On the Generalized Riesz Difference Sequence Space and Property

Journal of Inequalities and Applications Volume 2009, Article ID 385029,18pages doi:10.1155/2009/385029

Research Article

On the Generalized

B

m

-Riesz Difference Sequence

Space and

β

-Property

Metin Bas¸arir

_{and Mustafa Kayikc¸i}

1_{Department of Mathematics, Sakarya University, 54187 Sakarya, Turkey}

2_{Duzce MYO, Duzce University, 81010 Duzce, Turkey}

Correspondence should be addressed to Metin Bas¸arir,[email protected]

Received 1 May 2009; Accepted 17 July 2009

Recommended by Ramm Mohapatra

We introduce the generalized Riesz difference sequence space rq_{p, B}m _{which is defined by}

rq_{p, B}m _{{x x}

k ∈w : Bmx∈ rqp}whererqpis the Riesz sequence space defined by

Altay and Bas¸ar. We give some topological properties, compute theα , β duals, and determine the Schauder basis of this space. Finally; we study the characterization of some matrix mappings on this sequence space. At the end of the paper, we investigate some geometric properties ofrq_{p, B}m

and we have proved that this sequence space has propertyβforpk≥1.

Copyrightq2009 M. Bas¸arir and M. Kayikc¸i. 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.

1. Introduction

Letw be the space of all real valued sequences. We write l∞, c, c0 for the sequence spaces of all bounded, convergent, and null sequences, respectively. Also by cs, l1, and lp, we

denote the sequence spaces of all convergent, absolutely andp-absolutely, convergent series, respectively; where 1< p <∞.

Letqkbe a sequence of positive numbers and

Qn n

k0

qk, n∈N. 1.1

Then the matrixRq _rq

nkof the Riesz meanR, qn, which is triangle limitation matrix, is

given by

r_nkq

⎧ ⎪ ⎨ ⎪ ⎩

qk Qn

, 0≤k≤n,

0, k > n.

1.2

It is well known that the matrixRq _rq

Altay and Bas¸ar1,2 introduced the Riesz sequence spacerq_{p, r∞}q _{p, r}q

cp, and rcq0pof nonabsolute type which is the set of all sequences whose R

q_{-transforms are in the}

spacelp, l∞p, cp, andc0p; respectively. Here and afterwards,p pkwill be used as a bounded sequence of strictly positive real numbers with suppkHandMmax{1, H}and

Fdenotes the collection of all finite subsets ofN,whereN{0,1,2, . . .}. The Riesz sequence space introduced in1by Altay and Bas¸ar is

rqp

⎧ ⎨

⎩x xk∈w:

∞

k0 Q1k

k

j0

qjxj

pk

<∞

⎫ ⎬

⎭; with

0< pk≤H <∞

. 1.3

The difference sequence spacesl∞Δ, cΔ, andc0Δwere first defined and studied by Kızmaz in3and studied by several authors,4–9. Bas¸ar and Altay10have studied the sequence spacebvpas the set of all sequences such that theirΔ-transforms are in the spacelp;

that is,

bvp

x xk∈w:

∞

k0

|xk−xk−1|p<∞

, 1≤p <∞, 1.4

whereΔdenotes the matrixΔ Δnkdefined by

Δnk

⎧ ⎨ ⎩

−1n−k; n−1≤k≤n,

0; k < n−1ork > n. 1.5

The idea of difference sequences is generalized by C¸ olak and Et 11. They defined the sequence spaces:

Δm_{λ{x xk∈w:Δ}m_{x∈λ}, 1.6}

wherem∈N,Δ1_xx

k−xk 1, andΔmx ΔΔm−1x, whereΔmdenotes the matrixΔm Δm_nk defined by

Δm nk

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

−1n−k ⎛

⎝ m

n−k

⎞

⎠_{; max{0, n−m} ≤k≤n,}

0; 0≤k <max{0, n−m}ork > n,

1.7

for allk, n∈Nand for any fixedm∈N.

