Home On sequence spaces generated by binomial difference operator of fractional order
Article
Licensed
Unlicensed Requires Authentication

On sequence spaces generated by binomial difference operator of fractional order

  • Taja Yaying and Bipan Hazarika EMAIL logo
Published/Copyright: July 19, 2019
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 69 Issue 4

Abstract

In this article we introduce binomial difference sequence spaces of fractional order α, bpr,s (Δ(α)) (1 ≤ p ≤ ∞) by the composition of binomial matrix, Br,s and fractional difference operator Δ(α), defined by (Δ(α)x)k = ∑i=0∞(−1)iΓ(α+1)i!Γ(α−i+1)xk−i. We give some topological properties, obtain the Schauder basis and determine the α, β and γ-duals of the spaces. We characterize the matrix classes (bpr,s(Δ(α)), Y), where Y ∈ {ℓ∞, c, c0, ℓ1} and certain classes of compact operators on the space bpr,s(Δ(α)) using Hausdorff measure of non-compactness. Finally, we give some geometric properties of the space bpr,s(Δ(α)) (1 < p < ∞).

MSC 2010: Primary 46A45; Secondary 46A35
Keywords: binomial difference sequence space; difference operator Δ(α); Schauder basis; α, β, γ-duals; matrix transformation; compact operator; Hausdorff measure of non-compactness; geometric properties

  1. (Communicated by Werner Timmerman)

Acknowledgement

The authors thank the referee for the valuable suggestions for improvement of the article.

References

[1] Altay, B.—Polat, H: On some new Euler difference sequence spaces, Southeast Asian Bull. Math. 30 (2006), 209–220.Search in Google Scholar

[2] Altay, B.—Başar, F.—Mursaleen, M.: On the Euler sequence spaces which include the spacesℓpandℓ∞I, Inf. Sci. 176(10) (2006), 1450–1462.10.1016/j.ins.2005.05.008Search in Google Scholar

[3] Baliarsingh, P.—Dutta, S.: A unifying approach to the difference operators and their applications, Bol. Soc. Parana. Mat. 33 (2015), 49–57.10.5269/bspm.v33i1.19884Search in Google Scholar

[4] Baliarsingh, P.—Dutta, S.: 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

[5] Baliarsingh, P.: Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput. 219 (2013), 9737–9742.10.1016/j.amc.2013.03.073Search in Google Scholar

[6] Baliarsingh, P.—Dutta, S.: On an explicit formula for inverse of triangular matrices, J. Egyptian Math. Soc. 23 (2015), 297–302.10.1016/j.joems.2014.06.001Search in Google Scholar

[7] Bekatş, C.—Et, M.—Çolak, R.: Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl. 292 (2004), 423–432.10.1016/j.jmaa.2003.12.006Search in Google Scholar

[8] Beauzamy, B.: Banach-Saks properties and spreading models, Math. Scand. 44 (1997), 357–384.10.7146/math.scand.a-11818Search in Google Scholar

[9] Bişgin, M. C.: The binomial sequence spaces of nonabsolute type, J. Inequal. Appl. 309 (2016), 16 pp.10.1186/s13660-016-1256-0Search in Google Scholar

[10] Bişgin, M. C.: The binomial sequence spaces which include the spacesℓpandℓ∞and geometric properties, J. Inequal. Appl. 304 (2016) 15 pp.10.1186/s13660-016-1252-4Search in Google Scholar

[11] Chandra, P.—Tripathy, B. C.: On generalised Köthe-Toeplitz duals of some sequence spaces, Indian J. Pure Appl. Math. 33 (2002), 1301–1306.Search in Google Scholar

[12] Dutta, S.—Baliarsingh, P.: On some Toeplitz matrices and their inversion, J. Egyptian Math. Soc. 22 (2014), 420-423.10.1016/j.joems.2013.10.001Search in Google Scholar

[13] Dutta, S.—Baliarsingh, P.: A note on paranormed difference sequence spaces of fractional order and their matrix transformations, J. Egyptian Math. Soc. 22 (2014), 249–253.10.1016/j.joems.2013.07.001Search in Google Scholar

