DEFINED BY A MUSIELAK-ORLICZ FUNCTION

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DEFINED BY A MUSIELAK-ORLICZ FUNCTION

KULDIP RAJ and SUNIL K. SHARMA

In the present paper we introduce some vector-valued sequence spaces defined by a Musielak-Orlicz functionM= (Mk). We also study some topological properties and some inclusion relations between these spaces.

AMS 2010 Subject Classification: 40A05, 46C45, 40D05.

Key words: Orlicz function, Musielak-Orlicz function, paranorm space, lacunary sequence, statistical convergence.

1. INTRODUCTION

An Orlicz functionM is a function, which is continuous, non-decreasing and convex with M(0) = 0, M(x)>0 for x >0 and M(x)→ ∞ asx→ ∞.

Lindenstrauss and Tzafriri [15] used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x= (x_k), then

`_M =

x∈w:

∞

X

k=1

M|x_k| ρ

<∞

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

kxk= inf

ρ >0 :

∞

X

k=1

M|x_k| ρ

≤1

.

It is shown in [15] that every Orlicz sequence space`_M contains a subspace iso- morphic to `p (p≥1). The ∆2-condition is equivalent toM(Lx)≤kLM(x), for all values of x ≥0, and for L >1. An Orlicz function M can always be represented in the following integral form

M(x) = Z x

0

η(t)dt,

whereη is known as the kernel ofM, is right differentiable fort≥0,η(0) = 0, η(t)>0,η is non-decreasing andη(t)→ ∞ ast→ ∞.

REV. ROUMAINE MATH. PURES APPL.,57(2012),4, 383-399

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A sequence M = (Mk) of Orlicz function is called a Musielak-Orlicz function, see ([18], [26]). A sequence N = (N_k) defined by

N_k(v) = sup{|v|u−M_k(u) :u≥0}, k= 1,2, . . .

is called the complementary function of a Musielak-Orlicz function M. For a given Musielak-Orlicz function M, the Musielak-Orlicz sequence spacetM

and its subspacehM are defined as follows tM =

x∈w:IM(cx)<∞, for somec >0 , hM =

x∈w:IM(cx)<∞, for all c >0 , where IM is a convex modular defined by

IM(x) =

∞

X

k=1

Mk(xk), x= (xk)∈tM. We consider tM equipped with the Luxemburg norm

kxk= infn

k >0 :IM

x k

≤1o or equipped with the Orlicz norm

kxk⁰ = inf n1

k

1 +IM(kx)

:k >0 o

.

The Cesaro summable sequence spaces defined by an Orlicz function were studied by Parashar and Choudhary [27], Bhardwaj and Singh [1], Mursaleen et al. [23], Et et. al. [6] and many others.

Letq1 and q2 be seminorms on a vector spaceX. Then,q1 is said to be stronger than q₂ if whenever (x_n) is a sequence such that q₁(x_n) → 0, then q2(xn) → 0 also. If each is stronger than the others q1 and q2 are said to be equivalent see [38].

Let l∞, c and c₀ be the linear spaces of bounded, convergent and null sequences x = (x_k) with complex terms, respectively, normed by kxk_∞ = sup_k|x_k|, where k ∈ N, the set of positive integers. Throughout the paper, w(X), c(X), c0(X) and l∞(X) will represent the spaces of all, convergent, null and bounded X valued sequence spaces. ForX=C, the field of complex numbers, these represent the corresponding scalar valued sequence spaces.

The zero sequence is denoted by θ= (0,0, . . . ,0), where θis the zero element of X.

The studies on vector valued sequence spaces are done by Rath and Sri- vastava [28], Das and Choudhary [5], Leonard [14], Srivastava and Srivastava [36], Tripathy and Sen [37], Et et. al. [6] and many others.

Let u = (u_k) be a sequences of non-zero scalar. Then, for a sequence space E, the multiplier sequence space E(u), associated with the multiplier

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sequence u is defined as

E(u) ={(x_k)∈w: (u_kx_k)∈E}.

The studies on the multiplier sequence spaces are done by C¸ olak [4], Srivastava and Srivastava [36] and many others.

The notion of difference sequence spaces was introduced by Kızmaz [12], who studied the difference sequence spacesl∞(∆),c(∆) andc₀(∆). The notion was further generalized by Et and C¸ olak [7] by introducing the spacesl∞(∆ⁿ), c(∆ⁿ) and c0(∆ⁿ). Let m, n be non-negative integers, then, for Z = l∞, c and c₀, we have sequence spaces,

Z(∆^m_n) ={x= (x_k)∈w: (∆^m_nx_k)∈Z},

where ∆^m_nx = (∆^m_nx_k) = (∆^m−1_n x_k −∆^m−1_n x_k+n) and ∆⁰_nx_k = x_k, for all k∈N, which is equivalent to the following binomial representation

∆^m_nx_k=

m

X

v=0

(−1)^v m

v

x_k+nv.

