A Variation on Statistical Ward Continuity

Article
  • 156 Downloads

Abstract

A sequence \((\alpha _{k})\) of points in \(\mathbb {R}\), the set of real numbers, is called \(\rho \) -statistically convergent to an element \(\ell \) of \(\mathbb {R}\) if
$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\rho _{n}}|\{k\le n: |\alpha _{k}-\ell |\ge {\varepsilon }\}|=0 \end{aligned}$$
for each \(\varepsilon >0\), where \(\varvec{\rho }=(\rho _{n})\) is a non-decreasing sequence of positive real numbers tending to \(\infty \) such that \(\limsup _{n} \frac{\rho _{n}}{n}<\infty \), \(\Delta \rho _{n}=O(1)\), and \(\Delta \alpha _{n} =\alpha _{n+1}-\alpha _{n}\) for each positive integer n. A real-valued function defined on a subset of \(\mathbb {R}\) is called \(\rho \)-statistically ward continuous if it preserves \(\rho \)-statistical quasi-Cauchy sequences where a sequence \((\alpha _{k})\) is defined to be \(\rho \)-statistically quasi-Cauchy if the sequence \((\Delta \alpha _{k})\) is \(\rho \)-statistically convergent to 0. We obtain results related to \(\rho \)-statistical ward continuity, \(\rho \)-statistical ward compactness, ward continuity, continuity, and uniform continuity. It turns out that the set of uniformly continuous functions coincides with the set of \(\rho \)-statistically ward continuous functions not only on a bounded subset of \(\mathbb {R}\), but also on an interval.

Keywords

Summability Statistical convergent sequences Quasi-Cauchy sequences Boundedness Uniform continuity 

Mathematics Subject Classification

Primary: 40A05 Secondaries: 26A05 26A15 26A30 

Notes

Acknowledgments

The author would like to thank the referees for a careful reading and several constructive comments that have improved the presentation of the results.

