On Generalized Paranormed Statistically Convergent Sequence Spaces Defined by Orlicz Function

Volume 2009, Article ID 729045,13pages doi:10.1155/2009/729045

Research Article

On Generalized Paranormed

Statistically Convergent Sequence Spaces

Defined by Orlicz Function

Metin Bas¸arir and Selma Altunda ˘g

Department of Mathematics, Faculty of Science and Arts, Sakarya University, 54187 Sakarya, Turkey

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

Received 8 May 2009; Revised 3 August 2009; Accepted 26 August 2009

Recommended by Andrei Volodin

We define generalized paranormed sequence spacescσ, M, p, q, s,c0σ, M, p, q, s,mσ, M, p, q,

s, and m0σ, M, p, q, s defined over a seminormed sequence space X, q. We establish some

inclusion relations between these spaces under some conditions.

Copyrightq2009 M. Bas¸arir and S. Altunda ˘g. 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

wX, cX, c0X, cX, c0X, l∞X, mX, m0X will represent the spaces of all,

con-vergent, null, statistically concon-vergent, statistically null, bounded, bounded statistically convergent, and bounded statistically nullX-valued sequence spaces throughout the paper, where X, q is a seminormed space, seminormed byq.For X C,the space of complex numbers, these spaces represent the w, c, c0, c, c0, l∞, m, m0 which are the spaces of all,

convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent, and bounded statistically null sequences, respectively. The zero sequence is denoted byθ θ, θ, θ, . . ., whereθis the zero element ofX.

The idea of statistical convergence was introduced by Fast1and studied by various authors see 2–4. The notion depends on the density of subsets of the setN of natural numbers. A subsetEofNis said to have densityδEif

δE lim

n→ ∞

1

n

n

k1

χEkexists, 1.1

A sequencex xkis said to be statistically convergent to the numberLi.e.,xk∈

cif for everyε >0

δ{k∈N:|xk−L| ≥ε} 0. 1.2

In this case, we writexk stat→ Lor stat−limxL.

Let σ be a mapping of the set of positive integers into itself. A continuous linear functional φ on l∞, the space of real bounded sequences, is said to be an invariant mean

orσ-mean if and only if

1φx≥0 when the sequencex xnhasxn≥0 for alln∈N, 2φe 1, wheree 1,1, . . .,

3φxσn φxfor allx∈l∞.

The mappingsσare one to one and such thatσkn/nfor all positive integersnandk, whereσkndenotes thekth iterate of the mappingσatn. Thusφextends the limit functional onc, the space of convergent sequences, in the sense thatφx limxfor allx∈c. In that caseσis translation mappingn → n1, aσ-mean is often called a Banach limit, andVσ, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences5.

Ifx xn, setTx Txn xσn.It can be shown6that

Vσ

x xn: lim_m tmnx L euniformly inn, Lσ−limx

1.3

wheretmnx xnTxn. . .Tmxn/m1.

Several authors including Schaefer 7, Mursaleen 6, Savas 8, and others have studied invariant convergent sequences.

An Orlicz function is a function M : 0,∞ → 0,∞, which is continuous, nondecreasing, and convex withM0 0, Mx>0 forx >0 andMx → ∞asx → ∞. If the convexity of an Orlicz functionMis replaced by

Mxy≤Mx My, 1.4

then this function is called modulus function, introduced and investigated by Nakano9 and followed by Ruckle10, Maddox11, and many others.

Lindenstrauss and Tzafriri 12 used the idea of Orlicz function to construct the sequence space

lM

x∈w:

∞

k1

M

_|x

k|

ρ <∞, for someρ >0

1.5

The spacelMbecomes a Banach space with the norm

xinf

ρ >0 :

∞

k1

M

_|x

k|

ρ ≤1

. 1.6

The spacelMis closely related to the spacelpwhich is an Orlicz sequence space with

Mx xp_{for 1≤p <∞. Orlicz sequence spaces were introduced and studied by Parashar} and Choudhary13, Bhardwaj and Singh14, and many others.

It is well known that sinceMis a convex function andM0 0 thenMtx≤tMx for alltwith 0< t <1.

An Orlicz funtionMis said to satisfyΔ2-condition for all values ofu, if there exists

constantK > 0, such thatM2u ≤ KMu u ≥ 0. TheΔ2-condition is equivalent to the

inequalityMLu≤KLMufor all values ofuand forL >1 being satisfied15.

