Volume 2022, Issue 1 2779479

Research Article

Open Access

A Note on Lacunary Sequence Spaces of Fractional Difference Operator of Order (α, β)

Qing-Bo Cai

orcid.org/0000-0003-4759-7441

Fujian Provincial Key Laboratory of Data-Intensive Computing, Key Laboratory of Intelligent Computing and Information Processing, School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China qztc.edu.cn

Search for more papers by this author

Sunil K. Sharma,

Sunil K. Sharma

orcid.org/0000-0002-6549-3024

Department of Mathematics, Central University of Jammu, Bagla Suchani, Samba 181143, Jammu & Kashmir, India cujammu.ac.in

Search for more papers by this author

Mohammad Ayman Mursaleen,

Corresponding Author

Mohammad Ayman Mursaleen

[email protected]

orcid.org/0000-0002-2566-3498

School of Information and Physical Sciences, The University of Newcastle, Callaghan, New South Wales 2308, Australia newcastle.edu.au

Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia upm.edu.my

Search for more papers by this author

Qing-Bo Cai,

Qing-Bo Cai

orcid.org/0000-0003-4759-7441

Search for more papers by this author

Sunil K. Sharma,

Sunil K. Sharma

orcid.org/0000-0002-6549-3024

Department of Mathematics, Central University of Jammu, Bagla Suchani, Samba 181143, Jammu & Kashmir, India cujammu.ac.in

Search for more papers by this author

Mohammad Ayman Mursaleen,

Corresponding Author

Mohammad Ayman Mursaleen

[email protected]

orcid.org/0000-0002-2566-3498

School of Information and Physical Sciences, The University of Newcastle, Callaghan, New South Wales 2308, Australia newcastle.edu.au

Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia upm.edu.my

Search for more papers by this author

First published: 15 March 2022

https://doi.org/10.1155/2022/2779479

Citations: 2

Academic Editor: Mahmut isik

Share a link

Email
Wechat
Bluesky

Abstract

In the present paper, we defined lacunary sequence spaces of fractional difference operator of order (α, β) over n-normed spaces via Musielak-Orlicz function . Our aim in this paper is to study some topological properties and inclusion relation between the spaces , , and .

1. Introduction and Preliminaries

The concept of statistical convergence was introduced by Fast [1] and Schoenberg [2] independently. Many authors studied the concept of statistical convergence from the past few years we may refer to ([3–19]) and references therein.

The sequence ξ = (ξ_k) is statistically convergent of order α to ℓ (see Çolak) if there is a complex number ℓ such that

(1)

Let 0 < α ≤ β ≤ 1. We define the (α, β)-density of the subset E of ℕ by

(2)

provided the limit exists, where |{k ≤ n : k ∈ E}|^β denotes the βth power of number of elements of E not exceeding n ([20–22]).

By a lacunary sequence θ = (θ_r), we mean a sequence of positive integers such that θ₀ = 0, 0 < θ_r < θ_r+1, and ϕ_r = θ_r − θ_r−1⟶∞ as r⟶∞. The intervals determined by θ will be denoted by J_r = (θ_r−1, θ_r] and t_r = θ_r/θ_r−1. Freedman et al. [23] defined the space of lacunary strongly convergent sequences by

(3)

Definition 1. Let θ = (θ_r) be a lacunary sequence. The sequence ξ = (ξ_k) is -statistically convergent (or lacunary statistically convergent of order (α, β)) (see [20]) if there is a real number L such that

(4)

where J_r = (θ_r−1, θ_r] and

denotes the αth power

of ϕ_r, that is,

. In this case, we write

The set of all

-statistically convergent sequences is denoted by

. If α = β = 1 and θ = (2^r), then, we will write S instead of

A family of subsets of a nonempty set X is said to be an ideal in X if

(1)
(2)
imply
(3)
, B ⊂ A imply

while an admissible ideal

of X further satisfies

for each ξ ∈ X (see [24]).

