough difference triple sequence of a subset A of R is Sθα⁢(I)-sequentially compact if and only if it is subsequentially compact in the ordinary sense.

Proof.

“⟹”: Let a rough difference triple sequence of A be a subset of ℝ and Sθα⁢(I)-sequentially compact. Then any convergent subsequence of it is Sθα⁢(I)-convergent. So it is subsequentially compact in the ordinary sense.

“⟸”: Suppose that a rough triple difference sequence of A is Sθα⁢(I)-sequentially compact. If x=(xr⁢s⁢t) is a rough difference triple sequence of points in A, then it has an Sθα⁢(I)-convergent subsequence (xrm⁢sn⁢tk) of the triple sequence of x. By Lemma 3.5, the rough difference triple sequence (xrm⁢sn⁢tk) has a convergent subsequence (xrmi⁢snj⁢tkℓ) with limit in A. ∎

Corollary 3.7.

A rough difference triple sequence of the function f is Sθα⁢(I)-sequentially continuous if and only if it is continuous in the ordinary sense.

4 Conclusion

In this paper, we investigate rough ideal lacunary statistical ward continuity of order α, where a function f is called I-lacunary statistically ward continuous of order α if it preserves I-lacunary statistically quasi Cauchy triple difference sequences of order α, i.e. (f⁢(xr⁢s⁢t)) is an Sθα⁢(I)-quasi Cauchy triple difference sequence whenever (xr⁢s⁢t).

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Received: 2019-12-11

Revised: 2020-08-03

Accepted: 2020-08-05

Published Online: 2020-10-09

Published in Print: 2021-02-01

Rough variation on lacunary quasi Cauchy triple difference sequences

Abstract

1 Introduction and background

2 Definitions and preliminaries

Definition 2.1.

Definition 2.2.

Definition 2.3.

Definition 2.4.

Definition 2.5.

Definition 2.6.

Definition 2.7.

3 Main results

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

Corollary 3.3.

Theorem 3.4.

Proof.

Lemma 3.5.

Theorem 3.6.

Proof.

Corollary 3.7.

4 Conclusion

References

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