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Year 2019, Volume: 1 Issue: 1, 1 - 10, 15.06.2019

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References

  • [1] A.Aizpuru, M.C.Listán-Garcĭa and F.Rambla-Borreno, Density by moduli and statistical convergence, Quaest. Math. 37 4 (2014) 525--530.
  • [2] G. Aslim, G.Sh. Guseinov, Weak semirings, ω-semirings, and measures, Bull. Allahabad Math. Soc. 14 (1999) 1--20.
  • [3] A.Cabada and D.R.Vivero, Expression of the Lebesque Δ-integral on time scales as a usual Lebesque integral; application to the calculus of Δ-antiderivates, Math. Comput. Modelling, 43 (2006) 194--207.
  • [4] H. Fast, Sur la convergence statitique, Colloq. Math. 2 (1951) 241--244.
  • [5] J.A.Fridy, On statistical convergence, Analysis, 5 (1985) 301--313.
  • [6] M.Gürdal, M.O.Özgür, A generalized statistical convergence via moduli, Electron. J. Math. Anal. Applic. 3 2 (2015) 173--178.
  • [7] G.Sh.Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 1 (2003) 107--127.
  • [9] S.Hilger: Ein maßkettenkalkül mit anwendung auf zentrumsmanningfaltigkeilen Ph.D thesis, Universitat, Würzburg (1989).
  • [10] S.Hilger, Analysis on measure chains-A unified a approach to continuous and discrete calculus, Results Math. 18 (1990) 19--56.
  • [11] H. Cakalli, A new approach to statistically quasi Cauchy sequences, Maltepe Journal of Mathematics, 1, 1, (2019) 1--8.
  • [12] E.Kolk The statistical convergence in Banach spacess, Acta Comment. Univ. Tartu. Math. 928 (1991) 41--52.
  • [13] K.Li, S.Lin and Y. Ge, On statistical convergence in cone metric space, Topology Appl. 196 (2015) 641--651.
  • [14] G.Di.Maio and L.D.R. Kočinac, Statistical convergence in topology, Topology Appl. 156 (2008) 28--45
  • [15] H.Nakano, Concav modulus, J. Math. Soc. Jpn. 5 (1953) 29--49.
  • [16] W.H.Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Can. J. Math. 25 (1973) 973--978.
  • [17] T.Rzezuchowski, A note on measures on time scales, Demonstr. Math. 33 (2009) 27--40.
  • [18] M.S.Seyyidoğlu and N.O.Tan, A note on statistical convergence on time scales, J. Inequal. Appl. (2012) 219--227.
  • [19] H. Steinhaus, Sur la convergence ordinarie et la convergence asimptotique, Colloq. Math. 2 (1951) 73--74.
  • [20] I. Taylan, Abel statistical delta quasi Cauchy sequences of real numbers, Maltepe Journal of Mathematics, 1, 1, (2019)18--23.
  • [21] Ş. Yıldız, Lacunary statistical p-quasi Cauchy sequences, Maltepe Journal of Mathematics, 1, 1, (2019) 9--17.
  • [22] A. Zygmund, Trigonometric Series, United Kingtom: Cambridge Univ. Press (1979).
  • [23] Aulbach, B, Hilger, S: A unified approach to continuous and discrete dynamics. J.Qual. Theory Diff.Equ. (Szeged, 1988), Colloq. Math. Soc. J´anos Bolyai, North-Holland Amsterdam 53, 37--56 (1990).
  • [24] I. J. Maddox, Spaces of strongly summable sequences, Quarterly Journal of Mathematics: Oxford Journals, 18(2) (1967), 345-355
  • [25] M. Bohner and A. Peterson, Dynamic equations on time scales, an introduction with applications, (2001), Birkhauser, Boston.
  • [26] R. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, Dynamic equations on time scales: a survey, Journal of Computational and Applied Mathematics, 141(1--2) (2002), 1-26.
  • [27] I. J. Maddox, Statistical convergence in a locally convex space, Mathematical Proceedings of the Cambridge Philosophical Society, 104(1) (1988), 141--145.
  • [28] M. Mursaleen, -statistical convergence, Mathematica Slovaca, 50 (1) (2000), 111-115.
  • [29] F. Nuray, λ-strongly summable and λ-statistically convergent functions, Iranian Journal of Science and Technology; Transaction A Science, 34(4) (2010), 335--338.
  • [30] T. Salat, On statistically convergent sequences of real numbers, Mathematica Slovaca, 30 (1980), 139-150.
  • [31] C. Turan and O. Duman, Statistical convergence on time scales and its characterizations, Advances in Applied Mathematics and Approximation Theory, Springer, Proceedings in Mathematics & Statistics, 41 (2013), 57-71.
  • [32] Y. Altin, H. Koyunbakan and E. Yilmaz, Uniform Statistical Convergence on Time Scales, Journal of Applied Mathematics, Volume 2014, Article ID 471437, 6 pages.
  • [33] F. Moricz, Statistical limit of measurable functions, Analysis, 24 (2004), 1-18.
  • [34] E. Yilmaz, Y. Altin and H. Koyunbakan, λ- Statistical convergence on Time scales,Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 23 (2016) 69-78
  • [35] S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical convergence through de la Vall´ee-Poussin mean in locally solid Riesz spaces, Advances in Difference Equations, 2013, 2013:66.
  • [36] A. Cabada and D. R. Vivero, Expression of the Lebesque - integral on time scales as a usual Lebesque integral; application to the calculus of -antiderivates, Mathematical and Computer Modelling, 43 (2006), 194-207.
  • [37] A. D.Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32(1) (2002), 129-138.
  • [38] R. Çolak, Statistical convergence of order α, Modern Methods in Analysis and Its Applications, Anamaya Pub., New Delhi, India (2010) 121-129.
  • [39] E. Kayan and R. Çolak, λ_{d} -Statistical Convergence, λ_{d}-statistical Boundedness and Strong (V,λ)_{d}- summability in Metric Spaces, Mathematics and Computing. ICMC 2017. Communications in Computer and Information Science, vol 655 (2017) pp. 391- 403 Springer, doi: 10.1007/978-981-10-4642-1-33.
  • [40] I. J. Maddox, Sequence spaces de…ned by a modulus, Math. Proc. Camb. Philos. Soc., 100 (1986) 161-166.
  • [41] V.K. Bhardwaj, S. Dhawan f- statistical convergence of order α and strong Cesàro summability of order α with respect to a modulus, J. Inequal. Appl. 332 (2015) 14 pp. doi:10.1186/s13660-015-0850-x.
  • [42] Nihan Turan and Metin Başarır, A note on quasi-statistical convergence of order α in rectangular cone metric space, Konuralp J.Math., 7 (1) (2019) 91-96.
  • [43] Nihan Turan and Metin Başarır, On the Δ_{g}-statistical convergence of the function defined time scale, AIP conference proceding, 2019.