Recently, Bas¸arir and ¨Ozt ¨urk12introduced the Riesz difference sequence space as follows:

rq_{p,Δx xk∈w:Δx x}

k−xk−1∈rq

p; with0< pk≤H <∞

Bas¸ar and Altay defined the matrixB bnkwhich generalizes the matrixΔ Δnk. Now we define the matrixBm _bm

nkand if we taker1, s−1, then it corresponds to the matrix Δm _Δm

nk. We define

bm nk

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

m n−k

rm−n ksn−k; max{0, n−m} ≤k≤n,

0; 0≤k <max{0, n−m}ork > n.

1.9

The results related to the matrix domain of the matrix Bm _{are more general and more}

comprehensive than the corresponding consequences of matrix domain ofΔm_.

Our main subject in the present paper is to introduce the generalized Riesz difference sequence spacerq_{p, B}m_{which consists of all the sequences such that theirB}m_{-transforms are}

in the spacerq_{pand to investigate some topological and geometric properties with respect}

to paranorm on this space.

2. Basic Facts and Definitions

In this section we give some definitions and lemmas which will be frequently used.

Definition 2.1. Letλandμbe two sequence spaces and letA ankbe an infinite matrix of real numbersank,wheren, k∈N. Then, we say thatAdefines a matrix mapping fromλinto μ,and we denote it by writingA: λ → μif for every sequencex xk∈ λthe sequence

Ax{Ax_n},theA-transform ofx,is inμ; where

Ax_n ∞

k0

ankxk, n∈N. 2.1

Byλ:μ,we denote the class of all matricesAsuch thatA:λ → μ.Thus,A∈λ:μif and only if the series on the right side of2.1converges for eachn∈Nand everyx∈λ,and we haveAx {Ax_n}_n∈N ∈μfor allx∈λ.A sequencexis said to beA-summable toαifAx

converges toαwhich is called as theA-limit ofx.

Definition 2.2. For any sequence space λ, the matrix domain λA of an infinite matrix A is

defined by

λA{x xk∈w:Ax∈λ}. 2.2

Definition 2.3. If a sequence space λ paranormed by h contains a sequence bn with the property that for everyx∈λthere is a unique sequence of scalarsαnsuch that

lim

n→ ∞h

x− n

k0

αkbk

0, 2.3

thenbnis called a Schauder basisor briefly basisforλ. The series∞_k0αkbkwhich has the

Definition 2.4. For the sequence spacesλandμ, define the setSλ, μby

Sλ, μz zk∈w:xz xkzk∈μ∀x∈λ

. 2.4

With the notation of2.2, theα-, β-, γ-duals of a sequence spaceλ,which are, respectively, denoted byλα_{, λ}β_{, λ}γ_{,are defined by}

λαSλ, l1, λβSλ, cs, λγSλ, bs. 2.5

Now we give some lemmas which we need to prove our theorems.

Lemma 2.5see13. iLet1< pk≤H <∞for everyk∈N.ThenA∈lp:l1if and only if

there exists an integerK >1such that

sup

K∈F ∞

k0

n∈K ankK−1

pk

<∞. 2.6

iiLet0< pk≤1for everyk∈N.ThenA∈lp:l1if and only if

sup

K∈Fsupk∈N

n∈K ank

pk

<∞. 2.7

Lemma 2.6see14. iLet1< pk≤H <∞for everyk∈N.ThenA∈lp:l∞if and only if

there exists an integerK >1such that

sup

n∈N

∞

k0 _a−1

nkK−

1 pk

<∞. 2.8

iiLet0< pk≤1for everyk∈N.ThenA∈lp:l∞if and only if

sup

n,k∈N|ank|

pk _<∞_{. 2.9}

Lemma 2.7see14. Let0 < pk ≤ H < ∞for everyk ∈N.ThenA∈ lp: cif and only if 2.8,2.9hold, and

lim

n→ ∞ankβk fork∈N 2.10

3. Some Topological Properties of Generalized

B

_{-Riesz Difference Sequence Space}

Let us define the sequencey {ynq},which will be used for theRq_Bm_{-transform of a}

sequencex xk, that is,

yn

q RqBmx_n 1 Qn n−1 k0 _n ik m i−k

rm−i ksi−kqixk

rm Qn

qnxn. 3.1

After this, byRqBm,we denote the matrixRqBm rnkm, q, r, sdefined by

rnk

m, q, r, s

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 Qn n−1 k0 _n ik m i−k

rm−i ksi−kqi

, k < n,

rm Qn

qn, kn,

0, k > n,

3.2

for alln, k, m∈N.Then we define

rqp, Bmx xk∈w:yn

q∈lp

x xk∈w:

∞ k0 1 Qk k−1 n0 _k in m i−n

rm−i nsi−nqixn

rm Qkqkxk

pk <∞

.