The notion of paranormed space was introduced by Nakano16and Simons 17. Later on it was investigated by Maddox 18, Lascarides 19, Rath and Tripathy 20, Tripathy and Sen21, Tripathy22, and many others.

The following inequality will be used throughout this paper. Letp pkbe a sequence of positive real numbers with 0 < pk ≤ suppk G and let D max1,2G−1. Then for

ak, bk∈C, the set of complex numbers for allk∈N, we have23

|akbk|pk ≤D|ak|pk|bk|pk. 1.7

2. Definitions and Notations

A sequence spaceEis said to be solidor normalifαkxk∈E, wheneverxk∈Eand for all sequencesαkof scalars with|αk| ≤1 for allk∈N.

A sequence spaceEis said to be symmetric ifxk∈Eimpliesxπk∈E, whereπk

is a permutation ofN.

A sequence spaceEis said to be monotone if it contains the canonical preimages of its step spaces.

Throughout the paperp pkwill represent a sequence of positive real numbers and X, qa seminormed space over the fieldCof complex numbers with the seminormq. We define the following sequence spaces:

cσ, M, p, q, s

xk∈l∞X:k−s

M

q

_x

σk_n−L

ρ

pk

stat

−→0,

ask−→ ∞,uniformly inn, s≥0,for someρ >0, L∈X

,

c0

σ, M, p, q, s

xk∈l∞X:k−s

M

q

_x

σk_n

ρ

pk

stat

−→0,

ask−→ ∞,uniformly inn, s≥0,for someρ >0

l∞σ, M, p, q, s

xk∈l∞X: sup

k,n k

−s_Mqxσkn

ρ

pk

<∞,

s≥0, for someρ >0

,

Wσ, M, p, q, s

xk∈l∞X: lim_j 1_j

j

k1

k−s_Mqxσk_n−L

ρ

pk 0,

uniformly inn, s≥0, for someρ >0

.

2.1

We write

mσ, M, p, q, s cσ, M, p, q, s∩l_∞σ, M, p, q, s,

m0

σ, M, p, q, s c0

σ, M, p, q, s∩l∞σ, M, p, q, s.

2.2

IfMx x,qx |x|, s0, σn n1 for eachnandk0 then these spaces reduce to the spaces

cpxk∈w:|xk−L|pk −→stat 0, ask−→ ∞, L∈X

,

c0

pxk∈w:|xk|pk −→stat 0, ask−→ ∞

,

l_∞p

xk∈w: sup k |

xk|pk <∞

,

mpcp∩l_∞p, m0

pc0

p∩l_∞p,

2.3

defined by Tripathy and Sen21.

Firstly, we give some results; those will help in establishing the results of this paper.

Lemma 2.1 21. For two sequences pk and tk one has m0p ⊇ m0t if and only if

lim infk∈Kpk/tk>0, whereK⊆Nsuch thatδK 1.

Lemma 2.221. LethinfpkandGsuppk, then the followings are equivalent: iG <∞andh >0,

iimp m.

Lemma 2.321. LetK{n1, n2, n3, . . .}be an infinite subset ofNsuch thatδK 0.Let

T {xk:xk0 or1 forkni, i∈Nand xk0,otherwise}. 2.4

Lemma 2.424. If a sequence spaceEis solid thenEis monotone.

3. Main Results

Theorem 3.1. cσ, M, p, q, s,c0σ, M, p, q, s,mσ, M, p, q, s,m0σ, M, p, q, sare linear spaces.

Proof. Letxk,yk ∈ cσ, M, p, q, s. Then there exist ρ1,ρ2 positive real numbers andK,

L∈Xsuch that

k−s_Mqxσkn−K

ρ1

pk

stat

−→0, ask−→ ∞, uniformly inn,

k−s_Mqyσkn−L

ρ2

pk

stat

−→0, ask−→ ∞, uniformly inn.

3.1

Letα,βbe scalars and letρ3 max2|α|ρ1,2|β|ρ2. Then by1.7we have

k−s

M

q

αxσk_nβy_σk_n−αKβL

ρ3

pk

≤k−s_Mqxσk_n−K 2ρ1

q

_y

σk_n−L 2ρ2

pk

≤k−s 1 2pk

M

q

_x

σk_n−K

ρ1

M

q

_y

σk_n−L

ρ2

pk

≤D

k−s_Mqxσk_n−K

ρ1

pk

k−s_Mqyσk_n−L

ρ2

pk

stat

−→0 ask−→ ∞, uniformly inn.