Takingm=n= 1, we get the spaces l∞(∆),c(∆) andc0(∆) introduced and studied by Kızmaz [12].

A sequence of positive integers θ = (kr) is called lacunary if k0 = 0, 0< k_r < k_r+1 and h_r =k_r−kr−1 → ∞ asr → ∞. The intervals determined by θ will be denoted byI_r= (kr−1, k_r) and q_r = _k^k^r

r−1. The space of lacunary strongly convergent sequences N_θ was defined by Freedman et al. [9] as

N_θ=

x∈w: lim

r→∞

1 h_r

X

k∈Ir

|x_k−l|= 0, for somel

. The spaceN_θ is a BK-space with norm

kxk_θ = sup

r

h⁻¹_r X

k∈Ir

|(x_k)|

.

Freedman et al. [9] also gave some relation between N_θ and the space

|σ₁|of strongly Cesaro summable sequences, which is defined by

|σ₁|=

x= (x_k) : lim

n→∞

n

X

k=1

|x_k−l|= 0, for somel

.

Strongly almost convergent sequence was introduced and studied by Maddox [16] and Freedman [9]. Parashar and Choudhary [27] have introduced and examined some properties of four sequence spaces defined by using an Orlicz function M, which generalized the well-known Orlicz sequence spaces [C,1, p], [C,1, p]₀ and [C,1, p]∞. It may be noted here that the space of strongly summable sequences were discussed by Maddox [17]. Subsequently,

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difference sequence spaces have been discussed by several authors see ([2], [20], [21], [22], [23], [29], [30], [31], [32]) and references therein.

Let X be a linear metric space. A function p : X → R is called paranorm, if

(1)p(x)≥0, for allx∈X;

(2)p(−x) =p(x), for allx∈X;

(3)p(x+y)≤p(x) +p(y), for allx, y∈X;

(4) if (σn) is a sequence of scalars with σn → σ asn → ∞ and (xn) is a sequence of vectors with p(x_n−x) → 0 as n → ∞, then, p(σ_nx_n−σx) → 0 asn→ ∞.

A paranormp for whichp(x) = 0 implies x= 0 is called total paranorm and the pair (X, p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see [38], Theorem 10.4.2, p. 183).

Let M = (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of positive real numbers and u = (u_k) be a sequence of positive reals such that u_k 6= 0 for all k, X be a seminormed space over the field C of complex numbers with the seminorm q_k, for each k ∈ N. We define the following sequence spaces in the present paper:

(w₀,M, θ,∆^m_n, Q, u, p) =

=n

x= (x_k)∈w(X) : lim

r→∞

1 hr

X

k∈Ir

h M_k

q_ku_k∆^m_nx_k ρ

ipk

→0, asr→ ∞, for someρ >0

o ,

(w,M, θ,∆^m_n, Q, u, p) =

=n

r→∞

1 h_r

X

k∈Ir

h M_k

q_ku_k∆^m_nx_k−l ρ

ipk

→0, asr→ ∞, for someρ >0 and l∈X

o ,

and

(w∞,M, θ,∆^m_n, Q, u, p) =

=n

x= (x_k)∈w(X) : sup

r

1 hr

X

k∈Ir

h M_k

q_ku_k∆^m_nx_k ρ

ip_k

<∞, for someρ >0o .

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If we take M(x) =x,we get (w0, θ,∆^m_n, Q, u, p) =

n

x= (xk)∈w(X) : lim

r→∞

1 hr

X

k∈Ir

qk

uk∆^m_nxk

ρ

pk

→0, asr → ∞, for someρ >0o

,

(w, θ,∆^m_n, Q, u, p) = n

r→∞

1 h_r

X

k∈Ir

q_k

uk∆^m_nxk−l ρ

pk

→0, asr→ ∞, for someρ >0 andl∈Xo

, and

(w∞, θ,∆^m_n, Q, u, p) =

= n

x= (xk)∈w(X) : sup

r

1 hr

X

k∈Ir

qk

u_k∆^m_nx_k ρ

pk

<∞,for someρ >0 o

.

If we take p= (pk) = 1, for allk∈N,we have (w0,M, θ,∆^m_n, Q, u) =

n

r→∞

1 hr

X

k∈Ir

Mk

qk

uk∆^m_nxk

ρ

o ,

(w,M, θ,∆^m_n, Q, u) = n

r→∞

1 h_r

X

k∈Ir

M_k

q_k

uk∆^m_nxk−l ρ

, and

(w∞,M, θ,∆^m_n, Q, u) =

= n

x= (xk)∈w(X) : sup

r

1 hr

X

k∈Ir

Mk

qk

u_k∆^m_nx_k ρ

<∞, for someρ >0 o

.

