Some vector valued multiplier difference sequence spaces defined by a sequence of Orlicz functions
Автор: Dutta Hemen
Журнал: Владикавказский математический журнал @vmj-ru
Статья в выпуске: 2 т.13, 2011 года.
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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.
Difference sequence, 2-norm, orlicz function, paranorm, completeness, solidity, symmetricity, convergence free, monotone space.
Короткий адрес: https://sciup.org/14318344
IDR: 14318344
Текст научной статьи Some vector valued multiplier difference sequence spaces defined by a sequence of Orlicz functions
Throughout the paper w, ' ^ , c and c o denote the spaces of all bounded, convergent, and null sequences x = (x k ) with complex terms, respectively. The zero sequence is denoted by θ = (0, 0, 0, . . . ).
The notion of difference sequence spaces was introduced by Kizmaz [11] who studied the difference sequence spaces ' ^ (A), c(A) and c o (A). The notion was further generalized by Et and Colak [4] by introducing the spaces ' ^ (A s ), c(A s ) and c o (A s ). 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(A s r ) ) = n x = (x k ) G w : (A s r ) x k ) G Z 0
where A sr) X = (A sr ) X k ) = (A S — X k - A S -1 X k - r ) and A^x k = X k for all k G N and which is equivalent to the binomial representation A sr ) X k = ^2V =o ( — 1) v(V) X k - 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 c o (A).
Let Λ = ( λ k ) be a sequence of non-zero scalars. Then for a sequence space E the multiplier sequence space E (Λ), associated with the multiplier sequence Λ is defined as
E (Л) = { (x k ) G w : (X k X k ) G 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
(c) 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 G¨ahler [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 k· , ·k be a real valued function on X × X satisfying the following conditions:
-
(1) k x, y k = 0 if and only if x and y are linearly dependent vectors,
-
(2) k x, y k = k y, x k ,
-
(3) k αx, y k 6 | α | · k x, y k , for every α ∈ R
-
(4) k x,y+z k 6 k x, y k + k x, z k
then the function k· , ·k is called a 2-norm on X and the pair (X, k· , ·k ) is called a 2-normed linear space.
An Orlicz function is a function M : [0, ∞ ) → [0, ∞ ) which is continuous, non-decreasing and convex with M (0) = 0, M (x) > 0 , for x > 0 and M (x) → ∞ , as x → ∞ .
Lindenstrauss and Tzafriri [14] used the Orlicz function and introduced the sequence space ` M as follows:
∞
' m = j (xk) € w : X M k =1
| x k | ρ
< ∞ , for some ρ > 0 .
They proved that ` M is a Banach space normed by
k (x k ) k = inf
{P> 0^ M (V) 6 1}.
Remark 1. An Orlicz function satisfies the inequality M (λx) < λM (x), for all λ with 0 < λ < 1. The following inequality will be used throughout the article.
Let p = (p k ) be a positive sequence of real numbers with 0 < p k 6 sup p k = G, D = max ( 1, 2 G 1 ) . Then for all a k , b k € C for all k € N , we have
|ak + bk|pk 6 D {|ak|pk +|bk|pk} and for all A € C, |A|pk 6 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.
-
2. Definition and Preliminaries
A sequence space E is said to be: solid (or normal) if (x k ) ∈ E implies (α k x k ) ∈ E for all sequences of scalars (α k ) with | α k | 6 1 for all k ∈ N ; monotone if it contains the canonical preimages of all its step spaces; symmetric if (x n ( k ) ) € E whenever (x k ) € E , where n is a permutation on N ; convergence free if (y k ) ∈ E whenever (x k ) ∈ E and y k = 0 whenever x k = 0; sequence algebra if (x k , y k ) ∈ E whenever (x k ) ∈ E and (y k ) ∈ E.
A sequence (x k ) in a 2-normed space (X, k· , ·k ) is said to converge to some L ∈ X in the 2-norm if lim k→∞ k x k - L, u k = 0, for every u ∈ X , and is said to be Cauchy sequence with respect to the 2-norm if lim k,l →∞ k x k - x l , u k = 0, for every u ∈ X .
If every Cauchy sequence in X converges to some L ∈ 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 c g of real sequences. Let us define:
kx, yk = sup sup |xiyj - xjyi|, i∈N j∈N where x = (xi, x2, X3,...) and y — (yi, y2, Уз,...). Then Ц’, •k is a 2-norm on '^, c and cg.
Example 2. Let us take X = R 2 and Consider the function Ц’ , -| on X definded as:
k x i , X 2 | e = abs
x ii x i2
x 2i x 22
x i — (x i 1 ,x i 2 ) e R , i — 1, 2*
Then k· , ·k is a 2-norm on X.
Let p = (p k ) be any bounded sequence of positive real numbers and Λ = (λ k ) be a sequence of non-zero reals. Let m, n be non-negative integers, then for a real linear 2-normed space (X, k· , ·k ) and for a sequence M = (M k ) of Orlicz functions we define the following sequence spaces:
c g (M, lh •k, A m ) ,Л,Р) = x = (x k ) e w(X) :
lim k →∞

