Applications of four dimensional matrices to generating some …annalsmath/pdf-uri anale/F2... ·...

An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.)

Tomul LXIV, 2018, f. 2

Applications of four dimensional matrices to generatingsome roughly I2-convergent double sequence spaces

Kuldip Raj · Charu Sharma

Abstract In [3] Dundar introduced the notion of rough I2-convergence and obtained twocriteria for rough I2-convergence. In the present paper we introduce some new rough idealconvergent sequence spaces of Musielak-Orlicz functions and four dimensional bounded-regular matrices. We study some topological and algebraic properties of these spaces. Wealso establish some inclusion relations between these spaces. Finally, we examine thatthese spaces are normal as well as monotone and sequence algebras.

Keywords Ideal · Bounded-regular matrix · Musielak-Orlicz function · Rough I2-convergence · Double difference sequence space

Mathematics Subject Classification (2010) 40A05 · 40A35 · 46A45

1 Introduction

The notion of I-convergence was initially introduced by Kostyrko et al. [9]as a generalization of statistical convergence [5]. Later on, it was furtherinvestigated from the sequence spaces point of view and linked with thesummability theory by Salat [26], Tripathy and Hazarika [29], Savas andDas [27] and many others. The concept of rough I-convergence of singlesequences was introduced by Pal et al. [18] which is a generalization of theearlier concepts namely rough convergence [28] and rough statistical con-vergence [1] of single sequences. Recently, rough statistical convergence ofdouble sequences has been introduced by Malik and Maity [12] as a gen-eralization of rough convergence of double sequences [11] and investigatedsome basic properties of this type of convergence and also studied relationbetween the set of statistical cluster points and the set of rough limit pointsof a double sequence. Recently, the notion of rough I2-convergence for dou-ble sequences has been introduced by Dundar [3]. So in view of the recentapplications of ideals in the theory of convergence of sequences, it seemsvery natural to extend the interesting concept of rough convergence further

293

2 Kuldip Raj, Charu Sharma

by using Orlicz functions.Let X be a non empty set. Then a family of sets I ⊆ 2X (Power set of X)is said to be an ideal if I is additive, that is, A,B ∈ I ⇒ A ∪ B ∈ I and ifA ∈ I,B ⊆ A⇒ B ∈ I.A non-empty family of sets F ⊂ 2X is said to be a filter on X if and only if(i) φ ∈ F(ii) for all A,B ∈ F ⇒ A ∩B ∈ F(iii) A ∈ F , A ⊂ B ⇒ B ∈ F .An ideal I ⊆ 2X is called non-trivial if I 6= 2X . A non-trivial ideal in I iscalled admissible if and only if I ⊃ {{x} : x ∈ X}. A non-trivial ideal ismaximal if there does not exist any non-trivial ideal J 6= I, containing I asa subset. For each ideal I there is a filter F (I) corresponding to I, that is,F (I) = {K ⊆ N : Kc ∈ I}, where Kc = N \K.

An Orlicz function M : [0,∞) → [0,∞) is a continuous, non-decreasingand convex such that M(0) = 0, M(x) > 0 for x > 0 and M(x) −→ ∞as x −→ ∞. If convexity of Orlicz function is replaced by M(x + y) ≤M(x) + M(y), then this function is called modulus function. Let w denotethe set of all real or complex sequences. Lindenstrauss and Tzafriri [10] usedthe idea of Orlicz function to define the following sequence space,

`M ={x = (xk) ∈ w :

∞∑

k=1

M( |xk|ρ

)<∞, for some ρ > 0

}

is known as an Orlicz sequence space. The space `M is a Banach space withthe norm

||x|| = inf{ρ > 0 :

∞∑

k=1

M( |xk|ρ

)≤ 1}.

A sequence M = (Mk) of Orlicz functions is said to be Musielak-Orliczfunction (see [16], [17]). A Musielak-Orlicz function M = (Mk) is said tosatisfy ∆2-condition if there exist constants a, K > 0 and a sequencec = (ck)

∞k=1 ∈ l1+ (the positive cone of l1) such that the inequality

Mk(2u) ≤ KMk(u) + ck

hold for all k ∈ N and u ∈ R+, whenever Mk(u) ≤ a.