If we take M(x) =x andu=e= (1,1,1, . . . ,1) then, these spaces reduces to (w0, θ,∆^m_n, Q, p) =

n

r→∞

1 hr

X

k∈Ir

qk

∆^m_nxk

ρ

pk

o ,

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(w, θ,∆^m_n, Q, p) =n

r→∞

1 h_r

X

k∈Ir

q_k∆^m_nx_k−l ρ

pk

, and

(w∞, θ,∆^m_n, Q, p) =

=n

x= (x_k)∈w(X) : sup

r

1 h_r

X

k∈Ir

q_k∆^m_nx_k ρ

pk

<∞, for someρ >0o .

The following inequality will be used throughout the paper. If 0 ≤ pk ≤ supp_k=H,K = max(1,2^H−1) then,

(1.1) |a_k+b_k|^p^k ≤K{|a_k|^p^k+|b_k|^p^k}

for all k and a_k, b_k∈C. Also,|a|^p^k ≤max(1,|a|^H), for alla∈C.

The main motive of this paper is to study some topological properties and prove some inclusion relations between above defined sequence spaces.

2. MAIN RESULTS

Theorem 2.1. Let M= (M_k) be a Musielak-Orlicz function, p = (p_k) be a bounded sequence of positive real numbers and u= (u_k)be any sequence of strictly positive real numbers then, the classes of sequences (w0,M, θ,∆^m_n, Q, u, p),(w,M, θ,∆^m_n, Q, u, p)and(w∞,M, θ,∆^m_n, Q, u, p)are linear spaces over the field of complex number C.

Proof. Let x= (x_k), y = (y_k)∈(w₀,M, θ,∆^m_n, Q, u, p) andα, β ∈C.In order to prove the result we need to find some ρ₃ such that

r→∞lim 1 h_r

X

k∈Ir

h M_k

q_k

uk∆^m_n(αxk+βyk) ρ₃

ipk

= 0.

Sincex= (x_k), y = (y_k)∈(w₀,M, θ,∆^m_n, Q, u, p), there exist positive numbers ρ₁, ρ₂>0 such that

r→∞lim 1 hr

X

k∈Ir

h Mk

qk

uk∆^m_nxk

ρ1

ipk

= 0 and

r→∞lim 1 hr

X

k∈Ir

h M_k

q_ku_k∆^m_ny_k ρ2

ip_k

= 0.

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Defineρ3= max(2|α|ρ₁,2|β|ρ₂).SinceMkis non-decreasing convex function and so by using inequality (1),we have

1 hr

X

k∈Ir

h Mk

qk

u_k∆^m_n(αx_k+βy_k) ρ3

ipk

≤ 1 hr

X

k∈Ir

h Mk

qk

αu_k∆^m_nx_k ρ3

+βu_k∆^m_ny_k ρ3

ip_k

≤K 1 h_r

X

k∈Ir

1 2^p^k

h M_k

q_ku_k∆^m_nx_k ρ₁

ipk

+K 1 h_r

X

k∈Ir

1 2^p^k

h M_k

q_ku_k∆^m_ny_k ρ₂

ipk

≤K 1 h_r

X

k∈I_r

h M_k

ipk

+K 1 h_r

X

k∈I_r

h M_k

q_ku_k∆^m_ny_k ρ₂

ipk

→0 asr→ ∞.

Thus, we haveαx+βy∈(w0,M, θ,∆^m_n, Q, u, p). Hence, (w0,M, θ,∆^m_n, Q, u, p) is a linear space. Similarly, we can prove that (w,M, θ,∆^m_n, Q, u, p) and (w∞,M, θ,∆^m_n, Q, u, p) are linear spaces.

Theorem 2.2. Let M= (Mk) be a Musielak-Orlicz function, p = (pk) be a bounded sequence of positive real numbers and u = (u_k) be a sequence of strictly positive real numbers. Then, (w₀,M, θ,∆^m_n, Q, u, p) is a topological linear space paranormed by

g(x) =

m

X

i=1

q(x_i) + inf

ρ>0, s≥1

n

ρ^ps^H : sup

k

1 h_r

X

k∈Ir

h M_k

q_ku_k∆^m_nx_k ρ

ipk_H¹

≤1o ,

where H= max(1,sup_kpk)<∞.

Proof. (i) Clearly,g(x)≥0, forx= (x_k)∈(w₀,M, θ,∆^m_n, Q, u, p). Since Mk(0) = 0,we getg(0) = 0.

(ii)g(−x) =g(x).

(iii) Let x = (x_k), y = (y_k) ∈ (w₀,M, θ,∆^m_n, Q, u, p) then, there exist ρ1, ρ2>0 such that

sup

k

1 hr

X

k∈Ir

h M_k

q_k u_k∆^m_nx_k ρ1

ipk

≤1 and

sup

k

1 hr

X

k∈Ir

h M_k

q_k u_k∆^m_ny_k ρ2

ip_k

≤1.