A m ) X k X k ρ

∈ X, for some ρ > 0 ,
c(M, |. , •k , А Пт ) , Л,р)
x = (x k ) ∈ w(X) :
lim k →∞

∆ n λ x - L
( m ) k k
,z ρ

= 0, z ∈ X, L ∈ X, for some ρ > 0 ,
'^(M, II1 , •k , А Пт ) , Л,Р) = 1 x = (x k ) e w(X) :
sup k>1
An X k x k
,z ρ
p k
< ∞, z ∈ X, for some ρ > 0 , where (A^m)Xkxk) — (A^Xkxk - A^Xk-mxk-m) and Agm)Akxk — Xkxk for all k e N and which is equivalent to the binomial representation
n

λ k - mv x k - mv .
A’rnX- xk = Е<-^’ v=g
In the above expansion it is important to note that we take x k - mv = 0 and λ k - mv = 0, for non-positive values of k - mv.
It is obvious that cg(M, |., •k, A(m), Л,р) C c(M, k-, •k, A(m), Л,р) C '_(M, k^, •k, A(^), Л,р).
The inclusions are strict as follows from the following examples.
Example 3. Let m = 2, n = 2, M k (x) = x 2 for all k is odd and M k (x) = x 6 for all k is even, for all x ∈ [0 , ∞ ) and p k = 1 for all k > 1. Consider the 2-normed space as defined in Example 2 and let the sequences Л = (k 4 ) and x = (^ 2 , ^ 2) . Then x G c(M, Ц’ , •k , A 2 2 ) , Л,р), but x / c 0 (M , ||- , •k , A 22) , л,р).
Example 4. Let m = 2, n = 2, M k (x) = | x | , for all k > 1 and x ∈ [0, ∞ ) and p k = 2 for all k odd and p k = 3 for all k even. Consider the 2-normed space as defined in Example 1 and let the sequences Λ = (1, 1, 1, . . . ) and x = { 1, 3, 2, 4, 5, 7, 6, 8, 9, 11, 10, 12, . . . } . Then x G ' x (M, ||- , •k , A 22) , Л,Р), but x / c(M, Ц- , •k , A 22) , Л,Р)-
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 c o (M, ||-, -||, A nm ) , л,р), c(M ||-, -||, A nm ) , л,р) and ' ^ (M, ||- , - I , A nm ) , Л,р) are linear spaces.
Theorem 2. For Z = '^, c and cg, the spaces Z(M, Ц*, •I, Anm), Л,р) are paranormed sapces, paranormed by pk
g(x) = inf ρ H
: sup Mk k>1
A ( m ) X k x k ρ
z 6 1, z G X
where H = max(1,sup k >1 p k ).
C Clearly g(x) = g( - x); x = θ implies g(θ) = 0. Let (x k ) and (y k ) be any two sequences of the space c g (M, Ц^ , •I , A nm ) , Л,р). Then there exist p i , p 2 > 0 such that for every z in X,
sup M k k > i
A nm ) X k x k
,z pi

sup M k k > i
A nm ) ХУ
P 2

Let p = p i + p 2 . Then by the convexity of Orlicz functions, we have for every z in X
sup M k k
Hence we have,
A ( m ) X k x k + A ( m ) X k y k ρ

p i
6 sup Mk pi + p2 k
p 2
+ sup Mk pi + p2 k
p k
g(x + y) = inf ρ H : sup M k
k > i
6 inf
p k
P i H
: sup M k k > i
A ( m ) X k x k P i
p k
+ inf < p 2 H : sup M k k > i
A nm ) X k y k P 2

