STATISTICAL CONVERGENCE IN A BICOMPLEX VALUED METRIC SPACE

Subhajit Bera     (Tripura University, Suryamaninagar-799022, Agartala, India)
Binod Chandra Tripathy     (Tripura University, Suryamaninagar-799022, Agartala, India)

Abstract


In this paper, we study some basic properties of bicomplex numbers. We introduce two different types of partial order relations on bicomplex numbers, discuss bicomplex valued metric spaces with respect to two different partial orders, and compare them. We also define a hyperbolic valued metric space, the density of natural numbers, the statistical convergence, and the statistical Cauchy property of a sequence of bicomplex numbers and investigate some properties  in a bicomplex metric space and prove that a bicomplex metric space is complete if and only if two complex metric spaces are complete.

Keywords


Partial order, Bi-complex valued metric space, Statistically convergent

Full Text:

PDF

References


REFERENCES

  1. Azam A., Fisher B. and Khan M. Common fixed point theorems in complex valued metric spaces. Numer. Funct. Anal. Optim., 2011. Vol. 32, No. 3. P. 243–253. DOI:  10.1080/01630563.2011.533046
  2. Buck R.C. Generalized asymptotic density. Amer. J. Math., 1953. Vol. 75, No. 2. P. 335–346. DOI: 10.2307/2372456
  3. Beg I., Datta S.K., Pal D. Fixed point in bicomplex valued metric spaces. Int. J. Nonlinear Anal. Appl., 2021. Vol. 12, No. 2. P. 717–727. DOI: 10.22075/IJNAA.2019.19003.2049
  4. Choi J., Datta S.K., Biswas T., Islam Md N. Some fixed point theorems in connection with two weakly compatible mappings in bicomplex valued metric spaces. Honam Mathematical J., 2017. Vol. 39, No. 1. P. 115–126. DOI: 10.5831/HMJ.2017.39.1.115
  5. Das N.R., Dey R., Tripathy B.C. Statistically convergent and statistically Cauchy sequence in a cone metric space. TWMS J. Pure Appl. Math., 2014. Vol. 5, No. 1. P. 59–65.
  6. Fast H. Sur la convergence statistique. Colloq. Math., 1951. Vol. 2, No. 3–4. P. 241–244. (in French)
  7. Fridy J.A. On statistical convergence. Analysis, 1985. Vol. 5, No. 4. P. 301–313. DOI: 10.1524/anly.1985.5.4.301
  8. Rath D., Tripathy B.C. Matrix maps on sequence spaces associated with sets of integers. Indian J. Pure Appl. Math., 1996. Vol. 27, No. 2. P. 197–206.
  9. Rochon D., Shapiro M. On algebraic properties of bicomplex and hyperbolic numbers. An. Univ. Oradea Fasc. Mat., 2004. Vol. 11. P. 71–110.
  10. Sager N., Sağir B. On completeness of some bicomplex sequence spaces. Palestine J. Math., 2020. Vol. 9, No. 2. P. 891–902.
  11. Šalát T. On statistically convergent sequences of real numbers. Math. Slovaca, 1980. Vol. 30, No. 2. P. 139–150.
  12. Schoenberg I.J. The integrability of certain functions and related summability methods. Amer. Math Monthly, 1959. Vol. 66, No. 5. P. 361–375. DOI: 10.2307/2308747
  13. Segre C. Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici. Math. Ann., 1892. Vol. 40. P. 413–467. DOI: 10.1007/BF01443559 (in Italiano)
  14. Singh S. A Study of Bicomplex Space with a Topological View Point. Thesis. Punjab: Lovely Professional University, 2018. 120 p.
  15. Srivastava R.K., Srivastava N.K. On a class of entire bicomplex sequences. South East Asian J. Math. Math. Sci., 2007. Vol. 5, No. 3. p. 47–68.
  16. Tripathy B.C., Nath P.K. Statistical convergence of complex uncertain sequences. New Math. Nat. Comput., 2017. Vol. 13, No. 3. P. 359–374. DOI: 10.1142/S1793005717500090
  17. Tripathy B.C. Matrix transformations between some classes of sequences. J. Math. Anal. Appl., 1997. Vol. 206, No. 2. P. 448–450. DOI: 10.1006/jmaa.1997.5236
  18. Tripathy B.C., Sen M. On generalized statistically convergent sequences. Indian J. Pure Appl. Math., 2001. Vol. 32, No. 11. P. 1689–1694.
  19. Tripathy B.C. On generalized difference paranormed statistically convergent sequences. Indian J. Pure Appl. Math., 2004. Vol. 35, No. 5. P. 655–663.
  20. Wagh M.A. On certain spaces of bicomplex sequences. Inter. J. Phy. Chem. Math. Fundam., 2014. Vol. 7, No. 1. P. 1–6.




DOI: http://dx.doi.org/10.15826/umj.2023.1.004

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.