Gülsen Kılınç, Murat Candan

DOI Number
First page
Last page


This paper submits the sequence space $l\left( \widehat{F}\left( r,s\right)
,\mathcal{F},p,u\right) $ and $l_{\infty }\left( \widehat{F}\left(
r,s\right) ,\mathcal{F},p,u\right) $of non-absolute type under the domain of
the matrix$\widehat{\text{ }F}\left( r,s\right) $ constituted by using
Fibonacci sequence and non-zero real number $r$, $s$ and a sequence of
modulus functions. We study some inclusion relations, topological and
geometric properties of these spaceses. Further, we give the $\alpha $- $%
\beta $- and $\gamma $-duals of said sequence spaces and characterization of
the classes $\left( l\left( \widehat{F}\left( r,s\right) ,\mathcal{F}%
,p,u\right) ,X\right) $ and $\left( l_{\infty }\left( \widehat{F}\left(
r,s\right) ,\mathcal{F},p,u\right) ,X\right) $.


Sequence space, Fibonacci sequence, Modulus functions


Fibonacci numbers, modulus function, sequence spaces, matrix transformations, $\alpha -,\beta -,\gamma -$ duals, difference matrix.

Full Text:



bibitem{basar} F. Bac{s}ar, Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, xi+405 pp., .{I}stanbul, (2012), ISB:978-1-60805-252-3.

bibitem{m2} I. J. Maddox, Paranormed sequence spaces generated by infinite

matrices, Proc. Cambridge Philos. Soc., textbf{64}(1968), 335--340.

bibitem{m1} I. J. Maddox, Spaces of strongly summable sequences, Quart. J.

Math., Oxford, textbf{18}(2)(1967), 345--355.

bibitem{s} S. Simons, The sequence spaces $ell (p_{v})$ and $m(p_{v})$,

Proc. London Math. Soc., textbf{15}(3)(1965), 422--436.

bibitem{na} H. Nakano, Modulared sequence spaces, Proc. Japan Acad.,

textbf{27}(2)(1951), 508--512.

bibitem{albamu} B. Altay, F. Bac{s}ar, M. Mursaleen, On the Euler sequence

spaces which include the spaces $ell _{p}$ and $ell _{infty }$ I, Inform.

Sci., textbf{176}(10)(2006), 1450--1462.

bibitem{mubaal} M. Mursaleen, F. Bac{s}ar, B. Altay, On the Euler sequence

spaces which include the spaces $ell _{p}$ and $ell _{infty }$ II,

Nonlinear Anal., textbf{65}(3)(2006), 707--717.

bibitem{altba} B. Altay, F. Bac{s}ar, Generalization of the sequence space

$ell (p)$ derived by weighted mean, J. Math. Anal. Appl., textbf{330}%

(2007), 174--185.

bibitem{cafb1} C. Aydi n, F. Bac{s}ar, Some new sequence spaces which

include the spaces $ell _{p}$ and $ell _{infty }$, Demonstratio Math.,

textbf{38}(3)(2005), 641-656.

bibitem{cm} B. Choudhary, S. K Mishra, On K"{o}the-Toeplitz duals of

certain sequence spaces and their matrix transformations, Indian J. Pure

Appl. Math.,textbf{ 24, }no.5(1993), 291--301.

bibitem{murnom} M. Mursaleen, A. K. Noman , On some new sequence spaces of

non-absolute type related to the spaces $ell _{p}$ and $ell _{1}$I,

Filomat, textbf{25}(2)(2011), 33--51.

bibitem{mkfb} M. Kiri{c{s}}{c{c}}i, F. Bac{s}ar, Some new sequence

spaces derived by the domain of generalized difference matrix, Comput. Math.