A ( m ) X k x k + A ( m ) X k y k
,z ρ
A ( m ) X k x k
,z pi
A nm ) X k y k
P 2

6 1, z ∈ X
6 1, z ∈ X
6 1, z ∈ X = ⇒ g(x + y) 6 g(x) + g(y).
The continuity of the scalar multiplication follows from the following equality:
g(αx) = inf
p k
ρ H : sup Mk k>1
= inf (t | α | ) Hk : sup M k
I k > 1
A ( m ) aX k x k
,z ρ
A(m)Xk xk t ,z )
6 1, z G X
6 1, z G X
where t = Д Hence the spaces c o (M, ||- , •k , A^ m ) , Л,р) is a paranormed space, paranormed by g. The rest of the cases will follow similarly. B
Theorem 3. If (X, ||-, •k) is a 2-Banach space, then the spaces Z(M, ||-, •k, A^m), Л,р), for Z = '^, c and co are complete paranormed spaces, paranormed by pk
g(x) = inf ρ H : sup M k k > 1
A ( m ) X k x k
,z ρ
6 1, z G X
where H = max(1,sup k > 1 p k ).
<1 We prove the result for the space ' ^ (M, ||- , •k , A^ m ) , Л,р) and for other spaces it will follow on applying similar arguments.
Let (xi) be any Cauchy sequence in '^(M, ||-, •k, A^m), Л,р). Let xo > 0 be fixed and t > 0 be such that for 0 < e < 1, > 0 and xot > 0. Then there exists a positive integer x0t no such that g(xi — xj) < -Ц, for all i, j > no. Using the definition of paranorm, we get x0t inf
p k
ρ H : sup Mk k>1
A ( m X k (x k — x k ) ρ

z ∈ X
ε
< xot
( V i, j > n o ).
Then we get for every z in X
sup M k k > 1
A" m) X k ( x k - x k ) g(x i - x j )

( V i,j > n o ).
It follows that for every z G X and k > 1

A" m) X k ( x k - x k ) g(x i - x j )
( V i,j > n o ).
Now for t > 0 with M k ( tX j 0 ) > 1, for each k > 1

AX m) X k ( x k - x k ) g(x i - x j )
6 M k
txo \
2,
z ∈ X.
This implies that
|| A (- m ) X t x k — A? m , X k x , z || 6 ('x") ) = f. z G X.
2 tx o 2
Hence (A^ m ) X k xk) is a Cauchy sequence in 2-Banach space X for all k G N . ^ (A^ m ) X k xk) is convergent in X for all k G N . For simplicity, let lim i ,^ AT ( m ) X k x k = y k for each k G N . Let k = 1, we have
n lim Anm)Aix! = lim X(-1)v( )Ai—mvxi-m,
= lim A i x l = y 1 . i →∞
i→∞ i→∞ v=0
Similarly we have lim Anm)Ak xk = yk, k = 1, 2,...,nm.(2)
i →∞
Thus from (1) and (2) we have lim i ,^ x\ + nm exists. Let lim i ,^ x 1+n m = x 1+nm . Proceeding in this way inductively, we have lim i→∞ x i k = x k exists for each k ∈ N . Now we have for all
i, j > n o .
pk inf ρ H : sup Mk
I k > i
A ( m ) A k (x k — x k )
,z ρ
6 1, z ∈ X 6 ε
p k
=⇒ lim inf ρ H : sup Mk j 'x I k>1
A m A - A m A k . z ρ,
6 1, z ∈ X 6 ε
p k
=⇒ lim inf ρ H : sup Mk j^^ I k>1
A(m ) A k X k
-
ρ
A ( m ) A k x k
,z
6 1, z E X > 6 e ( V i > n o ).
It follows that (x i - x) E ' ^ (M, Ц- , •k , A nm ) , Л,р).
Since (x i ) E ' ^ (M, Ц- , •k , A nm ) , Л,р) and ' ^ (M, Ц- , •k , A nm ) , Л,р) is a linear space, so we have x = x i — (x i — x) E ' ^ (M, ||- , •k , A nm , Л,р . This completes the proof. B
Theorem 4. If 0 < P k 6 q k < to for each k, then Z ( M, Ц- , •k , A nm , Л,р ) C Z ( M, lh -lb A m) , ^q ) , for Z = c o and c.
C We prove the result for the case Z = c o and for the other case it will follow on applying similar arguments.
Let (xk) E co(M, ||-, •k, Anm), Л,р). Then there exist some p > 0 such that get
This implies that

