ome Topological and Geometrical Properties of a New Difference Sequence Space

Volume 2011, Article ID 213878,14pages doi:10.1155/2011/213878

Research Article

Some Topological and Geometrical Properties of

a New Difference Sequence Space

Serkan Demiriz

_{and Celal C}

_{¸ akan}

1_{Department of Mathematics, Faculty of Arts and Science, Gaziosmanpas¸a University,}

60250 Tokat, Turkey

2_{Faculty of Education, In¨on ¨u University, 44280 Malatya, Turkey}

Correspondence should be addressed to Serkan Demiriz,serkandemiriz@gmail.com

Received 9 August 2010; Accepted 25 January 2011 Academic Editor: Narcisa C. Apreutesei

Copyrightq 2011 S. Demiriz and C. C¸akan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the new difference sequence space ar

pΔ . Further, it is proved that the space arpΔ

is the BK-space including the spacebvp, which is the space of sequences of pbounded variation.

We also show that the spacesar_pΔ, and pare linearly isomorphic for 1≤ p < ∞. Furthermore,

the basis and theα-, β- and γ-duals of the space ar_pΔ are determined. We devote the final section of the paper to examine some geometric properties of the spacear_pΔ.

1. Preliminaries, Background, and Notation

Byω, we will denote the space of all real valued sequences. Any vector subspace of ω is called

as a sequence space. We will write∞, c, and c0for the spaces of all bounded, convergent and

null sequences, respectively. Also, bybs, cs, 1 andp; we denote the spaces of all bounded,

convergent, absolutely, andp-absolutely convergent series, respectively, where 1 < p < ∞.

A sequence spaceλ with a linear topology is called a K-space, provided each of the

mapspi : λ → C defined by pix  xi is continuous for alli ∈ N, where C denotes the

complex field andN  {0, 1, 2, . . .}. A K-space λ is called an FK-space, provided λ is a complete

linear metric space. AnFK-space whose topology is normable is called a BK-space see 1,

pages 272-273.

Letλ, μ be two sequence spaces and A  ank an infinite matrix of real or complex

numbersa_nk, wheren, k ∈ N. Then, we say that A defines a matrix mapping from λ into μ,

and we denote it by writingA : λ → μ; if for every sequence x  xk ∈ λ, the sequence

Ax  {Ax_n}, the A-transform of x, is in μ, where

Ax_n

k

For simplicity in notation, here and in what follows, the summation without limits runs from

0 to∞. By λ : μ we denote the class of all matrices A such that A : λ → μ. Thus, A ∈ λ : μ

if and only if the series on the right side of1.1 converges for each n ∈ N and every x ∈ λ,

and we haveAx  {Ax_n}_n∈N∈ μ for all x ∈ λ. A sequence x is said to be A-summable to α

ifAx converges to α which is called as the A-limit of x.

If a normed sequence spaceλ contains a sequence b_n with the property that for every

x ∈ λ, there is a unique sequence of scalars αn such that

lim

n−→∞x − α0b0 α1b1 · · · αnbn  0, 1.2

thenbn is called a Schauder basis or briefly basis for λ. The seriesαkbkwhich has the sum

x is then called the expansion of x with respect to bn and written as x αkbk.

For a sequence spaceλ, the matrix domain λ_Aof an infinite matrixA is defined by

λA  {x  xk ∈ ω : Ax ∈ λ}, 1.3

which is a sequence space. The new sequence spaceλAgenerated by the limitation matrixA

from the spaceλ either includes the space λ or is included by the space λ, in general; that is,

the spaceλ_Ais the expansion or the contraction of the original spaceλ.

We will defineB_r  br_nk by br nk ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1− r n 1rk, 0 ≤ k ≤ n − 1, rn ₁ n 1, k  n, 0, k > n, 0 < r < 1, 1.4

for alln, k ∈ N and denote the collection of all finite subsets of N by F. We will also use the

convention that any term with negative subscript is equal to naught.