3.3

If we takem1,then we have

rqp, B

⎧ ⎨

⎩x xk∈w:

∞

k0 Q1k

⎡ ⎣k−1

j0

qjr qj 1s

xj qkrxk

⎤ ⎦ pk <∞ ⎫ ⎬

⎭. 3.4

Here are some topological properties of the generalized Riesz difference sequence space.

Theorem 3.1. The sequence spacerq_{p, B}m_{is a complete linear metric space paranormed by}

gx

⎛ ⎝∞

k0 Q1k

k−1

j0 ⎡ ⎣k

ij

m

i−j

rm−i jsi−jqixj

⎤ ⎦ rmqk

Qk xk

pk⎞ ⎠

1/M

, 3.5

Proof. The linearity of rq_{p, B}m _{with respect to the co-ordinatewise addition and scalar}

multiplication follows from the inequalites which are satisfied foru, v∈rq_{p, B}m _15:

_∞

k0

|Rq_Bm_u

k RqBmvk|pk

1/M

≤

_∞

k0

|Rq_Bm_u k|pk

1/M _∞

k0

|Rq_Bm_v k|pk

1/M ,

3.6

and for anyα∈R16, we have

|α|pk ≤_{max 1,}|_α|M!_{. 3.7}

It is obvious thatgθ 0 andg−u gufor allu∈rq_{p, B}m_{. Letu}

k, vk∈rqp, Bm:

gu v

⎛ ⎝∞

k0 Q1k

k−1

j0 ⎡ ⎣k

ij

m i−j

rm−i jsi−jqi

uj vj

⎤_⎦ rm_q k Qk

uk vk

pk⎞ ⎠

1/M

≤

⎛ ⎝∞

k0 Q1k

k−1

j0 ⎡ ⎣k

ij

m i−j

rm−i jsi−jqi

uj

⎤_⎦ rm_q k Qk

uk

pk⎞ ⎠

1/M

⎛ ⎝∞

k0 Q1k

k−1

j0 ⎡ ⎣k

ij

m i−j

rm−i jsi−jqi

vj

⎤_⎦ rm_q k Qk

vk

pk⎞ ⎠

1/M

,

3.8

gu v≤gu gv. 3.9

Again the inequalities3.7and3.9yield the subadditivity ofgand

gαu≤max{1,|α|}gu. 3.10

Let{xn_{}be any sequence of the elements of the spacer}q_{p, B}m_{such that}

gxn−x−→0, 3.11

andλnalso be any sequence of scalars such thatλn → λ.Then, since the inequality

holds by subadditivity ofg,{gxn_{}is bounded, and thus we have}

gλnxn−λx

⎛ ⎝∞

k0 Q1k

k−1

j0 ⎡ ⎣k

ij

m

i−j

rm−i jsi−jqi

"

λnxnj −λxj

#⎤ ⎦ rmqk

Qk

λnxnk−λxk

pk⎞ ⎠

1/M

,

≤ |λn−λ|1/M

⎛ ⎝∞

k0 Q1k

k−1

j0 ⎡ ⎣k

ij

m

i−j

rm−i jsi−jqi

"

xn_j#

⎤ ⎦ rmqk

Qk xn_k

pk⎞ ⎠

1/M

|λ|1/M

⎛ ⎝∞

k0 Q1k

k−1

j0 ⎡ ⎣k

ij

m

i−j

rm−i jsi−jqi

"

xn_j −xj

#⎤ ⎦ rmqk

Qk

xn_k−xk

pk⎞ ⎠

1/M

,

≤ |λn−λ|1/Mgxn |λ|1/Mgxn−x,

3.13

which tends to zero asn → ∞.Hence the continuity of the scalar multiplication has shown. Finally; it is clear to say thatgis a paranorm on the spacerq_{p, B}m_.

