On a new class of generalized difference sequence spaces of fractional order defined by modulus function





Difference operator Δᵅᵥ, Paranormed sequence, Lacunary sequence, Modulus function


Recently Baliarsingh and Dutta [11], [12] introduced the fractional difference operator Δα , defined by Δα(xk) = art31proya.png and defined new classes of generalized difference sequence spaces of fractional order X(Γ, Δα, u) where X = {ℓ∞, c, c0} . More recently, Kadak [21] studied strongly Cesàro and statistical difference sequence space of fractional order involving lacunary sequences using the fractional difference operator art31proyb.png


is is any fixed sequence of positive real or complex numbers. Following Baliarsingh and Dutta [11], [12] and Kadak [21], we introduce paranormed difference sequence spaces art31proyd.png of fractional order involving lacunary sequence, θ and modulus function, f. We investigate topological structures of these spaces and examine various inclusion relations.

Author Biography

Taja Yaying, Dera Natung Govt. College.

Department of Mathematics.


H. Kizmaz, “On Certain Sequence Spaces”, Canadian Mathematical Bulletin, vol. 24, no. 2, pp. 169–176, Jun. 1981, doi: 10.4153/CMB-1981-027-5.

M. Et and R. Colak, “On some generalized difference sequence spaces”, Soochow Journal of Mathematics, vol. 21, no. 4, pp. 377-386, Oct. 1995. [On line]. Available: http://bit.ly/2yGK1P4

M. Et and A. Esi, “On Köthe-Toeplitz duals of generalized difference sequence spaces”, Bulletin of the Malaysian Mathematical Science Society (Second series), vol. 23, no. 1, pp. 25-32, 2000. [On line]. Available: http://bit.ly/2KhvMqx

M. Et and M. Basarir, “On some new generalized difference sequence spaces”, Periodica Mathematica Hungarica, vol. 35, no. 3, pp. 169–175, Dec. 1997, doi: 10.1023/A:1004597132128.

E. Malkowsky and S. D. Parashar, “Matrix transformations in spaces of bounded and convergent difference sequences of order m”, Analysis, vol. 17, no. 1, pp. 87–98, 1997, doi: 10.1524/anly.1997.17.1.87.

R. Colak, “Lacunary strong convergence of difference sequences with respect to a modulus function”, Filomat, vol. 17, pp. 9-14, 2003. [On line]. Available: http://bit.ly/2ZJD0sA

C. Aydın and F. Başar, “Some new difference sequence spaces”, Applied Mathematics and Computation, vol. 157, no. 3, pp. 677–693, Oct. 2004, doi: 10.1016/j.amc.2003.08.055.

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.

Ç. Bektaş, M. Et, and R. Çolak, “Generalized difference sequence spaces and their dual spaces”, Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 423–432, Apr. 2004, doi: 10.1016/j.jmaa.2003.12.006.

P. Baliarsingh, “Some new difference sequence spaces of fractional order and their dual spaces”, Applied Mathematics and Computation, vol. 219, no. 18, pp. 9737–9742, May 2013, doi: 10.1016/j.amc.2013.03.073.

P. Baliarsingh and S. Dutta, “A unifying approach to the difference operators and their applications”, Boletim da Sociedade Paranaense de Matemática, vol. 33, no. 1, pp. 49-57, 2013, doi: 10.5269/bspm.v33i1.19884.

P. Baliarsingh and S. Dutta, “On the classes of fractional order difference sequence spaces and their matrix transformations”, Applied Mathematics and Computation, vol. 250, pp. 665–674, Jan. 2015, doi: 10.1016/j.amc.2014.10.121.

U. Kadak and P. Baliarsingh, “On certain Euler difference sequence spaces of fractional order and related dual properties”, Journal of Nonlinear Sciences and Applications, vol. 08, no. 06, pp. 997–1004, Nov. 2015, doi: 10.22436/jnsa.008.06.10.

H. Fast, “Sur la convergence statistique”, Colloqium Mathematicum, vol. 2, no. 3-4, pp. 241-244, 1951. [On line]. Available: http://bit.ly/2M1DC9M

I. J. Schoenberg, “The Integrability of Certain Functions and Related Summability Methods”, The American Mathematical Monthly, vol. 66, no. 5, pp. 361–775, 1959, doi: 10.1080/00029890.1959.11989303.

I. Maddox, “Paranormed sequence spaces generated by infinite matrices”, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 64, no. 2, pp. 335–340, Apr. 1968, doi: 10.1017/S0305004100042894.

H. Nakano, “Concave modulars.”, Journal of the Mathematical Society of Japan, vol. 5, no. 1, pp. 29–49, 1953, doi: 10.2969/jmsj/00510029.

