In this paper, we introduced the concepts of (strongly) $ \nu $-conull FDK-spaces, which can be regarded as double-indexed versions of FK-spaces (sequence space with coordinate functionals), by utilizing the notion of $ \nu $-convergence for double sequences. We provided fundamental characterizations of these new spaces and established several inclusion relations among them. Furthermore, we investigated the conditions under which the summability domain $ E_A^{(\nu)} $ is (strongly) $ \nu $-conull, thereby providing new insights into its structural and topological properties.
Citation: Şeyda Sezgek, İlhan Daǧadur. $ \nu- $Conull FDK-spaces[J]. AIMS Mathematics, 2025, 10(11): 25708-25728. doi: 10.3934/math.20251138
In this paper, we introduced the concepts of (strongly) $ \nu $-conull FDK-spaces, which can be regarded as double-indexed versions of FK-spaces (sequence space with coordinate functionals), by utilizing the notion of $ \nu $-convergence for double sequences. We provided fundamental characterizations of these new spaces and established several inclusion relations among them. Furthermore, we investigated the conditions under which the summability domain $ E_A^{(\nu)} $ is (strongly) $ \nu $-conull, thereby providing new insights into its structural and topological properties.
| [1] |
A. Wilansky, An application of Banach linear functionals to summability, Trans. Amer. Math. Soc., 67 (1949), 59–68. https://doi.org/10.2307/1990418 doi: 10.2307/1990418
|
| [2] | E. I. Yurimyae, Einige Fragen über verallgemeinerte Matrixverfahren, co-regular und co-null verfahren, Eesti NSV Tead. Akad. Toim. Tehn. Füüs Mat., 8 (1959), 115–121. |
| [3] | A. K. Snyder, On a definition for conull and coregular FK spaces, Notices Amer. Math. Soc., 10 (1963), 183. |
| [4] |
G. Bennett, A new class of sequence spaces with applications in summability theory, J. Reine Angew. Math., 1974 (1974), 49–75. https://doi.org/10.1515/crll.1974.266.49 doi: 10.1515/crll.1974.266.49
|
| [5] |
H. G. İnce, Cesàro conull FK-spaces, Demonstr. Math., 33 (2000), 109–122. https://doi.org/10.1515/dema-2000-0114 doi: 10.1515/dema-2000-0114
|
| [6] |
İ. Daǧadur, $C_{\lambda}$-conull FK-spaces, Demonstr. Math., 35 (2002), 835–848. https://doi.org/10.1515/dema-2002-0414 doi: 10.1515/dema-2002-0414
|
| [7] | M. Zeltser, Investigation of double sequence spaces by soft and hard analytical methods, Tartu: Tartu University Press, 2001. |
| [8] |
F. Moricz, B. E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Cambridge Philos. Soc., 104 (1988), 283–294. https://doi.org/10.1017/S0305004100065464 doi: 10.1017/S0305004100065464
|
| [9] |
A. Pringsheim, Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann., 53 (1900), 289–321. https://doi.org/10.1007/BF01448977 doi: 10.1007/BF01448977
|
| [10] |
B. Altay, F. Başar, Some new spaces of double sequences, J. Math. Anal. Appl., 309 (2005), 70–90. https://doi.org/10.1016/j.jmaa.2004.12.020 doi: 10.1016/j.jmaa.2004.12.020
|
| [11] | F. Başar, M. Y. Savaşcı, Double sequence spaces and four-dimensional matrices, New York: Chapman and Hall/CRC, 2022. https://doi.org/10.1201/9781003285786 |
| [12] |
F. Başar, H. Çapan, On the paranormed spaces of regularly convergent double sequences, Results Math., 72 (2017), 893–906. https://doi.org/10.1007/s00025-017-0693-5 doi: 10.1007/s00025-017-0693-5
|
| [13] |
J. Boos, T. Leiger, K. Zeller, Consistency theory for SM-methods, Acta Math. Hungar., 76 (1997), 109–142. https://doi.org/10.1007/bf02907056 doi: 10.1007/bf02907056
|
| [14] | H. Çapan, Some paranormed double sequence spaces and backward difference matrix, PhD Thesis, Istanbul University, 2018. |
| [15] | M. Başarır, O. Sonalcan, On some double sequence spaces, J. Indian Acad. Math., 21 (1999), 193–200. |
| [16] |
O. Tuǧ, V. Rakočević, E. Malkowsky, On the spaces $\mathcal{BV}_0$ of double sequences of bounded variation, Asian Eur. J. Math., 15 (2022), 2250204. https://doi.org/10.1142/S1793557122502047 doi: 10.1142/S1793557122502047
|
| [17] | M. Zeltser, The solid topology in double sequence spaces, In: Seminarberichte des facbereichs mathematik der fernuniversität hagen, 1998, 53–67. https://doi.org/10.18445/20180823-144022-0 |
| [18] |
M. Yeşilkayagil, F. Başar, $AK(\nu)$-property of double series spaces, Bull. Malays. Math. Sci. Soc., 44 (2021), 881–889. https://doi.org/10.1007/s40840-020-00982-z doi: 10.1007/s40840-020-00982-z
|
| [19] | Ş. Sezgek, İ. Daǧadur, $\nu$-wedge FDK-Spaces, Math. Morav., 29 (2025), 67–-82. |
| [20] | P. K. Kamthan, M. Gupta, Sequence spaces and series, New York: Marcel Dekker, 1981. |
| [21] |
G. Bennett, The gliding hump technique for FK-spaces, Trans. Amer. Math. Soc., 166 (1972), 285–292. https://doi.org/10.1090/S0002-9947-1972-0296564-9 doi: 10.1090/S0002-9947-1972-0296564-9
|
| [22] |
M. Zeltser, On conservative matrix methods for double sequence spaces, Acta Math. Hungar., 95 (2002), 221–242. https://doi.org/10.1023/A:1015636905885 doi: 10.1023/A:1015636905885
|
| [23] | A. Wilansky, Summability through functional analysis, North-Holland, 1984. |
| [24] |
Y. Shi, X. Yang, The pointwise error estimate of a new energy-preserving nonlinear difference method for supergeneralized viscous Burgers' equation, Comp. Appl. Math., 44 (2025), 257. https://doi.org/10.1007/s40314-025-03222-x doi: 10.1007/s40314-025-03222-x
|
| [25] |
X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7 doi: 10.1007/s10915-024-02511-7
|
Şeyda Sezgek, İlhan Daǧadur. $ \nu- $Conull FDK-spaces[J]. AIMS Mathematics, 2025, 10(11): 25708-25728. doi: 10.3934/math.20251138
DownLoad: