Some vector valued multiplier difference sequence spaces defined by a sequence of Orlicz functions Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2011, Том 13, Выпуск 2, С. 26-34

SOME VECTOR VALUED MULTIPLIER DIFFERENCE SEQUENCE SPACES DEFINED BY A SEQUENCE OF ORLICZ FUNCTIONS

H. Dutta

In this article we introduce some new difference sequence spaces with a real 2-normed linear space as base space and which are defined using a sequence of Orlicz functions, a bounded sequence of positive real numbers and a sequence of non-zero reals as multiplier sequence. We show that these spaces are complete paranormed spaces when the base space is a 2-Banach space and investigate these spaces for solidity, symmetricity, convergence free, monotonicity and sequence algebra. Further we obtain some relation between these spaces as well as prove some inclusion results.

Mathematics Subject Classification (2000): 40A05, 46A45, 46E30, 46B20.

Key words: difference sequence, 2-norm, Orlicz function, paranorm, completeness, solidity, symmetricity, convergence free, monotone space.

1. Introduction

Throughout the paper w, , c and c0 denote the spaces of all bounded, convergent, and null sequences x = (xk) with complex terms, respectively. The zero sequence is denoted by d = (0,0,0,...).

The notion of difference sequence spaces was introduced by Kizmaz [11] who studied the difference sequence spaces ^(A), c(A) and c0(A). The notion was further generalized by Et and Colak [4] by introducing the spaces ^(As), c(As) and c0(As). Recently Dutta [2] introduced and studied the following difference sequence spaces:

Let r, s be non-negative integers, then for Z a given sequence space we have

z]

Z(A(r)) = {x = (xk) G w : (A^x*,) G z},

where A^x = (A|r)xk) = (A^x^ - A^x^) and A0r)xfc = xfc for all k € N and which is equivalent to the binomial representation A^xk = J2s=0(-1)^1^) xk—rv.

For s = 1, we get the difference operator A(r) introduced and studied by Dutta [3] for sequences of fuzzy numbers. Again r = s = 1, we get spaces ^(A), c(A) and c0(A).

Let A = (Ak) be a sequence of non-zero scalars. Then for a sequence space E the multiplier sequence space E(A), associated with the multiplier sequence A is defined as

E(A) = {(xk) € w : (Akxk) € E}.

The scope for the studies on sequence spaces was extended by using the notion of associated multiplier sequences. Goes and Goes [8] defined the differentiated sequence space dE and integrated sequence space E for a given sequence space E, using the multiplier sequences

© 2011 Dutta H.

(k-1) and (k) respectively. A multiplier sequence can be used to accelerate the convergence of the sequences in some spaces. In some sense, it can be viewed as a catalyst, which is used to accelerate the process of chemical reaction. Sometimes the associated multiplier sequence delays the rate of convergence of a sequence.

The concept of 2-normed spaces was initially developed by Gahler [6] in the mid of 1960's. Since then, Gunawan and Mashadi [10], Dutta [1] and many others have studied this concept and obtained various results.

Let X be a real linear space of dimension greater than one and let -|| be a real valued function on X x X satisfying the following conditions:

(1) ||x, y|| = 0 if and only if x and y are linearly dependent vectors,

(2) ||x,y| = ||y,x|

(3) ||ax,y|| ^ |a| ■ ||x, y||, for every a G R

(4) ||x, y + z|| ^ ||x, y|| + ||x, z||

then the function ||-, -|| is called a 2-norm on X and the pair (X, ||-, -||) is called a 2-normed linear space.

An Orlicz function is a function M : [0, to) ^ [0, to) which is continuous, non-decreasing and convex with M(0) = 0, M(x) > 0 , for x > 0 and M(x) ^ to, as x ^ to.

Lindenstrauss and Tzafriri [14] used the Orlicz function and introduced the sequence space lM as follows:

Remark 1. An Orlicz function satisfies the inequality M(Ax) < AM(x), for all A with 0 < A < 1. The following inequality will be used throughout the article.

Let p = (pk) be a positive sequence of real numbers with 0 < pk ^ suppk = G, D = max (1,2G-1). Then for all ak, bk G C for all k G N, we have

and for all A G C, |A|Pk < max (1, |A|G).

The studies on paranormed sequence spaces were initiated by Nakano [17] and Simons [20] at the initial stage. Later on it was further studied by Maddox [15], Nanda [18], Las-cardies [12], Lascardies and Maddox [13] and many others. Parasar and Choudhary [19], Mursaleen, Khan and Qamaruddin [16] and many others studied paranormed sequence spaces using Orlicz functions.