By l∞, c and c0 we denote the classes of all bounded, convergent and nullsequence spaces, respectively. The notion of difference sequence spaces wasintroduced by Kızmaz [8] who studied the difference sequence spaces l∞(∆),c(∆) and c0(∆). The notion was further generalized by Et and Colak [4] byintroducing the spaces l∞(∆m), c(∆m) and c0(∆m).Let m be a non-negative integer. Then for Z = c, c0 and l∞, we have thefollowing sequence spaces

Z(∆m) = {x = (xk) ∈ w : (∆mxk) ∈ Z},

294

Applications of four dimensional matrices 3

where ∆mx = (∆mxk) = (∆m−1xk − ∆m−1xk+1) and ∆0xk = xk for allk ∈ N, which is equivalent to the following binomial representation

∆mxk =m∑

v=0

(−1)v(mv

)xk+v.

Taking m = 1, we get the spaces studied by Et and Colak [4]. Similarly, wecan define difference operators on double sequence spaces as:

∆xk,l = (xk,l − xk,l+1)− (xk+1,l − xk+1,l+1)

= xk,l − xk,l+1 − xk+1,l + xk+1,l+1

and

∆mxk,l = ∆m−1xk,l −∆m−1xk,l+1 −∆m−1xk+1,l +∆m−1xk+1,l+1.

For more details about sequence spaces (see [13], [14], [15], [20], [22], [23],[24]) and references therein.

Let X be a linear space. A function p : X → R is called paranorm if(i) p(x) ≥ 0 for each x ∈ X,(ii) p(−x) = p(x) for any x ∈ X,(iii) p(x+ y) ≤ p(x) + p(y) for any x, y ∈ X,(iv) if (λn) is a sequence of scalars with λn → λ as n → ∞ and (xn) is asequence of vectors with p(xn − x) → 0 as n → ∞, then p(λnxn − λx) →0 as n→∞.A paranorm p for which p(x) = 0 implies x = 0 is called total paranorm onX and the pair (X, p) is called a total paranormed space. It is well knownthat the metric of any linear metric space is given by some total paranorm(see [30] Theorem 10.4.2, pp. 183).

By the convergence of double sequence x = (xkl) we mean the conver-gence in the Pringsheim sense, i.e. a double sequence x = (xkl) is said toconverges to the limit L in Pringsheim sense (denoted by, P - limx = L)provided that given ε > 0 there exists n ∈ N such that |xkl − L| < ε when-ever k, l > n (see [21]). We shall write more briefly as P -convergent. If, inaddition, x ∈ l2∞, then x is said to be bounded P -convergent to L. We shalldenote the space of all bounded convergent double sequences by c2∞. A dou-ble sequence x = (xkl) is said to be bounded if ‖x‖(∞,2) = supk,l |xkl| < ∞.

By l2∞, we denote the space of all bounded double sequences. Throughoutthe paper we take I2 as a nontrivial ideal in N× N.

Definition 1.1 A double sequence (xkl) ∈ w is said to be I2-convergent toa number L if for every ε > 0, {(k, l) ∈ N× N : |xkl − L| > ε} ∈ I2 and wewrite I2 − limxkl = L.

Definition 1.2 A double sequence (xkl) ∈ w is said to be I2-null if for everyε > 0, {(k, l) ∈ N× N : |xkl| > ε} ∈ I2 and we write I2 − limxkl = 0.

295


Definition 1.3 A double sequence (xkl) ∈ w is said to be I2-bounded if thereexists M > 0 such that {(k, l) ∈ N× N : |xkl| > M} ∈ I2.

Definition 1.4 A sequence space E is said to be solid or normal if (xkl) ∈ Eimplies αklxkl for all sequence of scalars (αkl) with |αkl| < 1 for all k, l.

Definition 1.5 A sequence space E is said to be convergence free if (ykl) ∈E, whenever (xkl) ∈ E and xkl = 0 implies ykl = 0 for all k, l.

