Hüseyin ÇAKALLI

8822

A New Approach to Statistically Quasi Cauchy Sequences

A New Approach to Statistically Quasi Cauchy Sequences

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,

PDF

___

  • [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)
Maltepe Journal of Mathematics-Cover
  • ISSN: 2667-7660
  • Yayın Aralığı: Yılda 2 Sayı
  • Başlangıç: 2019
  • Yayıncı: Hüseyin ÇAKALLI
Arşiv
Sayıdaki Diğer Makaleler

On the Existence and Multiplicity of Positive Radial Solutions for Nonlinear Elliptic Equation on Bounded Annular Domains via Fixed Point Index

Bouteraa NOUREDDİNE, Slimane BENAİCHA, Habib DJOURDEM

Abel Statistical Delta Quasi Cauchy Sequences of Real Numbers

İffet TAYLAN

More on Translationally Slowly Varying Sequences

Dragan DJURCİC, Ljubiša D. R. KOČİNAC

A New Approach to Statistically Quasi Cauchy Sequences

Hüseyin ÇAKALLI

Lacunary Statistical $p$-Quasi Cauchy Sequences

Şebnem YILDIZ