Some Algebraic and Topological Properties of New Lucas Difference Sequence Spaces

Year 2018, Volume: 10, 144 - 152, 29.12.2018

Hacer Bilgin Ellidokuzoğlu , Serkan Demiriz

Abstract

Karakaş and Karabudak [14], introduced the Lucas sequence spaces $X(E)$  and studied their some properties. The main purpose of this study is to introduce the Lucas difference sequence spaces $c_0(\hat{L},\Delta)$ and $c(\hat{L},\Delta)$  by using the Lucas sequence sequences. Also, the spaces $c_0(\hat{L},\Delta)$ and $c(\hat{L},\Delta)$, are linearly isomorphic to spaces $c_0$ and $c$, respectively, have been proved. Besides this, the $\alpha-,\beta-$ and $\gamma-$duals of this spaces have been computed, their bases have been constructed and some topological properties of these spaces have been studied. Finally, the classes of matrices $(c_0(\hat{L},\Delta) : \mu)$ and $(c(\hat{L},\Delta) : \mu)$ have been characterized, where $\mu$ is one of

the sequence spaces $\ell_\infty, c$ and $c_0$ and derives the other characterizations for the special cases of $\mu$.

Keywords

Lucas sequence spaces, matrix domain

References

• Ahmad, Z.U., Mursaleen, M., \textit{K\"{o}the-Toeplitz duals of some new sequence spaces and their matrix maps. Publ. Inst.}, Math. (Belgr.) \textbf{42}(1987), 57-61.
• Altay, B., Ba\c{s}ar, F., \textit{The fine spectrum and the matrix domain of the difference operator $\Delta$ on the sequence space $\ell_p, (0 < p < 1)$}, Commun.Math. Anal., \textbf{2}(2007), 1-11.
• Altay, B., Ba\c{s}ar, F., \textit{Certain topological properties and duals of the domain of a triangle matrix in a sequence space}, J. Math. Anal. Appl., \textbf{336}(2007), 632-645.
• Ba\c{s}ar, F., Domain of the composition of some triangles in the space of $p-$summable sequences for $0 < p \leq 1$, AIP Conference Proceedings, \textbf{1759}(2016), 020003-1-020003-6; doi: $10.1063/1.4959617.$
• Ba\c{s}ar, F., Braha, N.L., {\em Euler-Ces\'{a}ro difference spaces of bounded, convergent and null sequences}, Tamkang J. Math. \textbf{47(4)}(2016), 405-420.
• Ba\c{s}ar, F., \c{C}akmak, A.F., {\em Domain of triple band matrix $B(r, s, t)$ on some Maddox's spaces}, Ann. Funct. Anal., \textbf{3(1)}(2012), 32-48.
• Ba\c{s}ar, F., Kiri\c{s}\c{c}i, M., {\em Almost convergence and generalized difference matrix}, Comput. Math. Appl., \textbf{61(3)}(2011), 602-611.
• Ba\c{s}ar, F., Malkowsky, E., Altay, B., {\em Matrix transformations on the matrix domains of triangles in the spaces of strongly $C_1-$summable and bounded sequences}, Publ. Math. Debrecen, \textbf{73(1-2)}(2008), 193-213.
• Ye\c{s}ilkayagil, M., Ba\c{s}ar, F., {\em Spaces of $A_\lambda-$almost null and $A_\lambda-$almost convergent sequences}, J. Egypt. Math. Soc., \textbf{23(2)}(2015), 119-126.
• Ye\c{s}ilkayagil, M., Ba\c{s}ar, F., {\em On the paranormed N\"{o}rlund sequence space of non-absolute type}, Abstr. Appl. Anal., (2014), Article ID 858704, 9 pages, 2014. doi:$10.1155/2014/858704$.
• Ba\c{s}ar, F., Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, Istanbul, 2012.
• Ba\c{s}ar, F., Altay, B., \textit{On the space of sequences of $p-$bounded variation and related matrix mappings,} Ukrainian Math. J., \textbf{55}(2003), 136-147.
• Ba\c{s}ar, F., Altay, B., Mursaleen, M., \textit{Some generalizations of the space $bv_p$ of $p-$bounded variation sequences}, Nonlinear Anal. TMA, \textbf{68}(2008), 273-287. \c{C}apan, H., Ba\c{s}ar, F., \textit{Domain of the double band matrix defined by Fibonacci numbers in the Maddox's space $\ell(p)$}, Electron. J. Math. Anal. Appl., \textbf{3(2)}(2015), 31-45.