Definition 1.6 A sequence space E is said to be a sequence algebra if(xklykl) ∈ E whenever (xkl) ∈ E and (ykl) ∈ E.Definition 1.7 [19] A double sequence x = (xkl) is said to be rough con-vergent (r-convergent) to x∗ with the roughness degree r which is denoted by

xklr−→ x∗ provided that

∀ε > 0 ∃ kε ∈ N : k, l ≥ kε ⇒ ‖xkl − x∗‖ < r + ε,

or equivalent, iflim sup ‖xkl − x∗‖ < r.

Definition 1.8 [3] For some given real number r ≥ 0, a double sequencex = (xkl) is said to be rough I2-convergent to x∗ with the roughness degree

r which is denoted by xklr−I2−−−→ x∗ provided that

{(k, l) ∈ N× N : ‖xkl − x∗‖ ≥ r + ε} ∈ I2,for every ε > 0 or equivalent, if the condition

I2 − lim sup ‖xkl − x∗‖ < r

is satisfied. In addition, xklr−I2−−−→ x∗ if the inequality ‖xkl−x∗‖ < r+ε holds

for every ε > 0 and almost all k, l.

Lemma 1.9 [7] Every normal sequence space is monotone.

Let A = (anmkl) be a four-dimensional infinite matrix of scalars. For allm,n ∈ N0, where N0 := N ∪ {0}, the sum

ynm =∑

k,l

anmklxkl

is called the A-means of the double sequence (xkl). A double sequence (xkl)is said to be A-summable to the limit L if the A-means exist for all m,n inthe sense of Pringsheim’s convergence:

P - limp,q→∞

p,q∑

k,l=0,0

anmklxkl = ynm and P - limn,m→∞

ynm = L.

296


A four-dimensional matrix A is said to be bounded-regular (or RH-regular)if every bounded P -convergent sequence is A-summable to the same limitand the A-means are also bounded.The following is a four-dimensional analogue of the well-known Silverman-Toeplitz theorem [2].

Theorem 1.10 (Robison [25] and Hamilton [6]) The four dimensional ma-trix A is RH-regular if and only if

(RH1) P - limn,m

anmkl = 0 for each k and l,

(RH2) P - limn,m

∑

k,l

|anmkl| = 1,

(RH3) P - limn,m

∑

k

|anmkl| = 0 for each l,

(RH4) P - limn,m

∑

l

|anmkl| = 0 for each k,

(RH5)∑

k,l

|anmkl| <∞ for all n,m ∈ N0.

Let M = (Mkl) be a Musielak-Orlicz function, p = (pkl) be a boundeddouble sequence of real numbers such that pkl > 0 for all k, l, u = (ukl) be adouble sequence of strictly positive real numbers and let A = (anmkl) be anonnegative four-dimensional bounded-regular matrix. In the present paperwe define the following classes of sequence spaces:

w2Ic

[A,M, u, r,∆m, p] ={x = (xkl) :

{(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆mxkl − x∗ρ

∥∥∥)]pkl

≥ r + ε

}∈ I2, for some ρ > 0 and x∗ ∈ R

},

w2Ic0

[A,M, u, r,∆m, p] ={x = (xkl) :

{(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆mxklρ

∥∥∥)]pkl

≥ r + ε

}∈ I2, for some ρ > 0

},

297


w2Il∞

[A,M, u, r,∆m, p] ={x=(xkl) :∃K>0,

{(k, l)∈N×N :

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

≥ r +K

}∈ I2, for some ρ > 0 and x∗ ∈ R

}

and

w2l∞

[A,M, u, r,∆m, p] ={x = (xkl) :

{(k, l) ∈ N×N : sup

n,m

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

<∞,

for some ρ > 0

}.

The following results are obtained for some special cases:

(i) If we take Mkl(x) = x for all k, l, then the above classes of sequencesare denoted by w2

Ic[A, u, r,∆m, p], w2

Ic0[A, u, r,∆m, p], w2

Il∞[A, u, r,∆m, p], re-

spectively.