On the $\lambda _{h}^{\alpha }-$Statistical Convergence of the Functions Defined on the Time Scale

Year 2019, Volume: 1 Issue: 1, 1 - 10, 15.06.2019

Abstract

In this paper, we have introduced the concepts $\lambda _{h}^{\alpha }$% -density of a subset of the time scale $\mathbb{T}$ and $\lambda _{h}^{\alpha }$-statistical convergence of order $\alpha $ $(0<\alpha \leq 1) $ of $\Delta -$ measurable function $f$ \ defined on the time scale $% \mathbb{T}$ with the help of modulus function $h$ and $\lambda =(\lambda _{n})$ sequences. Later, we have discussed the connection between classical convergence, $\lambda $-statistical convergence and $\lambda _{h}^{\alpha }$% -statistical convergence. In addition, we have seen that $f$ is strongly $% \lambda _{h}^{\alpha }$-Cesaro summable on T then $f$ is $\lambda _{h}^{\alpha }$-statistical convergent of order $\alpha .$

References

  • [1] A.Aizpuru, M.C.Listán-Garcĭa and F.Rambla-Borreno, Density by moduli and statistical convergence, Quaest. Math. 37 4 (2014) 525--530.
  • [2] G. Aslim, G.Sh. Guseinov, Weak semirings, ω-semirings, and measures, Bull. Allahabad Math. Soc. 14 (1999) 1--20.
  • [3] A.Cabada and D.R.Vivero, Expression of the Lebesque Δ-integral on time scales as a usual Lebesque integral; application to the calculus of Δ-antiderivates, Math. Comput. Modelling, 43 (2006) 194--207.
  • [4] H. Fast, Sur la convergence statitique, Colloq. Math. 2 (1951) 241--244.
  • [5] J.A.Fridy, On statistical convergence, Analysis, 5 (1985) 301--313.
  • [6] M.Gürdal, M.O.Özgür, A generalized statistical convergence via moduli, Electron. J. Math. Anal. Applic. 3 2 (2015) 173--178.
  • [7] G.Sh.Guseinov, Integration on time scales, J. Math. Anal. Appl. 285 1 (2003) 107--127.
  • [9] S.Hilger: Ein maßkettenkalkül mit anwendung auf zentrumsmanningfaltigkeilen Ph.D thesis, Universitat, Würzburg (1989).
  • [10] S.Hilger, Analysis on measure chains-A unified a approach to continuous and discrete calculus, Results Math. 18 (1990) 19--56.
  • [11] H. Cakalli, A new approach to statistically quasi Cauchy sequences, Maltepe Journal of Mathematics, 1, 1, (2019) 1--8.
  • [12] E.Kolk The statistical convergence in Banach spacess, Acta Comment. Univ. Tartu. Math. 928 (1991) 41--52.
  • [13] K.Li, S.Lin and Y. Ge, On statistical convergence in cone metric space, Topology Appl. 196 (2015) 641--651.
  • [14] G.Di.Maio and L.D.R. Kočinac, Statistical convergence in topology, Topology Appl. 156 (2008) 28--45
  • [15] H.Nakano, Concav modulus, J. Math. Soc. Jpn. 5 (1953) 29--49.
  • [16] W.H.Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Can. J. Math. 25 (1973) 973--978.
  • [17] T.Rzezuchowski, A note on measures on time scales, Demonstr. Math. 33 (2009) 27--40.
  • [18] M.S.Seyyidoğlu and N.O.Tan, A note on statistical convergence on time scales, J. Inequal. Appl. (2012) 219--227.
  • [19] H. Steinhaus, Sur la convergence ordinarie et la convergence asimptotique, Colloq. Math. 2 (1951) 73--74.
  • [20] I. Taylan, Abel statistical delta quasi Cauchy sequences of real numbers, Maltepe Journal of Mathematics, 1, 1, (2019)18--23.
  • [21] Ş. Yıldız, Lacunary statistical p-quasi Cauchy sequences, Maltepe Journal of Mathematics, 1, 1, (2019) 9--17.
  • [22] A. Zygmund, Trigonometric Series, United Kingtom: Cambridge Univ. Press (1979).
  • [23] Aulbach, B, Hilger, S: A unified approach to continuous and discrete dynamics. J.Qual. Theory Diff.Equ. (Szeged, 1988), Colloq. Math. Soc. J´anos Bolyai, North-Holland Amsterdam 53, 37--56 (1990).
