On $I-$Deferred Statistical Convergence in Topological Groups

Year 2019, Volume: 1 Issue: 2, 48 - 55, 30.10.2019

Hacer Şengül Kandemir

Abstract

In this paper, the concepts of $I-$deferred statistical convergence of order $\alpha $ and $I-$deferred statistical convergence of order $\left( \alpha ,\beta \right) $ in topological groups were defined. Also some inclusion relations between $I-$statistical convergence of order $% \alpha $, $I-$deferred statistical convergence of order $\alpha $, $I-$% statistical convergence of order $\left( \alpha ,\beta \right) $ and $I-$% deferred statistical convergence of order $\left( \alpha ,\beta \right) $ in topological groups are given.

Keywords

Topological groups, statistical convergence, deferred statistical convergence

References

• [1] R. P. Agnew, On deferred Cesàro means, Ann. of Math. (2) 33(3) (1932), 413-421.
• [2] H. Çakallı, Lacunary statistical convergence in topological groups, Indian J. Pure Appl.Math. 26(2) (1995), 113-119.
• [3] H. Çakallı, Upward and downward statistical continuities, Filomat 29(10) (2015), 2265-2273.
• [4] H. Çakallı, Statistical quasi-Cauchy sequences, Math. Comput. Modelling 54(5-6) (2011),1620-1624.
• [5] H. Çakallı, Statistical ward continuity, Appl. Math. Lett. 24(10) (2011), 1724-1728.
• [6] H. Çakallı, A new approach to statistically quasi Cauchy sequences, Maltepe Journal ofMathematics, 1(1) 2019, 1-8.
• [7] M. Çınar, M. Karaka¸s, and M. Et, On pointwise and uniform statistical convergence oforder alpha for sequences of functions, Fixed Point Theory Appl. 2013, 2013:33, 11 pp.
• [8] R. Çolak, Statistical convergence of order alpha, Modern Methods in Analysis and Its Applica-tions, New Delhi, India: Anamaya Pub, 2010: 121-129.
• [9] M. Et, A. Alotaibi and S. A. Mohiuddine, On (Delta_m, I)-statistical convergence of order alpha,The Scientific World Journal, 2014, 535419 DOI: 10.1155/2014/535419.
• [10] M. Et, B. C. Tripathy and A. J. Dutta, On pointwise statistical convergence of order alpha ofsequences of fuzzy mappings. Kuwait J. Sci. 41(3) (2014), 17-30.
• [11] M. Et, R. Çolak and Y. Altın, Strongly almost summable sequences of order alpha, Kuwait J.Sci. 41(2), (2014), 35-47.
• [12] M. Et, S. A. Mohiuddine and H. Şengül, On lacunary statistical boundedness of order alpha,Facta Univ. Ser. Math. Inform. 31(3) (2016), 707-716.
• [13] H. Fast, Sur La Convergence Statistique, Colloq. Math. 2 (1951), 241-244.
• [14] J. A. Fridy, On Statistical Convergence, Analysis 5 (1985), 301-313.
• [15] J. A. Fridy and C. Orhan, Lacunary Statistical Convergence, Pacific J. Math. 160 (1993),43-51.
• [16] A. D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence,Rocky Mountain J. Math. 32(1) (2002), 129-138.
• [17] M. Işık and K. E. Akbaş, On lambda-statistical convergence of order alpha in probability, J. Inequal.Spec. Funct. 8(4) (2017), 57-64.
• [18] M. Küçükaslan andM. Yılmaztürk On deferred statistical convergence of sequences, Kyung-pook Math. J. 56 (2016), 357-366.
• [19] P. Kostyrko, T. Salat and W. Wilczy´nski, I-convergence. Real Anal. Exchange 26,(2000/2001), 669-686.
• [20] P. Kostyrko, M. Macaj, M. Sleziak, and T. Salat, I-convergence and extremal I-limitpoints. Math. Slovaca 55(4), (2005), 443-464.
• [21] F. Nuray and W. H. Ruckle, Generalized statistical convergence and convergence freespaces. J. Math. Anal. Appl. 245(2), (2000), 513-527.
• [22] T. Salat, On Statistically Convergent Sequences of Real Numbers, Math. Slovaca. 30 (1980),139-150.
• [23] E. Savaş, Lacunary statistical convergence of double sequences in topological groups, J.Inequal. Appl. 2014, 2014:480, 10 pp.
• [24] E. Savaş and M. Et, On (Delta^m_lambda, I)-statistical convergence of order alpha, Period. Math. Hungar.71(2) (2015), 135-145.
• [25] E. Savaş, On I-lacunary statistical convergence of order alpha for sequences of sets. Filomat29(6), (2015), 1223–1229.
• [26] T. Salat, B. C. Tripathy and M. Ziman, On I-convergence field. Ital. J. Pure Appl. Math.No. 17, (2005), 45-54.
• [27] T. Salat, B. C. Tripathy and M. Ziman, On some properties of I-convergence. Tatra Mt.Math. Publ. 28, part II, (2004), 279-286.
• [28] I. J. Schoenberg, The Integrability of Certain Functions and Related Summability Methods,Amer. Math. Monthly 66 (1959), 361-375.
• [29] H. M. Srivastava and M. Et, Lacunary statistical convergence and strongly lacunary sum-mable functions of order alpha, Filomat 31(6) (2017), 1573-1582.
• [30] H. Steinhaus, Sur La Convergence Ordinaire et la Convergence Asymptotique, ColloquiumMathematicum 2 (1951), 73-74.
• [31] H. Şengül and M. Et, On Lacunary Statistical Convergence of Order alpha, Acta Math. Sci.Ser. B Engl. Ed. 34(2) (2014), 473–482.
• [32] H. ¸Sengül, On Wijsman I􀀀lacunary statistical equivalence of order (eta, mu), J. Inequal. Spec.Funct. 9(2) (2018), 92-101.
• [33] H. Şengül, On S_alpha^beta(teta)-convergence and strong N_alpha^beta(teta,p)-summability, J. Nonlinear Sci.Appl. 10(9) (2017), 5108-5115.
• [34] H. Şengül and M. Et, On I-lacunary statistical convergence of order alpha of sequences of sets.Filomat 31(8) (2017), 2403-2412.
• [35] H. Şengül and M. Et, On (lambda, I)-statistical convergence of order alpha of sequences of function,Proc. Nat. Acad. Sci. India Sect. A 88 (2018), no. 2, 181-186.
• [36] U. Ulusu and E. Dündar, I-lacunary statistical convergence of sequences of sets. Filomat28(8), (2014), 1567-1574.
• [37] ¸S. Yıldız, Lacunary statistical p-quasi Cauchy sequences, Maltepe Journal of Mathematics,1, 1, 2019, pp. 9-17.
• [38] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, UK, (1779).