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PARANORMED FIBONACCI SEQUENCE SPACES. 917. bF10.p/ D. \. B>1 a D .ak/ 2 ! W 9.˛k/ R. 3 lim n!1 n. X. kD0. ˇ. ˇ. ˇ. ˇ n. X. jDk f 2. jC1. fkfkC1 aj. ˛k.
Miskolc Mathematical Notes Vol. 16 (2015), No. 2, pp. 907–923

HU e-ISSN 1787-2413 DOI: 10.181514/MMN.2015.1227

SOME NEW PARANORMED DIFFERENCE SEQUENCE SPACES DERIVED BY FIBONACCI NUMBERS EMRAH EVREN KARA AND SERKAN DEM˙IR˙IZ Received 06 May, 2014 Abstract. In this study, we define new paranormed sequence spaces derived by the sequences of Fibonacci numbers. Furthermore, we compute the ˛ ; ˇ and duals and obtain bases for these sequence spaces. Besides this, we characterize the matrix transformations from the new paranormed sequence spaces to the Maddox’s spaces c0 .q/; c.q/; `.q/ and `1 .q/. 2010 Mathematics Subject Classification: 46A45; 40C05; 40A05 Keywords: paranormed sequence spaces, matrix transformations, the sequences of Fibonacci numbers

1. P RELIMINARIES , BACKGROUND AND NOTATION By !, we shall denote the space of all real valued sequences. Any vector subspace of ! is called as a sequence space. We shall write `1 ; c and c0 for the spaces of all bounded, convergent and null sequences, respectively. Also by bs; cs; `1 and `p ; we denote the spaces of all bounded, convergent, absolutely and p absolutely convergent series, respectively; where 1 < p < 1. A linear topological space X over the real field R is said to be a paranormed space if there is a subadditive function g W X ! R such that g. / D 0; g.x/ D g. x/ and scalar multiplication is continuous, i.e., j˛n ˛j ! 0 and g.xn x/ ! 0 imply g.˛n xn ˛x/ ! 0 for all ˛ 0 s in R and all x’s in X, where  is the zero vector in the linear space X. Assume here and after that .pk / be a bounded sequences of strictly positive real numbers with sup pk D H and M D maxf1; H g. Then, the linear spaces c.p/; c0 .p/; `1 .p/ and `.p/ were defined by Maddox [20, 21] (see also Simons [27] and Nakano [25]) as follows:   c.p/ D x D .xk / 2 ! W lim jxk ljpk D 0 for some l 2 C ; k!1   pk c0 .p/ D x D .xk / 2 ! W lim jxk j D 0 ; k!1

c 2015 Miskolc University Press

EMRAH EVREN KARA AND SERKAN DEM˙IR˙IZ

908

  `1 .p/ D x D .xk / 2 ! W sup jxk jpk < 1 k2N

and   X pk `.p/ D x D .xk / 2 ! W jxk j < 1 ; k

which are the complete spaces paranormed by pk =M

h1 .x/ D sup jxk j k2N

iff inf pk > 0 k2N

and

h2 .x/ D

X

pk

jxk j

1=M ;

k 0

respectively. We shall assume throughout that pk 1 C .pk / 1 D 1 provided 1 < inf pk < H < 1. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to 1. By F and Nk , we shall denote the collection of all finite subsets of N and the set of all n 2 N such that n  k, respectively. We write by U for the set of all sequences u D .un / such that un ¤ 0 for all n 2 N. For u 2 U, let 1=u D .1=un /. For the sequence spaces X and Y , define the set S.X; Y / by S.X; Y / D f´ D .´k / 2 ! W x´ D .xk ´k / 2 Y for all x 2 Xg:

(1.1)

With the notation of (1.1), the ˛ ; ˇ and duals of a sequence space X , which are respectively denoted by X ˛ ; X ˇ and X , are defined by X ˛ D S.X; `1 /; X ˇ D S.X; cs/ and X D S.X; bs/: Let .X; h/ be a paranormed space. A sequence .bk / of the elements of X is called a basis for X if and only if, for each x 2 X, there exists a unique sequence .˛k / of scalars such that ! n X h x ˛k bk ! 0 as n ! 1: kD0