A sequence space E is said to be: solid (or normal) if (xk) G E implies (akxk) G E for all sequences of scalars (ak) with |ak | ^ 1 for all k G N; monotone if it contains the canonical preimages of all its step spaces; symmetric if (xn(k)) G E whenever (xk) G E, where n is a permutation on N; convergence free if (yk) G E whenever (xk) G E and yk = 0 whenever xk = 0; sequence algebra if (xk,yk) G E whenever (xk) G E and (yk) G E.

k=l

They proved that lM is a Banach space normed by

|ak + bkr < D {|akP + |bk^}

2. Definition and Preliminaries

A sequence ) in a 2-normed space (X, -||) is said to converge to some L e X in the 2-norm if limk^^ ||Xk — L, u|| = 0, for every u e X, and is said to be Cauchy sequence with respect to the 2-norm if ||xk — xi,u|| = 0, for every u e X.

If every Cauchy sequence in X converges to some L e X, then X is said to be complete with respect to the 2-norm. Any complete 2-normed space is said to be 2-Banach space.

Now we give the following two familiar examples of 2-norm which will be used in the next section to construct examples.

Example 1. Consider the spaces c and c0 of real sequences. Let us define:

||X,y|| = sup sup |xiyj — Xjyi|,

ieN jeN

where X = (x, X2, X3,...) and y = (y1; y2, y3, • • •). Then ||-, -|| is a 2-norm on c and c0. Example 2. Let us take X = R2 and Consider the function ||-, -|| on X definded as:

||xi, x2 ||e = abs

X11 X12 X21 X22

Xj = (Xi1,Xi2) G R2, i = 1, 2.

Then ||-, -|| is a 2-norm on X.

Let p = (pk) be any bounded sequence of positive real numbers and A = (Ak) be a sequence of non-zero reals. Let m, n be non-negative integers, then for a real linear 2-normed space (X, ||-,-||) and for a sequence M = (Mk) of Orlicz functions we define the following sequence spaces:

lim ( Mk

k—>oo

co(M, ||., -||, A^), A,p) = <J X = (Xk) g w(X) :

Pk

Afm)Ak Xk

P

, z

= 0, z G X, for some p > 0 >,

lim Mk k

c(M, ||-, -||, A(m), A,p) = <J X = (Xk) G w(X) :

Pk

Afm)AkXk - L

P

,z

= 0, z G X, L G X, for some p > 0 },

-MM, ||.,-||, A^m), A,p) = <| X = (Xk) G w(X) :

sup Mk k>1

Afm)Ak Xk

,z

Pk

< to, z G X, for some p > 0

where (A^ AkXk) = (A^AkXk — A™„l)1Ak-mXk-m) and A0m)AkXk = AkXk for all k e N and which is equivalent to the binomial representation

" /n\

A(m)AkXk = ^ y( —1) \ ^k-mvXk-mv v=0

In the above expansion it is important to note that we take Xk-mv = 0 and Ak-mv = 0, for non-positive values of k — mv. It is obvious that

Co(M, ||, -||, A(m), A,p) C c(M, ||., -||, A(m), A,p) C ^(M, ||-,-||, A(m), A,p).

p

The inclusions are strict as follows from the following examples.

Example 3. Let m = 2, n = 2, Mk(x) = x2 for all k is odd and Mk(x) = x6 for all k is even, for all x G [0, to) and pk = 1 for all k ^ 1. Consider the 2-normed space as defined in Example 2 and let the sequences A = (A;4) and x = (p-, p). Then x G c(M, ||-, -||, A^, A,p), but x G c0(M, 11-, -||, A22), A,p).

Example 4. Let m = 2, n = 2, Mk(x) = |x|, for all k ^ 1 and x G [0, to) and pk = 2 for all k odd and p& = 3 for all k even. Consider the 2-normed space as defined in Example 1 and let the sequences A = (1,1,1,...) and x = {1,3,2,4,5, 7,6,8,9,11,10,12,...}. Then x G -MM, ||.,-||, A22), A,p), but x G c(M, ||-, -||, A22), A,p).

Lemma 1. If a sequence space E is solid, then E is monotone.

3. Main Results

In this section we prove the main results of this article.

Proposition 1. The classes of sequences c0(M, ||-,-||, Anm), A,p), c(M, ||-,-||, Anm), A,p) and (M, 11-, -| |, Anm), A, p) are linear spaces.

