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On the domain of difference double sequence spaces of arbitrary orders

  • S. Samantaray , Pinakadhar Baliarsingh EMAIL logo and L. Nayak
Published/Copyright: October 10, 2021

Abstract

The prime objective of this paper is to define a new double difference operator with arbitrary order via which new classes of difference double sequences are introduced. Results on topological structures, dual spaces and four-dimensional matrix mappings related to the proposed difference double sequence spaces are discussed. As an application of this work, the proposed operator is being used to approximate partial derivatives of fractional orders. Some numerical examples are also given in support of the validity or the clear visualization of the results obtained.

Award Identifier / Grant number: 02011/7/2020/NBHM

Funding statement: This work is partially supported by NBHM, DAE, Mumbai, under grant no. 02011/7/2020/NBHM.

References

[1] Z. U. Ahmad and Mursaleen, Köthe–Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math. (Beograd) (N. S.) 42(56) (1987), 57–61. Search in Google Scholar

[2] B. Altay and F. Başar, Some new spaces of double sequences, J. Math. Anal. Appl. 309 (2005), no. 1, 70–90. 10.1016/j.jmaa.2004.12.020Search in Google Scholar

[3] Ç. Asma and R. Çolak, On the Köthe–Toeplitz duals of some generalized sets of difference sequences, Demonstratio Math. 33 (2000), no. 4, 797–803. 10.1515/dema-2000-0412Search in Google Scholar

[4] P. Baliarsingh, Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput. 219 (2013), no. 18, 9737–9742. 10.1016/j.amc.2013.03.073Search in Google Scholar

[5] P. Baliarsingh, On a fractional difference operator, Alexandria Eng. J. 55 (2016), no. 2, 1811–1816. 10.1016/j.aej.2016.03.037Search in Google Scholar

[6] P. Baliarsingh, On difference double sequence spaces of fractional order, Indian J. Math. 58 (2016), no. 3, 287–310. Search in Google Scholar

[7] P. Baliarsingh, On double difference operators via four dimensional matrices, J. Indian Math. Soc. (N. S.) 83 (2016), no. 3–4, 209–219. Search in Google Scholar

[8] P. Baliarsingh and S. Dutta, A unifying approach to the difference operators and their applications, Bol. Soc. Parana. Mat. (3) 33 (2015), no. 1, 49–57. 10.5269/bspm.v33i1.19884Search in Google Scholar

[9] P. Baliarsingh and S. Dutta, On the classes of fractional order difference sequence spaces and their matrix transformations, Appl. Math. Comput. 250 (2015), 665–674. 10.1016/j.amc.2014.10.121Search in Google Scholar

[10] P. Baliarsingh and L. Nayak, A note on fractional difference operators, Alexandria. Eng. J. 57 (2018), no. 2, 1051–1054. 10.1016/j.aej.2017.02.022Search in Google Scholar

[11] M. Başarır, On the strong almost convergence of double sequences, Period. Math. Hungar. 30 (1995), no. 3, 177–181. 10.1007/BF01876616Search in Google Scholar

[12] M. Başarır, A note on the rates of convergence of double sequences, Sarajevo J. Math. 10 (2014), 87–92. 10.5644/SJM.10.1.11Search in Google Scholar

[13] M. Başarır and Ş. Konca, On some lacunary almost convergent double sequences spaces and Banach limits, Abstr. Appl. Anal. 2012 (2012), Article ID 426357. 10.1155/2012/426357Search in Google Scholar

[14] M. Başarır and Ş. Konca, Weighted lacunary statistical convergence, Iran. J. Sci. Technol. Trans. A Sci. 41 (2017), no. 1, 185–190. 10.1007/s40995-017-0188-ySearch in Google Scholar

[15] M. Başarır and O. Sonalcan, On some double sequence spaces, J. Indian Acad. Math. 21 (1999), no. 2, 193–200. Search in Google Scholar

