Difference sequence spaces based on Lucas band matrix and modulus function


In this work, we define some difference sequence spaces based upon Lucas band matrix and associated with the idea of sequence of modulus functions and then establish that aforementioned spaces are BK-spaces of non absolute type.We further investigate that our spaces are linearly isomorphic to the space l(p), where \(l(p)=\{x=(x_{k}) \in w:\sum _{k}|x_{k}|^{p_{k}}<\infty \}\). Moreover, we demonstrate inclusion relationship between newly defined spaces as well as establish certain geometrical properties.

This is a preview of subscription content, access via your institution.


  1. 1.

    Altay, B., Başar, F.: On some Euler sequence spaces of nonabsolute type. Ukrainian Math. J. 57, 1–17 (2005)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Altay, B., Başar, F.: The matrix domain and the fine spectrum of the difference operator \(\Delta\) on the sequence space \(\ell _{p}\), \((0<p<1)\). Commun. Math. Anal. 2, 1–11 (2007)

    MathSciNet  Google Scholar 

  3. 3.

    Altay, B., Başar, F., Mursaleen, M.: On the Euler sequence spaces which include the spaces \(\ell _{p}\) and \(\ell _{\infty }\) I. Inform. Sci. 176, 1450–1462 (2006)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Altin, Y., Et, M.: Generalized difference sequence spaces defined by a modulus function in a locally convex space. Soochow. J. Math. 31, 233–243 (2005)

    MathSciNet  Google Scholar 

  5. 5.

    Clarkson, J.A.: Uniformly convex spaces. Trans. Am. Math. Soc. 40, 396–414 (1936)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Debnath, P.: Set-valued Meir–Keeler, Geraghty and Edelstein type fixed point results in \(b\)-metric spaces. Rend. Circ. Mat. Palermo II. Ser (2020). https://doi.org/10.1007/s12215-020-00561-y

    Article  Google Scholar 

  7. 7.

    Debnath, P., Neog, M., Radenović, S.: Set valued Reich type \(G\)-contractions in a complete metric space with graph. Rend. Circ. Mat. Palermo II. Ser 69, 917–924 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Debnath, P., Sen, M.D.L.: Contractive inequalities for some asymptotically regular set-valued mappings and their fixed points. Symmetry 12(3), 411 (2020)

    Article  Google Scholar 

  9. 9.

    Debnath, P., Srivastava, H.M.: New extensions of Kannan’s and Reich’s fixed point theorems for multivalued maps using Wardowski’s technique with application to integral equations. Symmetry 12(7), 1090 (2020)

    Article  Google Scholar 

  10. 10.

    Debnath, P., Srivastava, H.M.: Global optimization and common best proximity points for some multivalued contractive pairs of mappings. Axioms 9(3), 102 (2020)

    Article  Google Scholar 

  11. 11.

    Et, M.: Generalized Cesàro difference sequence spaces of non-absolute type involving lacunary sequences. Appl. Math. Comput. 219, 9372–9376 (2013)

    MathSciNet  Google Scholar 

  12. 12.

    Et, M., Altinok, H., Çolak, R.: On \(\lambda\)-statistical convergence of difference sequences of fuzzy numbers. Inform. Sci. 176, 2268–2278 (2006)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Et, M., Çolak, R.: On some generalized difference sequence spaces. Soochow J. Math. 21(4), 377–386 (1995)

    MathSciNet  Google Scholar 

  14. 14.

    Et, M., Mursaleen, M., Işik, M.: On a class of fuzzy sets defined by Orlicz functions. Filomat 27(5), 789–796 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    García-Falset, J.: Stability and fixed points for nonexpansive mappings. Houst. J. Math. 20, 495–506 (1994)

    MathSciNet  Google Scholar 

  16. 16.

    García-Falset, J.: The fixed point property in Banach spaces with the NUS-property. J. Math. Anal. Appl. 215, 532–542 (1997)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Gurarii, V.I.: On differential properties of the convexity moduli of Banach spaces. Mat. Issled. 2, 141–148 (1967)

    MathSciNet  Google Scholar 

  18. 18.

