Báo cáo hóa học: " Research Article Some Geometric Inequalities in a New Banach Sequence Space" docx

Mursaleen, Rifat C¸olak, and Mikail Et Received 11 July 2007; Accepted 18 November 2007 Recommended by Peter Yu Hin Pang The difference sequence space mφ, p,Δr, which is a generalization

Volume 2007, Article ID 86757, 6 pages

doi:10.1155/2007/86757

Research Article

Some Geometric Inequalities in a New Banach Sequence Space

M Mursaleen, Rifat C¸olak, and Mikail Et

Received 11 July 2007; Accepted 18 November 2007

Recommended by Peter Yu Hin Pang

The difference sequence space m(φ, p,Δ(r)), which is a generalization of the spacem(φ)

introduced and studied by Sargent (1960), was defined by C¸olak and Et (2005) In this paper we establish some geometric inequalities for this space

Copyright © 2007 M Mursaleen et al 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 and preliminaries

LetᏯ denote the space whose elements are finite sets of distinct positive integers Given

an elementσ ∈ Ꮿ, we write c(σ) for the sequence (c n(σ)) such that c n(σ) =1 forn ∈ σ,

andc n(σ) =0, otherwise Further

Ꮿs =



σ ∈Ꮿ :∞

n =1

c n(σ) ≤ s



that is,Ꮿsis the set of thoseσ whose support has cardinality at most s, where s is a natural

number

Letw be the set of all real sequences and

Φ=



φ =φ n

∈ w : φ1> 0, ∇ φ k ≥0,∇



φ k k



≤0 (k =1, 2, )



where∇ φ k = φ k − φ k −1 For φ ∈Φ, Sargent [1] introduced the following sequence space:

m(φ) =

x =x n

∈ w : sup

s ≥1

sup

σ ∈Ꮿs

1

φ s



n ∈ σ

| x n |



In [2], the spacem(φ) has been considered for matrix transformations and in [3] some

of its geometric properties have been considered Tripathy and Sen [4] extendedm(φ) to m(φ, p),1 ≤ p < ∞ Recently, C¸olak and Et [5] defined the spacem(φ, p,Δ(r)) by using the idea of difference sequences (see [6–8])

Letr be a positive integer throughout The operators Δ(r),Σ(r):w → w are defined by



Δ(1)

x

k =(Δx) k = x k − x k+1,



Σ(1)x

k =(Σx)k =∞

j = k

x j (k =1, 2, ),

Δ(r) =Δ(1)◦Δ(r −1), Σ(r) =Σ(1)◦Σ(r −1), (r ≥2),

Σ(r) ◦Δ(r) =Δ(r) ◦Σ(r) = id, the identity onw.

(1.4)

For 0≤ p < ∞, the spacem(φ, p,Δ(r)) is defined as follows:

m

φ, p,Δ(r)

=

x ∈ w : sup

s ≥1,σ ∈Ꮿs

1

φ s



n ∈ σ

Δ(r) x n p

which is a Banach space (1≤ p < ∞) with the norm

x m(φ,p,Δ(r))=r

i =1

| x i |+ sup

s ≥1,σ ∈Ꮿs

1

φ s



n ∈ σ

Δ(r) x n p 1/ p

and a completep-normed space (0 < p < 1) with the p-norm

x m p(φ,Δ(r))=r

i =1

| x i | p+ sup

s ≥1,σ ∈Ꮿs

1

φ s



n ∈ σ

Δ(r) x n p

In this paper, we will consider the case 1< p < ∞to study some geometric properties of

m(φ, p,Δ(r)) We will examine the Banach-Saks property of type p, strict convexity and

uniform convexity The spacem(φ, p),1 ≤ p < ∞was defined by Tripathy and Sen [4] which is in factm(φ, p,Δ) with Δ replaced by id.