(ii) If (pkl) = (1, 1, 1, ...) for all k, l, then we denote the above sequences byw2Ic

[A,M, u, r,∆m], w2Ic0

[A,M, u, r,∆m], w2Il∞

[A,M, u, r,∆m], respectively.

(iii) If we take A = (C, 1, 1) in w2Ic

[A,M, u, r,∆m, p], w2Ic0

[A,M, u, r,∆m, p],

w2Il∞

[A,M, u, r,∆m, p] then these sequences are reduced to w2Ic

[M, u, r,∆m],

w2Ic0

[M, u, r,∆m], w2Il∞

[M, u, r,∆m].

(iv) If r = 0, we get w2Ic

[A, M, u, ∆m], w2Ic0

[A, M, u, ∆m], w2Il∞

[A, M, u,

∆m], respectively.

Throughout the paper, we shall use the following inequality. Let (akl) and(bkl) be two double sequences, then

|akl + bkl|pkl ≤ K(|akl|pkl + |bkl|pkl), (1.1)

where K = max(1, 2H−1) and supk,l

pkl = H.

The main purpose of this paper is to introduce and study some roughI2-convergent sequence spaces by using four dimensional bounded-regular

298


(shortly, RH-regular) matrices and a Musielak-Orlicz functions in more gen-eral setting. We also make an effort to study some properties like linearity,paranorm, solidity and some interesting inclusion relations between thesespaces.

2 Main results

Theorem 2.1 LetM = (Mkl) be a Musielak-Orlicz function, p = (pkl) be abounded sequence of strictly positive real numbers and u = (ukl) be a sequenceof strictly positive real numbers. Then the spaces w2

Ic[A, M, u, r, ∆m, p],

w2Ic0

[A,M, u, r,∆m, p] and w2Il∞

[A,M, u, r,∆m, p] are linear spaces over the

complex field C.

Proof. We shall prove the result for the space w2Ic0

[A,M, u, r,∆m, p] only

and the others can be proved in a similar way. Let x = (xkl) and y = (ykl)belongs to w2

Ic0[A,M, u, r,∆m, p]. Then there exists ρ1 > 0 and ρ2 > 0 such

that

A ε

2={

(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆mxklρ1

∥∥∥)]pkl ≥ r +

ε

2

}∈ I2

and

B ε

2={

(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆myklρ2

∥∥∥)]pkl ≥ r +

ε

2

}∈ I2.

Let α, β be any scalars. Then by using the inequality (1.1) and continuityof the function M = (Mkl), we have

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆m(αxkl + βykl)

|α|ρ1 + |β|ρ2

∥∥∥)]pkl

≤ D∞,∞∑

k,l=1,1

anmkl

[ |α||α|ρ1 + |β|ρ2

Mkl


∥∥∥)]pkl

+D

∞,∞∑

k,l=1,1

anmkl

[ |β||α|ρ1 + |β|ρ2

Mkl


∥∥∥)]pkl

≤ DK∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

+DK

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

,

299


where K = max

{1,

(|α|

|α|ρ1+|β|ρ2

)H,

(|β|

|α|ρ1+|β|ρ2

)H}.

From the above relation we obtain the following:{

(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆m(αxkl + βykl)

|α|ρ1 + |β|ρ2

∥∥∥)]pkl ≥ r + ε

}⊆

{(k, l) ∈ N× N : DK

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl ≥ r +

ε

2

}

∪{

(k, l) ∈ N× N : DK

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl ≥ r +

ε

2

}∈ I2.

This completes the proof. ut

Theorem 2.2 LetM = (Mkl) be a Musielak-Orlicz function, p = (pkl) be abounded sequence of strictly positive real numbers and u = (ukl) be a sequenceof strictly positive real numbers. Then the spaces w2

Ic[A,M, u, r,∆m, p],

w2Ic0

[A,M, u, r, ∆m, p] and w2Il∞

[A,M, u, r,∆m, p] are paranormed spaces

with the paranorm g defined by

g(x) = inf

{(ρ)

pklH :

( ∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

) 1H

≤ r,

for some ρ > 0

},

where H = max{r, supk,l

pkl}.