The approach constructing a new sequence space by means of the matrix domain of

a particular limitation method has been recently employed by Wang 2, Ng and Lee 3,

Malkowsky 4, and Altay et al. 5. They introduced the sequence spaces pNq in 2,

pC1  Xp in3, ∞Rt  r

t

∞, cRt  r_ct andc₀_Rt  r₀t in4 and pEr  er_pin5; where

Nq, C1, RtandEr denote the N ¨orlund, arithmetic, Riesz and Euler means, respectively, and

1≤ p ≤ ∞.

Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In literature, there are many papers concerning the geometric properties of different sequence spaces. For example,

in6, Mursaleen et al. studied some geometric properties of normed Euler sequence space.

S¸ims¸ek and Karakaya7 investigated the geometric properties of sequence space _ρu, v, p

equipped with Luxemburg norm. Further information on geometric properties of sequence

space can be found in8,9.

The main purpose of the present paper is to introduce the difference sequence space

ar

pΔ together with matrix domain and is to derive some inclusion relations concerning with

ar

pΔ. Also, we investigate some topological properties of this new space and furthermore

2.

a

Δ Difference Sequence Space

In the present section, we introduce the difference sequence space ar

pΔ and emphasize its

some properties. Although the difference sequence space λΔ corresponding to the space λ

was defined by Kızmaz 10 as follows:

λΔ  {x  xk ∈ ω : xk− xk 1 ∈ λ}, 2.1

the difference sequence space corresponding to the space p was not examined, where λ

denotes the anyone of the spacesc0, c or ∞. So, Bas¸ar and Altay have recently studied the

sequence spacebv_p, the space ofp-bounded variation, in 11 defined by

bvpx  xk ∈ ω : xk− xk−1 ∈ p , 1≤ p < ∞, 2.2

which fills up the gap in the existing literature. Recently, Ayd´ın and Bas¸ar 12 studied the

sequence spacesar₀andar_c, defined by

ar 0  x  xk ∈ ω : lim_{n−→∞n 1}1 n  k0  1 rkxk 0 , ar c x  xk ∈ ω : lim_{n−→∞n 1}1 n  k0  1 rkx_k exists . 2.3

Ayd´ın and Bas¸ar 13 introduced the difference sequence spaces ar₀Δ and ar_cΔ, defined by

ar 0Δ  x  xk ∈ ω : lim_{n−→∞n 1}1 n  k0  1 rkx_k− x_k−1  0 , ar cΔ  x  xk ∈ ω : lim_{n−→∞n 1}1 n  k0  1 rkxk− xk−1 exists . 2.4

Ayd´ın 14 introduced ar_psequence space, defined by

ar p x  xk ∈ ω :  n    1 n 1 n  k0  1 rkxk    p < ∞ ; 1≤ p < ∞. 2.5

Define the matrixΔ  δ_nk by

δnk ⎧ ⎨ ⎩ −1n−k_{, n − 1 ≤ k ≤ n,} 0, 0≤ k < n − 1 or k > n. 2.6

As was made by Bas¸ar and Altay in11, we treat slightly more different than Kızmaz and

by the matrix domain of a triangle limitation method. We will introduce the sequence space

ar

pΔ which is a natural continuation of Ayd´ın and Bas¸ar 13, as follows:

ar pΔ  x  xk ∈ ω :  n    1 n 1 n  k0  1 rkx_k− x_k−1_ p < ∞ ; 1≤ p < ∞. 2.7

With the notation of1.3, we may redefine the space ar_pΔ by

ar pΔ   ar p  Δ. 2.8

Define the sequencey  {y_nr} which will be frequently used as the B_r-transform of

a sequencex  xk, that is,

ynr  n−1  k0 1 − rrk 1 n xk 1 rn 1 nxn, n ∈ N. 2.9 Now, we may begin with the following theorem which is essential in the text.