Moreover; we will prove the completeness of the spacerq_{p, B}m_.Letxi_{be any Cauchy}

sequence in the spacerq_{p, B}m_wherexi _{xi

k} {x0i, x1i, . . .} ∈rqp, Bm.Then, for a given

ε >0,there exists a positive integern0εsuch that

g"xi_−xj#_{< ε, 3.14}

for alli, j≥n0ε.If we use the definition ofg, we obtain for each fixedk∈Nthat

"

RqBmxi# k−

"

RqBmxj# k

_≤∞ k0

"

RqBmxi# k−

"

RqBmxj# k

pk 1/M

< ε, 3.15

fori, j≥n0εwhich leads us to the fact that "

RqBmx0# k,

"

RqBmx1# k, . . .

!

, 3.16

is a Cauchy sequence of real numbers for every fixedk∈N.SinceRis complete, it converges, so we writeRq_Bm_xi

k → RqBmxkasi → ∞.Hence by using these infinitely many limits Rq_Bm_x

0,RqBmx1, . . ., we define the sequence{RqBmx0,RqBmx1, . . .}. Since3.14holds for eachp∈Nandi, j ≥n0ε,

p

k0 "

Rq_Bm_xi# k−

"

Rq_Bm_xj# k

pk

Take anyi≥n0ε,first letj → ∞in3.17and thenp → ∞,to obtaingxi−x≤ε.Finally, takingε 1 in3.17and lettingi≥n01,we have Minkowski’s inequality for eachp ∈N, that is,

_p

k0 "

Rq_Bm_xi# k

pk 1/M

≤g"xi_−x# _g"_xi#_{≤1 g}"_xi#_{, 3.18}

which implies thatx∈rq_{p, B}m_{. Sincegx}i_{−x≤εfor alli≥n}

0εit follows thatxi → xas

i → ∞,sorq_{p, B}m_{is complete.}

Theorem 3.2. Let k_ij"_i−jm#rm−i jsi−jqi/0 for each k ∈ N. Then the difference sequence space rq_{p, B}m_{is linearly isomorphic to the spacelpwhere0< p}

k≤H <∞.

Proof. For the proof of the theorem, we should show the existence of a linear bijection between the spacesrqp, Bmandlpfor 0< pk≤H <∞.With the notation of

yk

1

Qk k−1

j0 ⎡ ⎣k

ij

m i−j

rm−i jsi−jqixj

⎤ ⎦ rm

Qkqkxk, 3.19

define the transformationT fromrq_{p, B}m_{tolpbyx → y Tx. However,T is a linear}

transformation, moreover; it is obviuos thatxθwheneverTxθand henceT is injective. Lety∈lpand define the sequencex xkby

xk k−1

n0 _{n 1}

in

−1k−n s

k−i

rm k−i

m k−i−1

k−i

1

qi Qnyn

Qk rm_q k

yk, fork∈N. 3.20

Then,

gx

⎛ ⎝∞

k0 Q1k

k−1

j0 ⎡ ⎣k

ij

m i−j

rm−i jsi−jqixj

⎤ ⎦ rm

Qk qkxk

pk⎞ ⎠

1/M

⎛ ⎝∞

k0

k

j0

δkjyj

pk⎞ ⎠

1/M

_∞

k0 _yk pk

1/M g1

y<∞,

3.21

where

δkj

⎧ ⎨ ⎩

1, kj,

0, k /j,

and g1y is a paranorm on lp. Thus, we have that x ∈ rqp, Bm. Consequently; T is surjective and is paranorm preserving. Hence,T is a linear bijection and this explains that the spacesrq_{p, B}m_{andlpare linearly isomorphic.}

Now, the Schauder basis for the spacerq_{p, B}m_{will be given in the following theorem.}

Theorem 3.3. Define the sequencebkq {bkn q}_n∈Nof the elements of the spacerqp, Bmfor

every fixedk∈Nby

bkn

q

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

k 1

ik

−1n−k s

n−i

rm n−i

m n−i−1

n−i

1

qi

Qk, n > k,

Qk

rm_qk, nk,

0, k > n.