P The series ˛k bk which has P the sum x is then called the expansion of x with respect to .bn / and written as x D ˛k bk . Let X; Y be any two sequence spaces and A D .ank / be an infinite matrix of real numbers ank , where n; k 2 N. Then, we say that A defines a matrix mapping from X into Y , and we denote it by writing A W X ! Y , if for every sequence x D .xk / 2 X the sequence Ax D ..Ax/n /, the A-transform of x, is in Y , where X .Ax/n D ank xk ; .n 2 N/: (1.2) k

By .X W Y /, we denote the class of all matrices A such that A W X ! Y . Thus, A 2 .X W Y / if and only if the series on the right-hand side of (1.2) converges for each n 2 N and every x 2 X, and we have Ax D f.Ax/n gn2N 2 Y for all x 2 X . A

PARANORMED FIBONACCI SEQUENCE SPACES

909

sequence x is said to be A- summable to ˛ if Ax converges to ˛ which is called as the A- limit of x. For a sequence space X, the matrix domain XA of an infinite matrix A is defined by XA D fx D .xk / 2 ! W Ax 2 X g: (1.3) The approach constructing a new paranormed sequence space by means of the matrix domain of a particular limitation method has recently been employed by Malkowsky [23], Et and C ¸ olak [15], Malkowsky and Savas¸ [24], Altay and Bas¸ar [3, 4], F. Bas¸ar et al., [9], Aydın and Bas¸ar [5, 6]. Define the sequence ffn g1 nD0 of Fibonacci numbers given by the linear recurrence relations f0 D f1 D 1 and fn D fn 1 C fn 2 ; n  2: In modern science and particularly physics, there is quite an interest in the theory and applications of Fibonacci numbers. The ratio of the successive Fibonacci numbers is as known golden ratio. There are many applications of golden ratio in many places of mathematics and physics, in general theory of high energy particle theory [26]. Also, some basic properties of Fibonacci numbers [19] are given as follows: p fnC1 1C 5 lim D D˛ .golden ratio/ n!1 fn 2 n X kD0

fk D fnC2

1

.n 2 N/ and

X 1 fk

converges

k

fn 1 fnC1 fn2 D . 1/nC1 .n  1/ .Cassini formula/: Substituting for fnC1 in Cassini’s formula yields fn2 1 C fn fn 1 fn2 D . 1/nC1 . Let fn be the nth Fibonacci number for every n 2 N. Then, the infinite Fibonacci bnk / is defined by b D .f matrix F 8 fnC1 ˆ ˆ .k D n 1/; ˆ ˆ < fn bnk D fn f .k D n/; ˆ ˆ ˆ f nC1 ˆ : 0 .0  k < n 1 or k > n/ where n; k 2 N [17]. Recently, a lot of papers have been studying by many researchers on Fibonacci sequences. For instance, see [11–13]. b ; p/; The main purpose of this study is to introduce the sequence spaces c0 .F b ; p/; `1 .F b ; p/ and `.F b ; p/ which are the set of all sequences whose F b trans c.F forms are in the spaces c0 .p/; c.p/; `1 .p/ and `.p/, respectively. Also, we have investigated some topological structures, which have completeness, the ˛ ; ˇ and

duals, and the bases of these sequence spaces. Besides this, we characterize some matrix mappings on these spaces.

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EMRAH EVREN KARA AND SERKAN DEM˙IR˙IZ

2. T HE PARANORMED F IBONACCI DIFFERENCE SEQUENCE SPACES b ; p/; c.F b ; p/; `1 .F b ; p/ In this section, we define the new sequence spaces c0 .F b ; p/ by using the sequences of Fibonacci numbers, and prove that these seand `.F quence spaces are the complete paranormed linear metric spaces and compute their b ; p/; c.F b ; p/ ˛ ; ˇ and duals. Moreover, we give the basis for the spaces c0 .F b ; p/. and `.F In [14], Choudhary and Mishra have defined the sequence space `.p/ which consists of all sequences such that S -transforms are in `.p/, where S D .snk / is defined by  1; .0  k  n/; snk D 0; .k > n/: Bas¸ar and Altay [8] have recently examined the space bs.p/ which is formerly defined by Bas¸ar in [7] as the set of all series whose sequences of partial sums are in `1 .p/. More recently, Altay and Bas¸ar have studied the sequence spaces t .p/ in [1] and r t .p/; r t .p/ in [2] which are derived by the Riesz means r t .p/; r1 c 0 from the sequence spaces `.p/; `1 .p/; c.p/ and c0 .p/ of Maddox, respectively. t .p/; r t .p/ and r t .p/ may With the notation of (1.3), the spaces `.p/; bs.p/; r t .p/; r1 c 0 be redefined by `.p/ D Œ`.p/S ; bs.p/ D Œ`1 .p/S ; r t .p/ D Œ`.p/Rt ; t .p/ D Œ`1 .p/Rt ; rct .p/ D Œc.p/Rt ; r0t .p/ D Œc0 .p/Rt : r1

