On Continuities via P-Statistical Convergence
DOI:
https://doi.org/10.15377/2409-5761.2024.11.8Keywords:
Continuity, Power series method, Statistical convergence, P-statistical convergence, P-statistical ward continuityAbstract
This study investigates the concepts of P-statistical continuity (statistical continuity with respect to power series method) and P-statistical ward continuity (statistical ward continuity with respect to power series method) within the framework of power series methods, which extend the scope of statistical convergence beyond classical matrix methods. In the background, the limitations of traditional methods in capturing generalized continuity behaviors are explored and the use of power series as a versatile tool is motivated. Connections between these specialized forms of continuity and standard continuity are established, providing proofs and detailed properties. The results include several foundational theorems characterizing P-statistical continuity and ward continuity under various settings. These findings contribute to a more profound comprehension of continuity concepts within the context of regular summability methods.
MSC: 26A15, 40A05, 40G10.
Downloads
References
Robbins H. Problem 4216. Amer Math Monthly. 1946; 53: 470-1. https://doi.org/10.2307/2306259 DOI: https://doi.org/10.2307/2306260
Problem 4216 (1946, 470), Amer. Math. Monthly, Propesed Robins H. Solution by Buck RC. Amer Math Monthly. 1948; 55: 36. https://doi.org/10.2307/2305332 DOI: https://doi.org/10.2307/2305332
Antoni J. On the A-continuity of real functions II. Math Slovaca. 1986; 36: 283-8.
Antoni J, Salat T. On the A-continuity of real functions. Acta Math Univ Comenian. 1980; 39: 159-64.
Bilalov B, Sadigova S. On -statistical convergence. Proc Am Math Soc. 2015; 143(9): 3869-78. https://doi.org/10.1090/S0002-9939-2015-12528-2 DOI: https://doi.org/10.1090/S0002-9939-2015-12528-2
Borsik J, Salat T. On F-continuity of real functions. Tatra Mountains Math Publ. 1993; 2: 37-42.
Demirci K, Yıldız S, Dirik F. On the Ka-continuity of real functions. Commun Fac Sci Univ Ank Ser A1 Math Stat. 2020; 69(1): 540-46. DOI: https://doi.org/10.31801/cfsuasmas.569892
Öztürk E. On almost continuity and almost -continuity of real functions. Commun Fac Sci Univ Ank Ser A1 Math Stat. 1983; 32: 25-30. https://doi.org/10.1501/Commua1_0000000218 DOI: https://doi.org/10.1501/Commua1_0000000218
Posner EC. Summability preserving functions. Proc Am Math Soc. 1961; 12: 73-6. https://doi.org/10.1090/S0002-9939-1961-0121591-X DOI: https://doi.org/10.1090/S0002-9939-1961-0121591-X
Savas¸ E, Das G. On the A-continuity of real functions. İstanbul Üniv Fen Fak Mat Derg. 1994; 53: 61-66.
