Abstract
In this article, we aim to define
is a closed subspace of
Acknowledgements
The authors would like to thank the reviewers for their careful reading, which improved the presentation of the paper.
References
[1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), no. 1, 87–96. 10.1016/S0165-0114(86)80034-3Search in Google Scholar
[2] F. Başar, Summability Theory and its Applications, 2nd ed., CRC Press/Taylor & Francis Group, Boca Raton, 2022. 10.1201/9781003294153Search in Google Scholar
[3] F. Başar and M. Yeşilkayagil Savaşcı, Double Sequence Spaces and Four Dimensional Matrices, Monogr. Research Notes in Math., CRC Press/Taylor & Francis Group, Boca Raton, 2022. 10.1201/9781003285786Search in Google Scholar
[4] C. Belen and M. Yildirim, On generalized statistical convergence of double sequences via ideals, Ann. Univ. Ferrara Sez. VII Sci. Mat. 58 (2012), no. 1, 11–20. 10.1007/s11565-011-0137-1Search in Google Scholar
[5] J. S. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis 8 (1988), no. 1–2, 47–63. 10.1524/anly.1988.8.12.47Search in Google Scholar
[6] S. Debnath and D. Rakshit, On I-statistical convergence, Iran. J. Math. Sci. Inform. 13 (2018), no. 2, 101–109. Search in Google Scholar
[7] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244. 10.4064/cm-2-3-4-241-244Search in Google Scholar
[8] J. A. Fridy, On statistical convergence, Analysis 5 (1985), no. 4, 301–313. 10.1524/anly.1985.5.4.301Search in Google Scholar
[9] J. A. Fridy and C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl. 173 (1993), no. 2, 497–504. 10.1006/jmaa.1993.1082Search in Google Scholar
[10] N. Harnpornchai and W. Wonggattaleekam, An application of neutrosophic set to relative importance assignment in AHP, Mathematics 9 (2021), no. 20, Article ID 2636. 10.3390/math9202636Search in Google Scholar
[11] B. Hazarika, A. Alotaibi and S. A. Mohiuddine, Statistical convergence in measure for double sequences of fuzzy-valued functions, Soft Comput. 24 (2020), 6613–6622. 10.1007/s00500-020-04805-ySearch in Google Scholar
[12] B. Hazarika, S. A. Mohiuddine and M. Mursaleen, Some inclusion results for lacunary statistical convergence in locally solid Riesz spaces, Iran. J. Sci. Technol. Trans. A Sci. 38 (2014), no. 1, 61–68. 10.1155/2013/507962Search in Google Scholar
[13] V. A. Khan, M. D. Khan and M. Ahmad, Some new type of lacunary statistically convergent sequences in neutrosophic normed space, Neutrosophic Sets Syst. 42 (2021), Paper No. 15. Search in Google Scholar
[14] M. Kirişçi and N. Şimşek, Neutrosophic normed spaces and statistical convergence, J. Anal. 28 (2020), no. 4, 1059–1073. 10.1007/s41478-020-00234-0Search in Google Scholar
[15]
P. Kostyrko, T. Šalát and W. Wilczyński,
[16] D. Koundal, S. Gupta and S. Singh, Applications of neutrosophic sets in medical image denoising and segmentation, Infinite Study, 2016. Search in Google Scholar
[17] S. Kumar, V. Kumar and S. S. Bhatia, On ideal version of lacunary statistical convergence of double sequences, Gen. Math. Notes 17 (2013), no. 1, 32–44. Search in Google Scholar
[18]
V. Kumar,
On I and
[19]
V. Kumar and M. Mursaleen,
On
[20] I. J. Maddox, Statistical convergence in a locally convex space, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 1, 141–145. 10.1017/S0305004100065312Search in Google Scholar
[21] P. Majumdar, Neutrosophic Sets and its Applications to Decision Making, Computational Intelligence for Big Data Analysis, Springer, Cham (2015), 97–115. 10.1007/978-3-319-16598-1_4Search in Google Scholar
[22] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 28 (1942), 535–537. 10.1073/pnas.28.12.535Search in Google Scholar PubMed PubMed Central
[23] S. A. Mohiuddine, B. Hazarika and A. Alotaibi, On statistical convergence of double sequences of fuzzy valued functions, J. Intell. Fuzzy Syst. 32 (2017), 4331–4342. 10.3233/JIFS-16974Search in Google Scholar
[24] F. Móricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994), no. 1, 83–96. 10.4064/sm-110-1-83-96Search in Google Scholar
[25] M. Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), no. 1, 223–231. 10.1016/j.jmaa.2003.08.004Search in Google Scholar
[26] M. Mursaleen and S. A. Mohiuddine, On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space, J. Comput. Appl. Math. 233 (2009), no. 2, 142–149. 10.1016/j.cam.2009.07.005Search in Google Scholar
[27] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos Solitons Fractals 22 (2004), 1039–1046. 10.1016/j.chaos.2004.02.051Search in Google Scholar
[28] U. Praveena and M. Jeyaraman, On generalized Cesaro summability method in neutrosophic normed spaces using two-sided Taubarian conditions, J. Algebraic Stat. 13 (2022), no. 3, 1313–1323. Search in Google Scholar
[29] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos Solitons Fractals 27 (2006), 331–344. 10.1016/j.chaos.2005.03.019Search in Google Scholar
[30] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980), no. 2, 139–150. Search in Google Scholar
[31] E. Savaş and P. Das, A generalized statistical convergence via ideals, Appl. Math. Lett. 24 (2011), no. 6, 826–830. 10.1016/j.aml.2010.12.022Search in Google Scholar
[32] E. Savaş and R. F. Patterson, On some double almost lacunary sequence spaces defined by Orlicz functions, Filomat 19 (2005), 35–44. 10.2298/FIL0519035SSearch in Google Scholar