3.23

Then; the sequence{bkq}_k∈Nis a basis for the spacerq_{p, B}m_{and anyx∈ r}q_{p, B}m_has

a unique representation of the form

x ∞

k0

μk

qbkq, 3.24

whereμkq Rq_Bm_x

kfor allk∈Nand0< pk≤H <∞.

Proof. This can be easily obtained by12, Theorem 5so we omit the proof.

Theorem 3.4. iLet1< pk≤H <∞for everyk∈N.Define the setQ1pas follows:

Q1

p

& K>1

⎧ ⎨

⎩aak∈w:sup_N∈F

∞

k0

n∈N

_{k 1}

ik

−1n−k sn−i

rm n−i

m n−i−1

n−i

1

qi anQk

an rm_qnQn

K−1

p_k

<∞

.

3.25

Then;rq_{p, B}mα_Q

iiLet0< pk≤1for everyk∈N.Define the setQ2pby

Q2

p

& K>1

aak∈w: sup

N∈Fsupk∈N

n∈N

_{k 1}

ik

−1n−k s

n−i

rm n−i

m n−i−1

n−i

1

qianQk an rm_qnQn

K−1

pk

<∞

.

3.26

Then;rq_{p, B}mα_Q

2p.

Proof. iLeta ak∈w.We easily derive with the notation

yk

1

Qk k−1

j0 ⎡ ⎣k

ij

m i−j

rm−i jsi−jqixj

⎤

⎦ 1

Qk

qkxk, 3.27

and the matrixU unkwhich is defined by

unk

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

k 1

ik

−1n−k sn−i

rm n−i

m n−i−1

n−i

1

qi

anQk, 0≤k≤n−1,

anQn rm_q

n

, kn,

0, k > n,

3.28

for all k, n ∈ N, thus, by using the method in 1,12 we deduce that ax anxn ∈ l1 wheneverx xk ∈ rq_{p, B}m _{if and only ifUy ∈ l}

1 whenever y yk ∈ lp.From Lemma 2.5i, we obtain the desired result that

'

rq_{p, B}m(α_Q

1

p. 3.29

Theorem 3.5. iLet1< pk≤H <∞for everyk∈N.Define the setQ3pas follow: Q3 p & K>1 ⎧ ⎪ ⎨ ⎪

⎩aak∈w:

∞ k0 ⎡ ⎣ ⎛ ⎝ ak

rm_q k

k 1

ik

−1n−k sn−i

rm n−i

m n−i−1

n−i

1

qi n

jk 1

aj

⎞ ⎠_Qk

⎤ ⎦K−1

p_k

<∞

.

3.30

Then;rq_{p, B}mβ_Q

3p∩cs.

iiLet0< pk≤1for everyk∈N.Define the setQ4pby

Q4 p ⎧ ⎨

⎩a ak∈w: sup_k∈N ⎡ ⎣ ⎛ ⎝ ak

rm_q k

k 1

ik

−1n−k s

n−i

rm n−i

m n−i−1

n−i

1

qi n

jk 1

aj

⎞ ⎠_Qk

⎤ ⎦ pk <∞ ⎫ ⎬ ⎭. 3.31

Then;rq_{p, B}mβ_Q

4p∩cs.

Proof. iIf we take the matrixT tnkby

tnk ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎛ ⎝ ak

rm_q k

k 1

ik

−1n−k s

n−i

rm n−i

m n−i−1

n−i

1

qi n

jk 1

aj

⎞

⎠_{Qk, 0≤k≤n,}

0, k > n,

3.32

fork, n∈Nand if we carry out the method which is used in1,12, we get thatax anxn∈ cswheneverx xk∈rq_{p, B}m_{if and only ifTy∈cwhenevery yk∈lp.Hence we}

deduce fromLemma 2.7that

∞ k0 ⎡ ⎣ ⎛ ⎝ ak

rm_qk k 1

ik

−1n−k sn−i

rm n−i

m n−i−1

n−i

1

qi n

jk 1

aj

⎞ ⎠Qk

⎤ ⎦K−1

p_k

<∞, 3.33

and limntnkexists which is shown that

'

rqp, Bm(βQ3

iiThis may be obtained in the similar way as in the proof ofiabove by using the second part of Lemmas2.6and2.7. So we omit the detail.

