×
Alotaibi, Abdullah; Mursaleen, Mohammad; Sharma, Sunil K.

Double sequence spaces over \(n\)-normed spaces defined by a sequence of Orlicz functions. (English) Zbl 1321.46005

J. Inequal. Appl. 2014, Paper No. 216, 12 p. (2014).
Summary: In the present paper we introduce the double sequence space \(m^2 (\mathcal M, A, \phi, p, ||\cdot, \dots, \cdot||)\) defined by a sequence of Orlicz functions over an \(n\)-normed space. We examine some of its topological properties and establish some inclusion relations.

Cited in 6 Documents

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
40A05 Convergence and divergence of series and sequences

Keywords:

double sequence spaces; paranormed space; Orlicz function; \(n\)-normed space
PDF BibTeX XML Cite
\textit{A. Alotaibi} et al., J. Inequal. Appl. 2014, Paper No. 216, 12 p. (2014; Zbl 1321.46005)
Full Text: DOI

References:

[1] Bromwich TJ: An Introduction to the Theory of Infinite Series. Macmillan Co., New York; 1965.
[2] Hardy GH: On the convergence of certain multiple series.Proc. Camb. Philos. Soc. 1917, 19:86-95.
[3] Moricz F: Extension of the spacescandfrom single to double sequences.Acta Math. Hung. 1991, 57:129-136. 10.1007/BF01903811 · Zbl 0781.46025
[4] Moricz F, Rhoades BE: Almost convergence of double sequences and strong regularity of summability matrices.Math. Proc. Camb. Philos. Soc. 1988, 104:283-294. 10.1017/S0305004100065464 · Zbl 0675.40004
[5] Başarır M, Sonalcan O: On some double sequence spaces.J. Indian Acad. Math. 1999, 21:193-200. · Zbl 0978.40002
[6] Zeltser, M., Diss. Math. Univ. Tartu 25 (2001), Tartu
[7] Mursaleen M, Edely OHH: Statistical convergence of double sequences.J. Math. Anal. Appl. 2003,288(1):223-231. 10.1016/j.jmaa.2003.08.004 · Zbl 1032.40001
[8] Mohiuddine, SA; Alotaibi, A.; Mursaleen, M., Statistical convergence of double sequences in locally solid Riesz spaces, No. 2012 (2012) · Zbl 1262.40005
[9] Mursaleen M: Almost strongly regular matrices and a core theorem for double sequences.J. Math. Anal. Appl. 2004,293(2):523-531. 10.1016/j.jmaa.2004.01.014 · Zbl 1043.40002
[10] Mursaleen M, Savas E: Almost regular matrices for double sequences.Studia Sci. Math. Hung. 2003, 40:205-212. · Zbl 1050.40003
[11] Altay B, Başar F: Some new spaces of double sequences.J. Math. Anal. Appl. 2005, 309:70-90. 10.1016/j.jmaa.2004.12.020 · Zbl 1093.46004
[12] Başar F, Sever Y:The space[InlineEquation not available: see fulltext.] of double sequences. Math. J. Okayama Univ. 2009, 51:149-157. · Zbl 1168.46300
[13] Raj K, Sharma SK: Some multiplier double sequence spaces.Acta Math. Vietnam. 2012, 37:391-406. · Zbl 1296.46004
[14] Pringsheim A: Zur Theorie der zweifach unendlichen Zahlenfolgen.Math. Ann. 1900, 53:289-321. 10.1007/BF01448977 · JFM 31.0249.01
[15] Limaye BV, Zeltser M: On the Pringsheim convergence of double series.Proc. Est. Acad. Sci. 2009, 58:108-121. 10.3176/proc.2009.2.03 · Zbl 1206.40004
[16] Cakan C, Altay B, Mursaleen M: Theσ-convergence andσ-core of double sequences.Appl. Math. Lett. 2006, 19:1122-1128. 10.1016/j.aml.2005.12.003 · Zbl 1122.40004
[17] Mursaleen M, Mohiuddine SA: Doubleσ-multiplicative matrices.J. Math. Anal. Appl. 2007, 327:991-996. 10.1016/j.jmaa.2006.04.081 · Zbl 1107.40004