Proof. Clearly, g(−x) = g(x) and g(0) = 0. Let x = (xkl) and y = (ykl)belongs to w2

Ic0[A,M, u, r,∆m, p]. Then for every ρ > 0 we write

A1 =

{ρ1 > 0 :

( ∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

) 1H

≤ r}

and

A2 =

{ρ2 > 0 :

( ∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

) 1H

≤ r}.

300


Let ρ1 ∈ A1 and ρ2 ∈ A2. If ρ = ρ1 + ρ2, then we get the following:( ∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆m(xkl + ykl)

ρ

∥∥∥)]pkl

)

≤( ρ1ρ1 + ρ2

)( ∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

)

+( ρ2ρ1 + ρ2

)( ∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

).

Thus, we have∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆m(xkl + ykl)

ρ

∥∥∥)]pkl ≤ r

and

g(x+ y) = inf{(ρ1 + ρ2)pklH : ρ1 ∈ A1, ρ2 ∈ A2}

≤ inf{(ρ1)pklH : ρ1 ∈ A1}+ inf{(ρ2)

pklH : ρ2 ∈ A2}

= g(x) + g(y).

Let tmkl → t, where tmkl, t ∈ C and let g(xmkl − xkl)→ 0 as m→∞. To provethat g(tmklx

mkl − txkl) → 0 as m → ∞. Let tkl → t, where tkl, t ∈ C and

g(xmkl − xkl)→ 0 as m→∞. We have

A3 =

{ρk > 0 :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆mxmklρk

∥∥∥)]pkl ≤ r

}

and

A4 =

{ρ′k > 0 :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆m(xmkl − x)

ρ′k

∥∥∥)]pkl ≤ r

}.

If ρk ∈ A3 and ρ′k ∈ A4 then by inequality (1.1) and continuity of thefunction M = (Mkl), we have

Mkl

(∥∥∥ukl∆m(tmxmkl − tx)

|tm − t|ρk + |t|ρ′k

∥∥∥)≤Mkl

(∥∥∥ukl∆m(tmxmkl − txkl)


∥∥∥)

+Mkl

(∥∥∥ukl∆m(txkl − tx)


∥∥∥)

≤ |tm − t|ρk|tm − t|ρk + |t|ρ′k

Mkl

(∥∥∥ukl∆mxmklρk

∥∥∥)

+|t|ρ′k

|tm − t|ρk + |t|ρ′kMkl

(∥∥∥ukl∆m(xmkl − xkl)ρ′k

∥∥∥).

301


From the above inequality it follows that

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆m(tmxmkl − tx)


∥∥∥)]pk ≤ r

and consequently,

g(tmxmkl + tx) = inf{(|tm − t|ρk + |t|ρ′k)}pklH : ρk ∈ A3, ρ

′k ∈ A4}

≤ |tm − t|pklH inf{(ρk)

pklH : ρk ∈ A3}+ |t|

pklH inf{(ρ′k)

pklH : ρ′k ∈ A4}

≤ max{1, |tm − t|pklH }g(xmkl) + max{1, |t|

pklH }g(xmkl − x)

→ 0 as m→∞.

This completes the proof. ut

Theorem 2.3 (i) Let 0 < inf pkl ≤ pkl ≤ 1. Thenw2Ic

[A,M, u, r,∆m, p] ⊆ w2Ic

[A,M, u, r,∆m], w2Ic0

[A,M, u, r,∆m, p] ⊆w2Ic0

[A,M, u, r,∆m].

(ii) Let 1 ≤ pkl ≤ sup pkl <∞. Thenw2Ic

[A,M, u, r,∆m] ⊆ w2Ic

[A,M, u, r,∆m, p], w2Ic0

[A,M, u, r,∆m] ⊆w2Ic0

[A,M, u, r,∆m, p].

Proof. (i) Let x = (xkl) ∈ w2Ic

[A,M, u, r,∆m, p]. Since 0 < inf pkl ≤ pkl ≤ 1,we have

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]

≤∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

.

Therefore,{(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]≥ r + ε

}

⊆{

(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl ≥ r + ε

}∈ I2.