Theorem 2. For Z = c and c0, the spaces Z(M, ||-, -||, Anm), A,p) are paranormed sapces, paranormed by

Ei

g(x) = inf p h : sup M fe>i

A"m) 'k xk

P

-, Z

< 1, z G X

where H = max(1, sup^ pk).

< Clearly g(x) = g(-x); x = 0 implies #(0) = 0. Let (xk) and (yk) be any two sequences of the space c0(M, ||-, -||, Anm), A,p). Then there exist pi, p2 > 0 such that for every z in X,

sup Mk fe>i

Afm) 'k Xk

Pi

, z

^ 1, sup Mk fe>i

A(m)

P2

-, Z

< 1.

Let p = p1 + p2. Then by the convexity of Orlicz functions, we have for every z in X

sup Mk k

A(m) 'k xk + A(m)Ak yk

(m)'

P

-, Z

<

Pi

+

P2

Pi+P2J k

sup Mk

Pi + P2J k A(m) 'kyk

sup Mk

P2

-, z

A(m)'k xk

Pi

Hence we have,

g(x + y) = inf p h : sup Mk k>1

A(m) 'k xk + A(m)Ak yk

(m)

^ inf <J Pi H : sup Mfc k>1

P

A(m) 'kxk

+ inf <; p2 H : sup Mk k>1

A(m) 'kyk

P2

-, z

-, z Pi

^ 1, z G X

^ 1, z G X ^ 1, z G X

+ y) < g(x)+ g(y).

z

The continuity of the scalar multiplication follows from the following equality:

< 1, z e X

n

g(ax) = inf p h : sup Mk k> 1

A™m) aXk Xk

= inf s (t\a\)^H : supMk k> 1

P

Afm)Ak xk

t

, z

^ 1, z G X

where t = ^ Hence the spaces cq(M,

A ,

• A(m) >

A,p) is a paranormed space, paranormed by g. The rest of the cases will follow similarly. i>

Theorem 3. If (X, -||) is a 2-Banach space, then the spaces Z(M, -||, A(m), A,p), for Z = i^, c and c0 are complete paranormed spaces, paranormed by

g(x) = inf < p h : sup Mk k>1

A(m)Ak xk

P

,z

^ 1, z e X

where H = max(1, supk^1 pk).

< We prove the result for the space ^(M, ||-, -||, A(m), A,p) and for other spaces it will follow on applying similar arguments.

Let (xl) be any Cauchy sequence in ¿^(M, ||-,-||, A(m) , A,p). Let x0 > 0 be fixed and t > 0 be such that for 0 < e < 1, ^ > 0 and xq t > 0. Then there exists a positive integer

no such that g(xl — x3) < for all i, j ^ no- Using the definition of paranorm, we get

n

inf p h : sup Mk k>1

A(m) Ak(xk - xk)

<1, zeX} <— (Vi,j^n0).

Then we get for every z in X

sup Mk k>1

AMAk (xk - xjk)

g(x* — xj)

< 1 (V^ no).

It follows that for every z e X and k ^ 1

Mk

AUXk ^ — xjk)

g(x* — xj)

< 1 (V^ no).

Now for i > 0 with Mk (ifi) ^ 1, for each k ^ 1

Mk

Afm) Ak (xk — xk )

g(x* — xj)

,z

ZGX.

This implies that

A(m) Akxk — A(m) AkxJk,z|| <

(m) Akxk,

tx0 ~2~

tx0

z X.

Hence (A(m) Akxk) is a Cauchy sequence in 2-Banach space X for all k e N. ^ (A(m) Akxk) is convergent in X for all k e N. For simplicity, let lim^^ A(m)Akxk = yk for each k e N. Let k = 1, we have

z

P

z

z

£

n

lim AnmAix! = lim > ,(-1)

' v=0

mvx1 —mv

= lim A1x1 = y1.

(1)

Similarly we have

lim Anm) Afcxk = yk, k = 1,2,..., nm.

(2)

Thus from (1) and (2) we have lim^^ x1+nm exists. Let lim^^ x1+nm = x1+nm. Proceeding in this way inductively, we have lim^^ xk = xk exists for each k G N. Now we have for all i, j ^ no.

inf <j p h : sup Mk k>1

P

-, z

^ 1, z G X ^ ^ e

lim inf < p h : sup Mk

I k>1

Anm) AkXk - Am) AkXk

n

(m)'

-, Z

^ 1, z G X ^ ^ e

Pk

lim inf i p-H : sup Mk

I k>1

Anm) AkXk - Anm)AkXk

P

^ 1, z G X ^ ^ e (Vi ^ no).