A sequence in X is said to be -convergent to ξ ∈ X (see [24]), if for each ε > 0.

A sequence in X is said to be -bounded to ξ ∈ X if there exists an K > 0 such that . Many authors studied the topological properties and applications of ideal, we refer to ([25–37]) and references therein.

The concept of difference sequence spaces was introduced in [38] and further generalized in [39].

In [40], Baliarsingh defined the fractional difference operator as follows:

Let ξ = (ξ_k) ∈ w and γ be a real number, then, the fractional difference operator Δ^(γ) is defined by

(5)

where (−γ)_i denotes the Pochhammer symbol defined as

(6)

The concept of difference sequences, Orlicz function, Musielak-Orlicz function, and n-normed spaces was used by many authors and proves some topological properties (see [41–50]) and references therein. For details about n-normed spaces, we refer to ([51–55]), difference sequence spaces ([38, 39]), Orlicz function ([56–58]). Ideal convergence and fractional difference operator Δ^α has been studied in [59, 60]. We continue in this connection and construct new sequence spaces as follows.

Let

be a Musielak-Orlicz function, u = (u_k) be a bounded sequence of positive real numbers, and 0 < α ≤ β ≤ 1. We define the following sequence spaces in the present paper

(7)

(8)

(9)

If we take , the above spaces reduces to , , and .

If we take u = (u_k) = 1, the above spaces reduces to , , and .

The following inequality will be used in the proceeding results. If 0 ≤ u_k ≤ supu_k = H, D = max(1, 2^H−1), then

(10)

for all k and r_k, s_k ∈ ℂ. Also

for all r ∈ ℂ.

2. Main Results

In this section, we study topological properties and prove some inclusion relations. In what follows, we will take a Musielak-Orlicz function and u = (u_k) a bounded sequence of positive real numbers.

Theorem 2. The spaces , , and are linear spaces.

Proof. Let and let μ, ν be scalars. Then, there exist two positive numbers ρ₁ and ρ₂ for ε > 0

(11)

(12)

Let ρ₃ = max{2|μ|ρ₁, 2|ν|ρ₂} and by inequality (1), we have

(13)

Now by (11) and (12), we get

(14)

Therefore, . Hence, is a linear space. On a similar way, we can prove that and are linear spaces.

Theorem 3. The inclusions hold.

Proof. The inclusion is obvious. We prove . For this, let . Then, there exists ρ₁ > 0 such that for every ε > 0

(15)

We put ρ = 2ρ₁ and is a Musielak-Orlicz function, we have

(16)

Suppose that r ∉ B₁. Hence by above inequality and (1), we have

(17)

By using , we have

(18)

Put It follows that

(19)

This shows that

, which completes the proof.

Theorem 4. The space is a paranormed space with paranorm defined by

(20)

Proof. Since g(ξ) = g(−ξ) and , we have g(0) = 0. Let . Let

(21)

(22)

Let ρ₁ ∈ B(ξ₁) and ρ₂ ∈ B(ξ₂) and ρ = ρ₁ + ρ₂, we have

(23)

Thus, and

(24)

Let σ^s⟶σ where σ, σ^s ∈ ℂ and let g(ξ^s − ξ)⟶0 as s⟶∞. We have to show that g(σ^sξ^s − σξ)⟶0 as s⟶∞. Let

(25)

(26)

If ρ_s ∈ B(ξ^s) and ; then, we have

(27)

From the above inequality, it follows that

(28)

and consequently,

(29)

which completes the proof.

Theorem 5. Let and be Musielak-Orlicz functions that satisfy the Δ₂-condition. Then

(30)

(31)

(32)

Proof. (i) Let . Then, there exists K₁ > 0 such that

(33)

for ρ > 0. Since

is a Musielak-Orlicz function which satisfies Δ₂-condition, we have

(34)

for K ≥ 1. By continuity of

, we have

(35)

Suppose r ∉ B₁. Then, by using (34) and (35), we have

(36)

Hence,

and so B₂ ⊂ B₁ which implies

. This shows that

. Hence,

. Similarly, we can prove (ii) and (iii) part.