Connor J, Grosse-Erdmann KG. Sequential definitions of continuity for real functions. Rocky Mt J Math. 2003; 93-121. https://doi.org/10.1216/rmjm/1181069988 DOI: https://doi.org/10.1216/rmjm/1181069988
Çakalli H. Statistical ward continuity. Appl Math Lett. 2011; 24: 1724-28. https://doi.org/10.1016/j.aml.2011.04.029 DOI: https://doi.org/10.1016/j.aml.2011.04.029
Çakalli H. Statistical-quasi-Cauchy sequences. Math Comput Model. 2011; 54: 1620-24. https://doi.org/10.1016/j.mcm.2011.04.037 DOI: https://doi.org/10.1016/j.mcm.2011.04.037
Çakalli H, Hazarika B. Ideal quasi-Cauchy sequences. J Inequal Appl. 2012; 2012: 234. https://doi.org/10.1186/1029-242X-2012-234 DOI: https://doi.org/10.1186/1029-242X-2012-234
Çakalli H, Aras CG, Sonmez A. Lacunary statistical ward continuity. Adv Math Sci. 2015; 1676(1): 020042. https://doi.org/10.1063/1.4930468 DOI: https://doi.org/10.1063/1.4930468
Çakalli H, Sonmez A, Aras CG. -statistical ward continuity. An Stiint Univ Al I Cuza Iasi Mat (N.S.); 2015. https://doi.org/10.1515/aicu-2015-0016 DOI: https://doi.org/10.1515/aicu-2015-0016
Basu S, Dey LK, Som S. Farthest point problem and partial statistical continuity in normed linear spaces. Quaest Math. 2022; 45(4): 595-604. https://doi.org/10.2989/16073606.2021.1886193 DOI: https://doi.org/10.2989/16073606.2021.1886193
Çakalli H. Abel statistical quasi Cauchy sequences. Filomat. 2019; 33(2): 535-41. https://doi.org/10.2298/FIL1902535C DOI: https://doi.org/10.2298/FIL1902535C
Çakalli H, Taylan I. A variation on Abel statistical ward continuity. AIP Conf Proc. 2015; 1676(1): 020076. https://doi.org/10.1063/1.4930502 DOI: https://doi.org/10.1063/1.4930502
Sengül H, Çakall H, Et M. A variation on strongly ideal lacunary ward continuity. Bol Soc Paran Mat. 2020; 38: 99-108. https://doi.org/10.5269/bspm.v38i7.46136 DOI: https://doi.org/10.5269/bspm.v38i7.46136
Taylan I. Abel statistical delta quasi Cauchy sequences of real numbers. Maltepe J Math. 2019; 1(1): 18-23. https://doi.org/10.1063/1.5095128 DOI: https://doi.org/10.1063/1.5095128
Yildiz S. Variations on lacunary statistical quasi Cauchy sequences. AIP Conf Proc. 2019; 2086(1): 030045. https://doi.org/10.1063/1.5095130 DOI: https://doi.org/10.1063/1.5095130
Bayram N , Yildiz S. Approximation by statistical convergence with respect to power series methods. Hacet J Math Stat. 2022; 51(4): 1108-20. https://doi.org/10.15672/hujms.1022072 DOI: https://doi.org/10.15672/hujms.1022072
Belen C, Yıldırım M, Sümbül C. On statistical and strong convergence with respect to a modulus function and a power series method. Filomat. 2020; 34(12): 3981-93. https://doi.org/10.2298/FIL2012981B DOI: https://doi.org/10.2298/FIL2012981B
Demirci K, Durić D, Kočinac LD, Yıldız S. A theory of variations via -statistical convergence. Publ Inst Math. 2021; 110(124): 11-27. https://doi.org/10.2298/PIM2124011D DOI: https://doi.org/10.2298/PIM2124011D
Demirci K, Yıldız S, Dirik F. Statistical convergence with respect to power series method on time scales. Filomat. 2024; 38(13): 4775-89.
Sahin Bayram N. Criteria for statistical convergence with respect to power series methods. Positivity. 2021; 25(3): 1097-105. https://doi.org/10.1007/s11117-020-00801-6 DOI: https://doi.org/10.1007/s11117-020-00801-6
Sümbül C, Belen C, Yıldırım M. Properties of J_p-Statistical Convergence. Cumhuriyet Sci J. 2022; 43(2): 294-8. https://doi.org/10.17776/csj.1064559 DOI: https://doi.org/10.17776/csj.1064559
Ünver M, Bayram NS. On statistical convergence with respect to power series methods. Positivity. 2022; 26(3): 55. https://doi.org/10.1007/s11117-022-00921-1 DOI: https://doi.org/10.1007/s11117-022-00921-1
Niven I, Zuckerman HS. An Introduction to the Theory of Numbers. 4th ed. New York: John Wiley and Sons; 1980.
