Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions
UDC 517.9 We introduced a class of generalized vector-valued paranormed sequence space $X[E,A,\Delta_v^m,M,p]$ by using a sequence of Orlicz functions $M=(M_k),$ a non-negative infinite matrix $A=[a_{nk}],$ generalized difference operator $\Delta_v^m$ and bounded sequence of positive real numbers $p...
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Institute of Mathematics, NAS of Ukraine
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| author | Verma, A. K. Kumar, S. Verma, A. K. Kumar, S. |
| author_facet | Verma, A. K. Kumar, S. Verma, A. K. Kumar, S. |
| author_sort | Verma, A. K. |
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| description | UDC 517.9
We introduced a class of generalized vector-valued paranormed sequence space $X[E,A,\Delta_v^m,M,p]$ by using a sequence of Orlicz functions $M=(M_k),$ a non-negative infinite matrix $A=[a_{nk}],$ generalized difference operator $\Delta_v^m$ and bounded sequence of positive real numbers $p_k$ with $\inf_k p_k>0.$ Properties related to this space are studied under certain conditions. Some inclusion relations are obtained and a result related to subspace with Orlicz functions satisfying $\Delta_2$-condition has also been proved. |
| doi_str_mv | 10.37863/umzh.v74i4.6549 |
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DOI: 10.37863/umzh.v74i4.6549
UDC 517.5
A. K. Verma, S. Kumar (Dr. Harisingh Gour Univ., Sagar, India)
GENERALIZED VECTOR-VALUED PARANORMED SEQUENCE SPACES
DEFINED BY A SEQUENCE OF ORLICZ FUNCTIONS
УЗАГАЛЬНЕНI ВЕКТОРНОЗНАЧНI ПАРАНОРМОВНI ПРОСТОРОВI
ПОСЛIДОВНОСТI, ЩО ВИЗНАЧАЮТЬСЯ ПОСЛIДОВНIСТЮ
ФУНКЦIЙ ОРЛIЧА
We introduced a class of generalized vector-valued paranormed sequence space X[E,A,\Delta m
v ,M, p] by using a sequence of
Orlicz functions M = (Mk), a non-negative infinite matrix A = [ank], generalized difference operator \Delta m
v and bounded
sequence of positive real numbers pk with \mathrm{i}\mathrm{n}\mathrm{f}k pk > 0. Properties related to this space are studied under certain conditions.
Some inclusion relations are obtained and a result related to subspace with Orlicz functions satisfying \Delta 2 -condition has
also been proved.
Введено клас узагальнених векторнозначних паранормовних послiдовностей простору X[E,A,\Delta m
v ,M, p] на базi
послiдовностi функцiй Орлiча M = (Mk), невiд’ємної нескiнченної матрицi A = [ank], узагальненого рiзницевого
оператора \Delta m
v та обмеженої послiдовностi додатних дiйсних чисел pk з \mathrm{i}\mathrm{n}\mathrm{f}k pk > 0. Властивостi, пов’язанi з
цим простором, вивчаються за наявностi деяких умов. Отримано деякi спiввiдношення включення та доведено
результати, якi вiдносяться до пiдпростору з функцiями Орлiча, що задовольняють \Delta 2 -умову.
1. Introduction. The theory of sequence spaces has been one of the most active area of research in
functional analysis. Generalization of \ell p, p \geq 1, c0 and c has been studied by many authors with
the help of difference operator, modulus function and Orlicz functions in the last five decades.
Kizmaz [9] introduced the notion of difference operator \Delta and studied difference sequence
spaces \ell \infty (\Delta ), c(\Delta ) and c0(\Delta ). Et and Çolak [4] generalized the operator by introducing the spaces
\ell \infty (\Delta m), c(\Delta m) and c0(\Delta
m) for non-negative integer m. Further, Et and Esi [5] generalized these
spaces by taking the sequence v = (vk) of non-zero complex numbers which are defined as follows:
X(\Delta m
v ) =
\bigl\{
x = (xk) \in w : \Delta m
v x \in X
\bigr\}
for X = \ell \infty , c \mathrm{a}\mathrm{n}\mathrm{d} c0,
where w is the space of all complex sequences, \Delta 0
vx = (vkxk) and
\Delta m
v xk =
m\sum
i=0
( - 1)i
\Biggl(
m
i
\Biggr)
vk+ixk+i for m \in \BbbN .