[18] Mursaleen M, Mohiuddine SA: Regularlyσ-conservative andσ-coercive four dimensional matrices.Comput. Math. Appl. 2008, 56:1580-1586. 10.1016/j.camwa.2008.03.007 · Zbl 1155.40303
[19] Mursaleen M, Mohiuddine SA: Onσ-conservative and boundedlyσ-conservative four dimensional matrices.Comput. Math. Appl. 2010, 59:880-885. 10.1016/j.camwa.2009.10.006 · Zbl 1189.40005
[20] Mursaleen M, Mohiuddine SA: Convergence Methods for Double Sequences and Applications. Springer, Berlin; 2014. · Zbl 1290.40001
[21] Mohiuddine, SA; Alotaibi, A., Some spaces of double sequences obtained through invariant mean and related concepts, No. 2013 (2013) · Zbl 1277.46003
[22] Mohiuddine, SA; Raj, K.; Alotaibi, A., Some paranormed double difference sequence spaces for Orlicz functions and bounded-regular matrices, No. 2014 (2014) · Zbl 1469.53022
[23] Sharma SK, Raj K, Sharma AK: Some new double sequence spaces overn-normed space.Int. J. Appl. Math. 2012, 25:255-269. · Zbl 1278.40001
[24] Lindenstrauss J, Tzafriri L: On Orlicz sequence spaces.Isr. J. Math. 1971, 10:345-355. · Zbl 0227.46042
[25] Musielak, J., Lecture Notes in Mathematics 1034 (1983)
[26] Maligranda, L., Seminars in Mathematics 5 (1989), Warsaw
[27] Raj K, Sharma SK: Some multiplier sequence spaces defined by a Musielak-Orlicz function inn-normed spaces.N.Z. J. Math. 2012, 42:45-56. · Zbl 1257.46005
[28] Raj K, Sharma SK: Some double sequence spaces defined by a sequence of Orlicz function.J. Math. Anal. 2012, 3:12-20. · Zbl 1312.40003
[29] Raj K, Sharma SK: Some generalized difference double sequence spaces defined by a sequence of Orlicz-function.CUBO 2012, 14:167-189. 10.4067/S0719-06462012000300011 · Zbl 1295.46003
[30] Wilansky, A., North-Holland Math. Stud. 85 (1984)
[31] Gähler S: Lineare 2-normierte Räume.Math. Nachr. 1965, 28:1-43. · Zbl 0142.39803
[32] Misiak A: n-Inner product spaces.Math. Nachr. 1989, 140:299-319. 10.1002/mana.19891400121 · Zbl 0673.46012
[33] Gunawan H: Onn-inner product,n-norms, and the Cauchy-Schwartz inequality.Sci. Math. Jpn. 2001, 5:47-54.
[34] Gunawan H: The space ofp-summable sequence and its naturaln-norm.Bull. Aust. Math. Soc. 2001, 64:137-147. 10.1017/S0004972700019754 · Zbl 1002.46007
[35] Gunawan H, Mashadi M: Onn-normed spaces.Int. J. Math. Math. Sci. 2001, 27:631-639. 10.1155/S0161171201010675 · Zbl 1006.46006
[36] Sargent WLC:Some sequence spaces related to the[InlineEquation not available: see fulltext.] spaces. J. Lond. Math. Soc. 1960, 35:161-171. · Zbl 0090.03703
[37] Malkowsky E, Mursaleen M:Matrix transformations between FK-spaces and the sequence spaces[InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.]. J. Math. Anal. Appl. 1995, 196:659-665. 10.1006/jmaa.1995.1432 · Zbl 0846.40004
[38] Tripathy BC, Sen M:On a new class of sequences related to the space[InlineEquation not available: see fulltext.]. Tamkang J. Math. 2002, 33:167-171. · Zbl 1005.46002
[39] Mursaleen M:On some geometric properties of a sequence space related to[InlineEquation not available: see fulltext.]. Bull. Aust. Math. Soc. 2003, 67:343-347. 10.1017/S0004972700033803 · Zbl 1029.46012
[40] Duyar, C.; Oǧur, O., On a new space [InlineEquation not available: see fulltext.] of double sequences, No. 2013 (2013)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.