Fast H. Sur la convergence statistique. Colloq Math. 1951; 2: 241-4. https://doi.org/10.4064/cm-2-3-4-241-244 DOI: https://doi.org/10.4064/cm-2-3-4-241-244
Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique. Colloq Math. 1951; 2: 73-4.
Kratz W, Stadtmüller U. Tauberian theorems for -summability. J Math Anal Appl. 1989; 139: 362-71. https://doi.org/10.1016/0022-247X(89)90113-3 DOI: https://doi.org/10.1016/0022-247X(89)90113-3
Stadtmüller U, Tali A. On certain families of generalized Nörlund methods and power series methods. J Math Anal Appl. 1999; 238: 44-66. https://doi.org/10.1006/jmaa.1999.6503 DOI: https://doi.org/10.1006/jmaa.1999.6503
Boos J. Classical and Modern Methods in Summability. Oxford University Press; 2000. https://doi.org/10.1093/oso/9780198501657.001.0001 DOI: https://doi.org/10.1093/oso/9780198501657.001.0001
Ünver M, Orhan C. Statistical convergence with respect to power series methods and applications to approximation theory. Numer Func Anal Opt. 2019; 40(5): 535-47. https://doi.org/10.1080/01630563.2018.1561467 DOI: https://doi.org/10.1080/01630563.2018.1561467
Ünver M. Abel summability in topological spaces. Monatsh Math. 2015; 178: 633-43. https://doi.org/10.1007/s00605-014-0717-0 DOI: https://doi.org/10.1007/s00605-014-0717-0
Açikgöz A, Çakall H, Esenbel F, Kocinac LD. A quest of -continuity in neutrosophic spaces. Math Method Appl Sci. 2021; 44(9): 7834-44. https://doi.org/10.1002/mma.7113 DOI: https://doi.org/10.1002/mma.7113
Mucuk O, Behram S. G-sequential methods in product spaces. AIP Conf Proc. 2022; 2483(1): 020007. https://doi.org/10.1063/5.0115533 DOI: https://doi.org/10.1063/5.0115533
Yaying T. Arithmetic continuity in cone metric space. Dera Natung Govern Coll Res J. 2020; 5(1): 55-62. https://doi.org/10.56405/dngcrj.2020.05.01.07 DOI: https://doi.org/10.56405/dngcrj.2020.05.01.07
Yaying T, Hazarika B, Mohiuddine SA. Some new types of continuity in asymmetric metric spaces. Ser Math Inform. 2020; 35(2): 485-93. https://doi.org/10.22190/FUMI2002485Y DOI: https://doi.org/10.22190/FUMI2002485Y
Yıldız S, Demirci K. On power series statistical convergence and new uniform integrability of double sequences. Appl Math. 2024; 39(3): 519-32. https://doi.org/10.1007/s11766-024-5119-z DOI: https://doi.org/10.1007/s11766-024-5119-z
Yıldız S, Demirci K, Dirik F. Korovkin theory via -statistical relative modular convergence for double sequences. Rend Circ Mat Palermo 2. 2023; 72(2): 1125-41. https://doi.org/10.1007/s12215-021-00681-z DOI: https://doi.org/10.1007/s12215-021-00681-z
Çakalli H, Das P. Fuzzy compactness via summability. Appl Math Lett. 2009; 22(11): 1665-69. https://doi.org/10.1016/j.aml.2009.05.015 DOI: https://doi.org/10.1016/j.aml.2009.05.015
Fang JX, Xue QY. Some properties of the space of fuzzy-valued continuous functions on a compact set. Fuzzy Sets Syst. 2009; 160(11): 1620-31. https://doi.org/10.1016/j.fss.2008.07.014 DOI: https://doi.org/10.1016/j.fss.2008.07.014
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Sevda Yıldız, Kamil Demirci
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
All the published articles are licensed under the terms of the Creative Commons Attribution Non-Commercial License (CC BY-NC 4.0) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided the work is properly cited.
