On Some Properties of m  -Statistical Convergence in a Paranormed Space

Year 2019, Volume: 1 Issue: 1, 40 - 47, 15.01.2019

Çiğdem Bektaş Emine Özçelik

Abstract

In this study, we introduce the concepts of strongly   m  ,p -Cesàro summability, m  -statistical Cauchy sequence and m  -statistical convergence in a paronormed space. We give some certain properties of these concepts and some inclusion relations between them. Fast [1] and Steinhaus [2] introduced the concept of statistical convergence for sequences of real numbers. Several authors studied this concept with related topics [3-5]. The asymptotic density of K N  is defined as,   n 1 (K) lim k n : k K  n     where K be a subset of the set of natural numbers N and denoted by  K. . indicates the cardinality of the enclosed set. A sequence xk  is called statistically covergent to L provided that  k  n 1 lim k n х L 0  n       for each  0 . It is denoted by lim k k st x L    . A sequence хk  is called statistically Cauchy sequence provided that there exist a number N N( )   such that

Keywords

Statistical convergence, statistical Cauchy, paranormed space, difference sequence

On Some Properties of m  -Statistical Convergence in a Paranormed Space

Abstract

In this study, we introduce the concepts of strongly  ($\Delta ^{m}$,p)-Cesàro summability,  $\Delta ^{m}-statistical Cauchy sequence and  $\Delta ^{m}-statistical convergence in a paronormed space. We give some certain properties of these concepts and some inclusion relations between them.

Keywords

Statistical convergence, statistical Cauchy, paranormed space

References

• 1. Fast, H., Sur la convergence statistique, Colloq. Math., 2, 241-244 (1951)
• 2. Steinhaus, H., Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2, 73-74 (1951)
• 3. Fridy, J. A., On statistical convergence, Analysis, 5, 301-313 (1985)
• 4. Šalát, T., On statisticaly convergent sequences of real numbers, Math Slovaca, 30, 139-150 (1980)
• 5. Kolk, E., The Statistical convergence in Banach spaces, Tartu Ül. Toimetised, 928, 41-52 (1991)
• 6. Alotaibi, A., Alroqi, M. A., Statistical convergence in a paranormed space, J. Inequal. Appl., 2012, 2012:39, 6 pp.
• 7. Nakano, H., Concave modulars, J. Math. Soc. Japan 5(1953), 29-49.
• 8. Mohammed, A., Mursaleen, M., λ-statistical convergence in paranormed space, Abstr. Appl. Anal. 2013, Art. ID 264520, 5 pp.
• 9. Çolak, R., Bektaş, Ç. A., Altınok, H., Ercan, S., On inclusion relations between some sequence spaces, Int. J. Anal. 2016, Art. ID 7283527, 4 pp.
• 10. Ercan, S., Altın, Y., Bektaş, Ç., On weak lambda-Statistical convergence of order alpha, U.P.B. Sci. Bull., Series A, 80(2), 215-226 (2018)
• 11. Et, M., Çolak, R., On some generalized difference sequence spaces, Soochow Journal of Mathematics, 21(4) 377-386 (1995).
• 12. Ercan, S., Bektaş, Ç., Some generalized difference sequence spaces of non absolute type, General Mahematics Notes, 27(2), 37-46 (2015).
• 13. F. Başar, Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monograph, İstanbul-2012, ISBN: 978-1-60805-420-6.
• 14. Ercan, S., Some Cesàro-type summability and statistical convergence of sequences generated by fractional difference operator, AKU J. Sci. Eng., 18 (2018) 011302 (125-130)