Appl., textbf{60}(5)(2010), 1299--1309.

bibitem{candan} M. Candan, Domain of the double sequential band matrix in

the classical sequence spaces, J. Inequal. Appl., textbf{281}(2012), 15 pp.

bibitem{fbmk2} F. Bac{s}ar, M. Kiri{c{s}}{c{c}}i, Almost convergence and

generalized difference matrix, Comput. Math. Appl., textbf{61}(3) (2011),


bibitem{candan2} M. Candan, Almost convergence and double sequential band

matrix, Acta. Math. Sci.,textbf{ 34B}(2)(2014), 354--366.

bibitem{candan3} M. Candan, A new sequence space isomorphic to the space $%

ell (p)$ and compact operators, J. Math. Comput. Sci., textbf{4}, No:

(2014), 306-334

bibitem{candan4} M. Candan, Domain of the double sequential band matrix in

the spaces of convergent and null sequences, Adv. Difference Edu.,

(2014)163, 18 pp.

bibitem{candan5} M. Candan, Some new sequence spaces derived from the

spaces of bounded, convergent and null sequences, Int. J. Mod. Math. Sci.,%

textbf{ 12}(2)(2014), 74-87.

bibitem{pokasi} H. Polat, V. Karakaya, N. c{S}imc{s}ek, Difference

sequence spaces derived by generalized weighted mean, Appl. Math. Lett.,

textbf{24}(5)(2011), 608--314.

bibitem{basarir1} M. Bac{s}ari r, On the generalized Riesz $B$-difference

sequence spaces, Filomat, textbf{24}(4)(2010), 35--52.

bibitem{basarir2} M. Bac{s}ari r, M. "{O}zt"{u}rk, On the Riesz

diference sequence space, Rend. Circ. Mat. Palermo, textbf{57}(2008),


bibitem{basarir3} M. Bac{s}ari r, Paranormed Ces`{a}ro difference

sequence space and related matrix transformation, Dou{g}a Tr. J. Math.,

textbf{15}(1991), 14--19.

bibitem{hk} H. Ki zmaz, On certain sequence spaces, Canad. Math. Bull.,%

textbf{ 24}(2)(1981), 169--176.

bibitem{rrbf} B. Altay, F. Bac{s}ar, The matrix domain and the fine

spectrum of the difference operator $Delta $ on the sequence space $ell

_{p}$, $(0

bibitem{fbba} F. Bac{s}ar, B. Altay, On the space of sequences of

p-bounded variation and related matrix mappings, Ukrainian Math., J, textbf{%

}(1)(2003), 136--147.

bibitem{cem} R. c{C}olak, M. Et, Malkowsky E, Some Topics of Sequence

Spaces, Lecture Notes in Mathematics, Fi rat Univ. Press (2004) 1-63 ISBN:


bibitem{cet} R. c{C}olak, M. Et, On some generalized difference sequence

spaces and related matrix transformations, Hokkaido Math., J, textbf{26}%

(3)(1997), 483--492.

bibitem{ml} E. Malkowsky, S. D. Parashar, Matrix transformations in space

of bounded and convergent difference sequence of order $m$, Analysis,

textbf{17}(1997), 87--97.

bibitem{ba} B. Altay, On the space of $p-$summable difference sequences of

order $m$, $(1leq p

(4)(2006), 387--402.

bibitem{e} K. G. Grosseerdmann, Matrix transformations between the sequence

spaces of Maddox, J. Math. Anal. Appl., textbf{180}(1993), 223--238.

bibitem{lasmad} C. G. Lascarides, I. J. Maddox, Matrix transformations

between some classes of sequences, Proc. Cambridge Philos. Soc., textbf{68}%

(1970), 99--104.

bibitem{shga1} N. A. Sheikh, A. H. Ganie, A new paranormed sequence space

and some matrix transformations, Acta Math. Acad. Paedago, Nyregy, textbf{28%

}(2012), 47-58.

bibitem{shga2} A. H. Ganie, N. A. Sheikh, New type of paranormed sequence

space of non-absolute type and a matrix transformation, Int, J of Mod, Math,

Sci., textbf{8}(2)(2013), 196-211.

bibitem{kara} E. E. Kara, Some topological and geometrical properties of

new Banach sequence spaces, J. Inequal. Appl., textbf{38}(2013), 1-15.

bibitem{kara et al} E. E. Kara, M. Bac{s}ari r, M. Mursaleen, Compact

operators on the Fibonacci difference sequence spaces $l_{p}left( widehat{F%

}right) $ and $l_{infty }left( widehat{F}right) $, 1st International

Eurasian Conf. on Math.Sci.and Appl. Prishtine-Kosovo, (2012), September 3-7.

bibitem{başarkara} M. Bac{s}ari r, F. Bac{s}ar, E. E. Kara, On the

Fibonacci Difference Null and Convergent Sequences, arXiv:1309.0150.

bibitem{murat} M. Candan, A new aproach on the spaces of generalized

Fibonacci difference null and convergent sequences, Math. Aeterna, textbf{1}%


bibitem{candan7} M. Candan, K. Kayaduman, Almost convergent sequence space

derived by generalized Fibonacci matrix and Fibonacci core, Brithish J.