  • [24] I. J. Maddox, Spaces of strongly summable sequences, Quarterly Journal of Mathematics: Oxford Journals, 18(2) (1967), 345-355
  • [25] M. Bohner and A. Peterson, Dynamic equations on time scales, an introduction with applications, (2001), Birkhauser, Boston.
  • [26] R. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, Dynamic equations on time scales: a survey, Journal of Computational and Applied Mathematics, 141(1--2) (2002), 1-26.
  • [27] I. J. Maddox, Statistical convergence in a locally convex space, Mathematical Proceedings of the Cambridge Philosophical Society, 104(1) (1988), 141--145.
  • [28] M. Mursaleen, -statistical convergence, Mathematica Slovaca, 50 (1) (2000), 111-115.
  • [29] F. Nuray, λ-strongly summable and λ-statistically convergent functions, Iranian Journal of Science and Technology; Transaction A Science, 34(4) (2010), 335--338.
  • [30] T. Salat, On statistically convergent sequences of real numbers, Mathematica Slovaca, 30 (1980), 139-150.
  • [31] C. Turan and O. Duman, Statistical convergence on time scales and its characterizations, Advances in Applied Mathematics and Approximation Theory, Springer, Proceedings in Mathematics & Statistics, 41 (2013), 57-71.
  • [32] Y. Altin, H. Koyunbakan and E. Yilmaz, Uniform Statistical Convergence on Time Scales, Journal of Applied Mathematics, Volume 2014, Article ID 471437, 6 pages.
  • [33] F. Moricz, Statistical limit of measurable functions, Analysis, 24 (2004), 1-18.
  • [34] E. Yilmaz, Y. Altin and H. Koyunbakan, λ- Statistical convergence on Time scales,Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 23 (2016) 69-78
  • [35] S. A. Mohiuddine, A. Alotaibi and M. Mursaleen, Statistical convergence through de la Vall´ee-Poussin mean in locally solid Riesz spaces, Advances in Difference Equations, 2013, 2013:66.
  • [36] A. Cabada and D. R. Vivero, Expression of the Lebesque - integral on time scales as a usual Lebesque integral; application to the calculus of -antiderivates, Mathematical and Computer Modelling, 43 (2006), 194-207.
  • [37] A. D.Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32(1) (2002), 129-138.
  • [38] R. Çolak, Statistical convergence of order α, Modern Methods in Analysis and Its Applications, Anamaya Pub., New Delhi, India (2010) 121-129.
  • [39] E. Kayan and R. Çolak, λ_{d} -Statistical Convergence, λ_{d}-statistical Boundedness and Strong (V,λ)_{d}- summability in Metric Spaces, Mathematics and Computing. ICMC 2017. Communications in Computer and Information Science, vol 655 (2017) pp. 391- 403 Springer, doi: 10.1007/978-981-10-4642-1-33.
  • [40] I. J. Maddox, Sequence spaces de…ned by a modulus, Math. Proc. Camb. Philos. Soc., 100 (1986) 161-166.
  • [41] V.K. Bhardwaj, S. Dhawan f- statistical convergence of order α and strong Cesàro summability of order α with respect to a modulus, J. Inequal. Appl. 332 (2015) 14 pp. doi:10.1186/s13660-015-0850-x.
  • [42] Nihan Turan and Metin Başarır, A note on quasi-statistical convergence of order α in rectangular cone metric space, Konuralp J.Math., 7 (1) (2019) 91-96.
  • [43] Nihan Turan and Metin Başarır, On the Δ_{g}-statistical convergence of the function defined time scale, AIP conference proceding, 2019.
There are 42 citations in total.

Details

Primary Language English
Subjects Software Engineering (Other)
Journal Section Articles
Authors

Name Tok

Metin Basarır

Publication Date June 15, 2019
Acceptance Date December 9, 2019
Published in Issue Year 2019 Volume: 1 Issue: 1

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