In 1971, Lindenstrauss and Tzafriri [11] used the idea of an Orlicz function M to construct the
sequence space lM as follows:
lM =
\Biggl\{
x \in \omega :
\infty \sum
k=1
M
\biggl(
| xk|
\rho
\biggr)
< \infty for some \rho > 0
\Biggr\}
.
They proved that lM is Banach space under the following norm:
\| x\| = \mathrm{i}\mathrm{n}\mathrm{f}
\Biggl\{
\rho > 0 :
\infty \sum
k=1
M
\biggl(
| xk|
\rho
\biggr)
\leq 1
\Biggr\}
.
c\bigcirc A. K. VERMA, S. KUMAR, 2022
486 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
GENERALIZED VECTOR-VALUED PARANORMED SEQUENCE SPACES . . . 487
In 1994, Parashar and Choudhary [16] generalized the space lM to lM (p) by using bounded sequence
of real numbers (pk) as follows:
lM (p) =
\Biggl\{
x \in \omega :
\infty \sum
k=1
\biggl[
M
\biggl(
| xk|
\rho
\biggr) \biggr] pk
< \infty for some \rho > 0
\Biggr\}
,
W0(M,p) =
\Biggl\{
x \in \omega : \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
n
n\sum
k=1
\biggl[
M
\biggl(
xk
\rho
\biggr) \biggr] pk
= 0 for some \rho > 0
\Biggr\}
and
W\infty (M,p) =
\Biggl\{
x \in \omega : \mathrm{s}\mathrm{u}\mathrm{p}
n
1
n
n\sum
k=1
\biggl[
M
\biggl(
xk
\rho
\biggr) \biggr] pk
< \infty for some \rho > 0
\Biggr\}
.
For M(x) = x, above sequence spaces become \ell (p), [C, 1, p]0 and [C, 1, p]\infty , respectively, studied
by Maddox [13].
Mursaleen et al. [15] introduced sequence spaces c0(M,\Delta , p) and \ell \infty (M,\Delta , p) as follows:
c0(M,\Delta , p) =
\biggl\{
x \in \omega : \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\biggl[
M
\biggl(
| \Delta xk|
\rho
\biggr) \biggr] pk
= 0 for some \rho > 0
\biggr\}
,
\ell \infty (M,\Delta , p) =
\biggl\{
x \in \omega : \mathrm{s}\mathrm{u}\mathrm{p}
k
\biggl[
M
\biggl(
| \Delta xk|
\rho
\biggr) \biggr] pk
< \infty for some \rho > 0
\biggr\}
.
In 2005, Tripathy and Sarma [22] introduced spaces c0(M,\Delta , p, q) and \ell \infty (M,\Delta , p, q) in semi-
normed space (E, q) as follows:
c0(M,\Delta , p, q) =
\biggl\{
x \in \omega (E) : \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow \infty
\biggl(
1
pk
\biggr) \biggl[
M
\biggl(
q(\Delta xk)
\rho
\biggr) \biggr] pk
= 0 for some \rho > 0
\biggr\}
,
\ell \infty (M,\Delta , p, q) =
\biggl\{
x \in \omega (E) : \mathrm{s}\mathrm{u}\mathrm{p}
k
\biggl(
1
pk
\biggr) \biggl[
M
\biggl(
q(\Delta xk)
\rho
\biggr) \biggr] pk
< \infty for some \rho > 0
\biggr\}
.
By using sequence of Orlicz functions, Bektaş [3] constructed space lM (\Delta m
v , p, q, s) as follows:
lM (\Delta m
v , p, q, s) =
\Biggl\{
x \in \omega (E) :
\infty \sum
k=1
k - s
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
< \infty for some \rho > 0, s \geq 0
\Biggr\}
.
If Mk = M and vk = 1 for all k, then above space becomes lM (\Delta m, p, q, s) which is discussed by
Tripathy et al. [17]. Further, for m = 0, the space lM (\Delta m, p, q, s) reduces to lM (p, q, s), which is
studied by Bektaş and Altin [2].
Esi [7] used non-negative regular matrix to introduce spaces W0(A,M, p) and W\infty (A,M, p) as
follows:
W0(A,M, p) =
\Biggl\{
x \in \omega : \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\sum
k
ank
\biggl[
M
\biggl(
xk
\rho
\biggr) \biggr] pk
= 0 for some \rho > 0
\Biggr\}
,
W\infty (A,M, p) =
\Biggl\{
x \in \omega : \mathrm{s}\mathrm{u}\mathrm{p}
n
\sum
k
ank
\biggl[
M
\biggl(
xk
\rho
\biggr) \biggr] pk
< \infty for some \rho > 0
\Biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
488 A. K. VERMA, S. KUMAR
If M(x) = x, then above spaces reduce to [A, p]0 and [A, p]\infty , studied by Maddox [14]. Orlicz
function and generalized difference operator were frequently used to introduce scalar and vector-
valued sequence spaces by the researchers in [1, 6, 18 – 21] and many others.