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Let ρ=ρ1+ρ2,then by Minkowski’s inequality, we have sup

k

1 hr

X

k∈Ir

h Mk

qk

u_k∆^m_n(x_k+y_k) ρ

ipk

= sup

k

1 hr

X

k∈Ir

h M_k

q_k u_k∆^m_n(x_k+y_k) ρ1+ρ2

ip_k

≤ ρ₁ ρ₁+ρ₂

sup

k

1 h_r

X

k∈Ir

h M_k

q_k u_k∆^m_nx_k ρ₁

ipk

+ ρ2

ρ₁+ρ₂

sup

k

1 h_r

X

k∈I_r

h M_k

q_k u_k∆^m_ny_k ρ₂

ipk

and thus, g(x+y) =

m

X

i=1

q(xi+yi) + infn

(ρ₁+ρ₂)^ps^H : sup

n

1 h_r

X

k∈I_r

h M_k

q_k u_k∆ⁿ_mx_k+u_k∆ⁿ_my_k ρ

ipk_H¹

≤1, ρ >0o

≤

m

X

i=1

q(xi) + inf n

(ρ1)^ps^H : sup

n

1 hr

X

k∈Ir

h Mk

qk

uk∆ⁿ_mxk

ρ1

ipk_H¹

≤1, ρ1>0 o

+

m

X

i=1

q(y_i) + infn

(ρ₂)^ps^H : sup

n

1 n

X

k∈Ir

h M_k

q_k u_k∆ⁿ_my_k ρ₂

ipk_H¹

≤1, ρ₂>0o

≤g(x) +g(y).

(iv) Finally, we prove that scalar multiplication is continuous. Letλbe any complex number by definition

g(λx) =

m

X

i=1

q(λx_i)+

+ infn

(ρ)^ps^H : sup

n

1 hr

X

k∈Ir

h M_k

q_k u_k∆ⁿ_mλx_k ρ

ip_k_H¹

≤1, ρ >0o

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=|λ|

m

X

i=1

q(xi)+

+ infn

(|λ|r)^ps^H : sup

n

1 hr

X

k∈Ir

h M_k

q_k u_k∆ⁿ_mx_k t

ipk_H¹

≤1, ρ >0o , where t= _|λ|^ρ .Hence, (w₀,M, θ,∆^m_n, Q, u, p) is a paranormed space.

Theorem 2.3. Let M = (Mk) be a Musielak-Orlicz function. If sup_k[M_k(x)]^p^k <∞, for all fixed x >0, then

(w₀,M, θ,∆^m_n, Q, u, p)⊆(w∞,M, θ,∆^m_n, Q, u, p).

Proof. Let x= (xk)∈(w0,M, θ,∆^m_n, Q, u, p), then, there exists positive number ρ₁ such that

r→∞lim 1 h_r

X

k∈I_r

h M_k

ipk

= 0.

Defineρ= 2ρ₁.Since M_k is non-decreasing and convex and so, by using inequality (1),we have

sup

r

1 hr

X

k∈Ir

h M_k

q_ku_k∆^m_nx_k ρ

ipk

= sup

r

1 h_r

X

k∈I_r

h M_k

q_ku_k∆^m_nx_k+L−L ρ

ipk

≤Ksup

r

1 h_r

X

k∈Ir

1 2^p^k

h M_k

q_k

uk∆^m_nxk−L ρ₁

ipk

+Ksup

r

1 hr

X

k∈Ir

1 2^p^k

h Mk

qk

L ρ1

ipk

≤Ksup

r

1 hr

X

k∈Ir

h M_k

q_ku_k∆^m_nx_k−L ρ1

ip_k

+Ksup

r

1 h_r

X

k∈Ir

h M_k

q_kL ρ₁

ipk

<∞.

Hence, x= (x_k)∈(w∞,M, θ,∆^m_n, Q, u, p).

Theorem 2.4. Let 0 <infp_k = h ≤p_k ≤ supp_k =H < ∞ and M= (M_k), M⁰ = (M_k⁰) be two Musielak-Orlicz functions satisfying ∆₂-condition, then we have

(i) (w₀,M⁰, θ,∆^m_n, Q, u, p)⊂(w₀,M ◦ M⁰, θ,∆^m_n, Q, u, p);

(ii) (w,M⁰, θ,∆^m_n, Q, u, p)⊂(w,M ◦ M⁰, θ,∆^m_n, Q, u, p);

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(iii) (w∞,M⁰, θ,∆^m_n, Q, u, p)⊂(w∞,M ◦ M⁰, θ,∆^m_n, Q, u, p).