Now we will characterize the matrix mappings from the spacerq_{p, B}m_{to the space} l∞. It can be proved by applying the method in1,12. So we omit the proof.

Theorem 3.6. iLet1< pk≤H <∞for everyk∈N.ThenA∈rqp, Bm;l∞if and only if there

exists an integerK >1such that

QK sup

n∈N

∞

k0 ⎡ ⎣

⎛ ⎝ ank

rm_qk k 1

ik

−1n−k sn−i

rm n−i

m n−i−1

n−i

1

qi n

jk 1

anj

⎞ ⎠Qk

⎤ ⎦K−1

p_k

<∞,

{ank}_k∈N∈cs,

3.35

for eachn∈N.

iiLet0< pk≤1for everyk∈N.ThenA∈rqp, Bm;l∞if and only if

sup

n,k∈N ⎡ ⎣

⎛ ⎝ ank

rm_q k

k 1

ik

−1n−k s

n−i

rm n−i

m n−i−1

n−i

1

qi n

jk 1

anj

⎞ ⎠_Qk

⎤ ⎦

pk

<∞,

{ank}k∈N∈cs ,

3.36

for eachn∈N.

4.

β

-Property of Generalized Riesz Difference Sequence Space

In the previous section; we show that the sequence spacerq_{p, B}m_{,which is the space of all}

real sequencesx xnsuch that∞_k0|Rq_Bm_x

k|pk < ∞,is a complete paranormed space.

It is paranormed bygx ∞_k0|Rq_Bm_x

k|pk1/Mfor allx xn ∈rqp, Bm,whereM

max{1, H};H sup_kpk.We recall that a paranormed space is total ifgx 0 impliesx

0. Every total paranormed space becomes a linear metric space with the metric given by

dx, y gx−y.It is clear thatrq_{p, B}m_{is a total paranormed space.}

In this section, we investigate some geometric properties ofrq_{p, B}m_{. First we give the}

definition of the propertyβin a paranormed space and we will use the method in17to prove the propertyβ.Consequently, we obtain thatrq_{p, B}m_{has propertyβforp}

k≥1.

From here, for a sequencex xn ∈ rq_{p, B}m _{and fori ∈ N, we use the notation} x|i x1, x2, . . . , xi,0,0, . . .andx|N−i 0,0, . . . ,0, xi 1, xi 2, . . ..

Now we give the definition of the propertyβin a linear metric space.

Definition 4.1. A linear metric spaceX, dis said to have the propertyβif for eachε > 0 andr >0,there existsδ >0 such that for each elementx∈B0, rand each sequencexnin

Lemma 4.2. Iflim infk→ ∞pk>0,then for anyL >0andε >0,and for anyu, v∈rqp, Bm,there

existsδδε, L>0such that

dMu v,0< dMu,0 ε, 4.1

wheneverdM_u,0≤LanddM_v,0≤δ.

Proof. Letε >0 andL >0 be given. Let 0< α0 <lim infk→ ∞pkandα0<1, there existsk0 ∈N such that 0< α0 ≤pkfor allk ≥k0.Letα min{pk :k 1,2, . . . , k0;α0}.Thusα≤pkfor all k∈N. There existsK0 ≥2 such that

dM2u,0≤K0dMu,0, 4.2

for allu∈rq_{p, B}m_{.Setβ 2}α_ε/2K

0L1/α.There existsK1≥K0such that

dM

2

βu,0

≤K1dMu,0, 4.3

for allu ∈rq_{p, B}m_{.Setδ 2}α_ε/2βα_K

1.Assume thatdMu,0≤ LanddMv,0 ≤ δ.We recall thatx|i x1, x2, . . . , xi,0,0, . . .andx|N−i 0,0, . . . ,0, xi 1, xi 2, . . .. With

