Kuvvetli Boşluklu Quasi-Cauchy Dizileri Üzerine Yeni Bir Çalışma

Bu çalışmada 2 N -quasi-Cauchy dizisi kavramı tanıtılmış ve bu dizilerle ilgili ilginç teoremler ispatlanmıştır. ( ) k  R nin bir A altkümesi üzerinde tanımlı bir dizi olmak üzere, ( ) 2 k   N - quasi-Cauchy oluyorsa ( ) k  dizisine 2 N -quasi-Cauchy dizisidir denir. Burada k k k k       2 1 2 2 dır. f , R nin bir A altkümesinde tanımlı reel değerli bir fonksiyon olsun. Eğer f ,A daki 2 N -quasi-Cauchy dizilerini koruyorsa, yani, ( ) k  A da 2 N -quasi-Cauchy dizisi iken ( ( )) k f  da 2 N -quasi-Cauchy oluyorsa f e A da 2 N -ward süreklidir denir.

A new study on the strongly lacunary quasi Cauchy sequences

In this paper, the concept of an 2 N -quasi-Cauchy sequence is introduced. We proved interesting theorems related to 2 N -quasi-Cauchy sequences. A real valued function f defined on a subset A of R , the set of real numbers, is 2 N -ward continuous on A if it preserves 2 N -quasi-Cauchy sequences of points in A , i.e. ( ( )) k f  is an 2 N -quasi-Cauchy sequences whenever ( ) k  is an 2 N -quasi-Cauchy sequences of points in A , where a sequence ( ) k  is called 2 N -quasi-Cauchy if ( ) 2 k   is an N - quasi-Cauchy sequence where k k k k       2 1 2 2 for each positive integer k.