Proof. Letx= (x_k)∈(w₀,M⁰, θ,∆^m_n, Q, u, p) then, we have

r→∞lim 1 hr

X

k∈Ir

h M_k⁰

q_ku_k∆^m_nx_k ρ

ipk

= 0.

Let >0 and chooseδ with 0< δ <1 such thatMk(t)< , for 0≤t≤δ.

Let y_k=M_k⁰ q_k_u

k∆^m_nxk

ρ

, for all k∈N.We can write 1

h_r X

k∈Ir

M_k[y_k]^p^k = 1 h_r

X

k∈Ir,yk≤δ

M_k[y_k]^p^k + 1 h_r

X

k∈Ir,yk≥δ

M_k[y_k]^p^k. So, we have

1 h_r

X

k∈Ir,yk≤δ

M_k[y_k]^p^k ≤[M_k(1)]^H 1 h_r

X

k∈Ir,yk≤δ

M_k[y_k]^p^k (2.1)

≤[M_k(2)]^H 1 h_r

X

k∈I_r,yk≤δ

M_k[y_k]^p^k.

For yk > δ, yk < ^y_δ^k < 1 + ^y_δ^k. Since M_k⁰s are non-decreasing and convex, it follows that

Mk(yk)< Mk

1 +yk

δ

< 1

2Mk(2) + 1 2Mk

2yk

δ

. Since M= (M_k) satisfies ∆₂-condition, we can write

M_k(y_k)< 1 2Ty_k

δ M_k(2) + 1 2Ty_k

δ M_k(2) =Ty_k

δ M_k(2).

Hence, (2.2) 1

h_r X

k∈I_r,yk≥δ

M_k[y_k]^p^k ≤max 1,

TM_k(2) δ

H! 1 h_r

X

k∈I_r,yk≥δ

[y_k]^p^k. From equation (2) and (3),we have x = (x_k) ∈ (w₀,M ◦ M⁰, θ,∆^m_n, Q, u, p).

This completes the proof of (i). Similarly, we can prove that (w,M⁰, θ,∆^m_n, Q, u, p)⊂(w,M ◦ M⁰, θ,∆^m_n, Q, u, p) and

(w∞,M⁰, θ,∆^m_n, Q, u, p)⊂(w∞,M ◦ M⁰, θ,∆^m_n, Q, u, p).

Theorem 2.5. Let 0< h= infp_k=p_k<supp_k=H <∞.Then, for a Musielak-Orlicz function M= (M_k) which satisfies∆₂-condition, we have

(i) (w0, θ,∆^m_n, Q, u, p)⊂(w0,M, θ,∆^m_n, Q, u, p);

(ii) (w, θ,∆^m_n, Q, u, p)⊂(w,M, θ,∆^m_n, Q, u, p);

(iii) (w∞, θ,∆^m_n, Q, u, p)⊂(w∞,M, θ,∆^m_n, Q, u, p).

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Proof. It is easy to prove so, we omit the details.

Theorem 2.6. Let M = (Mk) be a Musielak-Orlicz function and 0 <

h= infp_k. Then,(w∞,M, θ,∆^m_n, Q, u, p)⊂(w0, θ,∆^m_n, Q, u, p) if and only if

(2.3) lim

r→∞

1 hr

X

k∈Ir

Mk(t)^p^k =∞, for some t >0.

Proof. Let (w∞,M, θ,∆^m_n, Q, u, p) ⊂ (w₀, θ,∆^m_n, Q, u, p). Suppose that (4) does not hold. Therefore, there are subinterval I_r(j) of the set of interval I_r and a numbert₀ >0,where

t₀=q_ku_k∆^m_nx_k ρ

for allk, such that

(2.4) 1

h_r(j) X

k∈I_r(j)

M_k(t₀)^p^k ≤K <∞, m= 1,2,3, . . . . Let us definex= (x_k) as follows

∆^m_nx_k=

ρt0 k∈I_r(j), 0 k /∈I_r(j).

Thus, by (5),x∈(w∞,M, θ,∆^m_n, Q, u, p).Butx /∈(w0, θ,∆^m_n, Q, u, p).Hence, (4) must hold.

Conversely, suppose that (4) holds and let x ∈(w∞,M, θ,∆^m_n, Q, u, p).

Then, for each r,

(2.5) 1

hr

X

k∈Ir

h Mk

qk

u_k∆^m_nx_k ρ

ipk

≤K <∞.

Suppose that x /∈ (w0, θ,∆^m_n, Q, u, p).Then, for some number >0, there is a number k₀ such that for a subinterval I_r(j),of the set of intervalI_r,

qk

uk∆^m_nxk

ρ

> , fork≥k0. From properties of sequence of Orlicz function, we obtain

Mk

qk

u_k∆^m_nx_k ρ

p_k

≥Mk()^p^k which contradicts (4),by using (6).Hence, we get

(w∞,M, θ,∆^m_n, Q, u, p)⊂(w0, θ,∆^m_n, Q, u, p).

