A New Class of Sequences Related to the Spaces Defined by Sequences of Orlicz Functions

(1)

Journal of Inequalities and Applications Volume 2011, Article ID 539745,10pages doi:10.1155/2011/539745

Research Article

A New Class of Sequences Related to the

l

p

Spaces

Defined by Sequences of Orlicz Functions

Naim L. Braha

Department of Mathematics and Computer Sciences, University of Prishtina, Avenue Mother Theresa, 5, Prishtin¨e 10000, Kosovo

Correspondence should be addressed to Naim L. Braha,nbraha@yahoo.com

Received 5 November 2010; Revised 10 February 2011; Accepted 18 February 2011

Academic Editor: S. S. Dragomir

Copyrightq2011 Naim L. Braha. 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 new sequence space mM, φ, q,Λ defined by combining an Orlicz function, seminorms, andλ-sequences. We study its diﬀerent properties and obtain some inclusion relation involving the space mM, φ, q, λ. Inclusion relation between statistical convergent sequence spaces and Cesaro statistical convergent sequence spaces is also given.

1. Introduction

Byw, we denote the space of all real or complex valued sequences. Ifx∈w, then we simply

writex xkinstead ofx xkk∞1. Also, we will use the conventions thate 1,1, . . ..

Any vector subspace of w is called a sequence space. We will writel∞, c, and c0 for the

sequence spaces of all bounded, convergent, and null sequences, respectively. Further, by

lp 1 ≤ p < ∞, we denote the sequence space of allp-absolutely convergent series, that is,

lp{x xk∈w:∞k0|xk|

p_<_∞}_{for 1}_≤_{p <}_∞_{. Throughout the article,}_w_X_,_l

∞X, and

lpXdenote, respectively, the spaces of all, bounded, andp-absolutely summable sequences

with the elements inX, where X, q is a seminormed space. Byθ 0,0, . . ., we denote

the zero element inX.Psdenotes the set of all subsets ofN, that do not contain more thans

elements. Withφs, we will denote a nondecreasing sequence of positive real numbers such

thats−1φs−1 ≤ s−1φs and φs → ∞, ass → ∞. The class of all the sequences φs

satisfying this property is denoted byΦ.

In paper 1 , the notion of λ-convergent and bounded sequences is introduced as

follows: letλ λk∞k0 be a strictly increasing sequence of positive reals tending to infinity,

that is

(2)

We say that a sequencex xk∈wisλ-convergent to the numberl∈C, called as theλ-limit

ofx, ifΛnx → lasn → ∞, where

Λnx _λ1 n

n

k0

λk−λk−1xk, n∈N. 1.2

In particular, we say thatxis aλ-null sequence ifΛnx → 0 asn → ∞. Further, we

say thatxisλ-bounded if sup|Λnx|<∞. Here and in the sequel, we will use the convention

that any term with a negative subscript is equal to naught, for example,λ−10 andx−1 0.

Now, it is well known1 that if limnxnain the ordinary sense of convergence, then

lim

n

1

λn n

k0

λk−λk−1|xk−a|

0. 1.3

This implies that

lim

n |Λnx−a|limn

1

λn n

k0

λk−λk−1xk−a

0, 1.4

which yields that limnΛnx aand hencexisλ-convergent toa. We therefore deduce that

the ordinary convergence implies theλ-convergence to the same limit.

2. Definitions and Background

The spacemφintroduced and studied by Sargent2 is defined as follows:

mφ x xi∈s:xmφ sup s≥1,σ∈Ps

1

φs

i∈σ

|xi|<∞

. 2.1

Sargent2 studied some of its properties and obtained its relationship with the space

lp. Later on it was investigated from sequence space point of view by Rath 3 , Rath and

Tripathy4 , Tripathy and Sen5 , Tripathy and Mahanta6 , and others. Lindenstrauss and

Tzafriri7 used the idea of Orlicz function to define the following sequence spaces:

lM x xi∈s: ∞

i1 M

_|_x_i|

<∞, >0

, 2.2

which is called an Orlicz sequence space. The spacelMis a Banach space with the norm

xinf >0 :

∞

i1 M

_|_x_i|

≤1

. 2.3

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

(3)

continuous, nondecreasing, and convex withM0 0,Mx>0 forx >0 andMx → ∞

asx → ∞. It is well known that ifM is a convex function andM0 0, thenMλx ≤

λ·Mxfor allλwith 0< λ≤1.

