On the $q$-Cesaro bounded double sequence space

Yıl 2024, Cilt: 12 Sayı: 3, 145 - 154, 24.09.2024

Sezer Erdem

https://doi.org/10.36753/mathenot.1492238

Öz

In this article, the new sequence space $\tilde{\mathcal{M}}_u^q$ is acquainted, described as the domain of the 4d (4-dimensional) $q$-Cesaro matrix operator, which is the $q$-analogue of the first order 4d Cesaro matrix operator, on the space of bounded double sequences. In the continuation of the study, the completeness of the new space is given, and the inclusion relation related to the space is presented. In the last two parts, the duals of the space are determined, and some matrix classes are acquired.

Anahtar Kelimeler

$q$-analogue, 4d $q$-Cesaro matrix, Double sequence space, Duals, Matrix transformations

Kaynakça

• [1] Kac, V., Cheung, P.: Quantum Calculus. Springer: New York, NY, USA. (2002).
• [2] Pringsheim, A.: Zur Theorie der zweifach unendlichen Zahlenfolgen. Mathematische Annalen. 53, 289-321 (1900).
• [3] Hardy, G. H.: On the convergence of certain multiple series. Proceedings of the London Mathematical Society. 19 s2-1 (1), 124-128 (1904).
• [4] Zeltser, M.: Investigation of double sequence spaces by soft and hard analitic methods. Dissertationes Mathematicae Universtaties Tartuensis. Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 25, (2001).
• [5] Başar, F., Sever, Y.: The space Lq of double sequences. Mathematical Journal of Okayama University. 51, 149-157 (2009).
• [6] Zeltser, M.: On conservative matrix methods for double sequence spaces. Acta Mathematica Hungarica. 95(3), 225-242 (2002).
• [7] Hamilton, H. J.: Transformations of multiple sequences. Duke Mathematical Journal. 2, 29-60 (1936).
• [8] Robison, G. M.: Divergent double sequences and series. Amer. Math. Soc. Trans. 28, 50-73 (1926).
• [9] Altay, B., Ba¸sar, F.: Some new spaces of double sequences. Journal of Mathematical Analysis and Applications. 309(1), 70-90 (2005).
• [10] Adams, C.R.: On non-factorable transformations of double sequences. Proceedings of the National Academy of Sciences. 19(5), 564-567 (1933).
• [11] Aktuğlu, H., Bekar, ¸S.: q-Cesàro matrix and q-statistical convergence. Journal of Computational and Applied Mathematics. 235(16), 4717–4723 (2011).
• [12] Başarır, M.: On the strong almost convergence of double sequences. Periodica Mathematica Hungarica. 30(), 177–181 (1995).
• [13] Bekar, Ş.: q-matrix summability methods. Ph.D. Dissertation, Applied Mathematics and Computer Science, Eastern Mediterranean University. (2010).
• [14] Boss, J.: Classical and Modern Methods in Summability. Oxford University Press, Newyork. (2000).
• [15] Cooke, R.C.: Infinite Matrices and Sequence Spaces. Macmillan and Co. Limited, London. (1950).
• [16] Çapan, H., Başar, F.: On the paranormed space L(t) of double sequences. Filomat. 32(3), 1043-1053 (2018).
• [17] Çapan, H., Başar, F.: On some spaces isomorphic to the space of absolutely q-summable double sequences. Kyungpook Mathematical Journal. 58(2), 271-289 (2018).
• [18] Demiriz, S., Şahin, A.: q-Cesàro sequence spaces derived by q-analogue. Advances in Mathematics, 5(2), 97-110 (2016).
• [19] Demiriz, S., Erdem, S.: On the new double binomial sequence space. Turkish Journal of Mathematics and Computer Science. 12(2), 101–111 (2020).
• [20] Demiriz, S., Erdem, S.: Domain of binomial matrix in some spaces of double sequences. Punjab University Journal of Mathematics. 52(11), 65-79 (2020).
• [21] Dündar, E., Ulusu, U.: Asymptotically I-Cesaro equivalence of sequences of sets. Universal Journal of Mathematics and Applications. 1(2), 101-105 (2018).
• [22] Erdem, S., Demiriz, S.: Almost convergence and 4-dimensional binomial matrix. Konuralp Journal of Mathematics. 8(2), 329-336 (2020).
• [23] Erdem, S., Demiriz, S.: A new RH-regular matrix derived by Jordan’s function and its domains on some double sequence spaces. Journal of Function Spaces. 2021, Article ID 5594751, 9 pages, (2021).
• [24] İlkhan, M., Alp, P. Z., Kara, E. E.: On the spaces of linear operators acting between asymmetric cone normed spaces. Mediterranean Journal of Mathematics. 15(136) (2018).
• [25] İlkhan, M., Kara, E. E.: A new Banach space defined by Euler Totient matrix operator. Operators and Matrices. 13(2), 527-544 (2019).
• [26] Mòricz, F., Rhoades, B.E.: Almost convergence of double sequences and strong regularity of summability matrices. Mathematical Proceedings of the Cambridge Philosophical Society. 104, 283-294 (1988).
• [27] Mursaleen, M.: Almost strongly regular matrices and a core theorem for double sequences. Journal of Mathematical Analysis and Applications. 293(2), 523-531 (2004).
• [28] Ng, P. N., Lee, P. Y.: Cesaro sequences spaces of non-absolute type. Comment. Math. Prace Mat. 20(2), 429–433 (1978).
• [29] Nuray, F., Ulusu, U., Dündar, E.: Cesàro summability of double sequences of sets. General Mathematics Notes. 25(1), 8–18 (2014).
• [30] Tuğ, O.: Four-dimensional generalized difference matrix and some double sequence spaces. Journal of Inequalities and Applications. 2017(1), 149 (2017).
• [31] Ulusu, U., Dündar, E., Gülle, E.: I2-Cesàro summability of double sequences of sets. Palestine Journal of Mathematics. 9(1), 561–568 (2020).
• [32] Yaying, T., Hazarika, B., Mursaleen, M.: On sequence space derived by the domain of q-Cesàro matrix in ℓp space and the associated operator ideal. Journal of Mathematical Analysis and Applications. 493(1), 124453 (2021).
• [33] Yeşilkayagil, M., Ba¸sar, F.: Domain of Euler mean in the space of absolutely p-summable double sequences with 0 < p < 1. Analysis in Theory and Applications. 34(3), 241-252 (2018).
• [34] Yaying, T., Hazarika, B., Mursaleen, M.: On generalized (p, q)-Euler matrix and associated sequence spaces. Journal of Function Spaces. 2021, Article ID 8899960, 14 pages, (2021).
• [35] Erdem, S., Demiriz, S.: q-Cesàro double sequence space ˜ Lq s derived by q-Analog.Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica. 22, 111-126 (2023).
• [36] Çinar, M. Et, M.: q-double Cesaro matrices and q-statistical convergence of double sequences, National Academy Science Letters. 43(1), 73–76 (2020).
• [37] Mursaleen, M., Başar, F.: Domain of Cesàro mean of order one in some spaces of double sequences. Studia Scientiarum Mathematicarum Hungarica. 51(3), 335-356 (2014).
• [38] Zeltser, M., Mursaleen, M., Mohiuddine, S. A.: On almost conservative matrix methods for double sequence spaces. Publicationes Mathematicae Debrecen. 75, 387-399 (2009).
• [39] Yeşilkayagil, M., Başar, F.: Domain of Riesz mean in the space Lp. Filomat. 31(4), 925-940 (2017).