In this study, we discuss the idea of difference operators $ \Delta _{p}^{\alpha }, \Delta _{p}^{\left(\alpha \right) } $ $ \left(\alpha \in \mathbb{R}\right) $ and examine some properties of these operators. We also describe the concepts of ordered statistical convergence and lacunary statistical by using the $ \Delta _{p}^{\alpha } $-difference operator. We examine some features of these sequence spaces and present some inclusion theorems. We obtain the Caputo fractional derivative in this work.
Citation: Abdulkadir Karakaş. Statistical convergence of new type difference sequences with Caputo fractional derivative[J]. AIMS Mathematics, 2022, 7(9): 17091-17104. doi: 10.3934/math.2022940
In this study, we discuss the idea of difference operators $ \Delta _{p}^{\alpha }, \Delta _{p}^{\left(\alpha \right) } $ $ \left(\alpha \in \mathbb{R}\right) $ and examine some properties of these operators. We also describe the concepts of ordered statistical convergence and lacunary statistical by using the $ \Delta _{p}^{\alpha } $-difference operator. We examine some features of these sequence spaces and present some inclusion theorems. We obtain the Caputo fractional derivative in this work.
| [1] |
H. Kızmaz, On certain sequence spaces, Can. Math. Bull., 24 (1981), 169–176. https://doi.org/10.4153/CMB-1981-027-5 doi: 10.4153/CMB-1981-027-5
|
| [2] | M. Et, R. Çolak, On some generalized difference sequence spaces, Soochow Journal of Mathematics, 21 (1995), 377–386. |
| [3] | I. Tincu, On some p-convex sequences, Acta Universitatis Apulensis, 11 (2006), 249–257. |
| [4] |
A. Karakaş, Y. Altin, M. Et, $\Delta _{p}^{m}$-statistical convergence of order$\alpha$, Filomat, 32 (2018), 5565–5572. https://doi.org/10.2298/FIL1816565K doi: 10.2298/FIL1816565K
|
| [5] | A. Zygmund, Trigonometric series, Cambridge: Cambridge University Press, 1979. https://doi.org/10.1017/CBO9781316036587 |
| [6] | H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73–74. |
| [7] | H. Fast, Sur la convergence statistique, Colloq. Math., 2 (1951), 241–244. |
| [8] |
N. Aral, M. Et, Generalized difference sequence spaces of fractional order defined by Orlicz functions, Commun. Fac. Sci. Univ., 69 (2020), 941–951. https://doi.org/10.31801/cfsuasmas.628863 doi: 10.31801/cfsuasmas.628863
|
| [9] | M. Et, F. Nuray, $\Delta ^{m}$-statistical convergence, Indian J. Pure Appl. Math., 32 (2001), 961–969. |
| [10] |
J. Fridy, On statistical convergence, Analysis, 5 (1985), 301–314. https://doi.org/10.1524/anly.1985.5.4.301 doi: 10.1524/anly.1985.5.4.301
|
| [11] |
A. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (2002), 129–138. https://doi.org/10.1216/rmjm/1030539612 doi: 10.1216/rmjm/1030539612
|
| [12] | R. Çolak, Statistical convergence of order $ \alpha$, In: Modern methods in analysis and its applications, New Delhi: Anamaya Pub., 2010, 129. |
| [13] | A. Karakaş, $\Delta _{p}^{m}$-lacunary statistical convergence of order $ \beta$, Proceedings of 8th International Conference on Recent Advances in Pure and Applied Mathematics, 2021, 96. |
| [14] |
A. Freedman, J. Sember, M. Raphael, Some Cesàro-type summability spaces, P. Lond. Math. Soc., 37 (1978), 508–520. https://doi.org/10.1112/plms/s3-37.3.508 doi: 10.1112/plms/s3-37.3.508
|
| [15] | J. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math., 160 (1993), 43–51. |
| [16] |
J. Fridy, C. Orhan, Lacunary statistical summability, J. Math. Anal. Appl., 173 (1993), 497–504. https://doi.org/10.1006/jmaa.1993.1082 doi: 10.1006/jmaa.1993.1082
|
| [17] |
H. Sengül, M. Et, On lacunary statistical convergence of order $\alpha $, Acta Math. Sci., 34 (2014), 473–482. https://doi.org/10.1016/S0252-9602(14)60021-7 doi: 10.1016/S0252-9602(14)60021-7
|
| [18] |
H. Nakano, Concave modulars, J. Math. Soc. Japan, 5 (1953), 29–49. https://doi.org/10.2969/jmsj/00510029 doi: 10.2969/jmsj/00510029
|
| [19] |
P. Baliarsingh, Some new difference sequence spaces of fractional order and their dual spaces, Appl. Math. Comput., 219 (2013), 9737–9742. https://doi.org/10.1016/j.amc.2013.03.073 doi: 10.1016/j.amc.2013.03.073
|
| [20] |
P. Baliarsingh, On a fractional difference operator, Alex. Eng. J., 55 (2016), 1811–1816. https://doi.org/10.1016/j.aej.2016.03.037 doi: 10.1016/j.aej.2016.03.037
|
| [21] |
P. Baliarsingh, S. Dutta, On the classes of fractional order difference sequence spaces and their matrix transformations, Appl. Math. Comput., 250 (2015), 665–674. https://doi.org/10.1016/j.amc.2014.10.121 doi: 10.1016/j.amc.2014.10.121
|
| [22] |
E. Malkowsky, S. Parashar, Matrix transformations in spaces of bounded and convergent difference sequences of order $m$, Analysis, 17 (1997), 87–98. https://doi.org/10.1524/anly.1997.17.1.87 doi: 10.1524/anly.1997.17.1.87
|
| [23] |
M. Et, On some topological properties of generalized difference sequence spaces, International Journal of Mathematics and Mathematical Sciences, 24 (2000), 716581. https://doi.org/10.1155/S0161171200002325 doi: 10.1155/S0161171200002325
|
| [24] |
D. Baleanu, A. Fernandez, A. Akgül, On a fractional operator combining proportional and classical differintegrals, Mathematics, 8 (2020), 360. https://doi.org/10.3390/math8030360 doi: 10.3390/math8030360
|
| [25] |
R. Leake, Monotone resolution sequence spaces and mappings, IEEE T. Circuits, 27 (1980), 800–804. https://doi.org/10.1109/TCS.1980.1084892 doi: 10.1109/TCS.1980.1084892
|
| [26] |
H. Kawamura, A. Tani, M. Yamada, K. Tsunoda, Real time prediction of earthquake ground motions and structural responses by statistic and fuzzy logic, Proceedings of First International Symposium on Uncertainty Modeling and Analysis, 1990, 534–538. https://doi.org/10.1109/ISUMA.1990.151311 doi: 10.1109/ISUMA.1990.151311
|
| [27] | S. Regmi, Optimized iterative methods with applications in diverse disciplines, New York: Nova Science Publisher, 2021. |
| [28] | G. Anastassiou, O. Duman, Towards intelligent modeling: statistical approximation theory, Berlin: Springer, 2011. http://dx.doi.org/10.1007/978-3-642-19826-7 |
Abdulkadir Karakaş. Statistical convergence of new type difference sequences with Caputo fractional derivative[J]. AIMS Mathematics, 2022, 7(9): 17091-17104. doi: 10.3934/math.2022940
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