Following Choudhary and Mishra [14], Bas¸ar and Altay [8], Altay and Bas¸ar [1, 2], b ; p/; c.F b ; p/; `1 .F b ; p/ and `.F b ; p/ by we define the sequence spaces c0 .F ˇ ˇpn   ˇ fn ˇ fnC1 ˇ ˇ b c0 .F ; p/ D x D .xk / 2 ! W lim ˇ xn xn 1 ˇ D 0 n!1 fnC1 fn ˇ ˇ pn   ˇ fn ˇ f nC1 ˇ b ; p/ D x D .xk / 2 ! W 9l 2 C 3 lim ˇ c.F x x l D 0 n n 1 ˇ n!1 ˇ fnC1 fn ˇ ˇpn   ˇ ˇ b ; p/ D x D .xk / 2 ! W sup ˇ fn xn fnC1 xn 1 ˇ < 1 `1 .F ˇf ˇ fn nC1 n2N and ˇ  X ˇ fn ˇ b xn `.F ; p/ D x D .xk / 2 ! W ˇf nC1 n

fnC1 xn fn

ˇpn  ˇ ˇ g is a paranorm for the spaces c.F b b b b 0 when c.F ; p/ D c.F / and `1 .F ; p/ D `1 .F /. b / is a complete linear metric space paranormed by (ii) `p .F ˇpk 1=M Xˇ ˇ ˇ fk fkC1  ˇ xk xk 1 ˇˇ : g .x/ D ˇf fk kC1 k

b ; p/. The linearity of c0 .F b ; p/ Proof. We prove the theorem for the space c0 .F with respect to the coordinatewise addition and scalar multiplication follows from b ; p/ (see [22, p.30]): the following inequalities which are satisfied for x; ´ 2 c0 .F ˇ ˇpk =M ˇ fk ˇ fkC1 ˇ sup ˇ .xk C ´k / .xk 1 C ´k 1 /ˇˇ fk k2N fkC1 ˇ ˇpk =M ˇ ˇ fk ˇ fk ˇ fkC1 ˇ ˇ C sup ˇˇ  sup ˇ xk xk 1 ˇ ´k fk k2N fkC1 k2N fkC1

fkC1 ´k fk

ˇpk =M ˇ ˇ 1ˇ

(2.2)

and for any ˛ 2 R (see [20]), j˛jpk  maxf1; j˛jM g:

(2.3)

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EMRAH EVREN KARA AND SERKAN DEM˙IR˙IZ

b ; p/. Again the inequalities It is clear that g. / D 0 and g.x/ D g. x/ for all x 2 c0 .F (2.2) and (2.3) yield the subadditivity of g and g.˛x/  maxf1; j˛jgg.x/: b ; p/ such that g.x n x/ ! 0 and Let fx n g be any sequence of the points x n 2 c0 .F .˛n / also be any sequence of scalars such that ˛n ! ˛. Then, since the inequality g.x n /  g.x/ C g.x n

x/

fg.x n /g

holds by the subadditivity of g, is bounded and we thus have ˇ ˇ fk fkC1 g.˛n x n ˛x/ D sup ˇˇ .˛n xkn ˛xk / .˛n xkn 1 ˛xk f f kC1 k k2N  j˛n

˛j g.x n / C j˛j g.x n

ˇpk =M ˇ ˇ 1 /ˇ

x/;