[16] Ç. A. Bektaş, M. Et and R. Çolak, Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl. 292 (2004), no. 2, 423–432. 10.1016/j.jmaa.2003.12.006Search in Google Scholar

[17] S. Dutta and P. Baliarsingh, A note on paranormed difference sequence spaces of fractional order and their matrix transformations, J. Egyptian Math. Soc. 22 (2014), no. 2, 249–253. 10.1016/j.joems.2013.07.001Search in Google Scholar

[18] M. Et and R. Çolak, On some generalized difference sequence spaces, Soochow J. Math. 21 (1995), no. 4, 377–386. Search in Google Scholar

[19] A. Gökhan and R. Çolak, The double sequences spaces c 2 P ( p ) and c 2 P B ( p ) , Appl. Math. Comput. 157 (2004), no. 2, 491–501. 10.1016/j.amc.2003.08.047Search in Google Scholar

[20] A. Gökhan and R. Çolak, Double sequence space l 2 ( p ) , Appl. Math. Comput. 160 (2005), no. 1, 147–153. 10.1016/j.amc.2003.08.142Search in Google Scholar

[21] H. Kızmaz, On certain sequence spaces, Canad. Math. Bull. 24 (1981), no. 2, 169–176. 10.4153/CMB-1981-027-5Search in Google Scholar

[22] F. Móricz, Extensions of the spaces c and c 0 from single to double sequences, Acta Math. Hungar. 57 (1991), no. 1–2, 129–136. 10.1007/BF01903811Search in Google Scholar

[23] Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl. 293 (2004), no. 2, 523–531. 10.1016/j.jmaa.2004.01.014Search in Google Scholar

[24] Mursaleen and O. H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003), no. 1, 223–231. 10.1007/978-81-322-1611-7_7Search in Google Scholar

[25] Mursaleen and O. H. H. Edely, Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl. 293 (2004), no. 2, 532–540. 10.1016/j.jmaa.2004.01.015Search in Google Scholar

[26] L. Nayak and P. Baliarsingh, On difference double sequences and their applications, Current Trends in Mathematical Analysis and its Interdisciplinary Applications, Birkhäuser/Springer, Cham (2019), 809–829. 10.1007/978-3-030-15242-0_20Search in Google Scholar

[27] R. F. Patterson, Double sequence core theorems, Int. J. Math. Math. Sci. 22 (1999), no. 4, 785–793. 10.1155/S0161171299227858Search in Google Scholar

[28] R. F. Patterson, Comparison theorems for four dimensional regular matrices, Southeast Asian Bull. Math. 26 (2002), no. 2, 299–305. 10.1007/s100120200050Search in Google Scholar

[29] A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900), no. 3, 289–321. 10.1007/BF01448977Search in Google Scholar

[30] G. M. Robison, Divergent double sequences and series, Trans. Amer. Math. Soc. 28 (1926), no. 1, 50–73. 10.1090/S0002-9947-1926-1501332-5Search in Google Scholar

[31] E. Savaş, On strong double matrix summability via ideals, Filomat 26 (2012), no. 6, 1143–1150. 10.2298/FIL1206143SSearch in Google Scholar

[32] E. Savaş and R. F. Patterson, Double sequence spaces defined by a modulus, Math. Slovaca 61 (2011), no. 2, 245–256. 10.2478/s12175-011-0009-2Search in Google Scholar

[33] U. Ulusu and F. Nuray, On strongly lacunary summability of sequences of sets, J. Appl. Math. Bioinform. 3 (2013), no. 2, 75–88. 10.1155/2013/310438Search in Google Scholar

[34] 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

[35] M. Zeltser, M. Mursaleen and S. A. Mohiuddine, On almost conservative matrix methods for double sequence spaces, Publ. Math. Debrecen 75 (2009), no. 3–4, 387–399. 10.5486/PMD.2009.4396Search in Google Scholar

Received: 2019-08-21
Revised: 2021-06-12
Accepted: 2021-07-03
Published Online: 2021-10-10
Published in Print: 2021-11-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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