An Orlicz functionMis said to satisfy theΔ2-condition for all values ofu, if there exists

a constantL >0 such thatM2u≤LMu, u≥0see, Krasnoselskii and Rutitsky8 . In the

later stage, diﬀerent Orlicz sequence spaces were introduced and studied by Bhardwaj and

Singh9 , G ¨ung ¨or et al.10 , Tripathy and Mahanta6 , Esi11 , Esi and Et12 , Parashar

and Choudhary13 , and many others.

The following inequality will be used throughout the paper,

|aibi|pi ≤max

1,2H−1|ai|pi_|_b_i|pi_, _2.4

whereaiandbiare complex numbers, andHsuppi<∞,hinfpi. Tripathy and Mahanta

6 defined and studied the following sequence space. LetMbe an Orlicz function, then

mM, φ x xi∈s: sup

s≥1,σ∈Ps 1

φs

i∈σ

M

_|_x_i|

<∞, for some >0

. 2.5

Recently, Altun and Bilgin14 defined and studied the following sequence spaces:

mM, A, φ, p x xi∈s: sup s≥1,σ∈Ps

1

φs

i∈σ

M

_|_A

ix|

pi

<∞, for some >0

, 2.6

whereAix ∞k1aikxkand converges for eachi. In this paper, we will define the following

sequence spaces:

mM, φ, q,Λ x xi∈w: lim_n _φ1 s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

0, for some >0

.

2.7

3. Results

Since the proofs of the following theorems are not hard we omit them.

Theorem 3.1. The sequence spacesmM, φ, q,Λare linear spaces over the complex fieldC.

Theorem 3.2. The spacemM, φ, q,Λis a linear topological space paranormed by

gx

⎧ ⎨ ⎩pr/H:

sup

s

1

φs

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn1/H

≤1, r1,2, . . . ⎫ ⎬

⎭. 3.1

(4)

Theorem 3.3. mM, φ1_{, q,}_Λ_⊂_m_{M, φ}2_{, q,}_Λ_{if and only if}

sup

s≥1 φ1

s

φ2

s

<∞. 3.2

Proof. Letx∈mM, φ1_{, q,}_Λ_and_K_sup

s≥1φ1s/φs2<∞. Then we get

1

φ2

s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

≤sup

s≥1 φ1

s

φ2

s

1

φ1

s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

K· 1

φ1

s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

,

3.3

hencex∈mM, φ2_{, q,}_Λ_{. Conversely, let us suppose that}_m_{M, φ}1_{, q,}_Λ_⊂_m_{M, φ}2_{, q,}_Λ_and

x∈mM, φ1_{, q,}_Λ_{. Then there exists a}_>_{0 such that}

1

φ1

s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

< , 3.4

for every > 0. Suppose that sup_s_≥₁φ1

s/φs2 ∞, then there exists a sequence of natural

numberssisuch that limj→ ∞φs1j/φs2j ∞. Hence we can write

1

φ2

s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

≥sup

j≥1 φ1

sj

φ2

s ·

1

φ1

sj

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

∞. 3.5

Therefore,x /∈mM, φ2, q,Λ, which is contradiction.