Math. Comput. Sci.,textbf{7}(2)(2015), 150-167.

bibitem{cankar} M. Candan, E. E. Kara, A study on topological and

geometrical characteristics of new Banach sequence spaces, under


bibitem{rajetal} Kuldip Raj, Suruchi Pandoh, Seema Jamwal, Fibonacci

difference sequence spaces for modulus functions, Le Matematiche,LXX textbf{%


bibitem{AHMUR} ZU. Ahmad, M. Mursaleen, K"{o}the-Toeplitz duals of some

new sequence spaces and their matrix maps, Publ. Inst. Math. (Belgr.)


bibitem{malkowsky} E. Malkowsky, Absolute and ordinary K"{o}the-Toeplitz

duals of some sets of sequences and matrix transformations, Publ. Inst.

Math. (Belgr.) textbf{46}(1989), 97-103.

bibitem{m3} I. J. Maddox, Continuous and K"{o}the-Toeplitz duals of

certain sequence spaces, Proc. Camb. Philos. Soc. textbf{65}(1965) 431-435.

bibitem{chnan} B. Choudhary and S. Nanda, Functional Analysis with

applications, John Wiley & Sons, New Delhi, .{I}ndia, 1989

bibitem{candan8} M. Candan, G. Ki li nc{c}, A different look for

paranormed Riesz sequence space derived by Fibonacci Matrix, Konuralp

Journal of Mathematics, volume 3, No:2, 62-76, 2015

bibitem{wilansky} A. Wilansky, Summability through functional analysis,

North- Holland Math. Stud. 85,1984.

bibitem{stieglitz} M. Stieglitz, H. Tietz, Matrix transformationen von

folgenr"{a}umen eine ergebnis"{u}bersicht, Math. Z. textbf{154}(1977),


bibitem{etçolak} M. Et, R. c{C}olak, On some generalized difference

sequence spaces and related matrix transformations, Hokkaido Math J. textbf{%

}(1997), 483-492.

bibitem{koshy} T. Koshy, Fibonacci and Lucas numbers with applications,

Wiley, New York, 2001.

bibitem{mursaleen} M. Mursaleen, On some geometric properties of a sequence

space related to $l_{p},$Bull. Aust. Math. Soc. textbf{67}(2003), 343-347.

bibitem{Raj} K. Raj, S. K. Sharma, Difference sequence spaces defined by

sequence of modulus function, Proyecciones textbf{30}(2011), 189-199.

bibitem{Rajsharma} K. Raj, S. K. Sharma, A. Gupta, Some multiplier lacunary

sequence spaces defined by a sequence of modulus functions, Acta Univ.

Sapientiae Math. textbf{4}(2012), 117-131.

bibitem{savaş} E. Savac{s}, V. Karakaya, N. c{S}imc{s}ek, Some $l_{p}-$%

type new sequence spaces and their geometric properties, Abstr. Appl. Anal.

, Article ID 696971 (2009).

bibitem{Suzan} Z. Suzan, c{C}. A. Bektac{s}, Generalized difference

sequence spaces defined by a sequence of moduli, Kragujevac J. Math. textbf{%

}(2012), 83-91.

bibitem{diestel} J. Diestel, Sequences and Series in Banach Spaces, Vol:

, Springer, New York, NY, USA, 1984.

bibitem{Garcia} J. Garcia-Falset, Stability and fixed points for

nonexpansive mappings, Houst. J. Math. 20 (1994), 495-506.

bibitem{Garcia 2} J. Garcia- Falset, The fixed point property in Banach

Spaces with the NUS-property, J. Math. Anal. Appl. 215 (1997), 532-542.

bibitem{Garcia 3} H. Knaust, Orlicz sequence Spaces of Banach-Saks Type,

Arch. Math. 59 (1992), 562-565.

bibitem{etalbas} M. Bac{s}ari r, F. Bac{s}ar, E. E. Kara, On the

Fibonacci Difference Null and Convergent Sequences, arXiv:1309.0150



  • There are currently no refbacks.

© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)