3.2

Hencecσ, M, p, q, sis a linear space. The rest of the cases will follow similarly.

Theorem 3.2. The spacesmσ, M, p, q, sandm0σ, M, p, q, sare paranormed spaces, paranormed

by

gx inf

ρpm/H_{: sup} k k

−s_Mqxσkn

ρ

≤1,uniformly inn, s≥0, ρ >0, m∈N

, 3.3

Proof. We prove the theorem for the spacem0σ, M, p, q, s. The proof for the other space can

be proved by the same way. Clearlygx g−xfor allx∈ m0σ, M, p, q, sandgθ 0.

Letx, y∈m0σ, M, p, q, s.Then we haveρ1,ρ2>0 such that

sup k k

−s_Mqxσkn

ρ1

≤1, uniformly inn,

sup k

k−s_Mqyσkn

ρ2

≤1, uniformly inn.

3.4

Letρρ1ρ2. Then by the convexity ofM, we have

sup k k

−s_Mqxσknyσkn

ρ

≤sup k

k−s_M ρ1

ρ1ρ2

q

_x

σk_n

ρ1

ρ2

ρ1ρ2

q

_y

σk_n

ρ2

≤ ρ1

ρ1ρ2supk

k−s_Mqxσk_n

ρ1

_ρ ρ2

1ρ2

sup k k

−s_Mqyσkn

ρ2

≤1, uniformly inn.

3.5

Hence from above inequality, we have

gxy

inf

ρpm/H_{: sup} k k

−s_Mqxσknyσkn

ρ

≤1,

uniformly inn, ρ >0, m∈N

≤inf

ρpm/H₁ : sup k k

−s_Mqxσkn

ρ1

≤1,uniformly inn, ρ1>0

inf

ρpm/H₂ : sup k k

−s_Mqyσkn

ρ1

≤1,uniformly inn, ρ2>0

gx gy.

For the continuity of scalar multiplication letλ /0 be any complex number. Then by the definition ofgwe have

gλx inf

ρpm/H_{: sup} k

k−s_Mqλxσk_n

ρ

≤1,uniformly inn, ρ >0

inf

r|λ|pm/H: sup k

k−s_Mqxσk_n

r

≤1,uniformly inn, r >0

,

3.7

whererρ/|λ|.

Since|λ|pm ≤max1,|λ|H,we have|λ|pm/H≤max1,|λ|H1/H.Then

gλx≤max1,|λ|H1/Hinf

rpm/H: sup k k

−s_Mqxσkn

r

≤1,

uniformly inn, r >0

max1,|λ|H1/H·gx,

3.8

and thereforegλxconverges to zero whengxconverges to zero orλconverges to zero. Hence the spacesmσ, M, p, q, sandm0σ, M, p, q, sare paranormed byg.

Theorem 3.3. Let X, q be complete seminormed space, then the spaces mσ, M, p, q, s and

m0σ, M, p, q, sare complete.

Proof. We prove it for the casem0σ, M, p, q, sand the other case can be established similarly.

Letxs xs_σk_nbe a Cauchy sequence inm0σ, M, p, q, sfor allk, n∈N. Thengxi−xj → 0,

asi, j → ∞. For a givenε >0, letr >0 andδ >0 to be such thatε/rδ>0. Then there exists a positive integerNsuch that

gxi_−xj_< ε

rδ ∀i, j ≥N. 3.9

Using definition of paranorm we get

inf

⎧ ⎨

⎩ρpk/H: sup_k k−sM ⎛ ⎝_q

⎛

⎝xiσk_n−x j σk_n

ρ

⎞ ⎠

⎞

⎠_{≤1,uniformly inn, ρ >0} ⎫ ⎬ ⎭<

ε rδ.

3.10

Hencexiis a Cauchy sequence inX, q. Therefore for eachε0< ε <1there exists a positive integerNsuch that

Using continuity ofM, we find that

sup k k

−s_M

q

xi_−limjxj

ρ

≤1. 3.12

Thus

sup k k

−s_M

q

xi_−x

ρ

≤1. 3.13

Taking infimum of suchρs we get

inf

ρpk/H _{: sup} k

k−s_M

q

xi_−x

ρ

≤1

< ε 3.14

for alli ≥ Nand j → ∞. Sincexi ∈ m0σ, M, p, q, sand Mis continuous, it follows that

x∈m0σ, M, p, q, s. This completes the proof of the theorem.