Let 1< p < ∞ A Banach space X is said to have the Banach-Saks property of type p or property (BS) p if every weakly null-sequence (x k) has a subsequence (x k i) such that for someC > 0, the inequality







n



i =0

x k i







X

holds

The property (BS) pfor a Ces`aro sequence space was considered in [9]

We find uniform convexity and strict convexity of our space through the Gurarii’s modulus of convexity (see [10,11])

For a normed linear spaceX, the modulus of convexity defined by

β X(ε) =inf

1− inf

0≤ α ≤1 αx + (1 − α)y :x, y ∈ S(X), x − y = ε

is called the Gurarii’s modulus of convexity, whereS(X) denotes the unit sphere in X and

0< ε ≤2 If 0 < β X(ε) < 1, then X is uniformly convex and if β X(ε) ≤1, thenX is strictly

convex

2 Main results

Theorem 2.1 The space m(φ, p,Δ(r) ) has the Banach-Saks property of type p.

Proof We will prove the case r =1 and the general case can be followed on the same

Let (ε n) be a sequence of positive numbers for which∞

n =1ε n ≤1/2 Let (x n) be a weakly null sequence inB(m(φ, p,Δ)), the unit ball in m(φ, p,Δ) Set x0=0 andz1=

x n1= Δx1 Then there exists s1∈ Nsuch that









i ∈ τ1

z1(i)e i







m(φ,p,Δ)

whereτ1consists of the elements ofσ which exceed s1 Since x n −→ w 0⇒ x n →0 coordinate-wise, there isn2∈ Nsuch that







s1



i =1

x n(i)e i





m(φ,p,Δ)

Setz2= x n2= Δx2 Then there exists s2> s1such that









i ∈ τ2

z2(i)e i







m(φ,p,Δ)

whereτ2consists of the elements ofσ which exceed s2 Again using the fact x n →0 coordi-natewise, there existsn3> n2such that







s2



i =1

x n(i)e i







m(φ,p,Δ)

Continuing this process, we can find two increasing sequences (s i) and (n i) such that







s j



i =1

x n(i)e i







m(φ,p,Δ)

< ε j, whenn ≥ n j+1,









i ∈ τ j

z j(i)e i







m(φ,p,Δ)

< ε j,

(2.5)

wherez j = x n j = Δx jandτ jconsists of the elements ofσ which exceed s j Note that z j(i)

is a term in the sequence with fixedj and running i.

Sinceε j −1+ε j < 1, we have

1

φ s



n ∈ σ

z j(n) ≤

ε j −1+ε j



for allj ∈ Nands ≥1 Hence







n



j =1

z j





m(φ,p,Δ)

=





n



j =1

sj −1

i =1

z j(i)e i+

s j



i = s j −1 +1

z j(i)e i+

i ∈ τ j

z j(i)e i 

m(φ,p,Δ)

≤





n



j =1

sj −1

i =1

z j(i)e i 

m(φ,p,Δ)

+







n



j =1

 s j

i = s j −1 +1

z j(i)e i 

m(φ,p,Δ)

+





n



j =1



i ∈ τ j

z j(i)e i 

m(φ,p,Δ)

≤n

j =1







 s j



i = s j −1 +1

z j(i)e i 

m(φ,p,Δ)

+ 2

n



j =1

ε j,

n



j =1







s j



i = s j −1 +1

z j(i)e i







p

m(φ,p,Δ)

=n

j =1

sup

s ≥1

sup

τ j −1∈ Ꮿs

1

φ s



i ∈ τ j −1

z j(i) p

≤n

j =1

sup

s ≥1 sup

σ ∈Ꮿs

1

φ s



i ∈ σ

z

j(i) p ≤ n.

(2.7) Therefore by (2.7)







n



j =1

z j





m(φ,p,Δ)

sincen

j =1ε j ≤1/2.

Hencem(φ, p,Δ) has the Banach-Saks property of type p.

Remark 2.2 The above result can also be extended to the case when r =1 and so the proof should also work for a more general case withΔ replaced by a matrix operator (transformation)

Theorem 2.3 The Gurarii’s modulus of convexity for the space X = m(φ, p,Δ) is

β X(ε) ≤1−



1−



ε

2

p 1/ p

where 0 < ε ≤2.

Proof Let x ∈ m(φ, p,Δ) Then

x m(φ,p,Δ) = Δx m(φ,p) = x1 + sup

s ≥1,σ ∈Ꮿs

1

φ s



n ∈ σ

Δx n p

 1/ p

Let 0< ε ≤2 and consider the sequences

u =(u n)=

1−



ε

2

p 1/ p

,ε

2



, 0, 0,



,

v =(v n)=

1−



ε

2

p 1/ p

,

− ε

2



, 0, 0,



.