Theorem 2.1. The set ar

pΔ becomes the linear space with the coordinatewise addition and scalar

multiplication which is the BK-space with the norm

x_ar

pΔyp, 2.10

where 1≤ p < ∞.

Proof. Since the proof is routine, we omit the details of the proof.

Theorem 2.2. The space ar

pΔ is linearly isomorphic to the space p; that is,arpΔ ∼ p, where

1≤ p < ∞.

Proof. It is enough to show the existence of a linear bijection between the spacesar_pΔ and _p

for 1≤ p < ∞. Consider the transformation T defined, with the notation of 2.9, from ar_pΔ

topbyx → y  Tx. The linearity of T is clear. Furthermore, it is trivial that x  θ whenever

Tx  θ, and hence, T is injective.

We assume thaty ∈ pfor 1≤ p < ∞ and define the sequence x  xk by

xnr  n  k0 k  jk−1 −1k−j 1 j 1 rkyj; k ∈ N. 2.11 Then, since Δx_n n jn−1 −1n−j 1 j 1 rnyj; n ∈ N, 2.12

we get that  n   1 n1 n  k0  1 rk k  jk−1 −1k−j 1 j 1 rkyj    p  n yn p_{< ∞. 2.13}

Thus, we have thatx ∈ ar_pΔ. In addition, one can derive that

x_ar pΔ _ n    1 1 n n  k0  1 rkxk− xk−1    p1/p  ⎛ ⎝ n   1 n1 n  k0  1 rk k  jk−1 −1k−j 1 j 1 rkyj    p⎞ ⎠ 1/p y_p 2.14

which means thatT is surjective and is norm preserving. Hence, T is a linear bijection.

We wish to exhibit some inclusion relations concerning with the spacear_pΔ.

Theorem 2.3. The inclusion bvp⊂ arpΔ strictly holds for 1 < p < ∞.

Proof. To prove the validity of the inclusionbvp⊂ arpΔ for 1 < p < ∞, it suffices to show the

existence of a numberK > 0 such that x_ar

pΔ≤ Kxbvpfor everyx ∈ bvp.

Letx ∈ bvpand 1< p < ∞. Then, we obtain

 n    1 1 n n  k0  1 rkx_k− x_k−1_ p ≤ n  2 n  k0 |xk− xk−1| 1 n p < 2p p p − 1 _p n |xn− xn−1|p, 2.15 as expected, x_ar pΔ≤  _2p p − 1  x_bvp, 2.16 for 1< p < ∞.

Furthermore, let us consider the sequencex  {xnr} defined by

xnr  n  k0 −1n 1 rk, n ∈ N. 2.17

Lemma 2.4 see 11, Theorem 2.4. The inclusion p⊂ bvpstrictly holds for 1< p < ∞.

CombiningLemma 2.4andTheorem 2.3, we get the following corollary.

Corollary 2.5. The inclusion p⊂ arpΔ strictly holds for 1 < p < ∞.

3. The Basis for the Space

a

Δ

In the present section, we will give a sequence of the points of the spacear_pΔ which forms a

basis for the spacear_pΔ, where 1 ≤ p < ∞.

Theorem 3.1. Define the matrix Br  {bkn r}n∈N of elements of the spacearpΔ for every fixed

k ∈ N by bnkr  ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 k 1 rk − 1 k 1 rk 1, 0 ≤ k ≤ n − 1, 1 n 1 rn, k  n, 0, k > n, 3.1

for every fixedk ∈ N. Then, the sequence {bkr}_k∈N is a basis for the spacear_pΔ, and any x ∈ ar

pΔ has a unique representation of the form

x 

k

λkrbkr, 3.2

whereλkr  Brxkfor allk ∈ N and 1 ≤ p < ∞.