Theorem 3.4. LetM1andM2be two Orlicz functions satisfyingΔ2-condition. Then

iZσ, M1, p, q, s⊆Zσ, M2◦M1, p, q, s,

iiZσ, M1, p, q, s∩Zσ, M2, p, q, s⊆Zσ, M1M2, p, q, s,

whereZc,m,c0,andm0.

Proof. i We prove this part forZ c0 and the rest of the cases will follow similarly. Let

xk∈c0σ, M1, p, q, s. Then for a given 0< ε <1, there existsρ >0 such that there exists a

subsetKofNwithδK 1, where

K

k∈N:k−s

M1

q

_x

σk_n

ρ pk < ε B ,

Bmax

1,sup

M2

1 k−s1/pk

pk

.

3.15

If we takeak k−s1/pkM1qxσk_n/ρthen apk_k < ε/B < 1 implies thata_k < 1. Hence we have by convexity ofM,

M2◦M1

q

_x

σk_n

ρ

M2

ak k−s1/pk

≤akM2

1 k−s1/pk

. 3.16

Thus

k−s_M

2akpk ≤k−s

M2

ak k−s1/pk

pk

≤k−s_Ba

Hence by3.15it follows that for a givenε >0, there existsρ >0 such that

δ

k∈N:k−s

M2

M1

q

_x

σk_n

ρ

pk

< ε 1. 3.18

Thereforexk∈c0σ, M2◦M1, p, q, s.

iiWe prove this part for the caseZc0and the other cases will follow similarly.

Letxk∈c0σ, M1, p, q, s∩c0σ, M2, p, q, s. Then by using1.7it can be shown that

xk∈c0σ, M1M2, p, q, s. Hence

c0

σ, M1, p, q, s

∩c0

σ, M2, p, q, s

⊆c0

σ, M1M2, p, q, s

. 3.19

This completes the proof.

Theorem 3.5. For any sequencep pkof positive real numbers and for any two seminormsq1and

q2onXone has

Zσ, M, p, q1, s

∩Zσ, M, p, q2, s

/

∅, 3.20

whereZc,m,c0,andm0.

Proof. The proof follows from the fact that the zero sequence belongs to each of the classes the sequence spaces involved in the intersection.

The proof of the following result is easy, so omitted.

Proposition 3.6. LetMbe an Orlicz function which satisfiesΔ2−condition, and letq1andq2be two

seminorms onX. Then

ic0σ, M, p, q1, s⊆cσ, M, p, q1, s,

iim0σ, M, p, q1, s⊆mσ, M, p, q1, s,

iiiZσ, M, p, q1, s∩Zσ, M, p, q2, s⊆Zσ, M, p, q1q2, swhereZc,m,c0, andm0,

ivifq1is stronger thanq2, then

Zσ, M, p, q1, s

⊆Zσ, M, p, q2, s

, 3.21

whereZc,m,c0,andm0.

Theorem 3.7. The spacesZσ, M, p, q, sare not solid, whereZcandm.

Proof. To show that the spaces are not solid in general, consider the following example. Let

Mx xp_{1 ≤ p < ∞, pk 1/pfor all k,qx sup}

i|xi|, where x xi ∈ l∞ and

σn n1 for alln ∈ N. Then we haveσkn nk for all k, n ∈ N. Consider the sequencexk, where xk x_ki ∈ l∞is defined by xki k, k, k, . . ., k i2, i ∈ Nand xi

k 2,2,2, . . ., k /i2, i ∈ Nfor each fixedk ∈N. Hencexk ∈ Zσ, M, p, q, sforZ c andm. Letαk 1,1,1, . . .ifkis odd andαkθ, otherwise. Thenαkxk/∈Zσ, M, p, q, sfor

The proof of the following result is obvious in view ofLemma 2.4.

Proposition 3.8. The spaceZσ, M, p, q, sis solid as well as monotone forZc0andm0.

Theorem 3.9. The spacesZσ, M, p, q, sare not symmetric, whereZc,m,c0,andm0.

Proof. To show that the spaces are not symmetric, consider the following examples. Let

Mx xp_{1 ≤ p < ∞, p}

k 1/pfor all k,qx supi|xi|, where x xi ∈ l∞ and

σn n1 for all n ∈ N. Then we haveσkn nk for allk ∈ N. We consider the sequencexk defined byxk 1,1,1, . . . if k i2, i ∈ N, and xk θ,otherwise. Then xk∈Zσ, M, p, q, sforZc0andm0. Letykbe a rearrangement ofxk, which is defined asyk 1,1,1, . . .ifkis odd andyk θ, otherwise. Thenyk/∈Zσ, M, p, q, sforZ c0

andm0.