^ nm ) ^ k x k ρ
lim k →∞

lim k →∞
A m) A k x k
,z ρ

A nm ) A k x k
,z ρ

< ε (0 < ε 6 1) for sufficiently large k. Hence we
6 lim k →∞
A m) A k x k ρ

= ^ (x k ) E c o (m, k^ , •k , A nm) , Л,q).
Thus c o (M, !• , •k , A nm ) , Л, p) C c o (k M, • , •k , A nm ) , Л, q).
Similarly, c ( M, Ц- , •k , A nm ) , Л,р ) C c ( |M, • , •k , A nm ) , Л,q ) . This completes the proof. B
The following result is a consequence of Theorem 6.
Corollary 5. (a) If 0 < inf p k 6 p k 6 1 , for each k , then
Z(M, k^ , •k , A nm) , Л,р) C Z(M, k^ , •k , A nm) , Л), Z = c o ,c.
(b) If 1 6 p k 6 sup p k < ∞ , for each k , then
Z ( M, k^ , •k , A nm) , Л ) C Z ( M, k^ , •k , A nm) , л,р ) , Z = c o ,c.
Theorem 6. Z(M, Ц- , •k , A n— , Л,р) C Z(M, Ц- , ’k , A nm , Л,р) (in general for i = 1, 2,...,n - 1) Z (M, k’ , •k , A— , Л,р) C Z (M, k’ , •k , A nm , Л,р)), for Z = ' ^ , c and c g .
C Here we prove the result for Z = c g and for the other cases it will follow on applying similar arguments.
Let x = (xk) E cg (M, ||’, -k, An—, Л,р). Then there exist p > 0 such that lim k→∞
— m—k^k ρ

On considering 2ρ, by the convexity of Orlicz functions, we have
A ( m ) X k x k 2ρ
A nm) ^ k x k ρ
1 +2
ρ
"A,
( m ) k - m x k - m