This completes the proof.

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Theorem 2.7. Let 0≤pk≤sk, for allkand let(^s_p^k

k)be bounded. Then, (w,M, θ,∆^m_n, Q, u, s)⊆(w,M, θ,∆^m_n, Q, u, p).

Proof. Letx= (x_k)∈(w,M, θ,∆^m_n, Q, u, s),write t_k=M_k

q_ku_k∆^m_nx_k−l ρ

sk

and µk= ^p_s^k

k, for all k∈N.Then, 0< µk≤1, for all k∈N.Take 0< µ≤µk, for k∈N.Define sequences (u_k) and (v_k) as follows:

Fortk≥1,letuk=tkandvk= 0 and, fortk<1,letuk= 0 andvk=tk. Then, clearly, for all k∈N,we have

t_k=u_k+v_k, t^µ_k^k =u^µ_k^k+v^µ_k^k Now, it follows that u^µ_k^k ≤u_k≤t_k and v_k^µ^k ≤v^µ_k.Therefore,

1 h_r

X

k∈I_r

t^µ_k^k = 1 h_r

X

k∈I_r

(u^µ_k^k +v^µ_k^k)≤ 1 h_r

X

k∈I_r

t_k+ 1 h_r

X

k∈I_r

v_k^µ. Now, for each k,

1 hr

X

k∈Ir

v^µ_k = X

k∈Ir

1 hr

vk

µ1 hr

1−µ

≤ X

k∈I_r

h 1 h_rv_k

µi¹

µµ X

k∈I_r

h 1 h_r

1−µi_1−µ¹ 1−µ

= 1

h_r X

k∈I_r

v_k µ

and so,

1 hr

X

k∈Ir

t^µ_k^k ≤ 1 hr

X

k∈Ir

tk+ 1

hr

X

k∈Ir

vk

µ

.

Hence, x = (x_k) ∈ (w,M, θ,∆^m_n, Q, u, p). This completes the proof of the theorem.

Theorem 2.8. (i) If 0<infp_k≤p_k≤1, for allk∈N,then (w,M, θ,∆^m_n, Q, u, p)⊆(w,M, θ,∆^m_n, Q, u).

(ii)If 1≤p_k≤supp_k=H <∞,for allk∈N, then (w,M, θ,∆^m_n, Q, u)⊆(w,M, θ,∆^m_n, Q, u, p).

Proof. (i) Letx= (x_k)∈(w,M, θ,∆^m_n, Q, u, p),then

r→∞lim 1 hr

X

k∈Ir

h M_k

q_ku_k∆^m_nx_k−L ρ

ip_k

= 0.

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Since 0<infpk≤pk≤1.This implies that

r→∞lim 1 hr

X

k∈Ir

M_k

≤ lim

r→∞

1 hr

X

k∈Ir

M_k

p_k

, therefore,

r→∞lim 1 h_r

X

k∈Ir

M_k

= 0.

Therefore,

(w,M, θ,∆^m_n, Q, u, p)⊆(w,M, θ,∆^m_n, Q, u).

(ii) Letp_k≥1, for eachkand supp_k<∞.Letx= (x_k)∈(w,M, θ,∆^m_n, Q, u),then, for each ρ >0,we have

r→∞lim 1 h_r

X

k∈I_r

h M_k

q_k

u_k∆^m_nx_k−L ρ

i

= 0<1.

Since 1≤pk ≤suppk<∞,we have

r→∞lim 1 hr

X

k∈Ir

h M_k

ip_k

≤ lim

r→∞

1 hr

X

k∈Ir

M_k

= 0<1.

Therefore, x= (x_k)∈(w,M, θ,∆^m_n, Q, u, p),for eachρ >0.Hence, (w,M, θ,∆^m_n, Q, u)⊆(w,M, θ,∆^m_n, Q, u, p).

This completes the proof of the theorem.

Theorem 2.9. If 0 < infp_k ≤ p_k ≤ supp_k = H < ∞, for all k ∈ N, then

(w,M, θ,∆^m_n, Q, u, p) = (w,M, θ,∆^m_n, Q, u).

Proof. It is easy to prove so, we omit the details.

3. STATISTICAL CONVERGENCE

The notion of statistical convergence was introduced by Fast [8] and Schoenberg [35] independently. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Later on, it was further investigated from the sequence space point of view and linked with summability theory by Fridy [10], Connor [3], Salat [33], Isik [11], Sava¸s [34], Malkosky and Savas [19], Kolk [13], Maddox [16], Tripathy and Sen [37], Mursaleen et. al. ([24], [25]) and many others. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure

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of ideals of bounded continuous functions on locally compact spaces. Statisti- cal convergence and its generalizations are also connected with subsets of the stone-Cech compactification of natural numbers. Moreover, statistical convergence is closely related to the concept of convergence in probability. The notion depends on the density of subsets of the set Nof natural numbers.