Corollary 3.4. LetMbe an Orlicz function. ThenmM, φ1_{, q,}_Λ_m_{M, φ}2_{, q,}_Λ_{if and only if}

sup

s≥1 φ1

s

φ2

s

<∞, sup

s≥1 φ2

s

φ1

s

<∞. 3.6

Theorem 3.5. LetM,M1,M2be Orlicz functions which satisfy theΔ2-condition andq, q1, andq2

seminorms. Then

1mM1, φ, q,Λ⊂mM◦M1, φ, q,Λ,

2mM1, φ, q,Λ∩mM2, φ, q,Λ⊂mM1M2, φ, q,Λ,

3mM, φ, q1,Λ∩mM, φ, q2,Λ⊂mM, φ, q1q2,Λ,

4Ifq1is stronger thanq2, thenmM, φ, q1,Λ⊂mM, φ, q2,Λ, and

5Ifq1is equivalent toq2, thenmM, φ, q1,Λ mM, φ, q2,Λ.

(5)

Corollary 3.6. Let M be an Orlicz function which satisfy the Δ2-condition. Then mφ, q,Λ ⊂ mM, φ, q,Λ.

Theorem 3.7. Let Ω Mi be a sequence of Orlicz functions. Then the sequence space

mM, φ, q,Λis solid and monotone.

Proof. Letx∈mM, φ, q,Λ, then there exists >0 such that

1

φs

n∈σ,σ∈Ps

M

q

_|_Λ

nx|

pn

< , _3.7

for every > 0. Letλn be a sequence of scalars with |λn| ≤ 1 for all n ∈ N. Then from

properties of Orlicz functions and seminorm, we get

1

φs

n∈σ,σ∈Ps

Mn

q

_|_Λ

nλnx|

pn

_φ1 s

n∈σ,σ∈Ps

Mn

q

_|_λ_n||_Λ

nx|

pn

≤ 1

φs

n∈σ,σ∈Ps

|λn|Mn

q

_|_Λ

nx|

pn

≤ _φ1 s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

,

3.8

which proves thatmM, φ, q,Λis solid space and monotone.

4. Statistical Convergence

In15 , Fast introduced the idea of statistical convergence. This ideas was later studied by

Connor16 , Freedman and Sember17 , and many others. A sequence of positive integers

θ kris called lacunary ifk0 0, 0 < kr < kr1, andhr kr −kr−1 → ∞asr → ∞. A

sequencex xiis said to beSθφ,Λstatistically convergent tosif for any >0,

lim

i

1

hrk

i∈σ, σ∈Pr, r ≥1 :Λix−s≥

0, 4.1

for some > 0, wherekAdenotes the cardinality ofA. A sequence x xiis said to be

S0

θφ,Λstatistically convergent tosif for any >0,

lim

i

1

hrk

i∈σ, σ∈Pr, r≥1 :Λix≥

0, 4.2

for some >0.

Theorem 4.1. IfMis any Orlicz function,φnstrictly increasing sequence, thenmM, φ, q,Λ⊂

S0

(6)

Proof. Letx∈mM, φ, q,Λ. Then

1

φs

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

< 1, 4.3

for every1 > 0. Letks sφs be a sequence of positive numbers. Then it follows thatksis

lacunary sequence. Then we get the following relation:

1

φs

n∈σ,σ∈Ps

Mn

q|Λnx|

pn

≥ _sφ 1 s−s−1φs−1

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

_h1 s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

≥ 1

hs

1 Mn

q

_|_Λ

nx|

pn

≥ _h1 s

1

Mnqpn

≥ _h1 s

1

minMnqh, MnqH where the summation

1

is over

_|_Λ

nx|

≥

≥ _h1 sk

n∈σ, σ∈Ps, s≥1 :

_|_Λ

nx|

≥

·minMnqh, MnqH

.

4.4

Taking the limit asn → ∞, it follows thatx∈S0_θM, φ, q,Λ.

Theorem 4.2. If M is any Orlicz bounded function, φs strictly increasing sequence, then

mM, φs, q,Λ·/s S0_θφs,Λ·/s, for everys≥1.