The other part can be proved in a similar way.

302


(ii) Let x = (xkl) ∈ w2Ic

[A,M, u, r,∆m]. Since 1 ≤ pkl ≤ sup pkl < ∞.Then, for each 0 < ε < 1 there exists positive integer n0 such that

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]≤ ε < 1 for all n,m ≥ n0.

This implies that

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

≤∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)].

Therefore, we have{

(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl ≥ r + ε

}

⊆{

(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]≥ r + ε

}∈ I2.

The other part can be proved in a similar way. This completes the proof.ut

Theorem 2.4 Let M = (Mkl) be a Musielak-Orlicz function, p = (pkl)be a bounded sequence of strictly positive real numbers and u = (ukl) be asequence of strictly positive real numbers. Then

w2Ic0

[A,M, u, r,∆m, p] ⊂ w2Ic [A,M, u, r,∆m, p] ⊂ w2

Il∞[A,M, u, r,∆m, p].

Proof. The inclusion w2Ic0

[A,M, u, r,∆m, p] ⊂ w2Ic

[A,M, u, r,∆m, p] is ob-

vious. Let x = (xkl) ∈ w2Ic

[A,M, u, r,∆m, p]. Then there is some number x∗such that

{(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl ≥ r + ε

}∈ I2.

Now by inequality (1.1), we have

303


∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

≤ D∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

+D

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥x∗ρ

∥∥∥)]pkl

.

This implies that x = (xkl) ∈ w2Il∞

[A,M, u, r,∆m, p]. This completes the

proof. ut

Theorem 2.5 Let M = (Mkl) and M′ = (M ′kl) be Musielak-Orlicz func-tions. Then

w2Ic [A,M, u, r,∆m, p]∩w2

Ic [A,M′, u, r,∆m, p] ⊂ w2Ic [A,M+M′, u, r,∆m, p].

Proof. Let x = (xkl) ∈ w2Ic

[A,M, u, r,∆m, p] ∩ w2Ic

[A,M′, u, r,∆m, p] usingthe inequality (1.1), we have

∞,∞∑

k,l=1,1

anmkl

[(Mkl +M ′kl)


∥∥∥)]pkl

=

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)

+M ′kl(∥∥∥ukl∆

mxkl − x∗ρ

∥∥∥)]pkl

≤ D{ ∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

+

∞,∞∑

k,l=1,1

anmkl

[M ′kl


∥∥∥)]pkl

}.

Thus, x = (xkl) ∈ w2Ic

[A,M +M′, u, r,∆m, p]. This completes the proof.ut

Theorem 2.6 LetM = (Mkl) be a Musielak-Orlicz function which satisfiesthe ∆2-condition. Then w2

Ic[A, u, r,∆m, p] ⊆ w2

Ic[A,M, u, r,∆m, p].

Proof. Let x = (xkl) ∈ w2Ic

[A, u, r,∆m, p], that is,

{(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[(∥∥∥ukl∆mxkl − x∗ρ

∥∥∥)]pkl ≥ r + ε

}∈ I2.

304


Let ε > 0 and choose δ with 0 < δ < 1 such that Mkl(t) < ε for 0 ≤ t ≤ δ.

Write ykl = (‖ukl∆mxkl−x∗ρ ‖) and consider

∞,∞∑

k,l=1,1

anmkl

[Mkl(ykl)

]pkl

=∑

k,l:|ykl|≤δanmkl

[Mkl(ykl)

]pkl+

∑

k,l:|ykl|>δanmkl

[Mkl(ykl)

]pkl

= ε∑

k,l:|ykl|≤δanmkl +

∑

k,l:|ykl|>δanmkl

[Mkl(ykl)

]pkl.

For ykl > δ, we use the fact that ykl <yklδ < 1 + ykl

δ . Hence,

Mkl(ykl) < Mkl

(1 +

yklδ

)<Mkl(2)

2+

1

2Mkl

(2yklδ

).