Corollary 6. Let satisfy Δ₂-condition. Then,

(37)

(38)

(39)

Proof. If we put and in Theorem 5, the result follows.

Theorem 7. Let and be Musielak-Orlicz functions that satisfy the Δ₂-condition. Then,

(40)

(41)

(42)

Proof. (i) Let . Then, there exists K₁ > 0 and K₂ > 0 such that

(43)

(44)

for some ρ > 0. Let r ∉ B₁ ∪ B₂. Then, we have

(45)

. We have and so B ⊂ B₁ ∪ B₂ which implies . This shows that . Hence, . Similarly, we can prove (ii) and (iii) part of the theorem.

Theorem 8. Let 0 < u_k ≤ v_k and (v_k/u_k) be bounded. Then, the following inclusions hold

(46)

(47)

Proof. (i) Let . Write and λ_k = u_k/v_k, so that 0 < λ < λ_k ≤ 1. By using Hölder inequality, we have

(48)

Hence, for every ε > 0, we have

(49)

This implies that and so . Hence, . Similarly, we can prove .

Corollary 9. If 0 < infu_k ≤ 1. Then, the following inclusions hold:

(50)

(51)

Proof. The proof follows from Theorem 8.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to writing this paper. All authors read and approved the manuscript.

Acknowledgments

Corresponding author is supported by JADD program (by UPM-UoN) while the first author is supported by the Natural Science Foundation of Fujian Province of China (Grant no. 2020J01783), the Project for High-level Talent Innovation and Entrepreneurship of Quanzhou (Grant no. 2018C087R), and the Program for New Century Excellent Talents in Fujian Province University.

Open Research

Data Availability

No data were used to support this study.