Proof. It is clear that{bkr} ⊂ ar_pΔ, since

Brbkr  ek∈ p, k  0, 1, 2, . . . , 3.3

for 1≤ p < ∞; here, ekis the sequence whose only nonzero term is 1 in thekth place for each

k ∈ N. Let x ∈ ar

pΔ be given. For every nonnegative integer m, we set

xm_m k0

Then, by applyingBr to3.4, we obtain with 3.3 that Brxm  m  k0 λkrBrbkr  m  k0 Brxkek,  Br  x − xm i ⎧ ⎨ ⎩ 0, 0≤ i ≤ m, Brxi, i > m, 3.5

wherei, m ∈ N. For a given ε > 0, there is an integer m0such that

_∞ im |Brxi|p 1/p < ε 2, 3.6

for allm ≥ m0. Hence,

 x − xm ar pΔ _∞  im |Brxi|p _1/p ≤  _∞  im0 |Brxi|p _1/p < ε 2 < ε, 3.7

for allm ≥ m0, which proves thatx ∈ arpΔ is represented as in 3.2.

Let us show the uniqueness of representation forx ∈ ar_pΔ given by 3.2. Assume,

on the contrary, that there exists a representation x  _kμ_krbkr. Since the linear

transformationT, from ar_pΔ to p, used inTheorem 2.2is continuous, at this stage, we have

Brxn  k μkr  Brbkr  n   k μkrekn  μnr; n ∈ N, 3.8

which contradicts the fact thatBrxn λnr for all n ∈ N. Hence, the representation 3.2 of

x ∈ ar

pΔ is unique. This step concludes the proof.

4. The

α-, β-, and γ-Duals of the Space a

Δ

In this section, we state and prove theorems determining theα-, β-, and γ-duals of the space

ar

pΔ. Since the case p  1 can be proved by the same analogy and can be found in the

literature, we omit the proof of that case and consider only the case 1< p < ∞ in the proof of

Theorems4.4–4.6.

For the sequence spacesλ and μ, define the set Sλ, μ by

With the notation of4.1, α-, β- and γ-duals of a sequence space λ, which are, respectively,

denoted byλα, λβandλγ, are defined by

λα_{ Sλ, }

1, λβ Sλ, cs, λγ  Sλ, bs. 4.2

It is well-known for the sequence spacesλ and μ that λα ⊆ λβ ⊆ λγ andλη ⊃ μη whenever

λ ⊂ μ, where η ∈ {α, β, γ}.

We begin with to quoting the lemmas due to Stieglitz and Tietz15, which are needed

in the proof of the following theorems.

Lemma 4.1. A ∈ p:1 if and only if

sup K∈F  k     n∈K ank    q < ∞ 1< p ≤ ∞. 4.3

Lemma 4.2. A ∈ p:c if and only if

lim

n−→∞ank exists for eachk ∈ N, 4.4

sup

n∈N



k

|ank|q< ∞ 1 < p < ∞. 4.5

Lemma 4.3. A ∈ p:∞ if and only if 4.5 holds.

Theorem 4.4. Define the set ar qby ar q  a  ak ∈ ω : sup K∈F  k     n∈K cr nk    q < ∞ , 4.6

whereCr  cr_nk is defined via the sequence a  a_n by

cr nk ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩  1 k 1 rk − 1 k 1 rk 1  an, 0 ≤ k ≤ n − 1, 1 n 1 rnan, k  n, 0, k > n. 4.7

for alln, k ∈ N. Then, {ar

pΔ}α arq, where 1< p < ∞.

Proof. Bearing in mind the relation2.9, we immediately derive that

anxn n  k0 k  jk−1 −1k−j 1 j 1 rkanyj C r_y n, n ∈ N. 4.8

It follows from4.8 that ax  anxn ∈ 1 whenever x ∈ ar_pΔ if and only if Cry ∈ 1

whenevery ∈ _p. This means thata  a_n ∈ {ar_pΔ}αif and only ifCr ∈ _p:1. Then, we

derive byLemma 4.1withCrinstead ofA that

sup K∈F  k     n∈K cr nk    q < ∞. 4.9

This yields the desired consequence that{ar_pΔ}α ar_q.