• \c{C}olak, R., Et, M., Malkowsky, E., \textit{Some Topics of Sequence Spaces}, in: Lecture Notes in Mathematics, F{\i}rat Univ. Press, (2004), 1-63, ISBN: 975-394-0386-6.
• Das, A., Hazarika, B., \textit{Some properties of generalized Fibonacci difference bounded and $p-$absolutely convergent sequences}, Bol. Soc. Paran. Mat., \textbf{36(3)}(2018), 37-50.
• Das, A., Hazarika, B., {\em Matrix transformation of Fibonacci band matrix on generalized bv-space and its dual spaces}, Bol. Soc. Paran. Mat., \textbf{36(3)}(2018), 41-52.
• Kara, E.E., Ilkhan, M., \textit{Some properties of generalized Fibonacci sequence spaces,} Linear Multilinear Algebra, \textbf{64(11)}(2016), 2208-2223.
• Ba\c{s}ar{\i}r, M., Ba\c{s}ar, F., Kara, E.E., \textit{On the spaces of Fibonacci difference absolutely $p-$summable, null and convergent sequences}, Sarajevo J. Math., \textbf{12(25)}(2016), 167-182.
• Kara, E.E., Ba\c{s}ar{\i}r, M., Mursaleen, M., \textit{Compactness of matrix operators on some sequence spaces derived by Fibonacci numbers,} Kragujevac J. Math., \textbf{39(2)}(2015), 217-230.
• Kara, E.E., Demiriz, S., \textit{Some new paranormed difference sequence spaces derived by Fibonacci numbers}, Miskolc Math. Notes., \textbf{16(2)}(2015), 907-923.
• Karaka\c{s}, M., Karabudak, H., \textit{Lucas Sayilari ve Sonsuz Toeplitz Matrisleri \"{U}zerine Bir Uygulama}, Cumhuriyet Science Journal, \textbf{38(3)}(2017), 557-562.
• K{\i}zmaz, H., \textit{On certain sequence spaces}, Canad.Math. Bull., \textbf{24}(1981), 169-176.
• Kiri\c{s}\c{c}i, M., Ba\c{s}ar, F., \textit{Some new sequence spaces derived by the domain of generalized difference matrix,} Comput.Math. Appl., \textbf{60}(2010), 1299-1309.
• Koshy, T., Fibonacci and Lucas Numbers with applications, Wiley, 2001.
• Maddox, I.J., \textit{Continuous and K\"{o}the-Toeplitz duals of certain sequence spaces}, Proc. Camb. Philos. Soc., \textbf{65}(1965), 431-435.
• Malkowsky, E., {\em Klassen von Matrixabbildungen in paranormierten FK-R\"{a}umen}, Analysis (Munich), \textbf{7}(1987), 275-292.
• Malkowsky, E., \textit{Absolute and ordinary K\"{o}the-Toeplitz duals of some sets of sequences and matrix transformations}, Publ. Inst. Math. (Belgr.), 46(60)(1989), 97-103.
• Mursaleen, M., \textit{Generalized spaces of difference sequences,} J. Math. Anal. Appl., \textbf{203(3)}(1996), 738-745.
• Simons, S, \textit{The sequence spaces $\ell(p_v)$ and $m(p_v )$}, Proc. Lond. Math. Soc., \textbf{3(15)}(1965), 422-436.
• S\"{o}nmez, A., \textit{Some new sequence spaces derived by the domain of the triple bandmatrix}, Comput.Math. Appl., \textbf{62}(2011), 641-650.
• Stieglitz, M., Tietz, H., \textit{Matrix transformationen von folgenraumen eine ergebnisubersict,} Math. Z., \textbf{154}(1977), 1-16.
• Tripathy, B.C., {\em Matrix transformation between some classes of sequences}, J. Math. Anal. Appl., \textbf{2}(1997), 448-450.
• Tu\u{g}, O., Ba\c{s}ar, F., {\em On the domain of N\"{o}rlund mean in the spaces of null and convergent sequences}, TWMS J. Pure Appl. Math., \textbf{7(1)}(2016), 76-87.
• Tu\u{g}, O., Ba\c{s}ar, F., {\em On the spaces of N\"{o}rlund almost null and N\"{o}rlund almost convergent sequences}, Filomat, \textbf{30(3)}(2016), 773-783.
• Wilansky, A., Summability through Functional Analysis, North-HollandMathematics Studies 85, Amsterdam-Newyork-Oxford, 1984.