I. J. Maddox, “Spaces Of Strongly Summable Sequences”, The Quarterly Journal of Mathematics, vol. 18, no. 1, pp. 345–355, Jan. 1967, doi: 10.1093/qmath/18.1.345.

W. H. Ruckle, “FK Spaces in Which the Sequence of Coordinate Vectors is Bounded”, Canadian Journal of Mathematics, vol. 25, no. 5, pp. 973–978, Oct. 1973, doi: 10.4153/CJM-1973-102-9.

I. Maddox, Elements of functional analysis. 2nd ed. Cambridge: Cambridge Univ. Press, 1989.

U. Kadak, “Generalized Lacunary Statistical Difference Sequence Spaces of Fractional Order”, International Journal of Mathematics and Mathematical Sciences, vol. 2015, Article ID 984283, 6 pages, 2015, doi: 10.1155/2015/984283.

B. Tripathy and M. Et, “On generalized difference lacunary statistical convergence”, Studia Universitatis Babeş-Bolyai Mathematica, vol. 50, no. 1, pp. 119-130, Mar. 2015. [On line]. Available: http://bit.ly/2TdQaMk

B. Tripathy and S. Mahanta, “On a Class of Generalized Lacunary Difference Sequence Spaces Defined by Orlicz Functions”, Acta Mathematicae Applicatae Sinica, English Series, vol. 20, no. 2, pp. 231–238, Jun. 2004, doi: 10.1007/s10255-004-0163-1.

B. Tripathy and A. Baruah, “Lacunary statistically convergent and lacunary strongly convergent generalized difference sequences of fuzzy real numbers”, Kyungpook Mathematical Journal, vol. 50, no. 4, pp. 565-574, 2010. [On line]. Available: http://bit.ly/33d71TP

B. Tripathy and P. Chandra, “On some generalized difference paranormed sequence spaces associated with multiplier sequence defined by modulus function”, Analysis in Theory and Applications, vol. 27, no. 1, pp. 21–27, Mar. 2011, doi: 10.1007/s10496-011-0021-y.

B. Tripathy and H. Dutta, “On some lacunary difference sequence spaces defined by a sequence of Orlicz functions and q-lacunary ∆n m-statistical convergence”, Analele Stiintifice ale Universitatii Ovidius, Seria Matematica, vol. 20, no. 1, pp. 417-430, 2012. [On line]. Available: http://bit.ly/2Kfkzqx

B. Tripathy, B. Hazarika and B. Choudhary, “Lacunary I-convergent sequences”, Kyungpook Mathematical Journal, vol. 52, no. 4, pp. 473-482, 2012. [On line]. Available: http://bit.ly/2yEvinR

M. Et, “Generalized Cesàro difference sequence spaces of non-absolute type involving lacunary sequences”, Applied Mathematics and Computation, vol. 219, no. 17, pp. 9372–9376, May 2013, doi: 10.1016/j.amc.2013.03.039.

S. Dutta and P. Baliarsingh, “A note on paranormed difference sequence spaces of fractional order and their matrix transformations”, Journal of the Egyptian Mathematical Society, vol. 22, no. 2, pp. 249–253, Jul. 2014, doi: 10.1016/j.joems.2013.07.001.

S. Demiriz and O. Duyar, “On some new difference sequence of fractional order”, 2014, arXiv:1408.1269v1.

J. Fridy and C. Orhan, “Lacunary statistical convergence”, Pacific Journal of Mathematics, vol. 160, no. 1, pp. 43–51, Sep. 1993, doi: 10.2140/pjm.1993.160.43.

S. Pehlivan and B. Fisher, “On Some Sequence Spaces”, Indian Journal of Pure and Applied Mathematics, vol. 25, no. 10, pp. 1067-1071, Oct. 1994. [On line]. Available: http://bit.ly/2YEmSHK

H. Furkan, “On some λ difference sequence spaces of fractional order”, Journal of the Egyptian Mathematical Society, vol. 25, no. 1, pp. 37–42, Jan. 2017, doi: 10.1016/j.joems.2016.06.005.

A. Freedman, J. Sember and M. Raphael, “Some Cesàro-Type Summability Spaces”, Proceedings of the London Mathematical Society, vol. s3-37, no. 3, pp. 508–520, Nov. 1978, doi: 10.1112/plms/s3-37.3.508.



How to Cite

T. Yaying, “On a new class of generalized difference sequence spaces of fractional order defined by modulus function”, Proyecciones (Antofagasta, On line), vol. 38, no. 3, pp. 485-497, Aug. 2019.