References

  1. 1.
    Alghamdi, M.A., Mursaleen, M.: \(\lambda \)-Statistical convergence in paranormed space. Abstr. Appl. Anal. 2013, (2013). Article ID 264520Google Scholar
  2. 2.
    Alghamdi, M.A., Mursaleen, M., Alotaibi, A.: Logarithmic density and logarithmic statistical convergence. Adv. Differ. Equ. 2013, 227 (2013). doi: 10.1186/1687-1847-2013-227 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alotaibi, A., Mursaleen, M.: Generalized statistical convergence of difference sequences. Adv. Differ. Equ. 2013, 212 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alotaibi, A., Mursaleen, M.: Statistical convergence in random paranormed space. J. Comput. Anal. Appl. 17(2), 297–304 (2014)MathSciNetMATHGoogle Scholar
  5. 5.
    Aras, C.G., Sonmez, A., Çakallı, H.: On soft mappings, arXiv:1305.4545v1 (2013)
  6. 6.
    Buck, R.C.: Generalized asymptotic density. Am. J. Math. 75, 335–346 (1953)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Burton, D., Coleman, J.: Quasi-Cauchy sequences. Am. Math. Mon. 117, 328–333 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cakalli, H.: N-theta-ward continuity. Abstr. Appl. Anal. 2012 (2012). Article ID 680456Google Scholar
  9. 9.
    Connor, J.S.: The statistical and strong p-Cesaro convergence of sequences. Analysis 8, 47–63 (1988)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Connor, J.S.: R-type summability methods, Cauchy criteria, p-sets, and statistical convergence. Proc. Am. Math. Soc. 115, 319–327 (1992)MathSciNetMATHGoogle Scholar
  11. 11.
    Connor, J., Grosse-Erdmann, K.G.: Sequential definitions of continuity for real functions. Rocky Mt. J. Math. 33(1), 93–121 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Çakalli, H.: Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math. 26(2), 113–119 (1995)MathSciNetMATHGoogle Scholar
  13. 13.
    Çakalli, H.: Sequential definitions of compactness. Appl. Math. Lett. 21, 594–598 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Çakalli, H.: Slowly oscillating continuity. Abstr. Appl. Anal. 2008, (2008). Article ID 485706Google Scholar
  15. 15.
    Çakalli, H.: \(\delta \)-quasi-Cauchy sequences. Math. Comput. Model. 53, 397–401 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Çakalli, H.: On \(G\)-continuity. Comput. Math. Appl. 61, 313–318 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Çakalli, H.: Statistical ward continuity. Appl. Math. Lett. 24, 1724–1728 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Çakalli, H.: Statistical-quasi-Cauchy sequences. Math. Comput. Model. 54, 1620–1624 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Çakalli, H.: Forward continuity. J. Comput. Anal. Appl. 13, 225–230 (2011)MathSciNetMATHGoogle Scholar
  20. 20.
    Çakalli, H., Aras, C.G., Sonmez, A.: On lacunary statistically quasi-Cauchy sequences, submitted and in Algerian-Turkish. International days on mathematics 2013, ATIM 2013, September 12–19, (2013) arXiv:1102.1531
  21. 21.
    Çakalli, H., Hazarika, B.: Ideal quasi-Cauchy sequences. J. Inequal. Appl. 2012, (2012). Article 234Google Scholar
  22. 22.
    Çakalli, H., Khan, M.K.: Summability in topological spaces. Appl. Math. Lett. 24, 348–352 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Çakalli, H., Patterson, R.F.: Functions preserving slowly oscillating double sequences. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) (in press) Google Scholar
  24. 24.
    Çakalli, H., Das, Pratulananda: Fuzzy compactness via summability. Appl. Math. Lett. 22, 1665–1669 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Çakalli, H., Sonmez, A.: Slowly oscillating continuity in abstract metric spaces. Filomat 27, 925–930 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Çakalli, H., Sonmez, A., Aras, C.G.: \(\lambda \)-statistical ward continuity. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) doi: 10.1515/aicu-2015-0016 March (2015)
  27. 27.
    Çakallı, H., Sonmez, A., Genc, C.: On an equivalence of topological vector space valued cone metric spaces and metric spaces. Appl. Math. Lett. 25, 429–433 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Canak, I., Dik, M.: New types of continuities. Abstr. Appl. Anal. 201, (2010). Article ID 258980Google Scholar
  29. 29.
    Caserta, A., Kočinac, L.J.D.R.: On statistical exhaustiveness. Appl. Math. Lett. 25, 1447–1451 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Caserta, A., Di Maio, G., Kočinac, LJ.D.R.: Statistical convergence in function spaces. Abstr. Appl. Anal. 2011, (2011). Article ID 420419Google Scholar
  31. 31.
    Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kızmaz, H.: On certain sequence spaces. Can. Math. Bull. 24(2), 169–176 (1981)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Di Maio, G., Djurcic, D., Kocinac, L.J.D.R., Zizovic, M.R.: Statistical convergence, selection principles and asymptotic analysis. Chaos Solitons and Fractal 42(5), 2815–2821 (2009)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Di Maio, G., Kočinac, L.J.D.R.: Statistical convergence in topology. Topol. Appl. 156, 28–45 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mursaleen, M.: \(\lambda \)-statistical convergence. Math. Slovaca 50, 111–115 (2000)MathSciNetMATHGoogle Scholar
  37. 37.
    Mursaleen, M., Edely, Osama H.H.: On the invariant mean and statistical convergence. Appl. Math. Lett. 22, 1700–1704 (2009)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Mohiuddine, S.A., Alotaibi, A., Mursaleen, M.: Statistical convergence of double sequences in locally solid Riesz spaces, Abstr. Appl. Anal. 2012. Article ID 719729Google Scholar
  39. 39.
    Mohiuddine, S.A., Alotaibi, A., Mursaleen, M.: Statistical convergence through de la Vallee-Poussin mean in locally solid Riesz spaces. Adv. Differ. Equ. 2013, 66 (2013)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Mohiuddine, S.A., Alotaibi, A., Mursaleen, M.: A new variant of statistical convergence. J. Inequal. Appl. 2013, 309 (2013)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Mursaleen, M., Mohiuddine, S.A.: Banach limit and some new spaces of double sequences. Turk. J. Math. 36, 121–130 (2012)Google Scholar
  42. 42.
    Özgüç, I.S., Yurdakadim, T.: On quasi-statistical convergence. Commun. Fac. Sci. Univ. Ank. Series A1 61(1), 11–17 (2012)MathSciNetMATHGoogle Scholar
  43. 43.
    Pal, S.K., Savas, E., Cakalli, H.: \(I\)-convergence on cone metric spaces. Sarajevo J. Math. 9, 85–93 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Patterson, R.F., Savaş, E.: Rate of P-convergence over equivalence classes of double sequence spaces. Positivity 16(4), 739–749 (2012)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Patterson, R.F., Savas, E.: Asymptotic equivalence of double sequences. Hacet. J. Math. Stat. 41, 487–497 (2012)MathSciNetMATHGoogle Scholar
  46. 46.
    Salat, T.: On statistically convergent sequences of real numbers. Math. Slovaca 30, 139–150 (1980)MathSciNetMATHGoogle Scholar
  47. 47.
    Kostyrko, P., Macaj, M., Salat, T., Strauch, O.: On statistical limit points. Proc. Am. Math. Soc. 129, 2647–2654 (2001)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Schoenberg, I.J.: The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361–375 (1959)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Sonmez, A., Çakallı, H.: Cone normed spaces and weighted means. Math. Comput. Model. 52, 1660–1666 (2010)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Vallin, R.W.: Creating slowly oscillating sequences and slowly oscillating continuous functions (with an appendix by Vallin and H. Çakalli). Acta Math. Univ. Comen. 25, 71–78 (2011)MathSciNetMATHGoogle Scholar
  51. 51.
    Zygmund, A.: Trigonometric series. I, II. With a foreword by Robert A. Fefferman (3rd edn). Cambridge Mathematical Library. Cambridge University Press, Cambridge (2002)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Universiti Sains Malaysia 2015

Authors and Affiliations

  1. 1.Faculty of Arts and SciencesMaltepe UniversityMaltepe, IstanbulTurkey

Personalised recommendations