    Hazar, G.C., Sarigöl, M.A.: Absolute Cesàro series spaces and matrix operators. Acta Appl. Math. 154, 153–165 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Kadak, U., Mohiuddine, S.A.: Generalized statistically almost convergence based on the difference operator which includes the \((p, q)\)-gamma function and related approximation theorems. Results Math. 73, 9 (2018)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Kara, E.E.: On some topological and geometric properties of new Banach sequence spaces. J. Inequal. Appl. 2013, 38 (2013)

    Article  Google Scholar 

  21. 21.

    Karakas, M., Karakas, A.M.: A study on Lucas difference sequence spaces \(\ell _{p}({\hat{E}}(r, s))\) and \(\ell _{\infty }({\hat{E}}(r, s))\). Maejo Int. J. Sci. Technol. 12, 70–78 (2018)

    Google Scholar 

  22. 22.

    Kızmaz, H.: On certain sequence spaces. Canad. Math. Bull. 24, 169–176 (1981)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Kirişçi, M., Başar, F.: Some new sequence spaces derived by the domain of generalized difference matrix. Comput. Math. Appl. 60, 1299–1309 (2010)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Koshy, T.: Fibonacci and Lucas Numbers with Applications. Wiley, New York (2001)

    Google Scholar 

  25. 25.

    Maddox, I.J.: Spaces of strongly summable sequences. Quart. J. Math. Oxford 18, 345–355 (1967)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Mursaleen, M., Noman, A.K.: On some new sequence spaces of non-absolute type related to the spaces \(\ell _{p}\) and \(\ell _{\infty }\) I. Filomat 25, 33–51 (2011)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Mohiuddine, S.A., Alamri, B.A.S.: Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. Racsam 113(3), 1955–1973 (2019)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Mohiuddine, S.A., Asiri, A., Hazarika, B.: Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems. Int. J. Gen. Syst. 48(5), 492–506 (2019)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Mohiuddine, S.A., Hazarika, B.: Some classes of ideal convergent sequences and generalized difference matrix operator. Filomat 31(6), 1827–1834 (2017)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Mohiuddine, S.A., Raj, K., Mursaleen, M., Alotaibi, A.: Linear isomorphic spaces of fractional-order difference operators. Alexandria Eng. J. (2020). https://doi.org/10.1016/j.aej.2020.10.039

    Article  Google Scholar 

  31. 31.

    Raj, K., Sharma, C.: Applications of strongly convergent sequences to Fourier series by means of modulus functions. Acta Math. Hungar. 150, 396–411 (2016)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Sanchez, L., Ullan, A.: Some properties of Gurarii’s modulus of convexity. Archiv der Math. 71, 399–406 (1998)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Sarigöl, M.A.: On difference sequence spaces. J. Karadeniz Tech. Univ. Fac. Arts Sci. Ser. Math. Phys. 10, 63–71 (1987)

    MathSciNet  Google Scholar 

  34. 34.

    Srivastava, H.M., Shehata, A., Moustafa, S.I.: Some fixed point theorems for \(F(\psi,\varphi )\)-contractions and their application to fractional differential equations. Russian J. Math. Phys. 27, 385–398 (2020)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Yaying, T., Hazarika, B., Mohiuddine, S.A., Mursaleen, M., Ansari, K.J.: Sequence spaces derived by the triple band generalized Fibonacci difference operator. Adv. Differ. Equ. 2020, 639 (2020)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to S. A. Mohiuddine.

Ethics declarations

Conflict of interest

The authors declare that they have no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by H. M. Srivastava.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mohiuddine, S.A., Raj, K. & Choudhary, A. Difference sequence spaces based on Lucas band matrix and modulus function. São Paulo J. Math. Sci. (2021). https://doi.org/10.1007/s40863-020-00203-2

Download citation


  • Lucas number
  • Modulus function
  • Weak fixed point property
  • Modulus of convexity
  • BK-spaces

Mathematics Subject Classification

  • 40A05
  • 40D25