Hence we have

^ nm^ X k x k ρ
Then using (3), we get

А Пт ) X k X k 2 ρ

lim k →∞

A ( m ) X k x k
2 ρ , z

"A,
( m ) k - m x k - m
ρ

Thus C g (M, k’ , •k , A nm ) 1 , Л, p) C c g (M, k’ , ’k , A nm , Л— . B
The inclusion is strict as follows from the following example.
Example 5. Let m = 3, n = 2, M k (x) = x 10 , for all k > 1 and x E [0, to ) and p k = 2 for all k odd and p k = 3 for all k even. Consider the 2-normed space as defined in Example 2 and let the sequences Л = (— and x = (x k ) = (k 2 , k 2 ). Then A 23) X k x k = 0, for all k E N . Then x E c g (M, k’ , ’k , A 23) , Л,р). Again we have A^ ) X k x k = — 3, for all k E N . Hence x E c g (M, k’ , •k , A 13) , Л,р). Thus the inclusion is strict.
Theorem 7. The following spaces c g ( M, k’ , ’k , A nm ) , M- c ( M, |h ’k , A, л, p ) and ' ^ Mi k k’ , ’k , A nm ) , Л,р) are not monotone and as such are not solid in general.
C The proof follows from the following example. B
Example 6. Let n = 2, m = 3, p k = 1 for all k odd and p k = 2 for all k even and M k (x) = x 2 , for all k > 1 and for all x E [0, to ). consider the 2-normed space as defined in Example 1. Then A 23) X k x k = X k x k — 2X k - 3 x k - 3 + X k 6 x k 6 - for all k E N . Consider the J th step space of a sequence space E defined as, for (x k ), (y k ) ∈ E J implies that y k = x k for k odd and y k = 0 for k even. Consider the sequences Л = (k 3 ) and x = (—). Then x E Z(M, k’ , ’k , A 23) , Л,р) for Z = ' ^ , c and c o , but its J th canonical pre-image does not belong to Z (k M, • , ’k , A 23) , Л,р) for Z = ' ^ , c and c o . Hence the spaces Z(M, k’ , ’k , A nm ) , Л,р) for Z = ' ^ , c and c o are not monotone and as such are not solid in general.
Theorem 8. The following spaces are not symmetric in general: c g (M, k’ , ’k , A nm ) , Л,р) , c ( M, k’ , ’k , A nm ) , Л,р ) , ' ^ ( M, k’ , ’k , A nm ) , Л,р ) .
C The proof follows from the following example. B
Example 7. Let n = 2, m = 2, pk = 2 for all k odd and pk = 3 for all k even and Mk(x) = x2, for all x ∈ [0, ∞) and for all k > 1. Consider the 2-normed space as defined in Example 1. Then ∆(22)λkxk = λkxk - 2λk-2xk-2 + λk-4xk-4, for all k ∈ N.
Consider the sequences Λ = (1, 1, 1, . . .) and x = (x k ) defined as x k = k for k odd and X k = 0 for k even. Then A^ X k X k = 0, for all k E N . Hence (x k ) E Z ( M, ||- , •k , Д 22) , Л,р ) , for Z = ` ∞ , c and c o . Consider the rearranged sequence, ( y k ) of ( x k ) defined as ( y k ) = ( x 1 ,x 3 ,x 2 ,x 4 ,x 5 ,x 7 ,x 6 ,x 8 ,x 9 ,x 11 ,x 10 ,x 12 , . . . ). Then ( y k ) ∈ / Z ( M, k· , ·k , ∆ ( 2 2) , Λ , p ), for Z = ` ∞ , c and c o . Hence the spaces Z M, k· , ·k , ∆ ( n m ) , Λ ,p , for Z = ` ∞ , c and c o are not symmetric in general.
Theorem 9. The following spaces are not convergence free in general:
c 0 M, k· , ·k , ∆ ( n m ) , Λ ,p , c M, k· , ·k , ∆ ( n m ) , Λ ,p , ` ∞ M, k· , ·k , ∆ ( n m ) , Λ ,p .
C The proof follows from the following example. B
Example 8. Let m = 3, n = 1, p k = 6 for all k and M k ( x ) = x 5 , for k is even and M k ( x ) = | x | , for k is odd, for all x ∈ [0 , ∞ ). Then ∆ ( 1 3) λ k x k = λ k x k - λ k - 3 x k - 3 , for all k ∈ N . Consider the 2-normed space as defined in Example 2. Let Λ = k 5 and consider the sequences ( x k ) and ( y k ) defined as x k = 4 5 k, 4 5 k for all k ∈ N and y k = 1 5 k 3 , 1 5 k 3 for all k E N . Then (x k ) E Z(M, Ц- , •k , А Ц3) ,Л,р ) , but (y k ) / Z(M, Ц- , •k , A^,Л,р ) , for Z = ' ^ , c and c o . Hence the spaces Z M, k· , ·k , ∆ n m ) , Λ ,p , for Z = ` ∞ , c and c o are not convergence free in general.
Theorem 10. The following spaces are not sequence algebra in general:
c 0 M, k· , ·k , ∆ n m ) , Λ ,p , c M, k· , ·k , ∆ n m ) , Λ ,p , ` ∞ M, k· , ·k , ∆ n m ) , Λ ,p .
C The proof follows from the following example. B
Example 9. Let n = 2, m = 1, p k = 1 for all k and M k ( x ) = x 2 , for each k ∈ N and x ∈ [0 , ∞ ). Then ∆ 2 1) λ k x k = λ k x k - 2 λ k - 1 x k - 1 + λ k - 2 x k - 2 , for all k ∈ N . Consider the 2-normed space as defined in Example 2. Consider Λ = k 1 4 and let x = ( k 5 , k 5 ) and y = (k 6 , k 6 ). Then x,y E Z(M, Ц- , •k , A 21) , Л,р ) , Z = ' ^ and c, but x,y / Z(M, Ц- , •k , A 21) , Л,р ) , for Z = c o Hence the spaces c M, k· , ·k , ∆ n m ) , Λ ,p , ` ∞ M, k· , ·k , ∆ n m ) , Λ ,p are not sequence algebra in general.
Example 10. Let n = 2, m = 1, p k = 3 for all k and M k ( x ) = x 7 , for each k ∈ N and x ∈ [0 , ∞ ). Then ∆ 2 1) λ k x k = λ k x k - 2 λ k - 1 x k - 1 + λ k - 2 x k - 2 , for all k ∈ N . Consider the 2-normed space as defined in Example 1. Consider Λ = k 1 6 and let x = ( k 7 ) and y = ( k 7 ). Then x,y E c o ( M, Ц- , •k , A 21) ,Л,р ) , but x,y / Z(M, Ц- , •k , A 21) ,Л,р ) , for Z = ' ^ , c. Hence the space c 0 M, k· , ·k , ∆ 2 1) , Λ , p is not sequence algebra in general.
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