Since M satisfies the ∆2-condition, we have

Mkl(ykl) < Kykl2δMkl(2) +K

ykl2δMkl(2) = K

yklδMkl(2),

and hence,

∑

k,l:|ykl|>δanmkl

[Mkl(ykl)

]pkl

≤ KMkl

δ(2)

∞,∞∑

k,l=1,1

anmkl


∥∥∥)]pkl

.

Since A is RH-regular and x ∈ w2Ic

[A, u, r,∆m, p], we get x ∈ w2Ic

[A,M,u, r,∆m, p]. ut

Theorem 2.7 Let M = (Mkl) be a Musielak-Orlicz function, A = (anmkl)be a nonnegative four-dimensional RH-regular matrix such that

supn,m

∞,∞∑

k,l=0,0

anmkl <∞ and β = limt→∞

Mkl(t)

t<∞. Then

w2Ic [A, u, r,∆

m, p] = w2Ic [A,M, u, r,∆m, p].

Proof. In order to prove that w2Ic

[A, u, r,∆m, p] = w2Ic

[A,M, u, r,∆m, p]. It

is sufficient to show that w2Ic

[A,M, u, r,∆m, p] ⊂ w2Ic

[A, u, r,∆m, p]. Now,let β > 0. By definition of β, we have Mkl(t) ≥ βt for all t ≥ 0. Since β > 0,we have t ≤ 1

βMkl(t) for all t ≥ 0. Let x = (xkl) ∈ w2Ic

[A,M, u, r,∆m, p].

305


Thus, we have

∞,∞∑

k,l=1,1

anmkl


∥∥∥)]pkl

≤ 1

β

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

which implies that x = (xkl) ∈ w2Ic

[A, u, r,∆m, p]. This completes the proof.utTheorem 2.8 The spaces w2

Ic0[A,M, u, r,∆m, p] and w2

Ic[A,M, u, r,∆m, p]

are normal as well as monotone.

Proof. We shall prove the result for w2Ic0

[A,M, u, r,∆m, p]. Let x = (xkl) ∈w2Ic0

[A,M, u, r,∆m, p], then we have

{(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl ≥ r + ε

}∈ I2.

Let (αkl) be a sequence scalar with |αkl| ≤ 1 for all k, l. Then we get∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥αklukl∆mxkl

ρ

∥∥∥)]pkl]

≤ |αkl|∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

≤∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl

.

The spaces w2Ic0

[A,M, u, r,∆m, p] is normal and hence monotone follows

from Lemma 1.9. Similarly, we can prove other. utTheorem 2.9 The sequence spaces w2

Ic[A,M, u, r,∆m, p] and w2

Ic0[A,M,

u, r,∆m, p] are sequence algebras.

Proof. Let x = (xkl) and y = (ykl) belongs to w2Ic

[A,M, u, r,∆m, p]. Letε > 0 be given. Then

A(ε) ={

(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl


∥∥∥)]pkl ≥ r + ε

}∈ I2,

306


B(ε) ={

(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆mykl − x∗ρ

∥∥∥)]pkl ≥ r + ε

}∈ I2

and

C(ε) =

{(k, l) ∈ N× N :

∞,∞∑

k,l=1,1

anmkl

[Mkl

(∥∥∥ukl∆m(xklykl)− x∗

ρ

∥∥∥)]pkl

≥ r + ε

}∈ I2.

Now it can be easily shown that C(ε) ⊂ A(ε) ∪ B(ε) and C(ε) ∈ w2Ic

[A,M,u, r,∆m, p]. utTheorem 2.10 The spaces w2

Ic[A,M, u, r, ∆m, p] and w2

Ic0[A, M, u, r,

∆m, p] are not convergence free in general.

Proof. The proof is easy so we omit it. ut

Acknowledgements We would like to express our sincere thanks for the reviewer(s) forthe kind remarks which improved the presentation of the paper.

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Received: 11.IV.2017 / Accepted: 28.XII.2017

Authors

Kuldip Raj (Corresponding author),School of Mathematics,Shri Mata Vaishno Devi University,Katra-182320, J & K India,E-mail : [email protected]

Charu Sharma,School of Mathematics,Shri Mata Vaishno Devi University,Katra-182320, J & K India,E-mail : [email protected]

308

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