A subsetE of Nis said to have the natural densityδ(E) if the following limit exists: δ(E) = lim

n→∞

1 n

Pn

k=1χE(k),where χE is the characteristic function of E. It is clear that any finite subset of Nhas zero natural density and δ(E^c) = 1−δ(E).

In this section, we introduce lacunary ∆^m_nuq-statistical convergent sequences and give some relations between lacunary ∆^m_nu_q-statistical convergent sequences and (w,M, θ,∆^m_n, q, u, p)-summable sequences.

A sequencex= (x_k) is said to be lacunary ∆^m_nuq-statistically convergent to l, if for every > 0 lim

r 1 hr

k ∈ I_r : q(u_k∆^m_nx_k−l) ≥ = 0. In this case, we write x_k →l S_θ ∆^m_nu_q

. The set of all lacunary ∆^m_nu_q-statistically convergent sequences is denoted by S_θ ∆^m_nu_q

.

Theorem 3.1. Let M= (M_k) be Musielak-Orlicz function and 0< h= inf_kp_k≤p_k≤sup_kp_k=H <∞. Then,(w,M, θ,∆^m_n, Q, u, p)⊂S_θ(∆^m_nu_q).

Proof. Letx∈(w,M, θ,∆^m_n, Q, u, p) and >0 be given. Then, 1

hr

X

k∈Ir

h M_k

qu_k∆^m_nx_k−l ρ

ip_k

≥ 1 hr

X

k∈I_r,q ^uk^∆^m^{n xk}_ρ ^−l

≥

h M_k

ip_k

≥ 1 h_r

X

k∈Ir,q ^uk^∆^m^{n xk}_ρ ^−l

≥

M_k()pk

≥ 1 h_r

X

k∈Ir,q ^uk^∆^m^{n xk}_ρ ^−l

≥

min

M_k()h

,

M_k()H

≥ 1 h_r

n

k∈I_r:qu_k∆^m_nx_k−l ρ

≥o min

M_k()h

,

M_k()H .

Hence, x∈S_θ(∆^m_nu_q).

Theorem 3.2. Let M = (M_k) be a Musielak-Orlicz function and 0 <

h= infkpk≤pk≤sup_kpk=H <∞.Then,Sθ(∆^m_nuq)⊂(w,M, θ,∆^m_n, Q, u, p).

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Proof. Suppose thatM = (Mk) is bounded. Then, there exists an integer K such thatM_k(t)< K, for allt≥0.Then,

1 hr

X

k∈Ir

h M_k

ipk

= 1 hr

X

k∈Ir,q

uk∆m n xk−l

ρ

≥

h M_k

ipk

+ 1 h_r

X

k∈Ir,q

uk∆m n xk−l

ρ

<

h M_k

q

u_k∆^m_nx_k−l ρ

ipk

≤ 1 hr

X

k∈Ir,q

uk∆m n xk−l

ρ

≥

max(K^h, K^H) + 1 hr

X

k∈Ir,q

uk∆m n xk−l

ρ

<

[Mk()]^p^k

≤max(K^h, K^H) 1 h_r

n

k∈I_r :qu_k∆^m_nx_k−l ρ

≥o

+ max

M_k()h

,

M_k()H . Hence, x∈(w,M, θ,∆^m_n, Q, u, p).

REFERENCES

[1] V.K. Bhardwaj and N. Singh,Some sequence spaces defined by Orlicz functions. Demon- stratio Math.33(2000), 571–582.

[2] T. Bilgin,Some new difference sequences spaces defined by an Orlicz function. Filomat 17(2003), 1–8.

[3] J.S. Connor,The statistical and strongp-Cesaro convergence of sequeces. Analysis (Mu- nich)8(1988), 47–63.

[4] R. C¸ olak,On some generalized sequence spaces. Commun. Fac. Sci. Univ. Ank. S´er. A1 Math. Stat.38(1989), 35–46.

[5] N.R. Das and A. Choudhary, Matrix transformation of vector valued sequence spaces.

Bull. Calcutta Math. Soc.84(1992), 47–54.

[6] M. Et, Y. Altin, B. Choudhary and B.C. Tripathy,On some classes of sequences defined by sequences of Orlicz functions. Math. Inequal. Appl.9(2006), 335–342.

[7] M. Et and R. C¸ olak,On some generalized difference sequence spaces. Soochow. J. Math.

21(1995), 377–386.

[8] H. Fast,Sur la convergence statistique. Colloq. Math.2(1951), 241–244.

