Serkan Demiri, Hacer Bilgin Ellidokuzoğlu

Some Algebraic and Topological Properties of New Lucas Difference Sequence Spaces

Karakas¸ and Karabudak [22], introduced the Lucas sequence spaces X(E) and studied their someproperties. The main purpose of this study is to introduce the Lucas difference sequence spaces c0(Lˆ, ∆) and c(Lˆ, ∆)by using the Lucas sequence. Also, we prove that the spaces c0(Lˆ, ∆) and c(Lˆ, ∆), are linearly isomorphic to spacesc0 and c, respectively. Besides this, the α−, β− and γ−duals of this spaces have been computed, their bases havebeen constructed and some topological properties of these spaces have been studied. Finally, the classes of matrices(c0(Lˆ, ∆) : µ) and (c(Lˆ, ∆) : µ) have been characterized, where µ is one of the sequence spaces `∞, c and c0 andderives the other characterizations for the special cases of µ.

PDF

___

[1] Ahmad, Z.U., Mursaleen, M., K ¨othe-Toeplitz duals of some new sequence spaces and their matrix maps. Publ. Inst., Math. (Belgr.) 42(1987), 57-61.
[2] Altay, B., Başar, F., The fine spectrum and the matrix domain of the difference operator ∆ on the sequence space `p, (0 < p < 1), Commun.Math. Anal., 2(2007), 1-11.
[3] Altay, B., Başar, F., Certain topological properties and duals of the domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336(2007), 632-645.
[4] Başar, F., Domain of the composition of some triangles in the space of p−summable sequences for 0 < p ≤ 1, AIP Conference Proceedings, 1759(2016), 020003-1-020003-6; doi: 10.1063/1.4959617.
[5] Başar, F., Braha, N.L., Euler-Ces´aro difference spaces of bounded, convergent and null sequences, Tamkang J. Math. 47(4)(2016), 405-420.
[6] Başar, F., C¸ akmak, A.F., Domain of triple band matrix B(r, s, t) on some Maddox’s spaces, Ann. Funct. Anal., 3(1)(2012), 32-48.
[7] Başar, F., Kirişc¸i, M., Almost convergence and generalized difference matrix, Comput. Math. Appl., 61(3)(2011), 602-611.
[8] Başar, F., Malkowsky, E., Altay, B., Matrix transformations on the matrix domains of triangles in the spaces of strongly C1−summable and bounded sequences, Publ. Math. Debrecen, 73(1-2)(2008), 193-213.
[9] Yeşilkayagil, M., Başar, F., Spaces of Aλ−almost null and Aλ−almost convergent sequences, J. Egypt. Math. Soc., 23(2)(2015), 119-126.
[10] Yeşilkayagil, M., Başar, F., On the paranormed N¨orlund sequence space of non-absolute type, Abstr. Appl. Anal., (2014), Article ID 858704, 9 pages, 2014. doi:10.1155/2014/858704.
[11] Başar, F., Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, Istanbul, 2012. 1
[12] Başar, F., Altay, B., On the space of sequences of p−bounded variation and related matrix mappings, Ukrainian Math. J., 55(2003), 136-147.
[13] Başar, F., Altay, B., Mursaleen, M., Some generalizations of the space bvp of p−bounded variation sequences, Nonlinear Anal. TMA, 68(2008), 273-287.
[14] C¸ apan, H., Başar, F., Domain of the double band matrix defined by Fibonacci numbers in the Maddox’s space `(p), Electron. J. Math. Anal. Appl., 3(2)(2015), 31-45.
[15] C¸ olak, R., Et, M., Malkowsky, E., Some Topics of Sequence Spaces, in: Lecture Notes in Mathematics, Fırat Univ. Press, (2004), 1-63, ISBN: 975-394-0386-6.
[16] Das, A., Hazarika, B., Some properties of generalized Fibonacci difference bounded and p−absolutely convergent sequences, Bol. Soc. Paran. Mat., 36(3)(2018), 37-50.
[17] Das, A., Hazarika, B., Matrix transformation of Fibonacci band matrix on generalized bv-space and its dual spaces, Bol. Soc. Paran. Mat., 36(3)(2018), 41-52.
[18] Kara, E.E., Ilkhan, M., Some properties of generalized Fibonacci sequence spaces, Linear Multilinear Algebra, 64(11)(2016), 2208-2223.
[19] Başarır, M., Başar, F., Kara, E.E., On the spaces of Fibonacci difference absolutely p−summable, null and convergent sequences, Sarajevo J. Math., 12(25)(2016), 167-182.
[20] Kara, E.E., Başarır, M., Mursaleen, M., Compactness of matrix operators on some sequence spaces derived by Fibonacci numbers, Kragujevac J. Math., 39(2)(2015), 217-230.
[21] Kara, E.E., Demiriz, S., Some new paranormed difference sequence spaces derived by Fibonacci numbers, Miskolc Math. Notes., 16(2)(2015), 907-923.
[22] Karakaş, M., Karabudak, H., Lucas Sayilari ve Sonsuz Toeplitz Matrisleri Uzerine Bir Uygulama, Cumhuriyet Science Journal, 38(3)(2017), 557-562. (document),
[23] Kızmaz, H., On certain sequence spaces, Canad.Math. Bull., 24(1981), 169-176.
[24] Kirişc¸i, M., Başar, F., Some new sequence spaces derived by the domain of generalized difference matrix, Comput.Math. Appl., 60(2010), 1299-1309.
[25] Koshy, T., Fibonacci and Lucas Numbers with applications, Wiley, 2001.
[26] Maddox, I.J., Continuous and K ¨othe-Toeplitz duals of certain sequence spaces, Proc. Camb. Philos. Soc., 65(1965), 431-435.
[27] Malkowsky, E., Klassen von Matrixabbildungen in paranormierten FK-R¨aumen, Analysis (Munich), 7(1987), 275-292.
[28] Malkowsky, E., Absolute and ordinary K ¨othe-Toeplitz duals of some sets of sequences and matrix transformations, Publ. Inst. Math. (Belgr.), 46(60)(1989), 97-103.
[29] Mursaleen, M., Generalized spaces of difference sequences, J. Math. Anal. Appl., 203(3)(1996), 738-745.
[30] Simons, S, The sequence spaces `(pv) and m(pv), Proc. Lond. Math. Soc., 3(15)(1965), 422-436.
[31] Sönmez, A., Some new sequence spaces derived by the domain of the triple bandmatrix, Comput.Math. Appl., 62(2011), 641-650.
[32] Stieglitz, M., Tietz, H., Matrix transformationen von folgenraumen eine ergebnisubersict, Math. Z., 154(1977), 1-16.
[33] Tripathy, B.C., Matrix transformation between some classes of sequences, J. Math. Anal. Appl., 2(1997), 448-450. 1
[34] Tug, O., Başar, F., ˘ On the domain of N¨orlund mean in the spaces of null and convergent sequences, TWMS J. Pure Appl. Math., 7(1)(2016), 76-87.
[35] Tuğ, O., Başar, F., On the spaces of N ¨orlund almost null and N ¨orlund almost convergent sequences, Filomat, 30(3)(2016), 773-783.
[36] Wilansky, A., Summability through Functional Analysis, North-HollandMathematics Studies 85, Amsterdam-Newyork-Oxford, 1984.