[14] Djolović, I, —Malkowsky, E.: A note on compact operators on matrix domains, J. Math. Anal. Appl. 340(1) (2008), 291–303.10.1016/j.jmaa.2007.08.021Search in Google Scholar

[15] Djolović, I.: Compact operators on the spacesa0r(Δ) andacr(Δ), J. Math. Anal. Appl. 318 (2006), 658–666.10.1016/j.jmaa.2005.05.085Search in Google Scholar

[16] Ercan, S.—Bektaş, Ç.A.: On sequence spaces of non-absolute type generated by the fractional order generalized difference matrix, J. Math. 4(2) (2015), 22–29.10.26634/jmat.4.2.3498Search in Google Scholar

[17] Et, M.—Çolak, R.: On generalized difference sequence spaces, Soochow J. Math. 21 (1995), 377–386.Search in Google Scholar

[18] Et, M.—Esi, A.: On Köthe-Toeplitz duals of generalized difference sequence spaces, Bull. Malays. Math. Sci. Soc. 231 (2000), 25–32.Search in Google Scholar

[19] Et, M.—Başarir, M.: On some new generalized difference sequence spaces, Period. Math. Hungar. 35 (1997), 169–175.10.1023/A:1004597132128Search in Google Scholar

[20] Garcia-Falset, J.: Stability and fixed points for nonexpansive mappings, Houston J. Math. 20 (1994), 495–505.10.1007/978-94-017-1748-9_7Search in Google Scholar

[21] Garcia-Falset, J.: The fixed point property in Banach spaces with NUS-property, J. Math. Anal. Appl. 215(2) (1997), 532–542.10.1006/jmaa.1997.5657Search in Google Scholar

[22] Kadak, U.—Baliarsingh, P.: On certain Euler difference sequence spaces of fractional order and related dual properties, J. Nonlinear Sci. Appl. 8 (2015), 997–1004.10.22436/jnsa.008.06.10Search in Google Scholar

[23] Kara, E. E.—Başarir, M.: On compact operators and some EulerB(m)difference sequence spaces, J. Math. Anal. Appl. 76 (2010), 87–100.10.1016/j.jmaa.2011.01.028Search in Google Scholar

[24] Karakaya, V.—Polat, H.: Some new paranormed sequence spaces defined by Euler and difference operators, Acta Sci. Math. 61(1) (2012), 1–12.10.1007/BF03549822Search in Google Scholar

[25] Kizmaz, H.: On certain sequence spaces, Canad. Math. Bull. 24 (1981), 169–176.10.4153/CMB-1981-027-5Search in Google Scholar

[26] Köthe, G.—Toeplitz, O.: Linear Raume mit unendlich vielen Koordinaten and Ringe unenlicher Matrizen, J. Reine Angew. Math. 171 (1934), 193–226.10.1515/crll.1934.171.193Search in Google Scholar

[27] Knaust, H.: Orlicz sequence spaces of Banach-Saks type, Arch. Math. (Basel) 59(6) (1998), 562–565.10.1007/BF01194848Search in Google Scholar

[28] Malkowsky, E.—Parashar, S.D.: Matrix transformations in spaces of bounded and convergent difference sequences of orderm, Analysis 17 (1997), 87–97.10.1524/anly.1997.17.1.87Search in Google Scholar

[29] Malkowsky, E.—Rakočević, V.: On matrix domains of triangles, Appl. Math. Comput. 189 (2007), 1146–1163.10.1016/j.amc.2006.12.024Search in Google Scholar

[30] Malkowsky, E.—Rakočević, V.: An introduction into the theory of sequence spaces and measure of noncompactness, Zb. Rad. (Beogr.) 9(17) (2000), 143–234.Search in Google Scholar

[31] Meng, J.—Song, M.: Binomial difference sequence space of orderm, Adv. Difference Equ. 241 (2017), 10 pages.10.1186/s13662-017-1291-2Search in Google Scholar

[32] Meng, J.—Song, M.: Some normed binomial difference sequence spaces related to theℓp spaces, J. Inequal. Appl. 128(2017) 10 pages.Search in Google Scholar