(2.11)

Then Δu m(φ,p) = u m(φ,p,Δ) =1, Δv m(φ,p) = v m(φ,p,Δ) =1, that is,u,v ∈ S(m(φ, p,Δ))

and Δu − Δv m(φ,p) = u − v m(φ,p,Δ) = ε.

For 0≤ α ≤1,

αu + (1 − α)vp

m(φ,p,Δ) =αΔu + (1 − α)Δvp

m(φ,p) =1−



ε

2

p

+|2α −1|



ε

2

p

(2.12) Hence

inf

0≤ α ≤1

αu + (1 − α)vp

m(φ,p,Δ) =1−



ε

2

p

Therefore, forp ≥1

β X(ε) ≤1−



1−



ε

2

p 1/ p

Corollary 2.4 (i) If ε = 2, then β X(ε) ≤ 1 and hence m(φ, p,Δ) is strictly convex (ii) If 0 < ε < 2, then 0 < β X(ε) < 1 and hence m(φ, p,Δ) is uniformly convex.

Remark 2.5 Note that these results are best possible for the time being, that is, they

cannot be readily generalized to the general case because our results also hold for general matrix transformation

Acknowledgments

The present paper was completed when Professor Mursaleen visited Firat University (May-June, 2007) The author is very much grateful to the Firat University for provid-ing hospitalities This research was supported by FUBAP (The Management Union of the Scientific Research Projects of Firat University) when the first author visited Firat Univer-sity under the Project no 1179

[1] W L C Sargent, “Some sequence spaces related to thel p spaces,” Journal of the London

Mathe-matical Society, vol 35, no 2, pp 161–171, 1960.

[2] E Malkowsky and M Mursaleen, “Matrix transformations between FK-spaces and the sequence spacesm(φ) and n(φ),” Journal of Mathematical Analysis and Applications, vol 196, no 2, pp.

659–665, 1995.

[3] M Mursaleen, “Some geometric properties of a sequence space related tol p ,” Bulletin of the

Australian Mathematical Society, vol 67, no 2, pp 343–347, 2003.

[4] B C Tripathy and M Sen, “On a new class of sequences related to the spacel p ,” Tamkang Journal

of Mathematics, vol 33, no 2, pp 167–171, 2002.

[5] R C¸olak and M Et, “On some difference sequence sets and their topological properties,” Bulletin

of the Malaysian Mathematical Sciences Society, vol 28, no 2, pp 125–130, 2005.

[6] M Et and R C¸olak, “On some generalized difference sequence spaces,” Soochow Journal of

Math-ematics, vol 21, no 4, pp 377–386, 1995.

[7] H Kı zmaz, “On certain sequence spaces,” Canadian Mathematical Bulletin, vol 24, no 2, pp.

169–176, 1981.

[8] E Malkowsky, M Mursaleen, and S Suantai, “The dual spaces of sets of difference sequences

of orderm and matrix transformations,” Acta Mathematica Sinica, vol 23, no 3, pp 521–532,

2007.

[9] Y Cui and H Hudzik, “On the Banach-Saks and weak Banach-Saks properties of some Banach

sequence spaces,” Acta Scientiarum Mathematicarum, vol 65, no 1-2, pp 179–187, 1999.

[10] V I Gurari˘ı, “Differential properties of the convexity moduli of Banach spaces,” Matematicheskie

Issledovaniya, vol 2, no 1, pp 141–148, 1967.

[11] L S´anchez and A Ull´an, “Some properties of Gurarii’s modulus of convexity,” Archiv der

Math-ematik, vol 71, no 5, pp 399–406, 1998.

M Mursaleen: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Email address:mursaleenm@gmail.com

Rifat C¸olak: Department of Mathematics, Firat University, 23119 Elazı˘g, Turkey

Email address:rcolak@firat.edu.tr

Mikail Et: Department of Mathematics, Firat University, 23119 Elazı˘g, Turkey

Email address:mikailet@yahoo.com

2007.

[9] Y Cui and H Hudzik, “On the Banach- Saks and weak Banach- Saks properties of some Banach< /small>

sequence spaces,” Acta Scientiarum Mathematicarum,...

is a term in the sequence with fixedj and running i.