A New Approach to Statistically Quasi Cauchy Sequences

Year 2019, Volume: 1 Issue: 1, 1 - 8, 09.04.2019

Hüseyin Çakallı

Abstract

A sequence $(\alpha _{k})$ of points in $\mathbb{R}$, the set of real numbers, is called $\rho$-statistically $p$ quasi Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{\rho _{n}}|\{k\leq n: |\Delta_{p}\alpha _{k} |\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$, where $\rho=(\rho_{n})$ 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 $\Delta_{p} \alpha _{k+p} =\alpha _{k+p}-\alpha _{k}$ for each positive integer $k$. A real-valued function defined on a subset of $\mathbb{R}$ is called $\rho$-statistically $p$-ward continuous if it preserves $\rho$-statistical $p$-quasi Cauchy sequences. $\rho$-statistical $p$-ward compactness is also introduced and investigated. We obtain results related to $\rho$-statistical $p$-ward continuity, $\rho$-statistical $p$-ward compactness, $p$-ward continuity, continuity, and uniform continuity.

Keywords

Statistical convergence, Summability, Quasi-Cauchy sequences, Continuity

References

• [1] C.G. Aras, A. Sonmez, H. Çakallı, J. Math. Anal. 8 2 (2017) 129-138.
• [2] R.C. Buck, Generalized asymptotic density, Amer. J. Math. 75 (1953) 335-346.
• [3] D. Burton, J. Coleman, Quasi-Cauchy sequences, Amer. Math. Monthly 117 (2010) 328-333.
• [4] H. Çakalli, N-theta-ward continuity, Abstr. Appl. Anal. 2012 (2012) Article ID 680456 8pages. doi:10.1155/2012/680456 .
• [5] H. Çakalli, Lacunary statistical convergence in topological groups, Indian J. Pure Appl.Math. 26 2 (1995) 113-119.
• [6] H. Çakalli, Sequential denitions of compactness, Appl. Math. Lett. 21 (2008) 594-598.
• [7] H. Çakalli, Slowly oscillating continuity, Abstr. Appl. Anal. 2008 (2008), Article ID 485706,5 pages. . https://doi.org/10.1155/2008/485706 .
• [8] H.Çakalli, $\delta$-quasi-Cauchy sequences, Math. Comput. Modelling 53 (2011) 397-401.
• [9] H. Çakalli, On G-continuity, Comput. Math. Appl. 61 (2011) 313-318.
• [10] H. Çakalli, Statistical ward continuity. Appl. Math. Lett. 24 (2011) 1724-1728.
• [11] H. Çakalli, Statistical-quasi-Cauchy sequences, Math. Comput. Modelling 54 (2011) 1620-1624.
• [12] H. Cakallı, Forward continuity, J. Comput. Anal. Appl. 13 (2011) 225-230.
• [13] H. Cakallı, Upward and downward statistical continuities, Filomat, 29, 10, 2265-2273,(2015).
• [14] H. Cakalli, A variation on statistical ward continuity, Bull. Malays. Math. Sci. Soc. 40 (2017)1701-1710. https://doi.org/10.1007/s40840-015-0195-0
• [15] H. Cakalli, More results on quasi Cauchy sequences, 2nd International Conference of Mathematical Sciences, 31 July 2018-6 August 2018, (ICMS 2018) Maltepe University, Istanbul, Turkey, page 67; Variations on rho statistical quasi Cauchy sequences, AIP Conference Proceedings 2086, 030010 (2019); https://doi.org/10.1063/1.5095095 Published Online: 02 April 2019
• [16] H. Çakalli, and B. Hazarika, Ideal quasi-Cauchy sequences, J. Inequal. Appl. 2012 (2012)Article 234, 11 pages. https://doi.org/10.1186/1029-242X-2012-234 .
• [17] H. Çakalli, and M.K. Khan, Summability in topological spaces, Appl. Math. Lett. 24 (2011)348-352.
• [18] H. Çakalli and R.F. Patterson, Functions preserving slowly oscillating double sequences, An.Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) Tomul LXII, 2 2 (2016) 531-536.
• [19] H. Çakalli, and Pratulananda Das, Fuzzy compactness via summability, Appl. Math. Lett.22 (2009) 1665-1669.
• [20] H. Çakalli and A. Sonmez, Slowly oscillating continuity in abstract metric spaces, Filomat27 (2013) 925-930.
• [21] H. Çakalli, A. Sonmez, and C.G. Aras, -statistical ward continuity, An. Stiint. Univ. Al. I.Cuza Iasi. Mat. (N.S.) DOI: 10.1515/aicu-2015-0016 March 2015.
• [22] H. Çakall, A. Sonmez, and C. Genc, On an equivalence of topological vector space valuedcone metric spaces and metric spaces, Appl. Math. Lett. 25 (2012) 429-433.
• [23] I. Canak and M. Dik, New types of continuities, Abstr. Appl. Anal. 2010 (2010), Article ID258980, 6 pages. https://doi.org/10.1155/2010/258980 .
• [24] A. Caserta, and Lj.D.R. Kocinac, On statistical exhaustiveness, Appl. Math. Lett. 25(2012) 1447-1451.
• [25] A. Caserta, G. Di Maio, and Lj.D.R. Kocinac, Statistical convergence in functionspaces, Abstr. Appl. Anal. 2011 (2011), Article ID 420419, 11 pages.https://doi.org/10.1155/2011/420419 .
• [26] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241-244.
• [27] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301-313.
• [28] G. Di Maio, and Lj.D.R. Kocinac, Statistical convergence in topology, Topology Appl. 156(2008) 28-45.
• [29] M. Mursaleen, lambda-statistical convergence, Math. Slovaca 50 (2000) 111-115.
• [30] I.S. Ozgüc and T. Yurdakadim, On quasi-statistical convergence, Commun. Fac. Sci. Univ.Ank. Series A1 61 1 (2012) 11-17.
• [31] S.K. Pal, E. Savas, and H. Cakalli, I-convergence on cone metric spaces, Sarajevo J. Math.9 (2013) 85-93.
• [32] R.F. Patterson, and E. Savaş, Rate of P-convergence over equivalence classes of doublesequence spaces, Positivity 16 4 (2012) 739-749.
• [33] T. Salat, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980)139-150.
• [34] I.J. Schoenberg, The integrability of certain functions and related summability methods,Amer. Math. Monthly 66, (1959), 361-375.
• [35] A. Sonmez, and H. Çakallı, Cone normed spaces and weighted means, Math. Comput.Modelling 52 (2010) 1660-16660.
• [36] R.W. Vallin, Creating slowly oscillating sequences and slowly oscillating continuous functions(with an appendix by Vallin and H. Çakalli), Acta Math. Univ. Comenianae 25 (2011) 71-78.
• [37] Ş. Yıldız, _Istatistiksel bosluklu delta 2 quasi Cauchy dizileri, Sakarya University Journal ofScience, 21 6 (2017) 1408-1412.
• [38] S. Yildiz, A new variation on lacunary statistical quasi Cauchy sequences, AIP Conf. Proc.1978 (2018) Article Number:380002. https://doi.org/10.1063/1.5043979
• [39] A. Zygmund, Trigonometric series. I, II. Third edition. With a foreword by Robert A. Fefferman.Cambridge Mathematical Library. Cambridge University Press, Cambridge, (2002)