A relation of Banach limit and difference matrix to generate some Orlicz sequence spaces





Almost convergence, Paranorm space, Orlicz function, β-and γ-duals, Matrix transformations


In the present paper we study the applications of generalized difference matrices and Orlicz function to generate some spaces of almost convergent sequences. We make an effort to study some algebraic and topological properties of these sequence spaces. Some inclusión relations between these spaces are establish. Furthermore, we study matrix transformations and compute β−, γ− duals of these spaces.

Author Biographies

Kuldip Raj, Shri Mata Vaishno Devi University.

School of Mathematics.

Charu Sharma, Shri Mata Vaishno Devi University.

School of Mathematics.


F. Başar, “Matrix transformations between certain sequence spaces of Xp and lp”, Soochow journal of mathematics, vol. 26, no. 2, pp. 191–204, 2000.

F. Başar, Summability theory and its applications. Istanbul: Bentham ebooks, 2012, doi: 10.2174/97816080545231120101

F. Başar and M. Kirişçi, “Almost convergence and generalized difference matrix”, Computers & mathematics with applications, vol. 61, no. 3, pp. 602–611, Feb. 2011, doi: 10.1016/j.camwa.2010.12.006

H. Dutta and F. Başar, “A generalization of Orlicz sequence spaces by Cesàro mean of order one”, Acta mathematica universitatis comenianae, vol. 80, no. 2, pp. 185-200, (2011).[On line] Available: http://bit.ly/2nbb9Uk

B. Kuttner, “On dual summability methods”, Mathematical proceedings of the cambridge philosophical society, vol. 71, no. 1, pp. 67–73, Jan. 1972, doi: 10.1017/S0305004100050283

J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces”, Israel journal of mathematics, vol. 10, no. 3, pp. 379–390, Sep. 1971, doi: 10.1007/BF02771656

G. Lorentz, “A contribution to the theory of divergent sequences”, Acta mathematica, vol. 80, pp. 167–190, 1948, doi: 10.1007/BF02393648

G. Lorentz and K. Zeller, “Summation of sequences and summation of series”, Proceedings of the american mathematical society, vol. 15, no. 5, pp. 743–746, Oct. 1964, doi: 10.2307/2034590

L. Maligranda, Orlicz spaces and interpolation, vol. 5. Campinas, SP: Universidade Estadual de Campinas, 1989.

M. Mursaleen, S. Sharma, S. Mohiuddine, and A. Kılıçman, “New difference sequence spaces defined by Musielak-Orlicz function”, Abstract and applied analysis, vol. 2014, ID 691632, Jul. 2014, doi:10.1155/2014/691632

M. Mursaleen, “Generalized spaces of difference sequences”, Journal of mathematical analysis and applications, vol. 203, no. 3, pp. 738–745, Nov. 1996, doi: 10.1006/jmaa.1996.0409

J. Musielak, Orlicz spaces and modular spaces, vol. 1034. Berlin: Springer, 1983, doi: 10.1007/BFb0072210

G. Petersen, Regular matrix transformations. London: McGraw-Hill, 1966.

K. Raj, A. Choudhary, and C. Sharma, “Almost strongly Orlicz double sequence spaces of regular matrices and their applications to statistical convergence”, Asian-European journal of mathematics, vol. 11, no. 05, ID 1850073, 2018, doi: 10.1142/S1793557118500730

K. Raj and A. Kılıçman, “On certain generalized paranormed spaces”, Journal of inequalities and applications, vol. 2015, no. 1, Jan. 2015, doi: 10.1186/s13660-015-0565-z

K. Raj and C. Sharma, “Applications of strongly convergent sequences to Fourier series by means of modulus functions”, Acta mathematica hungarica, vol. 150, no. 2, pp. 396–411, Aug. 2016, doi: 10.1007/s10474-016-0655-5

K. Raj and S. Sharma, “Some multiplier generalized difference sequence spaces over n-normed spaces defined by a Musielak-Orlicz function”, Siberian advances in mathematics, vol. 24, no. 3, pp. 193–203, Aug. 2014, doi: 10.3103/S1055134414030067

A. Sönmez, “Almost convergence and triple band matrix”, Mathematical and computer modelling, vol. 57, no. 9-10, pp. 2393–2402, May 2013, doi: 10.1016/j.mcm.2011.11.079

A. Wilansky, Ed., Summability through functional analysis, vol. 85. Amsterdam: Elsevier, 1984. Available: https://bit.ly/2npkCHC

M. Yeşilkayagil and F. Başar, “Spaces of Aλ-almost null and Aλ-almost convergent sequences”, Journal of the egyptian mathematical society, vol. 23, no. 1, pp. 119–126, Apr. 2015 , doi: 10.1016/j.joems.2014.01.013

Z. Zararsız and M. Şengönül, "On the almost convergence", Doctoral Thesis, Nevşehir Hacı Bektaş Veli University, 2019.

Z. Zararsiz, “On the extensions of the almost convergence idea and core theorems”, Journal of nonlinear sciences and applications, vol. 09, no. 01, pp. 112–125, Jan. 2016, doi: 10.22436/jnsa.009.01.11



How to Cite

K. Raj and C. Sharma, “A relation of Banach limit and difference matrix to generate some Orlicz sequence spaces”, Proyecciones (Antofagasta, On line), vol. 38, no. 4, pp. 637-652, Sep. 2019.