Above development motivated us to introduce a class of vector-valued sequence spaces
X[E,A,\Delta m
v ,M, p] by using non-negative matrix A = [ank], generalized difference operator \Delta m
v
and a sequence of Orlicz functions M = (Mk) which generalizes many known scalar and vector-
valued sequence spaces.
1.1. A new sequence space \bfitX [\bfitE ,\bfitA ,\bfDelta \bfitm
\bfitv ,\bfitM , \bfitp ]. Let M = (Mk) be a sequence of Orlicz
functions, v = (vk) be any fixed sequence of non-zero complex numbers, A = [ank] be a non-
negative infinite matrix, i.e., ank \geq 0 for all n, k \in \BbbN and (pk) be a bounded sequence of positive
real numbers such that \mathrm{i}\mathrm{n}\mathrm{f}k pk > 0. Further, let (E, q) be a seminormed space and X be a normal
(or solid) sequence space. We define
X
\bigl[
E,A,\Delta m
v ,M, p
\bigr]
=
=
\Biggl\{
x = (xk) \in W (E) :
\Biggl( \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk\Biggr)
\in X for some \rho > 0
\Biggr\}
,
where \Delta 0
vxk = vkxk and
\Delta m
v xk =
m\sum
i=0
( - 1)i
\Biggl(
m
i
\Biggr)
vk+ixk+i for m \in \BbbN .
Class of vector-valued sequences
\bigl[
X[E,A,\Delta m
v ,M, p]
\bigr]
is also defined by\bigl[
X[E,A,\Delta m
v ,M, p]
\bigr]
=
=
\Biggl\{
x = (xk) \in W (E) :
\Biggl( \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk\Biggr)
\in X for every \rho > 0
\Biggr\}
.
Clearly,
\bigl[
X[E,A,\Delta m
v ,M, p]
\bigr]
is a subspace of X[E,A,\Delta m
v ,M, p].
1.2. Particular cases:
(i) If we choose X = \ell \infty , E = \BbbC ,m = 0, ank = 1 for n \geq k and 0 otherwise, pk = 1 for all
k, Mk = M for all k and vk = 1 for all k, the space X[E,A,\Delta m
v ,M, p] reduces to lM [11].
(ii) If we choose X = c0 and \ell \infty , E = \BbbC ,m = 0, ank =
1
k
for n \geq k and 0 otherwise,
Mk = M for all k and vk = 1 for all k, the space X[E,A,\Delta m
v ,M, p] becomes W0(M,p) and
W\infty (M,p), respectively [16].
(iii) If we choose X = c0 and \ell \infty , E = \BbbC , m = 0, Mk = M for all k and vk = 1 for all k,
the space X[E,A,\Delta m
v ,M, p] reduces to W0(A,M, p) and W\infty (A,M, p), respectively [7].
(iv) If we choose X = c0 and \ell \infty , E = \BbbC ,m = 1, A = [ank] such that ank = 1 for n = k
and 0 otherwise, Mk = M for all k and vk = 1 for all k, the space reduces to c0(M,\Delta , p) and
\ell \infty (M,\Delta , p), respectively [15].
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
GENERALIZED VECTOR-VALUED PARANORMED SEQUENCE SPACES . . . 489
(v) If we choose X = c0 and \ell \infty , m = 1, ank =
1
pk
for n = k and 0 otherwise, Mk = M for all
k and vk = 1 for all k, the space X[E,A,\Delta m
v ,M, p] reduces to c0(M,\Delta , p, q) and \ell \infty (M,\Delta , p, q),
respectively [22].
(vi) If we choose X = \ell \infty , m = 0, ank = k - s for all n, Mk = M for all k and vk = 1 for all
k, the space X[E,A,\Delta m
v ,M, p] reduces to lM (p, q, s) [2].
(vii) If we choose X = \ell \infty and ank = k - s for all n, the space X[E,A,\Delta m
v ,M, p] reduces to
lM (\Delta m
v , p, q, s) [3].