which tends to zero as n ! 1. That is to say that the scalar multiplication is conb ; p/. tinuous. Hence, g is a paranorm on the space c0 .F b ; p/. Let fx i g be any It remains to prove the completeness of the space c0 .F .i / b ; p/, where x i D fx ; x .i / ; :::g. Then, for a given Cauchy sequence in the space c0 .F 0 1 " > 0 there exists a positive integer n0 ."/ such that " g.x i x j / < 2 for all i; j  n0 ."/. We obtain by using definition of g for each fixed k 2 N that ˇ ˇ ˇ ˇ ˇfF b x i gk fF b x j gk ˇpk =M  sup ˇfF b x i gk fF b x j gk ˇpk =M < " (2.4) 2 k2N b x 0 /k ; .F b x 1 /k ; :::g is a for every i; j  n0 ."/, which leads us to the fact that f.F Cauchy sequence of real numbers for every fixed k 2 N. Since R is complete, it converges, say b x i gk ! f F b xgk fF b x/0 ; .F b x/1 ; :::, we define the seas i ! 1. Using these infinitely many limits .F b x/0 ; .F b x/1 ; :::g. We have from (2.4) with j ! 1 that quence f.F ˇ ˇ ˇfF b x i gk fF b xgk ˇpk =M  " .i  n0 ."// (2.5) 2 .i / b ; p/, for every fixed k 2 N. Since x i D fxk g 2 c0 .F ˇ ˇ ˇfF b x i gk ˇpk =M < " 2 for all k 2 N. Therefore, we obtain (2.5) that ˇ ˇ ˇ ˇ ˇ ˇ ˇfF b x i gk ˇpk =M b xgk ˇpk =M  ˇfF b xgk fF b x i gk ˇpk =M C ˇfF 1 such that ˇ ˇp 0 ˇ k X ˇˇ X 1ˇ sup ank B ˇ < 1: (2.11) ˇ ˇ ˇ K2F k

n2K

(ii) Let 0 < pk  1 for all k 2 N. Then, A 2 .`.p/ W `1 / if and only if ˇ ˇ pk ˇX ˇ ˇ ˇ sup sup ˇ ank ˇ < 1: ˇ ˇ K2F k2N

(2.12)

n2K

Lemma 5 ([16], Theorem 1 (i)-(ii)). (i) Let 1 < pk  H < 1 for all k 2 N. Then, A 2 .`.p/ W `1 / if and only if there exists an integer B > 1 such that X 0 sup jank B 1 jpk < 1: (2.13) n2N k

(ii) Let 0 < pk  1 for all k 2 N. Then, A 2 .`.p/ W `1 / if and only if sup jank jpk < 1:

(2.14)

n;k2N

Lemma 6 ([16], Corollary for Theorem 1). Let 0 < pk  H < 1 for all k 2 N. Then, A 2 .`.p/ W c/ if and only if (2.13), (2.14) hold, and lim ank D ˇk ;

n!1

.k 2 N/

(2.15)

also holds. Theorem 3. Let K  D fk 2 N W 0  k  ng \ K for K 2 F and B 2 N2 . Define b 1 .p/; F b 2 .p/; F b 3 .p/; F b 4 .p/; F b 5 .p/; F b 6 .p/; F b 7 .p/ and F b 8 .p/ as follows: the sets F ˇ ˇ   2 [ X ˇ X fnC1 ˇ 1=p k ˇ ˇ1

K2F

n

k2K 

k kC1

PARANORMED FIBONACCI SEQUENCE SPACES

915

ˇ  n 2 X ˇˇ X ˇ fnC1 ˇ ˇ b 2 .p/ D a D .ak / 2 ! W F an ˇ < 1 ˇ f f 

n

b 3 .p/ D F

[

kD0

a D .ak / 2 ! W sup

k kC1

n ˇX n X

ˇ ˇ ˇ

n2N kD0 j Dk

B>1

fj2C1 fk fkC1

ˇ 1 ˇ  ˇ X fj2C1 ˇ ˇ b F 4 .p/ D a D .ak / 2 ! W ˇ aj ˇˇ < 1 fk fkC1 j Dk  [ b 5 .p/ D F a D .ak / 2 ! W 9.˛k /  R

ˇ ˇ aj ˇˇB

1=pk

 1

3 sup

n ˇX X ˇ n fj2C1 ˇ aj ˇ f f

n2N kD0 j Dk

ˇ ˇ ˛k ˇˇB

k kC1

1=pk

 n; where n; k 2 N. Bearing in mind (2.1) we immediately derive that an xn D

n 2 X fnC1 an yk D .Cy/n I fk fkC1

kD0

.n 2 N/:

(2.16)

EMRAH EVREN KARA AND SERKAN DEM˙IR˙IZ

916

b ; p/ if We therefore observe by (2.16) that ax D .an xn / 2 `1 whenever x 2 c0 .F b ; p/g˛ and only if Cy 2 `1 whenever y 2 c0 .p/. This means that a D .an / 2 fc0 .F b ; p/ if and only if C 2 .c0 .p/ W `1 /. Then, we derive by whenever x D .xn / 2 c0 .F Lemma 1 that b ; p/g˛ D F b 1 .p/: fc0 .F Consider the following equations for n 2 N,  X n n k 2 X X fkC1 ak xk D ak yj fj fj C1 kD0

