Abel Statistical Delta Quasi Cauchy Sequences of Real Numbers

Year 2019, Volume: 1 Issue: 1, 18 - 23, 09.04.2019

İffet Taylan

Abstract

In this paper, we investigate the concept of Abel statistical delta quasi Cauchy sequences. A real function $f$ is called Abel statistically delta ward continuous it preserves Abel statistical delta quasi Cauchy sequences, where a sequence $(\alpha_{k})$ of points in $\mathbb{R}$ is called Abel statistically delta quasi Cauchy if $\lim_{x \to 1^{-}}(1-x)\sum_{k:|\Delta^{2} \alpha_{k}|\geq\varepsilon}^{}x^{k}=0$ for every $\varepsilon>0$, where $\Delta^{2}  \alpha_{k}=\alpha_{k+2}-2\alpha_{k+1}+\alpha_{k}$ for every $k\in{\mathbb{N}}$. Some other types of continuities are also studied and interesting results are obtained.

Keywords

Abel statistical convergence , summability , quasi-Cauchy sequences , continuity

References

• [1] N.H.Abel, Recherches sur la srie 1+$m/x$+$[m(m-1)/ 1.2] x^{2}$+..., J., fr Math., 1 (1826), 311-339.
• [2] C.G. Aras, A. Sonmez, H. Cakalli, An approach to soft functions, J. Math. Anal. 8, 2, 129-138,(2017).
• [3] A.J.Badiozzaman and B.Thorpe, Some best possible Tauberian results for Abel and Cesaro summability, Bull. London Math. Soc., 28 3 (1996), 283-290. MR 97e:40003
• [4] Naim L. Braha, H. Cakalli, A new type continuity for real functions, J. Math. Anal. 7, 6, 68-76, (2016).
• [5] D. Burton, and J. Coleman, Quasi-Cauchy Sequences, Amer. Math. Monthly 117, 4, 328-333, (2010).
• [6] H. Cakalli, A Variation on Statistical Ward Continuity, Bull. Malays. Math. Sci. Soc. (2015).https://doi.org/10.1007/s40840-015-0195-0
• [7] H. Çakalli, C.G. Aras, and A. Sonmez, Lacunary statistical ward continuity, AIP Conf. Proc.1676, Article Number: 020042, (2015). doi: 10.1063/1.4930468
• [8] H. Cakalli and H. Kaplan, A variation on strongly lacunary ward continuity, J. Math. Anal.7, 3, 13-20, (2016).
• [9] J.Connor, K.-G.Grosse-Erdmann, Sequential definitions of continuity for real functions,Rocky Mountain J. Math. 33, 1, 93-121, (2003).
• [10] H. Çakallı, Slowly oscillating continuity, Abstr. Appl. Anal. Hindawi Publ. Corp. New York,ISSN 1085-3375, Volume 2008, Article ID 485706, (2008). doi:10.1155/2008/485706
• [11] H. Çakallı, Sequential definitions of compactness, Appl. Math. Lett. 21, 6, 594-598, (2008).
• [12] H. Çakallı, Forward continuity, J. Comput. Anal. Appl. 13, 2, 225-230, (2011).
• [13] H. Çakallı, On $\Delta$-quasi-slowly oscillating sequences, Comput. Math. Appl. 62, 9, 3567-3574,(2011).
• [14] H. Çakallı, Statistical quasi-Cauchy sequences, Math. Comput. Modelling, 54, no. 5-6, 1620-1624, (2011).
• [15] H. Çakallı, $\delta$-quasi-Cauchy sequences, Math. Comput. Modelling, 53, no. 1-2, 397-401, (2011).
• [16] H. Çakallı, Statistical ward continuity, Appl. Math. Lett. 24, 10, 1724-1728, (2011).
• [17] H. Çakallı, On G-continuity, Comput. Math. Appl. 61, 2, 313-318, (2011).
• [18] H. Çakallı, N-theta-Ward continuity, Abstr. Appl. Anal. Hindawi Publ. Corp., New York,Volume 2012, Article ID 680456, 8 pp, (2012). doi:10.1155/2012/680456.
• [19] H. Çakallı, Variations on quasi-Cauchy sequences, Filomat, 29, 1, 13-19, (2015).
• [20] H. Çakallı, Upward and downward statistical continuities, Filomat, 29, 10, 2265-2273, (2015).
• [21] H. Cakalli, A new approach to statistically quasi Cauchy sequences, Maltepe Journal of Mathematics,1, 1, 1-8, (2019).
• [22] H. Çakallı, A. Sonmez, and C. Genç, On an equivalence of topological vector space valuedcone metric spaces and metric spaces, Appl. Math. Lett. 25, 3, 429-433, (2012).
• [23] H. Çakalli and Pratulananda Das, Fuzzy compactness via summability, Appl. Math. Lett. 22,11, 1665-1669, (2009).