2. Some definitions and known results.
Result 1 [12]. For ak and bk in \BbbC , the following inequalities hold:
| ak + bk| pk \leq T
\bigl\{
| ak| pk + | bk| pk
\bigr\}
, (2.1)
| \lambda | pk \leq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl(
1, | \lambda | H
\bigr)
, (2.2)
where (pk) is a bounded sequence of real numbers with 0 < pk \leq \mathrm{s}\mathrm{u}\mathrm{p}k pk = H, T = \mathrm{m}\mathrm{a}\mathrm{x}(1, 2H - 1)
and \lambda in \BbbC .
Definition 1 [8]. A sequence space X is called normal (or solid) space if
x = (xk) \in X and | \lambda k| \leq 1 for each k \in \BbbN \Rightarrow \lambda x = (\lambda kxk) \in X,
where \lambda = (\lambda k) is a scalar sequence of real or complex numbers.
Definition 2 [11]. An Orlicz function is a function M : [0,\infty ) \rightarrow [0,\infty ), which is continuous,
non-decreasing and convex with M(0) = 0,M(x) > 0 for x > 0 and M(x) \rightarrow \infty as x \rightarrow \infty .
Remark 1 [10]. An Orlicz function M is said to satisfy the \Delta 2-condition for all values of u if
there exists a constant K > 0 such that M(2u) \leq KM(u), u \geq 0.
The \Delta 2-condition is equivalent to the inequality M(Lu) \leq KLM(u) which holds for all values
of u and for L > 1.
3. Results on sequence space \bfitX [\bfitE ,\bfitA ,\bfDelta \bfitm
\bfitv ,\bfitM , \bfitp ].
Theorem 1. X[E,A,\Delta m
v ,M, p] is a linear space over \BbbC .
Proof. Let x, y \in X[E,A,\Delta m
v ,M, p] and \alpha , \beta \in \BbbC . Then there exist some positive numbers \rho 1
and \rho 2 such that\Biggl( \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho 1
\biggr) \biggr] pk\Biggr)
\in X and
\Biggl( \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v yk)
\rho 2
\biggr) \biggr] pk\Biggr)
\in X.
Let \rho 3 = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl(
2| \alpha | \rho 1, 2| \beta | \rho 2
\bigr)
. By using the subadditive property of seminorm q, non-decreasing
and convexity of Orlicz functions, for each n, we have
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v (\alpha xk + \beta yk))
\rho 3
\biggr) \biggr] pk
\leq T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho 1
\biggr) \biggr] pk
+
+T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v yk)
\rho 2
\biggr) \biggr] pk
, by using (2.1).
Since X is a normal space, so
\biggl( \sum \infty
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v (\alpha xk + \beta yk))
\rho 3
\biggr) \biggr] pk\biggr)
\in X. Thus,
X[E,A,\Delta m
v ,M, p] is a linear space.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
490 A. K. VERMA, S. KUMAR
Theorem 2. The sequence space X[E,A,\Delta m
v ,M, p] is a paranormed space under paranorm g
defined by
g(x) =
m\sum
k=1
q(xk) + \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \rho
pn
H : \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk\Biggr] 1
H
\leq 1, n \in \BbbN
\right\} ,
where H = \mathrm{m}\mathrm{a}\mathrm{x}(1, \mathrm{s}\mathrm{u}\mathrm{p}k pk).
Proof. As q(\theta ) = 0 and Mk(0) = 0 for all k \in \BbbN , so \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
\rho
pn
H
\bigr\}
= 0 which implies that
g(\theta ) = 0 for x = \theta . Clearly, g(x) \geq 0 and g( - x) = g(x) for any x \in X[E,A,\Delta m
v ,M, p]. To show
that g(x + y) \leq g(x) + g(y), let x, y \in X[E,A,\Delta m
v ,M, p]. Then there exist \rho 1 > 0, \rho 2 > 0 such
that
\mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho 1
\biggr) \biggr] pk\Biggr] 1
H
\leq 1 and \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho 2
\biggr) \biggr] pk\Biggr] 1
H
\leq 1.
Let \rho = \rho 1 + \rho 2. Then by using convexity of Orlicz function and Minkowski’s inequality, we have
\mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v (xk + yk))
\rho
\biggr) \biggr] pk\Biggr] 1
H
\leq \rho 1
\rho
\mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho 1
\biggr) \biggr] pk\Biggr] 1
H
+
+
\rho 2
\rho
\mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho 2
\biggr) \biggr] pk\Biggr] 1
H
\leq 1.