these notations, letA{k∈N−i:pk<1}andC{k∈N−i:pk ≥1}.By using convexity

of the functionft |t|pk _{for allp}

k ≥ 1 and the fact thata bpk ≤ apk bpk forpk < 1 and

0< βpk _{< β}α_whereβ∈_0,1andk∈N_{, we have}

dMu v,0 dM$1−βu β"u β−1v#,0%

∞ i0

_Rq_Bm$_{1−βui β}"_{ui β}−1_vi#% pi

≤∞

i0

_Rq_Bm'_1−βui( _Rq_Bm$_β"_{ui β}−1_vi#% pi

i∈A

_Rq_Bm'_1−βui( _Rq_Bm$_β"_{ui β}−1_vi#% pi

i∈C

_Rq_Bm'_1−βui( _Rq_Bm$_β"_{ui β}−1_vi#% pi

≤1−β i∈A

|RqBmui|pi i∈A

_Rq_Bm_β$_{ui β}−1_vi% pi

1−β i∈C

|Rq_Bm_ui|pi i∈C

_Rq_Bm_β$_{ui β}−1_vi% pi

≤

i∈A

|RqBmui|pi _βα i∈A

_Rq_Bm$_{ui β}−1_vi% pi

i∈C

|RqBmui|pi _βα i∈C

≤∞

i0

|RqBmui|pi _βα ∞

i0

_Rq_Bm$_{ui β}−1_vi% pi

≤dMu,0 βα ∞

i0

₂−1"_2Rq_Bm$_{ui β}−1_vi%# pi

≤dMu,0 βα i∈A

₂−1"_2Rq_Bm$_{ui β}−1_vi%# pi

βα i∈C

₂−1"_2Rq_Bm$_{ui β}−1_vi%# pi

≤dMu,0 βα i∈A

₂−1$_2Rq_Bm_ui "_2Rq_Bm_β−1_vi#% pi

βα i∈C

₂−1$_2Rq_Bm_ui "_2Rq_Bm_β−1_vi#% pi

≤dM_{u,0 β}α i∈A

₂−1_2Rq_Bm_ui pi

βα i∈A

₂−1$_2Rq_Bm_β−1_vi% pi

1 2β

α

i∈C

|2RqBmui|pi

1 2β

α

i∈C

_2Rq_Bm_β−1_vi pi

≤dMu,0

1 2β

α∞

i0

|2RqBmui|pi

1 2β

α∞

i0

_2Rq_Bm_β−1_vi pi

≤dMu,0 1 2α

2α_ε

2K0L

dM2u,0 1 2αβ

α_dM"_2β−1_v,0#

≤dMu,0 ε 2

1 2αβ

α_K

1 2

α_ε

2βα_K

1

,

dMu v,0≤dMu,0 ε.

4.4

Lemma 4.3. Iflim infn→ ∞pn > 0,then for anyx∈ rqp, Bm,there existsk0 ∈Nandθ ∈ 0,1

such that

dM

_x

|N−k

2 ,0

≤ 1−θ

2 d

M_x

|N−k,0 4.5

Proof. Letαbe a real number such that 1< α <lim infn→ ∞pn.Then there existsk0 ∈Nsuch thatα≤pkfor allk≥k0.Letθ∈0,1be a real number such that1/2α<1−θ/2.Then for eachx∈rq_{p, B}m_andk≥k

0,we have

dM

_x

|N−k

2 ,0

∞ ik 1

RqBm₂xi pi

≤

1 2

α∞

ik 1

|RqBmxi|pi

≤ 1−θ

2

∞

ik 1

|RqBmxi|pi

1−θ

2 d

M_x |N−k,0.

4.6

Theorem 4.4. Ifpk≥1,thenrqp, Bmhas propertyβ.