Theorem 4.5. Define the sets ar

1, ar2, andar3by ar 1 a  ak ∈ ω : sup n∈N  k er nkq< ∞ , ar 2 ⎧ ⎨ ⎩a  ak ∈ ω : ∞  jk

aj exists for each fixedk ∈ N

⎫ ⎬ ⎭, ar 3 ! a  ak ∈ ω : ! 1 n 1 rnan " ∈ cs " , 4.10 whereE  er_nk is defined by er nk ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k 1 ⎡ ⎣ ak 1 rk  1 1 rk − 1 1 rk 1 _n jk 1 aj ⎤ ⎦, 0 ≤ k ≤ n − 1, 1 n 1 rnan, k  n, 0, k > n, 4.11

for alln, k ∈ N. Then, {ar_pΔ}β a₁r∩ ar₂∩ ar₃, where 1< p < ∞. Proof. Consider the equation

n  k0 akxk n  k0 ⎧ ⎨ ⎩ k  j0 ⎡ ⎣j ij−1 −1i−j 1 i 1 rjyj ⎤ ⎦ ⎫ ⎬ ⎭ak n−1 k0 k 1 ⎡ ⎣ ak 1 rk  1 1 rk − 1 1 rk 1 _n jk 1 aj ⎤ ⎦yk 1 n 1 rnanyn  Ey_n n ∈ N. 4.12

Thus, we deduce fromLemma 4.2with4.12 that ax  akxk ∈ cs whenever x  xk ∈

ar

pΔ if and only if Ey ∈ c whenever y  yk ∈ p. That is to say thata  ak ∈ {arpΔ}β

if and only ifE ∈ p:c. Therefore, we derive from 4.4 and 4.5 that {aprΔ}β ar1∩ ar2∩

ar 3.

Theorem 4.6. {ar

pΔ}γ  ar1, where 1< p < ∞.

Proof. It is natural that the present theorem may be proved by the same technique used in the

proof of Theorems4.4and4.5, above. But, we prefer here the following classical way.

Let a  a_k ∈ ar₁ and x  x_k ∈ ar_pΔ. Then, we obtain by applying H¨older’s inequality that    n  k0 akxk        n  k0 ⎧ ⎨ ⎩ k  j0 ⎡ ⎣j ij−1 −1i−j 1 i 1 rjyj ⎤ ⎦ ⎫ ⎬ ⎭ak        n  k0 er nkyk    ≤ _n k0e r nkq 1/q_n k0yk p 1/p , 4.13

which gives us by taking supremum overn ∈ N that

sup n∈N    n  k0 akxk    ≤sup_n∈N ⎡ ⎣ _n k0 er nkq 1/q_n k0 ykp 1/p⎤ ⎦ ≤y_p·  sup n∈N n  k0 er nkq 1/q < ∞. 4.14

This means thata  ak ∈ {arpΔ}γ. Hence,

ar 1⊂  ar pΔ γ . 4.15

Conversely, leta  a_k ∈ {ar_pΔ}γ and x  x_k ∈ ar_pΔ. Then, one can easily see that

n_k0er

nkykn∈N ∈ ∞wheneverakxk ∈ bs. This shows that the triangle matrix E  ernk,

defined by4.11, is in the class p:∞. Hence, the condition 4.5 holds with e_nkr instead of

ankwhich yields thata  ak ∈ ar1. That is to say that

 ar pΔ _γ ⊂ ar 1. 4.16

Therefore, by combining the inclusions4.15 and 4.16, we deduce that the γ-dual of the

spacear_pΔ is the set ar₁, and this step completes the proof.

5. Some Geometric Properties of the Space

a

Δ

A Banach spaceX is said to have the Banach-Saks property if every bounded sequence

xn in X admits a subsequence zn such that the sequence {tkz} is convergent in the norm

inX 16, where

tkz  _{k 1}1 z0 z1 · · · zk k ∈ N. 5.1

A Banach spaceX is said to have the weak Banach-Saks property whenever given any weakly

null sequencex_n ⊂ X and there exists a subsequence z_n of x_n such that the sequence

{tkz} strongly convergent to zero.