References

1 Fast H., Sur la convergence statistique, Colloquium Mathematicum. (1951) 2, 241–244.
10.4064/cm-2-3-4-241-244
Google Scholar
2 Schoenberg I. J., The integrability of certain functions and related summability methods, The American mathematical monthly. (1959) 66, 361–375.
10.1080/00029890.1959.11989303
Google Scholar
3 Esi A. and Tripathy B. C., Strongly almost convergent generalized difference sequences associated with multiplier sequences, Mathematica Slovaca. (2007) 57, no. 4, 339–348, https://doi.org/10.2478/s12175-007-0028-1, 2-s2.0-85026103940.
10.2478/s12175-007-0028-1
Web of Science® Google Scholar
4 Esi A., Tripathy B. C., and Sarma B., On some new type generalized difference sequence spaces, Mathematica Slovaca. (2007) 57, no. 5, 475–482, https://doi.org/10.2478/s12175-007-0039-y, 2-s2.0-85014159472.
10.2478/s12175-007-0039-y
Web of Science® Google Scholar
5 Esi A., Some classes of generalized difference paranormed sequence spaces associated with multiplier sequences, Journal of Computational Analysis and Applications. (2009) 11, no. 3, 536–545.
Google Scholar
6 Esi A., Strongly generalized difference [V_λ, Δ^M, P]-summable sequence spaces defined by a sequence of moduli, Nihonkai Mathematical Journal. (2009) 20, no. 2, 99–108.
Google Scholar
7 Esi A., Generalized difference sequence spaces defined by Orlicz functions, General Mathematics. (2009) 17, no. 2, 53–66.
Google Scholar
8 Connor J. S., W. O. Bray and Č. V. Stanojević, A topological and functional analytic approach to statistical convergence, 1999, Birkhäuser Boston, Analysis of Divergence, Applied and Numerical Harmonic Analysis, https://doi.org/10.1007/978-1-4612-2236-1_23.
Google Scholar
9 Fridy J. A., On the statistical convergence, Analysis. (1985) 5, 301–303.
10.1524/anly.1985.5.4.301
Google Scholar
10 Isik M., On statistical convergence of generalized difference sequence spaces, Soochow Journal of Mathematics. (2004) 30, 197–205.
Google Scholar
11 Kolk E., The statistical convergence in Banach spaces, Acta et Commentationes Universitatis Tartuensis de Mathematica. (1991) 928, 41–52.
Google Scholar
12 Maddox I. J., A new type of convergence, Mathematical Proceedings of the Cambridge Philosophical Society. (1978) 83, 61–64, https://doi.org/10.1017/S0305004100054281, 2-s2.0-0010678144.
10.1017/S0305004100054281
Web of Science® Google Scholar
13 Maddox I. J., Statistical convergence in a locally convex space, Mathematical Proceedings of the Cambridge Philosophical Society. (1988) 104, 141–145, https://doi.org/10.1017/S0305004100065312, 2-s2.0-84975988138.
10.1017/S0305004100065312
Web of Science® Google Scholar
14 Mursaleen M., λ-Statistical convergence, Mathematica Slovaca. (2000) 50, 111–115.
Google Scholar
15 Salat T., On statistical convergent sequences of real numbers, Mathematica Slovaca. (1980) 30, 139–150.
Google Scholar
16 Savas E., On some generalized sequence spaces defined by a modulus, Indian Journal of Pure and Applied Mathematics. (1999) 30, 459–464.
Web of Science® Google Scholar
17 Tripathy B. C., Esi A., and Balakrushna T., On a new type of generalized difference Cesaro sequence spaces, Soochow, Journal of Mathematics. (2005) 31, no. 3.
Google Scholar
18 Tripathy B. C. and Esi A., Generalized lacunary difference sequence spaces defined by Orlicz functions, Matimyás matematika. (2005) 28, no. 1-2, 50–57.
Google Scholar
19 Tripathy B. C. and Esi A., A new type of difference sequence spaces, International Journal of Science and Technology. (2006) 1, no. 1, 11–14.
Google Scholar
20 Sengül H., Some Cesàro-type summability spaces defined by a modulus function of order (α, β), Commun.Fac.Sci.Univ.Ank.Series A1. (2017) 66, no. 2, 80–90.
Google Scholar
21 Sengül H., On -convergence and strong -summability, Journal of Nonlinear Sciences and Applications. (2017) 10, no. 9, 5108–5115.
Google Scholar
22 Sengül H., On Wijsman I-lacunary statistical equivalence of order (η, μ), Journal of Inequalities and Special Functions. (2018) 9, no. 2, 92–101.