To show forZ candm, letpk1 for allkodd andpk2−1for allkeven. LetXR3 andqx max{|x1_|,|x2_|,|x3_{|}, wherex x}1_{, x}2_{, x}3_∈R3_{. LetMx x}4_{andσn n1}

for alln∈N. Then we haveσkn nkfor allk, n∈N. We consider

xk

⎧ ⎨ ⎩

1,1,1, i2_{≤k < i}2_{2i−1, i∈N,}

3,−3,5, otherwise. 3.22

Thenxk∈Zσ, M, p, q, sforZcandm. We consider the rearrengementykofxkas

yk

⎧ ⎨ ⎩

1,1,1, k is odd,

3,−3,5, k is even. 3.23

Thenyk/∈Zσ, M, p, q, sforZcandm. Thus the spacesZσ, M, p, q, sare not symmetric in general, whereZ c,m,c0andm0.

Proposition 3.10. For two sequencespkandtkone hasm0σ, M, p, q, s ⊇m0σ, M, t, q, sif

and only iflim infk∈Kpk/tk>0, whereK⊆Nsuch thatδK 1.

Proof. The proof is obvious in view ofLemma 2.1.

The following result is a consequence of the above result.

Corollary 3.11. For two sequencespkandtkone hasm0σ, M, p, q, s m0σ, M, t, q, sif and

only iflim infk∈Kpk/tk>0andlim infk∈Ktk/pk>0, whereK⊆Nsuch thatδK 1.

The following result is obvious in view ofLemma 2.2.

Proposition 3.12. LethinfpkandGsuppk, then the followings are equivalent:

iG <∞andh >0,

Theorem 3.13. Letp pkbe a sequence of nonnegative bounded real numbers such thatinfpk>0. Then

mσ, M, p, q, sWσ, M, p, q, s∩l_∞σ, M, p, q, s. 3.24

Proof. Letxk∈Wσ, M, p, q, s∩l∞σ, M, p, q, s. Then for a givenε >0, we have

j

k1

k−s_Mqxσkn−L

ρ

pk

≥!!!!_!

k≤j:k−s

M

q

_x

σk_n−L

ρ

pk

≥ε!!!!_!·ε,

3.25

where the vertical bar indicates the number of elements in the enclosed set. From the above inequality it follows thatxk∈mσ, M, p, q, s. Conversely letxk∈mσ, M, p, q, s. Letρ >0 such that

k−s_Mqxσkn−L

ρ

pk

stat

−→0, ask−→ ∞,uniformly inn. 3.26

For a givenε >0, letBsup_kk−sMqxσk_n−L/ρpk1/h<∞. LetLj{k≤j:k−sMqxσk_n−L/ρpk ≥ε/2}.

Sincexk∈mσ, M, p, q, s, so|{Lj}|/j → 0, uniformly inn, asj → ∞. There exits a positive integern0such that|{Lj}|/j < ε/2Bhfor allj > n0. Then for allj > n0, we have

1

j

j

k1

k−s_Mqxσk_n−L

ρ

pk

1_j k /∈Lj

k−s_Mqxσkn−L

ρ

pk

1_j k∈Lj

k−s_Mqxσkn−L

ρ

pk

≤ j−!!_jLj!!ε

2

!!Lj!!

j Bh< ε

2

ε

2 ε.

3.27

Hencexk∈Wσ, M, p, q, s∩l∞σ, M, p, q, s.

This completes the proof of the theorem.

The following result is a consequence of the above theorem.

Corollary 3.14. Letpkandtkbe two bounded sequences of real numbers such thatinfpk > 0 andinftk>0.Then

Since the inclusion relationsmσ, M, p, q, s ⊂ l∞σ, M, p, q, sandm0σ, M, p, q, s ⊂

l_∞σ, M, p, q, sare strict, we have the following result.

Corollary 3.15. The spaces mσ, M, p, q, s and m0σ, M, p, q, s are nowhere dense subsets of

l∞σ, M, p, q, s.

The following result is obvious in view ofLemma 2.3.

Proposition 3.16. The spacesmσ, M, p, q, sandm0σ, M, p, q, sare not separable.

Acknowledgment

The authors would like to express their gratitude to the reviewers for their careful reading and valuable suggestions which improved the presentation of the paper.

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