PDF

___

J. Antoni, and T. Salat, “ On the A-continuity of real functions ’’, Acta Math. Univ. Comenian, vol. 39, pp. 159-164, 1980.
C.G. Aras, A. Sonmez, H. Çakallı, “ An approach to soft functions ’’, J. Math. Anal., vol. 8, no.2, pp. 129-138
J. Borsik, and T. Salat, “ On F-continuity of real functions’’, Tatra Mt. Math. Publ., vol. 2, pp. 37-42, 1993.
Naim L. Braha, H. Cakalli, “ A new type continuity for real functions’’,J. Math. Anal., vol.7, no. 6, pp. 68-76, 2016.
R.C. Buck, “ Solution of problem 4216’’, Amer. Math. Monthly, vol.55, pp. 36, 1948.
D. Burton, and J. Coleman, “ Quasi-Cauchy Sequences’’, Amer. Math. Monthly, vol.117, no. 4, pp. 328-333, 2010.
H. Cakalli, “ N -ward continuity’’, Abstr. Appl. Anal. 2012 Article ID 680456, 8 pages. 2012.
H. Cakalli, “ A Variation on Statistical Ward Continuity ’’, Bull. Malays. Math. Sci. Soc. DOI10.1007/s40840-015-0195-0
H. Çakalli, C.G. Aras, and A. Sonmez, “ Lacunary statistical ward continuity’’ , AIP Conf. Proc.1676 Article Number: 020042 doi: 10.1063/1.4930468, 2015.
H. Cakalli, and H. Kaplan, “ Strongly lacunary delta ward continuity”, AIP Conf. Proc. 1676, Article Number:020063 http://dx.doi.org/10.1063/1.4930489, 2015.
I. Canak, and M. Dik, “ New Types of Continuities’’, Abstr. Appl. Anal. Article ID 258980, 6 pages. doi:10.1155/2010/258980, 2010.
J. Connor, and K.-G. Grosse-Erdmann, “ Sequential defnitions of continuity for real functions’’, Rocky Mountain J. Math., vol.33, no. 1 , pp. 93-121, 2003.
H. Çakallı, “ Sequential defnitions of compactness’’, Appl. Math. Lett., vol.21, no. 6, pp 594-598, 2008.
H. Çakallı, “ Slowly oscillating continuity’’, Abstr. Appl. Anal. 2008 ,Article ID 485706, 5 pages. doi:10.1155/2008/485706, 2008. [15] H. Çakallı, “ -quasi-Cauchy sequences’’, Math. Comput. Modelling, vol.53, no.1-2, pp. 397-401, 2011.
H. Çakallı, “ Forward continuity’’ , J. Comput. Anal. Appl. vol.13, no.2, pp. 225-230, 2011.
H. Çakallı, “ Statistical ward continuity’’, Appl. Math. Lett. Vol. 24, no. 10, pp. 1724- 1728, 2011.
H. Çakallı, “ Statistical-quasi-Cauchy sequences’’, Math. Comput. Modelling, vol.54, no. 5-6 , pp. 1620-1624, 2011.
H. Çakallı, “ On  -quasi-slowly oscillating sequences’’ , Comput. Math. Appl.vol. 62 no. 9, pp. 3567-3574, 2011.
H. Çakallı, “ Sequential defnitions of connectedness’’, Appl. Math. Lett. Vol.25, no. 3 pp. 461-465, 2012.
H. Çakallı, I. Canak, M. Dik, “  -quasislowly oscillating continuity’’, Appl. Math. Comput.,vol. 216, no. 10, pp. 2865-2868, 2010.
H. Çakallı, and Pratulananda Das, “Fuzzy compactness via summability’’ , Appl. Math. Lett. vol. 22 , no.11, pp. 1665-1669, 2009.
H. Çakallı, and B. Hazarika, “ Ideal Quasi-Cauchy sequences’’, J. Inequal. Appl. 2012: 234. doi:10.1186/1029-242X-2012-234, 2012.
H. Çakallı, and H. Kaplan, “ A study on N - quasi-Cauchy sequences’’ , Abstr. Appl. Anal. ArticleID836970,4pages http://dx.doi.org/10.1155/2013/836970
H. Cakalli and H. Kaplan, “A variation on strongly lacunary ward continuity’’, J. Math. Anal.vol. 7, no. 3 , pp. 13-20, 2016.
H. Çakallı, and O. Mucuk, “ connectedness via a sequential method’’ , Rev. Un. Mat. Argentina, vol. 54 , no.2, pp. 101-109, 2013.
H. Çakallı, and E. Savas, “Statistical convergence of double sequences in topological , J. Comput. Anal. Appl. , vol.12, no.2, pp. 421-426, 2010.
H. Çakallı, A. Sonmez, and C.G. Aras, “ - statistically ward continuity’’, An. Stiint.Univ. Al.I. Cuza Iasi. Mat.,vol. 63, no. 2, pp.313- 322, 2017. DOI: 10.1515/aicu-2015-0016
H. Çakallı, A. Sonmez, and Ç. Genç, “ On an equivalence of topological vector space valued cone metric spaces and metric spaces’’, Appl. Math. Lett., vol.25, no.3, pp. 429- 433, 2012.
H. Çakalli, and E.I. Taylan,“ On Absolutely Almost Convergence’’, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) DOI: https://doi.org/10.2478/aicu-2014-0032 .
A. Esi, “ Asymptotically double lacunary equivalent sequences defined by Orlicz functions’’, Acta Scientiarum-Technology, vol.36, no.2, pp 323-329, 2014.
A. Esi, M. Acikgoz, “On almost lambdastatistical convergence of fuzzy numbers’’, Acta Scientiarum-Technology, vol.36, no.1, pp.129- 133, 2014.
A.R. Freedman, J.J. Sember, and M. Raphael, “Some Cesaro-type summability spaces’’, Proc.London Math. Soc., vol.3, no. 37, pp. 508- 520, 1978.
G. Das, and E. Sava_s, “On the A-continuity of real functions’’, Istanbul Univ. Fen Fak. Mat Derg. Vol.53, pp.61-66, 1994.
H. Kaplan, H. Cakalli, “Variations on strongly lacunary quasi Cauchy sequences’’, AIP Conf. Proc., vol.1759, no. 020051, 2016 . doi: 10.1063/1.4959665
H. Kaplan, H. Cakalli, “Variations on strong lacunary quasi-Cauchy sequences’’, J. Nonlinear Sci. Appl. vol.9, pp. 4371-4380, 2016.
M. Keane, “ Understanding Ergodicity’’, Integers 11B. 1-11, 2011.
D. Djurcic, Ljubia D.R. Kocinac, M.R. Zizovic, “ Double Sequences and Selections’’, Abstr. Appl. Anal. Article ID:497594, 6 pp. 2012. DOI: 10.1155/2012/497594,
O. Mucuk, T. Sahan, “ On G-Sequential Continuity’’, Filomat, vol.28, no.6, pp1181- 1189, 2014.
H. S. Ozarslan, and Ş. Yıldız, “ A new study on the absolute summability factors of Fourier series’’, J. Math. Anal. vol.7, no. 2, pp.31-36, 2016.
R.F. Patterson and H. Cakalli, “ Quasi Cauchy double sequences’’, Tbilisi Mathematical Journal, vol. 8, no.2, pp. 211-219, 2015.
E.C. Posner, “ Summability preserving functions’’, Proc.Amer.Math.Soc. vol.12, pp.73- 76, 1961.
E. Spigel, and N. Krupnik, “ On the Acontinuity of real functions’’, J. Anal. vol. 2, pp.145- 155, 1994.
A. Sonmez, and H. Çakalli, “ Cone normed spaces and weighted means’’, Math. Comput. Modelling, vol. 52, no. 9-10, pp.1660-1666, 2010.
R.W. Vallin, “ Creating slowly oscillating sequences and slowly oscillating continuous functions’’, With an appendix by Vallin and H. Cakalli, Acta Math. Univ. Comenianae, vol. 25 no.1, pp.71-78, 2011.
P. Winkler, “ Mathematical Puzzles: A Connoisseurs Collection’’, A.K.Peters LTD, ISBN 1-56881-201-9, 2004.
Ş. Yıldız, “ A new theorem on local properties of factored Fourier series’’, Bull. Math. Anal. Appl., vol. 8, no. 2, pp.1-8, 2016.
Ş. Yıldız, “ On Absolute Matrix Summability Factors of Infnite Series and Fourier Series’’, Gazi University Journal of Science, vol.30, no.1, pp. 363-370, 2017.
Ş. Yıldız, “Istatistiksel boşluklu delta 2 quasi Cauchy dizileri”, Sakarya University Journal of Science, vol. 21, no. 6, pp.1408-1412, (2017). DOI: 10.16984/saufenbilder.336128 , http://www.saujs.sakarya.edu.tr/issue/26999/3361 28
R.F. Patterson, F. Nuray, M. Basarir, “Inclusion theorems of double Deferred Cesar means II”, Tbilisi Mathematical Journal, vol. 9, no.2, pp. 15-23, 2016. DOI: 10.1515/tmj-2016- 0016