Google Scholar
23 Freedman A. R., Sember J. J., and Raphael M., Some Cesàro-type summability spaces, Proceedings of the London Mathematical Society. (1978) 37, 508–520.
10.1112/plms/s3-37.3.508
Web of Science® Google Scholar
24 Kostyrko P., Salat T., and Wilczynski W., I-convergence, Real Analysis Exchange. (2000) 26, no. 2, 669–686.
10.2307/44154069
Google Scholar
25 Das P., Kostyrko P., Wilczynski W., and Malik P., I and I^∗ convergence of double sequences, Mathematica Slovaca. (2008) 58, 605–620.
10.2478/s12175-008-0096-x
Web of Science® Google Scholar
26 Das P. and Malik P., On the statistical and I- variation of double sequences, Real Analysis Exchange. (2007) 33, no. 2, 351–364.
Google Scholar
27 Kara E. E. and Ïlkhan M., Lacunary -convergent and lacunay -bounded sequence spaces defined by an Orlicz function, Journal of Mathematical Analysis and Applications. (2016) 4, 150–159.
Google Scholar
28 Kumar V., On I and I^∗ convergence of double sequences, Mathematical Communications. (2007) 12, 171–181.
Google Scholar
29 Mursaleen M. and Alotaibi A., On I-convergence in radom 2-normed spaces, Mathematica Slovaca. (2011) 61, no. 6, 933–940.
10.2478/s12175-011-0059-5
Web of Science® Google Scholar
30 Mursaleen M., Mohiuddine S. A., and Edely O. H. H., On ideal convergence of double sequences in intuitionistic fuzzy normed spaces, Computers & Mathematcs with Applications. (2010) 59, 603–611.
10.1016/j.camwa.2009.11.002
Web of Science® Google Scholar
31 Mursaleen M. and Mohiuddine S. A., On ideal convergence of double sequences in probabilistic normed spaces, Mathematical reports. (2010) 12, no. 64, 359–371.
Web of Science® Google Scholar
32 Mursaleen M. and Mohiuddine S. A., On ideal convergence in probabilistic normed spaces, Mathematica Slovaca. (2012) 62, 49–62.
10.2478/s12175-011-0071-9
Web of Science® Google Scholar
33 Mursaleen M. and Sharma S. K., Spaces of ideal convergent sequences, The Scientific World Journal. (2014) 6.
Google Scholar
34 Raj K., Mohiuddine S. A., and Mursaleen M. A., Some generalized sequence spaces of invariant means defined by ideal and modulus functions in n-normed space, Italian Journal of Pure and Applied Mathematics. (2018) 39, 579–659.
Google Scholar
35 Raj K. and Sharma S. K., Ideal convergence sequence spaces defined by a Musielak-Orlicz function, Thai Journal of Mathematics. (2013) 11, 577–587.
Google Scholar
36 Sahiner A., Gürdal M., Saltan S., and Gunawan H., On ideal convergence in 2-normed spaces, Taiwanese Journal of Mathematics. (2007) 11, 1477–1484.
10.11650/twjm/1500404879
Web of Science® Google Scholar
37 Tripathy B. C. and Hazarika B., Some I-convergent sequence spaces defined by Orlicz functions, Acta Mathematicae Applicatae Sinica. (2011) 27, 149–154.
10.1007/s10255-011-0048-z
Google Scholar
38 Kizmaz H., On certain sequence spaces, Canadian Mathematical Bulletin. (1981) 24, 169–176, https://doi.org/10.4153/CMB-1981-027-5.
10.4153/CMB-1981-027-5
Web of Science® Google Scholar
39 Et M. and Çolak R., On some generalized difference sequence spaces, Soochow Journal of Mathematics. (1995) 21, 377–386.
Google Scholar
40 Baliarsingh P., Some new difference sequence spaces of fractional order and their dual spaces, Applied Mathematics and Computation. (2013) 219, 9737–9742.
10.1016/j.amc.2013.03.073
Web of Science® Google Scholar
41 Ahmad N., Sharma S. K., and Mohiuddine S. A., Generalized entire sequence spaces defined by fractional difference operator and sequence of modulus functions, TWMS Journal of Applied and Engineering Mathematics. (2020) 10, no. Special Issue, 63–72.
Web of Science® Google Scholar
42 Malkowsky E. and Veličković V., Topologies of some new sequence spaces, their duals, and the graphical representations of neighbourhoods, Topology and its Applications. (2011) 158, 1369–1380.
10.1016/j.topol.2011.05.011
Web of Science® Google Scholar