[9] A.R. Freedman, J.J. Sember and M. Raphael, Some Cesaro-type summability spaces.

Proc. Lond. Math. Soc. (3)37(1978), 508–520.

[10] J.A. Fridy,On the statistical convergence. Analysis5(1985), 301–303.

(16)

[11] M. Isik,On statistical convergence of generalized difference sequence spaces. Soochow J.

Math.30 (2004), 197–205.

[12] H. Kızmaz,On certain sequence spaces. Canad. Math. Bull.24(1981), 169–176.

[13] E. Kolk, The statistical convergence in Banach spaces. Acta. Comment. Univ. Tartu.

Math.928(1991), 41–52.

[14] I.E. Leonard,Banach sequence spaces. J. Math. Anal. Appl.54(1976), 245–265.

[15] J. Lindenstrauss and L. Tzafriri,On Orlicz sequence spaces. Israel J. Math.10(1971), 379–390.

[16] I.J. Maddox, Spaces of strongly summable sequences. Quart. J. Math.18(1967), 345–

355.

[17] I.J. Maddox, On strong almost convergence. Math. Proc. Cambridge Philos. Soc. 85 (1979), 345–350.

[18] L. Maligranda,Orlicz spaces and interpolation. Seminars in Mathematics5, Polish Aca- demy of Science, 1989.

[19] E. Malkowsky and E. Savas,Someλ-sequence spaces defined by a modulus. Arch. Math.

(Brno)36(2000), 219–228.

[20] E. Malkowsky, M. Mursaleen and S. Suantai,The dual spaces of sets of difference sequences of ordermand matrix transformations. Acta. Math. Sin. (Engl. Ser.)23(2007), 3, 521–532.

[21] M. Mursaleen, Generalized spaces of difference sequences. J. Math. Anal. Appl. 203 (1996),2, 738–745.

[22] M. Mursaleen,Almost strongly regular matrices and a core theorem for double sequences.

J. Math. Anal. Appl.293(2004),2, 523–531.

[23] M. Mursaleen, M.A. Khan and Qamaruddin,Difference sequence spaces defined by Orlicz functions. Demonstratio Math.XXXII(1999), 145–150.

[24] M. Mursaleen, Q.A. Khan and T.A. Chishti, Some new convergent sequence spaces defined by Orlicz functions and statistical convergence. Ital. J. Pure Appl. Math. 9 (2001), 25–32.

[25] M. Mursaleen and O.H.H. Edely, Statistical convergence of double sequences. J. Math.

Anal. Appl.288(2003),1, 223–231.

[26] J. Musielak,Orlicz spaces and modular spaces. Lecture Notes in Math.1034(1983).

[27] S.D. Parashar and B. Choudhary, Sequence spaces defined by Orlicz functions. Indian J. Pure Appl. Math.25(1994), 419–428.

[28] A. Rath and P.D. Srivastava,On some vector valued sequence spacesl^(p)∞(Ek,Λ). Ganita 47(1996), 1–12.

[29] K. Raj, A.K. Sharma and S.K. Sharma, A sequence space defined by Musielak-Orlicz functions. Int. J. Pure Appl. Math.67(2011), 475–484.

[30] K. Raj, S.K. Sharma and A.K. Sharma, Some difference sequence spaces inn-normed spaces defined by Musielak-Orlicz function. Armen. J. Math.3(2010), 127–141.

[31] K. Raj and S.K. Sharma,Some sequence spaces in2-normed spaces defined by Musielak- Orlicz functions. Acta Univ. Sapientiae Math.3(2011), 97–109.

[32] K. Raj and S.K. Sharma,Some Multiplier Double Sequence spaces. Acta Mathematica Vietnam. To Appear.

[33] T. Salat,On statictical convergent sequences of real numbers. Math. Slovaca30(1980), 139–150.

[34] E. Sava¸s, Strong almost convergence and almost λ-statistical convergence. Hokkaido Math. J.29(2000), 531–566.

[35] I.J. Schoenberg,The integrability of certain functions and related summability methods.

Amer. Math. Monthly66(1959), 361–375.

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[36] J.K. Srivastava and B.K. Srivastava, Generalized sequence space c0(X, λ, p). Indian J.

Pure Appl. Math.27(1996), 73–84.

[37] B.C. Tripathy and M. Sen,Vector valued paranormed bounded and null sequence spaces associated with multiplier sequences. Soochow J. Math.29(2003), 379–391.

[38] A. Wilansky, Summability through Functional Analysis. North-Holland, Mathematics Stud. Tartu85, Amsterdam, New York, Oxford, 1984.

Received 10 April 2012 Shri Mata Vaishno Devi University School of Mathematics Katra-182320, J&K, India

kuldipraj68@gmail.com sunilksharma42@yahoo.co.in