It follows that (x* - x) G ¿^(M, -||, A^, A,p).

Since (x*) G ¿^ (M, ||-, -||, Anm), A,p) and ¿^,(M, ||-, -||, Anm), A,p) is a linear space, so we have x = x* — (x* — x) G ¿^(M, ||-, -||, Anm), A,p). This completes the proof. >

Theorem 4. If 0 < pk ^ qk < to for each k, then Z(M, ||-,-||, Anm), A,p) C Z(M, ||-,-||, Anm), A,q(, for Z = co and c.

< We prove the result for the case Z = c0 and for the other case it will follow on applying similar arguments.

Let (xk) G c0(M, ||-, -||, Anm), A,p). Then there exist some p > 0 such that

lim Mk k

Am) Akxk

Pk

0.

This implies that Mk get

lim Mk k

A"m)XkXk

,z

Pk

1 1 < e (0 < e ^ 1) for sufficiently large k. Hence we

A(m) Akxk

P

?k

^ lim ( Mk k

Anm) Akxk

Pk

(xk) G co (M, ||.,-||, Anm),A,q).

Thus co(M,||.,-||, A^m), A,p) C Co(||M, ■,-||, A^m), A,q) Similarly, c(M, ||-,-||, A^), A,p) C c(||M, ■,-||, A^) The following result is a consequence of Theorem 6. Corollary 5. (a) If 0 < inf pk ^ Pk ^ 1, for each k, then

Anm), A,p) C c(||M, ■,-||, Anm), A, q). This completes the proof. >

Z(M, ||-, -||, Anm), A,p) C Z(M, ||-, -||, Anm), A), Z = co,c. (b) If 1 ^ pk ^ suppk < to, for each k, then

Z(M, ||-, -||, Anm), A C Z(M, ||-, -||, Anm), A,P, Z = co,c.

v

P

z

z

P

p

0

z

z

P

Theorem 6. Z(M, ||-,-||, A^;1, A,p) C Z(M, ||-,-||, A^;, A,p) (in general for i =

1,2,...,n - 1) Z(M, ||., -||, A(;), A,p) C Z(M, ||., -||, Anm), A,p)), for Z = c and Cq.

< Here we prove the result for Z = c0 and for the other cases it will follow on applying similar arguments.

x

Let x = (xk) £ c0(M, -||, A"—, A,p). Then there exist p > 0 such that

lim ( Mk

k—>00

AMAk xk

p

, Z

Pk

(3)

On considering 2p, by the convexity of Orlicz functions, we have

Mk

A™m) Afexk

2p

Hence we have

An-)lAk xk

Mk

A™m) Ak xk

2p

Pk

A(m) Ak—mxk—

^ D

Mk

Then using (3), we get

Pk

+

lim k

Mk

A™m) Ak xk

2p

Mk

, z

A(m) Ak—mxk—

Pk

Pk

0.

Thus co(M, ||., -||, A^-)1, A,p) C co(M, ||-, -||, A^m), A,p). >

The inclusion is strict as follows from the following example.

Example 5. Let m = 3, n = 2, Mk(x) = x10, for all k ^ 1 and x G [0, to) and pk = 2 for all k odd and pk = 3 for all k even. Consider the 2-normed space as defined in Example 2 and let the sequences A = Q) and x = (irk) = (k2,k2). Then A^AkXk = 0, for all k G N. Then x G c0(M, ||-,-||, A?3), A,p). Again we have A13)Akxk = -3, for all k G N. Hence

x / Co (M, II-, -II, A1

(3), A,p). Thus the inclusion is strict. Theorem 7. The following spaces c0(M, ||-,-||, A";, A,p), c(M, ||-,-||, A";, A,p) and (M, ||-, -||, An;), A,p) are not monotone and as such are not solid in general.

< The proof follows from the following example. >

Example 6. Let n = 2, m = 3, pk = 1 for all k odd and pk = 2 for all k even and Mk(x) = x2, for all k ^ 1 and for all x G [0,to). consider the 2-normed space as defined in Example 1. Then A23)Akxk = Akxk — 2Ak_3xk_3 + Ak_6xk_6, for all k G N. Consider the Jstep space of a sequence space E defined as, for (xk), (yk) G EJ implies that yk = xk for k odd and yk = 0 for k even. Consider the sequences A = (k3) and

x = Then x G Z(M.......

pre-image does not belong to Z | M, Z(M, ||-, -||, An general.

Theorem 8. The following spaces are not symmetric in general: c0 (M.