Now,
g(x+ y) =
=
m\sum
k=1
q(xk + yk) + \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \rho
pn
H : \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\Biggl[
Mk
\Biggl(
q
\bigl(
\Delta m
v (xk + yk)
\bigr)
\rho
\Biggr) \Biggr] pk\Biggr] 1
H
\leq 1, n \in \BbbN
\right\} \leq
\leq
m\sum
k=1
q(xk) +
m\sum
k=1
q(yk) + \mathrm{i}\mathrm{n}\mathrm{f}
\Biggl\{
(\rho 1 + \rho 2)
pn
H : \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho 1
\biggr) \biggr] pk\Biggr] 1
H
\leq 1,
\mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v yk)
\rho 2
\biggr) \biggr] pk\Biggr] 1
H
\leq 1
\Biggr\}
\leq
\leq g(x) + g(y).
To prove continuity of scalar multiplication, let \lambda is fixed number in \BbbC . Then
g(\lambda x) =
m\sum
k=1
q(\lambda xk) + \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \rho
pn
H : \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\Biggl[
Mk
\Biggl(
q
\bigl(
\Delta m
v (\lambda xk)
\bigr)
\rho
\Biggr) \Biggr] pk\Biggr] 1
H
\leq 1, n \in \BbbN
\right\} =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
GENERALIZED VECTOR-VALUED PARANORMED SEQUENCE SPACES . . . 491
= | \lambda |
m\sum
k=1
q(xk) + \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ (r| \lambda | )
pn
H : \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl[ \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
r
\biggr) \biggr] pk\Biggr] 1
H
\leq 1, n \in \BbbN
\right\} \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}(1, | \lambda | )g(x),
where r =
\rho
| \lambda |
. Thus g(\lambda x) \rightarrow 0 as x \rightarrow 0.
Now, we will prove that g(\lambda ix) \rightarrow 0 as \lambda i \rightarrow 0 for a fixed x. As \lambda i \rightarrow 0, there exists a positive
integer m0 such that | \lambda i| < 1 for all i \geq m0. By non-decreasing property of Orlicz function, for all
i \geq m0, we have
\infty \sum
k=1
ank
\Biggl[
Mk
\Biggl(
q
\bigl(
\Delta m
v (\lambda ixk)
\bigr)
\rho
\Biggr) \Biggr] pk
\leq
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
< \infty ,
which implies that for every \varepsilon > 0, there exists a positive integer k0 such that
\infty \sum
k=k0
ank
\Biggl[
Mk
\Biggl(
q
\bigl(
\Delta m
v (\lambda ixk)
\bigr)
\rho
\Biggr) \Biggr] pk
<
\varepsilon
2
. (3.1)
Now, we define a function f by
f(t) =
k0\sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v (txk))
\rho
\biggr) \biggr] pk
.
Clearly, f(t) is continuous at 0 and f(0) = 0. This implies that for any \varepsilon > 0, there exists a \delta > 0
such that
\bigm| \bigm| f(t)\bigm| \bigm| < \varepsilon
2
whenever | t| < \delta . Since \lambda i \rightarrow 0, so there exists positive integer m1 such that
| \lambda i| < \delta for all i \geq m1. Which gives us
\bigm| \bigm| f(\lambda i)
\bigm| \bigm| < \varepsilon
2
for i \geq m1, i.e.,
k0\sum
k=1
ank
\Biggl[
Mk
\Biggl(
q
\bigl(
\Delta m
v (\lambda ixk)
\bigr)
\rho
\Biggr) \Biggr] pk
<
\varepsilon
2
. (3.2)
By inequalities (3.1) and (3.2), for i \geq m1, we have
\infty \sum
k=1
ank
\Biggl[
Mk
\Biggl(
q
\bigl(
\Delta m
v (\lambda ixk)
\bigr)
\rho
\Biggr) \Biggr] pk
< \varepsilon .
Using above inequality, we can obtain g(\lambda ix) \rightarrow 0 as \lambda i \rightarrow 0.
Theorem 2 is proved.
Remark 2. Sequence space X[E,A,\Delta m
v ,M, p] is not a total paranormed space because g(x) = 0
need not imply x = \theta due to seminorm q.
Theorem 3. Let M = (Mk) and T = (Tk) be any two sequences of Orlicz functions. If each Tk
satisfies \Delta 2-condition, then X[E,A,\Delta m
v ,M, p] \subseteq X[E,A,\Delta m
v , T \circ M,p], where T \circ M = (Tk\circ Mk).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
492 A. K. VERMA, S. KUMAR
Proof. Let x \in X[E,A,\Delta m
v ,M, p], i.e.,
\biggl( \sum \infty
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk\biggr)
\in X.