Proof. Inclusion mM, φs, q,Λ·/s ⊂ S_θ0φs,Λ·/s, is valid from Theorem 4.1. In what

follows, we will show converse inclusion. Letx∈S0

θφs,Λ·/s, sinceMnis bounded, there

exists a constantKsuch thatMnq|Λnx/s|/< K. Then for every given >0, we have

1

φs

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx/s|

pn

1

φs

n∈σ,σ∈Ps

Mn

q

1

s·

|Λnx|

pn

≤ 1_s·_φ1 s

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

.

(7)

Let us denote byks s·φs, as we know this sequence is lacunary and finally we get the

following relation:

1

ks

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

≤ _k 1 s−ks−1

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn _h1 s 1 Mn q _|_Λ

nx|

pn _h1 s 2 Mn q _|_Λ

nx|

pn

≤KH_· 1

hsk

n∈σ, σ∈Ps, s≥1 :

_|_Λ

nx|

≥

maxMnqh, MnqH

,

4.6

where the summation₁is over|Λnx|/≥and the summation2is over|Λnx|/≤

. Taking the limit as → 0 andn → ∞, it follows thatx∈mM, φs, q,Λ·/s.

5. Cesaro Convergence

In this paragraph, we will consider that φsis a nondecreasing sequence of positive real

numbers such thatφs≤s,φs → ∞, ass → ∞. Let us denote by

mc θ

M, φ, q,Λ x xi: lim_n → ∞

1

n1

n

k1 Mk

q

_|_Λ

kx|

pk

0, for some >0

.

5.1

Theorem 5.1. IfMis an Orlicz function. Thenmc_θM, φ, q,Λ⊂mM, φ, q,Λ.

Proof. From the definition of the sequencesφn, it follows that infnn1/n1−φn≥1.

Then there exist aδ >0, such that

n1

φn ≤

1δ

δ . 5.2

Then we get the following relation:

1

φs

n∈σ,σ∈Ps

Mn

q|Λnx|

pn

n_φ1 s

1

n1

k1 Mk

q

_|_Λ

kx|

pk

−_φ1 s

k∈{1,2,...,n1}\σ,σ∈Ps

Mk

q

_|_Λ

kx|

pk

≤ s_φ1 s

1

n1

k1 Mk

q

_|_Λ

kx|

pk

−_φ1 sMn0

q

_|_Λ

n0x|

pn₀

≤ 1δ

δ

1

n1

k1 Mk

q

_|_Λ

kx|

pk

− 1

φsMn0

q

_|_Λ

n0x|

pn0 ,

(8)

where n0 ∈ {1,2, . . . , n 1} \ σ, σ ∈ Ps. Knowing that x ∈ mc_θM, φ, q,Λ and Mi are

continuous, lettingn → ∞on last relation, we obtain

1

φs

n∈σ,σ∈Ps

Mn

q

_|_Λ

nx|

pn

−→0. 5.4

Hencex∈mM, φ, q,Λ.

Theorem 5.2. Let sup_sφs/φs−1 < ∞. Then for any Orlicz function, M,mM, φ, q,Λ ⊂ mc

θM, φ, q,Λ.

Proof. Suppose that sup_sφs/φs−1 < ∞, then there existsB > 0 such thatφs/φs−1 < Bfor all s≥1. Letx∈mM, φ, q,Λand >0, there existR >0 such that for everyk≥R

1

φs

k∈σ,σ∈Ps

Mk

q

_|_Λ

kx|

pk

< . 5.5

We can also find a constantK >0 such that

1

φs

k∈σ,σ∈Ps

Mk

q

_|_Λ

kx|

pk

< K, 5.6

for allk∈N. Letnbe any integer withφs−1 < n1≤φs , for everys > R. Then

1

n1

n

k1 Mk

q

_|_Λ

kx|

pk

≤ _φ1 s−1

φs

k1 Mk

q

_|_Λ

kx|

pk

_φ1 s−1

⎛ ⎝φs

k1 Mk

q

_|_Λ

kx|

pk

φs

Mk

q

_|_Λ

kx|

pk

· · ·

φs

φs−1 Mk

q

_|_Λ

kx|

pk

⎞ ⎠

≤ φ1

φs−1

⎛

⎝ 1

φ1

k∈σ,σ∈P1

Mk

q

_|_Λ

kx|

pk

⎞ ⎠

φ2 φs−1

⎛

⎝1

φ2

k∈σ,σ∈P2

Mk

q

_|_Λ

kx|

pk

(9)