C(M, |h 4 A(m), H> 4 A"m), A,P(-

< The proof follows from the following example. >

A23), A,p) for Z = c and co, but its Jih canonical -||, A23), A,p) for Z = c and co. Hence the spaces , A,p) for Z = c and co are not monotone and as such are not solid in

(m)

>A,p),

0

z

z

z

p

p

z

z

z

p

p

Example 7. Let n = 2, m = 2, p, = 2 for all k odd and = 3 for all k even and Mk(x) = x2, for all x G [0, to) and for all k ^ 1. Consider the 2-normed space as defined in Example 1. Then A22)Akxk = Akxk — 2Ak-2xk-2 + Ak-4xk-4, for all k G N. Consider the sequences A = (1,1,1,...) and x = (xk) defined as xk = k for k odd and xk = 0 for k even. Then A22)Akxk = 0, for all k G N. Hence (xk) G Z(M, A22), A,p), for Z = i^, c and co. Consider the rearranged sequence, (yk) of (xk) defined as (yk) = (xi, x3,x2,x4,x5,x7,x6,x8,x9,xn,xio,xi2,...). Then (yk) G Z(M, ||-, -||, A22), A,p), for Z = i^, c and co. Hence the spaces Z(M, ||-, -||, A?), A,p), for Z = i^, c and co are not symmetric

in general.

Theorem 9. The following spaces are not convergence free in general:

co (M |h 4 A(m), A,p > c(M H> 4 Am), A,p > ^ (M H> 4 A(m), A,P •

< The proof follows from the following example. >

Example 8. Let m = 3, n = 1, = 6 for all k and Mk(x) = x5, for k is even and Mk(x) = |x|, for k is odd, for all x G [0,to). Then A(3)Akxk = Akxk — Ak-3xk-3, for all k G N. Consider the 2-normed space as defined in Example 2. Let A = (|) and consider the sequences (xk) and (y&) defined as Xk = (f ft, for all k G N and yk = Qk3, |A;3) for all k G N. Then (xk) G Z(M, ||a-||, A(3), A,p), but (yk) G Z(M, ||-,-||, A(3), A,p), for Z = i^, c and co. Hence the spaces Z(M, ||-, -||, Anm), A,p), for Z = i^, c and co are not convergence free in general.

Theorem 10. The following spaces are not sequence algebra in general:

co (M |h 4 A(m), A,P > c(M H> 4 A(m), A,P > (M H> 4 A(m), A,P .

< The proof follows from the following example. >

Example 9. Let n = 2, m = 1, = 1 for all k and Mk(x) = x2, for each k G N and x G [0,to). Then A21)Akxk = Akxk — 2Ak-1xk_1 + Ak-2xk-2, for all k G N. Consider the 2-normed space as defined in Example 2. Consider A = and let ir = (k5,k5) and y =

21), A,p, Z = i^ and c, but x,yGZ(M, ||-,-||, A21),

(,

(m),

algebra in general.

Example 10. Let n = 2, m = 1, = 3 for all k and Mk(x) = x7, for each k G N and x G [0,to). Then A21)Akxk = Akxk — 2Ak-1 xk-1 + Ak-2xk-2, for all k G N. Consider the 2-normed space as defined in Example 1. Consider A = and let x = (k7) and y = (k7). Then x, y G c0(M, ||-, -||, A^, A,p), but x,y G Z(M, ||-, -||, A21), A,p), for Z = i^, c. Hence the space c0 (M, ||-, -||, A21), A,p) is not sequence algebra in general.

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Received August 3, 2009. Hemen Dutta

Department of Mathematics, Gauhati University INDIA, 781 014, Kokrajhar Campus, Assam E-mail: [email protected]

ВЕСОВЫЕ ПРОСТРАНСТВА ВЕКТОРНОЗНАЧНЫХ РАЗНОСТНЫХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ, ОПРЕДЕЛЯЕМЫЕ ПОСЛЕДОВАТЕЛЬНОСТЬЮ ФУНКЦИЙ ОРЛИЧА

Дутта Х.

Вводятся новые классы разностных последовательностей со значениями в 2-нормированном векторном пространстве с помощью последовательности функций Орлича, ограниченной последовательности положительных чисел и весовой последовательности ненулевых вещественных чисел. Устанавливается, что эти классы являются полными паранормированными пространствами и изучаются некоторые их свойства.

Ключевые слова: разностная последовательность, 2-норма, паранорма, функция Орлича, полнота, солидность, симметричность, монотонность.