On Some New Paranormed Lucas Sequence Spaces and Lucas Core

Yıl 2018, Cilt: 1 Sayı: 1, 32 - 35, 14.12.2018

Serkan Demiriz , Hacer Bilgin Ellidokuzoğlu

Öz

The sequence spaces $c_{0}(\hat{L}), c(\hat{L}), \ell_{\infty}(\hat{L}) $ and $\ell_{p}(\hat{L})$ have been recently introduced and studied by Karakaç and Karabudak. The aim of this paper is to extend the results of Karakaç and Karabudak to the paranormed case and is to work the spaces $c_{0}(\hat{L},p), c(\hat{L},p), \ell_{\infty}(\hat{L},p) $ and $\ell(\hat{L},p)$. Furthermore, Lucas core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.

Anahtar Kelimeler

Lucas numbers , Lucas core , Matrix transformations , Paranormed sequence spaces

Kaynakça

• [1] V. Steven, Fibonacci and Lucas numbers, and the golden section theory and applications, Courier Corporation, 2008.
• [2] M. Karakaş and H. Karabudak, Lucas sayilari ve sonsuz Toeplitz matrisleri üzerine bir uygulama, Cumhuriyet Sci. J., 38(3) (2017), 557-562.
• [3] M. Başarır, F. Başar, E. E. Kara, On the spaces of Fibonacci difference absolutely p-summable, null and convergent sequences, Sarajevo J. Math., 12(25)(2016), 167-182.
• [4] A. Das, B. Hazarika, Matrix transformation of Fibonacci band matrix on generalized $bv-$space and its dual spaces, Bol. Soc. Paran. Mat., 36(3)(2018), 41-52.
• [5] A. Das, B. Hazarika, Some new Fibonacci difference spaces of nonabsolute type and compact operators, Linear Multilinear Algebra, 65(12) (2017), 2551-2573.
• [6] S. Debnath, S. Saha, Some newly defined sequence spaces using regular matrix of Fibonacci numbers, AKU J. Sci. Eng., 14 (2014), 1-3.
• [7] E. E. Kara, M. Basarir, An application of Fibonacci numbers into infinite Toeplitz matrices, Caspian J. Math. Sci. (CJMS), 1(1) (2012), 43-47.
• [8] E. E. Kara, M. Ilkhan. Some properties of generalized Fibonacci sequence spaces, Linear Multilinear Algebra, 64(11) (2016), 2208-2223.
• [9] E. E. Kara, S. Demiriz, Some new paranormed difference sequence spaces derived by Fibonacci numbers, Miskolc Math. Notes. 16(2) (2015), 907-923.
• [10] M. Karakaş, A new regular matrix defined by Fibonacci numbers and its applications, BEU J. Sci., 4(2) (2015), 205-210.
• [11] M. Karakaş, A. M. Karakaş, New Banach sequence spaces that is defined by the aid of Lucas numbers, I˘gdır Univ. J. Inst. Sci. Tech., 7(4) (2017), 103-111.
• [12] M. Karakaş, A. M. Karaka¸s, A study on Lucas difference sequence spaces $\ell_p(E(r,s))$ and $\ell_\infty(E(r,s))$}, Maejo Int. J. Sci. Technol., 12(01) (2018), 70-78.
• [13] I. J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Camb. Phios. Soc., 64 (1968), 335-340 .
• [14] I. J.Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford, 18(2) (1967), 345-355.
• [15] H. Nakano, Modulared sequence spaces, Proc. Jpn. Acad., 27(2) (1951), 508-512 .
• [16] B. Altay, F. Başar, On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 26 (2002), 701-715 .
• [17] B. Altay, F. Başar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 30 (2006), 591-608.
• [18] B. Altay, F. Başar, Generalization of the sequence space $\ell(p)$ derived by weighted mean, J. Math. Anal. Appl., 330 (2007), 174-185.
• [19] F. Başar, B. Altay, Matrix mappings on the space $bs(p)$ and its $\alpha-, \beta-$ and $\gamma-$duals, Aligarh Bull. Math., 21(1) (2002), 79-91 .
• [20] M. Başarır, E. E. Kara, On some difference sequence spaces of weighted means and compact operators, Ann. Funct. Anal., 2(2) (2011), 116-131.
• [21] H. Bilgin Ellidokuzoğlu, S. Demiriz, A. Köseo˜glu, On the paranormed binomial sequence spaces, Univers. J. Math. Appl. 1(3) (2018), 137-147.
• [22] M. Candan, Some new sequence spaces defined by a modulus function and an infinite matrix in a seminormed space, J. Math. Anal., 3(2) (2012), 1-9.
• [23] M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, J. Inequal. Appl., 2012 (2012), 15 pages, doi.org/10.1186/1029-242X-2012-281.
• [24] M. Candan, Almost convergence and double sequential band matrix, Acta. Math. Sci., 34B(2) (2014), 354-366.
• [25] M. Candan, A new sequence space isomorphic to the space `(p) and compact operators, J. Math. Comput. Sci., 4(2) (2014), 306-334.
• [26] M. Candan, Domain of the double sequential band matrix in the spaces of convergent and null sequences, Adv. Difference Edu., 2014(2014), 18 pages, doi.org/10.1186/1687-1847-2014-163.
• [27] M. Candan, A. Güne¸s, Paranormed sequence space of non-absolute type founded using generalized difference matrix, Proc. Nat. Aca. Sci., India section A: Physical Sciences, 85(2) (2015), 269-276.
• [28] U. Ulusu, E. Dündar, Asymptotically I-Cesaro equivalence of sequences of sets, Univers. J. Math. Appl., 1(2) (2018), 101-105.
• [29] H. Bilgin Ellidokuzo˜glu, S. Demiriz, A. Köseo˜glu, On the paranormed binomial sequence spaces, Univers. J. Math. Appl., 1(3) (2018), 1-11.
• [30] H. Polat, Some new Pascal sequence spaces, Fundam. J. Math. Appl., 1(1) (2018), 61-68.
• [31] S. Altundağ, On generalized difference lacunary statistical convergence in a paranormed space, J. Inequal. Appl., 2013(2013), 7 pages, doi.org/10.1186/1029-242X-2013-256.
• [32] S. Altundağ, M. Ba¸sarır, On generalized paranormed statistically convergent sequence spaces defined by Orlicz function, J. Inequal. Appl., 2009(2009), Article ID 729045, 13 pages, doi:10.1155/2009/729045.
• [33] R. G. Cooke , Infinite matrices and sequence spaces, Macmillan, London, 1950.