[33] I. J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. 10.2307/2308747Search in Google Scholar
[34] A. Sharma and V. Kumar, Some remarks on generalized summability using difference operators on neutrosophic normed spaces, J. Ramanujan Soc. Math. Math. Sci. 9 (2022), no. 2, 153–164. Search in Google Scholar
[35]
A. Sharma, V. Kumar and I. R. Ganaie,
Some remarks on
[36]
A. Sharma, S. Murtaza and V. Kumar,
Some remarks on
[37] F. Smarandache, Neutrosophic set—a generalization of the intuitionistic fuzzy set, Int. J. Pure Appl. Math. 24 (2005), no. 3, 287–297. Search in Google Scholar
[38] M. Yeşilkayagil and F. Başar, Some topological properties of the spaces of almost null and almost convergent double sequences, Turkish J. Math. 40 (2016), no. 3, 624–630. 10.3906/mat-1504-52Search in Google Scholar
[39] M. Yeşilkayagil and F. Başar, A note on Abel summability of double series, Numer. Funct. Anal. Optim. 38 (2017), no. 8, 1069–1076. 10.1080/01630563.2017.1316991Search in Google Scholar
[40] M. Yeşilkayagil and F. Başar, Domain of Riesz mean in some spaces of double sequences, Indag. Math. (N.S.) 29 (2018), no. 3, 1009–1029. 10.1016/j.indag.2018.03.006Search in Google Scholar
[41] M. Yeşilkayagil and F. Başar, On the paranormed space of bounded variation double sequences, Bull. Malays. Math. Sci. Soc. 43 (2020), 2701–2712. 10.1007/s40840-019-00829-2Search in Google Scholar
[42] M. Yeşilkayagil and F. Başar, AK(ϑ)-property of the spaces of double series, Bull. Malays. Math. Sci. Soc. 44 (2021), 881–889. 10.1007/s40840-020-00982-zSearch in Google Scholar
[43] L. A. Zadeh, Fuzzy sets, Information Control 8 (1965), 338–353. 10.1016/S0019-9958(65)90241-XSearch in Google Scholar
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Articles in the same Issue
- Frontmatter
- New analytical technique to solve fractional-order Sharma–Tasso–Olver differential equation using Caputo and Atangana–Baleanu derivative operators
- Nondensely defined partial neutral functional integrodifferential equations with infinite delay under the light of integrated resolvent operators
- On the number of limit cycles coming from a uniform isochronous center with continuous and discontinuous quartic perturbations
- On ℐ2(𝒮θ p,r )-summability of double sequences in neutrosophic normed spaces
- Earthquake convexity and some new related inequalities
- On λ-statistically φ-convergence
- Multivalued relation-theoretic weak contractions and applications
- Titchmarsh’s theorem with moduli of continuity in Laguerre hypergroup
- Application of Touchard wavelet to simulate numerical solutions to fractional pantograph differential equations
- Existence and linear independence theorem for linear fractional differential equations with constant coefficients
- A study on δ‐ℐ‐compactness in a mixed fuzzy ideal topological space
- Lie symmetry, exact solutions and conservation laws of time fractional Black–Scholes equation derived by the fractional Brownian motion
- On statistical convergence of order α of sequences in gradual normed linear spaces
- Uniqueness of meromorphic functions and their powers in set sharing
- Multiplicative 𝔪-metric space, fixed point theorems with applications in multiplicative integrals equation and numerical results
- Note on bounds on the coefficients of a subclass of m-fold symmetric bi-univalent functions
- Multiple solitons, periodic solutions and other exact solutions of a generalized extended (2 + 1)-dimensional Kadomstev--Petviashvili equation
Articles in the same Issue
- Frontmatter
- New analytical technique to solve fractional-order Sharma–Tasso–Olver differential equation using Caputo and Atangana–Baleanu derivative operators
- Nondensely defined partial neutral functional integrodifferential equations with infinite delay under the light of integrated resolvent operators
- On the number of limit cycles coming from a uniform isochronous center with continuous and discontinuous quartic perturbations
- On ℐ2(𝒮θ p,r )-summability of double sequences in neutrosophic normed spaces
- Earthquake convexity and some new related inequalities
- On λ-statistically φ-convergence
- Multivalued relation-theoretic weak contractions and applications
- Titchmarsh’s theorem with moduli of continuity in Laguerre hypergroup
- Application of Touchard wavelet to simulate numerical solutions to fractional pantograph differential equations
- Existence and linear independence theorem for linear fractional differential equations with constant coefficients
- A study on δ‐ℐ‐compactness in a mixed fuzzy ideal topological space
- Lie symmetry, exact solutions and conservation laws of time fractional Black–Scholes equation derived by the fractional Brownian motion
- On statistical convergence of order α of sequences in gradual normed linear spaces
- Uniqueness of meromorphic functions and their powers in set sharing
- Multiplicative 𝔪-metric space, fixed point theorems with applications in multiplicative integrals equation and numerical results
- Note on bounds on the coefficients of a subclass of m-fold symmetric bi-univalent functions
- Multiple solitons, periodic solutions and other exact solutions of a generalized extended (2 + 1)-dimensional Kadomstev--Petviashvili equation