Proof. Letε >0 andxn⊂B0, rwithdxn, xm≥εform /n.Take 0< ε0 < εM.There exists

δ > 0 such thatεM_{−δ ≥ ε}

0.Letx ∈B0, r.Since for eachj ∈N,xnj∞_n1 is bounded, by using the diagonal method, we have that for eachq∈N, we can find a subsequencexnaof xnsuch thatxnajconverges for allj ∈Nwith 1≤j ≤q.Sincexnajis Cauchy sequence for all 1≤j ≤q,there existstq∈Nsuch that

q

k0

|RqBmxnak−R q_Bm_x

nbk| pk

q

k0

|RqBmxnak−xnbk|

pk _{< δ, 4.7}

for allna, nb ≥tq.Then we see that

ε < dxna, xnb

_∞

k0

|Rq_Bm_x

nak−xnbk| pk

1/M ,

εM≤ q

k0

|RqBmxnak−xnbk| pk

∞

kq 1

|RqBmxnak−xnbk| pk_,

εM≤δ ∞

kq 1

|RqBmxnak−xnbk| pk_.

4.8

Therefore, for eachq∈N,there existstq∈Nsuch that

dM"_x

na|N−q, xnb|N−q #

≥εM_−δ≥ε

for allna, nb ≥ tq.Hence, there is a sequence of positive integersσq∞_q1 withσ1 < σ2 < · · · such that

dM"x_σ

q|N−q,0 #

∞ kq 1

_Rq_Bm"_x σqk

# pk

≥ ε0

2, 4.10

for allq∈N.ByLemma 4.3, there existsq0∈Nandθ∈0,1such that

dM

_u| N−q

2 ,0

≤ 1−θ

2 d

M"_u

|N−q,0

#

, 4.11

for allu∈rq_{p, B}m_andq≥q

0.Letδ0be a real number corresponding toLemma 4.2with

ε θ

4 ·

ε0

2, 4.12

andLrM_{,that is}

dMu v,0< dMu,0 θ 4 ·

ε0

2, 4.13

wheneverdMu,0 ≤ rManddMv,0≤ δ0.Sincex∈ B0, r,we have thatdMx,0≤ rM. Letq≥q0be such that

dM"_x

|N−q,0

#

≤δ0. 4.14

Putuxσq|N−qandvx|N−q.Then

dM"u

2,0 #

dM

_x

σq|N−q 2 ,0

∞ kq 1

_Rq_Bm"_x σqk

# pk

< rM,

dM"v

2,0 #

dM"x_|N−q,0# ∞

kq 1|

RqBmxk|pk _{< δ} 0.

Hence;

dM"u v

2 ,0 #

∞ kq 1

R

q_Bm"_x

σqk xk # 2 pk ≤ ∞

kq 1 Rq_Bm_x

σqk R

q_Bm_xk

2

pk

≤dM"u

2,0

# _θ

4 ·

ε0 2

≤ 1−θ

2 d

M_u,0 θ

4 ·

ε0 2,

4.16

dM"u v

2 ,0 #

1−θ

2

∞

kq 1

_Rq_Bm_x σqk

pk θ 4 ·

ε0

2. 4.17

By using4.17and convexity of the functionft |t|pk_{, k}∈N_{, we have}

dM

_{x x}

σq 2 ,0

∞ k0 R

q_Bm"_x

σqk xk # 2 pk ∞ k0 Rq_Bm_x

σqk RqBmxk 2 pk ≤ q k0 Rq_Bm_x

σqk R

q_Bm_xk

2

pk

∞

kq 1

RqBmxσqk R

q_Bm_xk

2 pk ≤ 1 2 q k0

|RqBmxk|pk 1 2

q

k0

_Rq_Bm_x σqk

pk

1−θ

2

∞

kq 1

_Rq_Bm_x σqk

pk θ 4 · ε0 2 ≤ 1 2 q k0

|RqBmxk|pk 1 2

∞

k0

_Rq_Bm_x σqk

pk

− θ

2

∞

kq 1

_Rq_Bm_x σqk

pk θ 4 ·

ε0 2

≤ rM

2 rM 2 − θ 2 · ε0 2 θ 4 · ε0 2

≤rM−θ

4 ·

ε0 2.

HencedM_{x x}

σq/2,0≤rM−θ/4·ε0/2 1/M

.So this implies that

dM""x xσq #

/2,0#≤r−δ 4.19

for someδ >0.Finally; we can say that the sequence spacerq_{p, B}m_{has propertyβ.}

Acknowledgment

We would like to express our gratitude to the reviewer for his/her careful reading and valuable suggestions which improved the presentation of the paper.

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