In17, Garc´ıa-Falset introduce the following coefficient:

RX  sup ! lim inf n−→∞ xn− x : xn ⊂ BX, xn ω −→ 0, x ∈ BX " , 5.2

whereBX denotes the unit ball of X.

Remark 5.1. A Banach spaceX with RX < 2 has the weak fixed point property, 18.

Let 1< p < ∞. A Banach space is said to have the Banach-Saks type p or property BS_p,

if every weakly null sequencexk has a subsequence xkl such that for some C > 0,

   n  l0 xkl    < Cn 11/p, 5.3

for alln ∈ N see 19.

Now, we may give the following results related to the some geometric properties,

mentioned above, of the spacear_pΔ.

Theorem 5.2. The space ar

pΔ has the Banach-Saks type p.

Proof. Letεn be a sequence of positive numbers for whichεn≤ 1/2, and also let xn be a

weakly null sequence inBar_pΔ. Set b0 x0 0 and b1 xn1 x1. Then, there existsm1∈ N

such that    ∞  im1 1 b1iei    ar pΔ < ε1. 5.4

Sincexn is a weakly null sequence implies xn → 0 coordinatewise, there is an n2 ∈ N such

that    m1  i0 xniei    ar pΔ < ε1, 5.5

wheren ≥ n2. Setb2  xn2. Then, there exists anm2> m1such that    ∞  im2 1 b2iei    ar pΔ < ε2. 5.6

By using the fact thatx_n → 0 coordinatewise, there exists an n3> n2such that

   m2  i0 xniei    ar pΔ < ε2, 5.7 wheren ≥ n3.

If we continue this process, we can find two increasing subsequencesm_i and n_i

such that    mj  i0 xniei    ar pΔ < εj, 5.8

for eachn ≥ nj 1and

   ∞  imj 1 bjiei    ar pΔ < εj, 5.9 wherebj  xnj. Hence,    n  j0 bj    ar pΔ     n  j0 ⎛ ⎝mj−1 i0 bjiei mj  imj−1 1 bjiei ∞  imj 1 bjiei ⎞ ⎠  ar pΔ ≤    n  j0 _mj−1 i0 bjiei    ar pΔ    n  j0 ⎛ ⎝ mj imj−1 1 bjiei ⎞ ⎠  ar pΔ    n  j0 ⎛ ⎝ ∞ imj 1 bjiei ⎞ ⎠  ar pΔ ≤    n  j0 ⎛ ⎝ mj imj−1 1 bjiei ⎞ ⎠  ar pΔ 2n j0 εj. 5.10

On the other hand, it can be seen thatxnar pΔ< 1. Therefore, xn p ar pΔ< 1. We have    n  j0 ⎛ ⎝ mj imj−1 1 bjiei ⎞ ⎠  p ar pΔ n j0 mj  imj−1 1    i−1  k0 1 − rrk 1 i xjk 1 ri 1 ixik    p ≤n j0 ∞  i0    i−1  k0 1 − rrk 1 i xjk 1 ri 1 ixik    p ≤ n 1. 5.11 Hence, we obtain    n  j0 ⎛ ⎝ mj imj−1 1 bjiei ⎞ ⎠  ar pΔ ≤ n 11/p. 5.12

By using the fact 1≤ n 11/pfor alln ∈ N, we have

   n  j0 bj    ar pΔ ≤ n 11/p _{1 ≤ 2n 1}1/p_{. 5.13}

Hence,ar_pΔ has the Banach-Saks type p. This completes the proof of the theorem.

Remark 5.3. Note thatRar_pΔ  Rp  21/p, sincearpΔ is linearly isomorphic to p.

Hence, by the Remarks5.1and5.3, we have the following.

Theorem 5.4. The space ar

pΔ has the weak fixed point property, where 1 < p < ∞.

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