_φφR s−1

⎛

⎝ 1

φR

k∈σ,σ∈PR

Mk

q

_|_Λ

kx|

pk

⎞ ⎠_{· · ·}

_φφs s−1

⎛

⎝1

φs

k∈σ,σ∈Ps

Mk

q

_|_Λ

kx|

pk

⎞

⎠_,

5.7

wherePtare sets of integer numbers which have more thanφt elements fort∈ {1,2, . . . , s}.

Passing by limit on last relation, wherek → ∞sinces → ∞,φs → ∞andn → ∞, we get

that

1

n1

n

k1 Mk

q

_|_Λ

kx|

pk

−→0; 5.8

from this, it follows thatx∈mc_θM, φ, q,Λ.

Theorem 5.3. Let sup_sφs/φs−1 < ∞. Then for any Orlicz function, M,mM, φ, q,Λ mc

θM, φ, q,Λ.

References

1 M. Mursaleen and A. K. Noman, “On the spaces ofλ-convergent and bounded sequences,” Thai Journal of Mathematics, vol. 8, no. 2, pp. 311–329, 2010.

2 W. L. C. Sargent, “Some sequence spaces related to thelpspaces,”Journal of the London Mathematical

Society, vol. 35, pp. 161–171, 1960.

3 D. Rath, “Spaces ofr-convex sequences and matrix transformations,”Indian Journal of Mathematics, vol. 41, no. 2, pp. 265–280, 1999.

4 D. Rath and B. C. Tripathy, “Characterization of certain matrix operators,” Journal of Orissa Mathematical Society, vol. 8, pp. 121–134, 1989.

5 B. C. Tripathy and M. Sen, “On a new class of sequences related to the spacelp,”Tamkang Journal of Mathematics, vol. 33, no. 2, pp. 167–171, 2002.

6 B. C. Tripathy and S. Mahanta, “On a class of sequences related to thelpspace defined by Orlicz

functions,”Soochow Journal of Mathematics, vol. 29, no. 4, pp. 379–391, 2003.

7 J. Lindenstrauss and L. Tzafriri, “On Orlicz sequence spaces,”Israel Journal of Mathematics, vol. 10, pp. 379–390, 1971.

8 M. A. Krasnoselskii and Y. B. Rutitsky,Convex Function and Orlicz Spaces, P.Noordhoﬀ, Groningen, The Netherlands, 1961.

9 V. K. Bhardwaj and N. Singh, “Some sequence spaces defined by Orlicz functions,” Demonstratio Mathematica, vol. 33, no. 3, pp. 571–582, 2000.

10 M. G ¨ung ¨or, M. Et, and Y. Altin, “StronglyVσ, λ, q-summable sequences defined by Orlicz functions,”

Applied Mathematics and Computation, vol. 157, no. 2, pp. 561–571, 2004.

11 A. Esi, “On a class of new type diﬀerence sequence spaces related to the spacelp,”Far East Journal of

Mathematical Sciences, vol. 13, no. 2, pp. 167–172, 2004.

12 A. Esi and M. Et, “Some new sequence spaces defined by a sequence of Orlicz functions,”Indian Journal of Pure and Applied Mathematics, vol. 31, no. 8, pp. 967–973, 2000.

13 S. D. Parashar and B. Choudhary, “Sequence spaces defined by Orlicz functions,”Indian Journal of Pure and Applied Mathematics, vol. 25, no. 4, pp. 419–428, 1994.

(10)

15 H. Fast, “Sur la convergence statistique,”Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.

16 J. S. Connor, “The statistical and strongp-Ces`aro convergence of sequences,”Analysis, vol. 8, no. 1-2, pp. 47–63, 1988.