43 Malkowsky E. and Veličković V., Some new sequence spaces, their duals and a connection with Wulff's crystal, MATCH Communications in Mathematical and in Computer Chemistry. (2012) 67, 589–607.
CAS Web of Science® Google Scholar
44 Mursaleen M., Sharma S. K., and KiliÇman A., Difference Sequence Spaces Defined by Orlicz Functions, 2013, Abstarct and Applied Analysis.
Google Scholar
45 Mursaleen M. and Sharma S. K., Entire sequence spaces defined on locally convex Hausdorff topological space, Iranian Journal of Science and Technology. (2014) 38, 105–109.
Google Scholar
46 Mursaleen M., Sharma S. K., Mohiuddine S. A., and KiliÇman A., New Difference Sequence Spaces Defined by Musielak-Orlicz Function, 2014, 2014, article 691632, Abstract and Applied Analysis, https://doi.org/10.1155/2014/691632, 2-s2.0-84934990770.
10.1155/2014/691632
Google Scholar
47 Raj K. and Sharma S. K., Some sequence spaces in 2-normed spaces defined by Musielak-Orlicz function, Acta Universitatis Sapientiae, Mathematica. (2011) 3, 97–109.
Google Scholar
48 Raj K., Sharma A. K., and Sharma S. K., A sequence space defined by Musielak-Orlicz functions, International Journal of Pure and Applied Mathematics. (2011) 67, 475–484.
Google Scholar
49 Raj K., Sharma S. K., and Sharma A. K., Difference sequence spaces in n-normed spaces defined by Musielak-Orlicz function, Armenian journal of Mathematics. (2010) 3, 127–141.
Google Scholar
50 Sharma S. K., Mohiuddine S. A., Sharma A. K., and Sharma T. K., Sequence spaces over n-normed spaces defined by Musielak-Orlicz function of order (α, β), Facta Universitatis. (2018) 33, 721–738.
Google Scholar
51 Gähler S., Linear 2-normietre Rume, Mathematische Nachrichten. (1965) 28, 1–43.
10.1002/mana.19640280102
Google Scholar
52 Gunawan H., On n-inner product, n-norms, and the Cauchy-Schwarz inequality, Japanese Journal of Mathematics. (2001) 5, 47–54.
Google Scholar
53 Gunawan H., The space of p-summable sequence and its natural n-norm,, Bulletin of the Australian Mathematical Society. (2001) 64, 137–147.
10.1017/S0004972700019754
Web of Science® Google Scholar
54 Gunawan H. and Mashadi M., On n-normed spaces, International Journal of Mathematics and Mathematical Sciences. (2001) 27, 639.
10.1155/S0161171201010675
Google Scholar
55 Misiak A., n-Inner product spaces, Mathematische Nachrichten. (1989) 140, 299–319.
10.1002/mana.19891400121
Web of Science® Google Scholar
56 Lindenstrauss J. and Tzafriri L., On Orlicz sequence spaces, Israel Journal of Mathematics. (1971) 10, 345–355.
10.1007/BF02771656
Web of Science® Google Scholar
57 Maligranda L., Orlicz Spaces and Interpolation. Seminars in Mathematics 5, 1989, Polish Academy of Science.
Google Scholar
58 Musielak J., Orlicz spaces and modular spaces Lecture Notes in Mathematics, 1983, 1034, Springer-Verlag.
10.1007/BFb0072210
Web of Science® Google Scholar
59 Aral N. D. and Et M., Generalized difference sequence spaces of fractional order defined by Orlicz functions, Communications Faculty Of Science University of Ankara Series A1 Mathematics and Statistics. (2020) 69, no. 1, 941–951, https://doi.org/10.31801/cfsuasmas.628863.
10.31801/cfsuasmas.628863
Google Scholar
60 Aral N. D. and Kandemir H. S., I -lacunary statistical convergence of order β of difference sequences of fractional order, Facta Universitatis. (2021) 36, no. 1, 43–55.
10.22190/FUMI200117004A
Google Scholar

Citing Literature

All articles

A Note on Lacunary Sequence Spaces of Fractional Difference Operator of Order (α, β)

Abstract

1. Introduction and Preliminaries

2. Main Results

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

A Note on Lacunary Sequence Spaces of Fractional Difference Operator of Order (α, β)

Abstract

1. Introduction and Preliminaries

2. Main Results

Conflicts of Interest

Authors’ Contributions

Acknowledgments

Open Research

Data Availability

References

Citing Literature

References

Related

Information