[33] Meng, J.—Song, M.: On some binomial difference sequence spaces, Kyungpook Math. J. 57 (2017), 631–640.Search in Google Scholar

[34] Meng, J.—Song, M.: On some binomialB(m)-difference sequence spaces, J. Inequal. Appl. 194 (2017), 11 pages.10.1186/s13660-017-1470-4Search in Google Scholar

[35] Mursaleen, M.—Başar, F.—Altay, B.: On the Euler sequence spaces which include the spacesℓpandℓ∞II, Nonlinear Anal. 65(3) (2006), 707–717.10.1016/j.na.2005.09.038Search in Google Scholar

[36] Polat, H.—Başar, F.: Some Euler spaces of difference sequences of orderm, Acta Math. Sci. 27 (2007), 254–266.10.1016/S0252-9602(07)60024-1Search in Google Scholar

[37] Stieglitz, M.—Tietz, H.: Matrixtransformationen von Folgenräumen eine Ergebnisübersicht, Math. Z. 154(1977), 1-16.10.1007/BF01215107Search in Google Scholar

[38] Wilansky, A.: Summability through Functional Analysis. North-Holland Mathematics Studies 85, Elsevier, Amsterdam, 1984.Search in Google Scholar

Received: 2018-08-07
Accepted: 2018-11-05
Published Online: 2019-07-19
Published in Print: 2019-08-27

© 2019 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular papers
  2. On the finite embeddability property for quantum B-algebras
  3. A dual Ramsey theorem for finite ordered oriented graphs
  4. On EMV-Semirings
  5. A nonsymmetrical matrix and its factorizations
  6. On weakly 𝓗-permutable subgroups of finite groups
  7. The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions
  8. A geometrical version of Hardy-Rellich type inequalities
  9. On a Choquet-Stieltjes type integral on intervals
  10. Uniqueness of meromorphic functions sharing four small functions on annuli
  11. A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator
  12. Functions of bounded variation related to domains bounded by conic sections
  13. Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables
  14. Oscillatory criteria for the second order linear ordinary differential equations
  15. When deviation happens between rough statistical convergence and rough weighted statistical convergence
  16. The retraction of certain banach right modules associated to a character
  17. On sequence spaces generated by binomial difference operator of fractional order
  18. Some refinements of young type inequality for positive linear map
  19. On the Gromov-Hausdorff limit of metric spaces
  20. The beta exponentiated Nadarajah-Haghighi distribution: theory, regression model and application
  21. Hilbert algebras with supremum generated by finite chains
https://doi.org/10.1515/ms-2017-0276
Keywords for this article
binomial difference sequence space; difference operator Δ(α); Schauder basis; α, β, γ-duals; matrix transformation; compact operator; Hausdorff measure of non-compactness; geometric properties

Articles in the same Issue

  1. Regular papers
  2. On the finite embeddability property for quantum B-algebras
  3. A dual Ramsey theorem for finite ordered oriented graphs
  4. On EMV-Semirings
  5. A nonsymmetrical matrix and its factorizations
  6. On weakly 𝓗-permutable subgroups of finite groups
  7. The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions
  8. A geometrical version of Hardy-Rellich type inequalities
  9. On a Choquet-Stieltjes type integral on intervals
  10. Uniqueness of meromorphic functions sharing four small functions on annuli
  11. A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator
  12. Functions of bounded variation related to domains bounded by conic sections
  13. Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables
  14. Oscillatory criteria for the second order linear ordinary differential equations
  15. When deviation happens between rough statistical convergence and rough weighted statistical convergence
  16. The retraction of certain banach right modules associated to a character
  17. On sequence spaces generated by binomial difference operator of fractional order
  18. Some refinements of young type inequality for positive linear map
  19. On the Gromov-Hausdorff limit of metric spaces
  20. The beta exponentiated Nadarajah-Haghighi distribution: theory, regression model and application
  21. Hilbert algebras with supremum generated by finite chains
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 11.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0276/html
Scroll to top button