• [24] H. Çakallı, and H. Kaplan, A study on N-theta quasi-Cauchy sequences, Abstr. Appl. Anal.,Hindawi Publ. Corp., New York, Volume 2013, Article ID 836970 Article ID 836970, 4pages,(2013). doi:10.1155/2013/836970
• [25] H. Cakalli, and H. Kaplan, A variation on lacunary statistical quasi Cauchy sequences, Communications Faculty of Sciences University of Ankara-Series A1 Mathematics and Statistics,66, 2, 71-79, (2017). 10.1501/Commua1 0000000802
• [26] H. Çakalli, and M.K. Khan, Summability in topological spaces, Appl. Math. Lett. 24, 348-352,(2011).
• [27] H. Cakalli, R.F. Patterson, Functions preserving slowly oscillating double sequences, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 62, 2, vol. 2. 531-536, (2016).http://www.math.uaic.ro/ annalsmath/pdf-uri
• [28] H. Cakalli, A. Sonmez, Slowly oscillating continuity in abstract metric spaces Filomat, 27,5, 925-930, (2013).
• [29] H. Çakallı, A. Sönmez , and C.G. Aras, $\lambda$-statistically ward continuity, An. Stiint. Univ. Al.I. Cuza Iasi. Mat. Tomul LXII, 2017, Tom LXIII, f. 2, 308-3012, (2017). DOI: 10.1515/aicu-2015-0016
• [30] H. Cakalli, İffet Taylan, A variation on Abel statistical ward continuity, AIP Conf. Proc.1676 Article number 020076
• [31] I. Canak and M. Dik, New types of continuities, Abstr. Appl. Anal. 2010, Article ID 258980,6 pages, (2010).
• [32] D. Djurcic, Ljubisa D.R. Kocinac, M.R. Zizovic, Double sequences and selections, Abstr.Appl. Anal. Art. ID 497594, 6 pp, (2012).
• [33] A.E. Coskun, C.G Aras, H. Cakalli, and A. Sonmez, Soft matrices on soft multisets in an optimal decision process, AIP Conference Proceedings, 1759, 1, 020099 (2016); doi:10.1063/1.4959713
• [34] Fridy, J.A., On statistical convergence, Analysis, 5, 301-313 (1985)
• [35] J.A.Fridy and M.K.Khan, Statistical extensions of some classical Tauberian theorems, Proc.Amer. Math. Soc., 128 8 (2000), 2347-2355. MR 2000k:40003
• [36] H. Kaplan, H. Cakalli, Variations on strong lacunary quasi-Cauchy sequences, J. Nonlinear Sci. Appl. 9, 4371-4380, (2016).
• [37] H. Kaplan, H. Cakalli, Variations on strongly lacunary quasi Cauchy sequences, AIP Conf.Proc. 1759 (2016) Article Number: 020051
• [38] Ljubisa D.R. Kocinac, Selection properties in fuzzy metric spaces, Filomat, 26, 2, 305-312,(2012).
• [39] S.K. Pal, E. Savas, and H. Cakalli, I-convergence on cone metric spaces, Sarajevo J. Math.9, 85-93, (2013).
• [40] R.F. Patterson and H. Cakalli, Quasi Cauchy double sequences, Tbilisi Math. J., 8, 2, 211-219,(2015).
• [41] A. Sonmez, On paracompactness in cone metric spaces, Appl. Math. Lett. 23, 494-497,(2010).
• [42] I. Taylan, and H. Cakalli, Abel statistical delta quasi Cauchy sequences, AIP Conference Proceedings 2086, 030043 (2019); https://doi.org/10.1063/1.5095128 Published Online: 02 April 2019
• [43] M. Unver, Abel summability in topological spaces, Monatsh Math 178 (2015) 633-643.https://doi.org/10.1007/s00605-014-0717-0
• [44] R.W. Vallin, Creating slowly oscillating sequences and slowly oscillating continuous functions, With an appendix by Vallin and H. Cakalli, Acta Math. Univ. Comenianae, 25, 1, 71-78, (2011).
• [45] T. Yaying, B. Hazarika, H. Cakalli, New results in quasi cone metric spaces, J. Math. ComputerSci. 16, 435-444, (2016).
• [46] S. Yıldız, Istatistiksel boşluklu delta 2 quasi Cauchy dizileri, Sakarya University Journal of Science, 21, 6, (2017). DOI: 10.16984/saufenbilder.336128 , http://www.saujs.sakarya.edu.tr/issue/26999/336128 (2017).https://doi.org/10.1002/mma.4635
• [47] S. Yildiz Variations on lacunary statistical quasi Cauchy sequences, International Conference of Mathematical Sciences, (ICMS 2018), Maltepe University, Istanbul, Turkey
• [48] Ş. Yıldız, Lacunary statistical p-quasi Cauchy sequences, Maltepe Journal of Mathematics,1, 1, 9-17, (2019).