D

j D0 kD0  n n X X fj2C1 kD0

j Dk

fk fkC1

 aj yk

D .Dy/n

(2.17)

where D D .dnk / is defined by 8 n X fj2C1 ˆ < aj ; 0  k  n; fk fkC1 dnk D j Dk ˆ : 0; k > n; where n; k 2 N. Thus, we deduce from Lemma 2 with (2.17) that ax D .ak xk / 2 cs b ; p/ if and only if Dy 2 c whenever y D .yk / 2 c0 .p/. whenever x D .xk / 2 c0 .F b ; p/gˇ whenever x D .xn / 2 c0 .F b ; p/ if and only This means that a D .an / 2 fc0 .F if D 2 .c0 .p/ W c/. Therefore we derive from Lemma 2 that b 3 .p/ \ F b 4 .p/ \ F b 5 .p/: b ; p/gˇ D F fc0 .F As this, we deduce from Lemma 3 with (2.17) that ax D .ak xk / 2 bs whenever b ; p/ if and only if Dy 2 `1 whenever y D .yk / 2 c0 .p/. This x D .xk / 2 c0 .F b ; p/g whenever x D .xn / 2 c0 .F b ; p/ if and only if means that a D .an / 2 fc0 .F D 2 .c0 .p/ W `1 /. Therefore we obtain Lemma 3 that b ; p/g D F b 3 .p/ fc0 .F 

and this completes the proof.

Theorem 4. Let K  D fk 2 N W 0  k  ng \ K for K 2 F and B 2 N2 . Define b 8 .p/; F b 9 .p/; F b 10 .p/ and F b 11 .p/ as follows: the sets F ˇ ˇ  n \ Xˇ X X fj2C1 ˇ 1=pk ˇ ˇ b aj B F 8 .p/ D a D .ak / 2 ! W sup ˇ1

k2K j Dk

ˇ  n ˇX \ X ˇ n fj2C1 ˇ 1=p k ˇ ˇ b F 9 .p/ D a D .ak / 2 ! W sup aj ˇB 1

n2N kD0 j Dk

k kC1

PARANORMED FIBONACCI SEQUENCE SPACES

b 10 .p/ D F

917

\ a D .ak / 2 ! W 9.˛k /  R B>1 n ˇX X ˇ n fj2C1 ˇ 3 lim aj ˇ n!1 f f kD0 j Dk

b 11 .p/ D F

k kC1

ˇ  ˇ 1=p k ˇ D0 ˛k ˇB

ˇ  n ˇX \ X ˇ 1=p ˇ n fj2C1 k ˇ ˇ a D .ak / 2 ! W sup aj ˇB 1

k kC1

Then b ; p/g˛ D F b 8 .p/ (i) f`1 .F b ; p/gˇ D F b 9 .p/ \ F b 10 .p/ (ii) f`1 .F b ; p/g D F b 11 .p/. (iii) f`1 .F Proof. This may be obtained in the similar way, as mentioned in the proof of Theorem 3 with Lemmas 4(i), 5(i), 6 instead of Lemmas 1-3. So, we omit the details.  Theorem 5. Let K  D fk 2 N W 0  k  ng \ K for K 2 F and B 2 N2 . Define b 12 .p/; F b 13 .p/; F b 14 .p/; F b 15 .p/ and F b 16 .p/ as follows: the sets F ˇ X X ˇpk   n fj2C1 ˇ ˇ ˇ ˇ b F 12 .p/ D a D .ak / 2 ! W sup sup ˇ aj 1

b 14 .p/ D F

[ B>1

ˇ n Xˇ X X fj2C1 ˇ a D .ak / 2 ! W sup aj B ˇ fk fkC1 K2F k

n2K j Dk

n ˇX X ˇ n fj2C1 ˇ a D .ak / 2 ! W sup aj B ˇ f f n2N kD0 j Dk

k kC1

k kC1

 n X fj2C1 b aj F 16 .p/ D a D .ak / 2 ! W lim n!1 fk fkC1

 exists

j Dk

Then (i) ( b ; p/g˛ D f`.F

ˇ