Case (i) : If Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr)
\leq 1, then by convexity of Orlicz functions, for each n \in \BbbN ,
\infty \sum
k=1
ank
\biggl[
(Tk \circ Mk)
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
\leq
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr)
Tk(1)
\biggr] pk
\leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}
\bigl(
1, [T (1)]H
\bigr) \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
,
where T (1) = \mathrm{s}\mathrm{u}\mathrm{p}k Tk(1).
Case (ii): If Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr)
> 1. Then by \Delta 2-condition of Orlicz function, for each n \in \BbbN ,
we have
\infty \sum
k=1
ank
\biggl[
(Tk \circ Mk)
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
\leq
\infty \sum
k=1
ank
\biggl[
KMk
\biggl(
q(\Delta m
v xk)
\rho
\biggr)
Tk(1)
\biggr] pk
\leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}(1, [KT (1)]H)
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
, where K > 0.
As X is a normal space, so x \in X[E,A,\Delta m
v , T \circ M,p] in both cases. Hence, required inclusion
follows.
Theorem 4. Let M = (Mk), T = (Tk) be any two sequences of Orlicz functions. Then
(i) X[E,A,\Delta m
v ,M, p] \cap X[E,A,\Delta m
v , T, p] \subseteq X[E,A,\Delta m
v ,M + T, p]
and
(ii) X[E,A,\Delta m
v , T, p] \subseteq X[E,A,\Delta m
v ,M, p], if \mathrm{s}\mathrm{u}\mathrm{p}u
\biggl[
Mk(u)
Tk(u)
\biggr]
< \infty for each k \in \BbbN .
Proof. (i) Let x \in X[E,A,\Delta m
v ,M, p] \cap X[E,A,\Delta m
v , T, p]. By using inequality (2.1), for each
n, we obtain
\infty \sum
k=1
ank
\biggl[
(Mk + Tk)
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
\leq
\leq T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
+ T
\infty \sum
k=1
ank
\biggl[
Tk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
.
Since X is a normal space, so x \in X[E,A,\Delta m
v ,M + T, p]. Thus, we get the required result.
(ii) Let x \in X[E,A,\Delta m
v , T, p]. Then
\biggl( \sum \infty
k=1
ank
\biggl[
Tk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk\biggr)
\in X. Since
\mathrm{s}\mathrm{u}\mathrm{p}
u
\biggl[
Mk(u)
Tk(u)
\biggr]
< \infty for each k \in \BbbN , so there exists \eta > 0 such that Mk(u) \leq \eta Tk(u) for each
k \in \BbbN and for all u > 0. Now,
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
\leq
\infty \sum
k=1
ank
\biggl[
\eta Tk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
\leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
GENERALIZED VECTOR-VALUED PARANORMED SEQUENCE SPACES . . . 493
\leq \mathrm{m}\mathrm{a}\mathrm{x}(1, \eta H)
\infty \sum
k=1
ank
\biggl[
Tk
\biggl(
q(\Delta m
v xk)
\rho
\biggr) \biggr] pk
, by using (2.2).
Since X is a normal space, so x \in X[E,A,\Delta m
v ,M, p] and thus inclusion follows.
Theorem 5. Let X1 and X2 be two normal sequence spaces with X1 \subseteq X2. Then
X1[E,A,\Delta m
v ,M, p] \subseteq X2[E,A,\Delta m
v ,M, p].
Proof. Inclusion follows by the definition of X[E,A,\Delta m
v ,M, p].
Theorem 6. Let A = [ank] be non-negative infinite matrix such that ank \leq an(k+1) for all
n, k \in \BbbN and m \geq 1. Suppose (Mk) is non-decreasing sequence of Orlicz functions, i.e., Mk(x) \leq
\leq Mk+1(x) for all x \geq 0. Then
X[E,A,\Delta l
v,M, p] \subset X[E,A,\Delta l+1
v ,M, p] for any l \in \{ 1, 2, . . . ,m - 1\} .
Proof. Let x \in X[E,A,\Delta l
v,M, p]. Then\Biggl( \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk)
\rho
\biggr) \biggr] pk\Biggr)
\in X for some \rho > 0.
Since seminorm q is subadditive and each Mk is non-decreasing convex function, so we have
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l+1
v xk)
2\rho
\biggr) \biggr] pk
=
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk - \Delta l
vxk+1)
2\rho
\biggr) \biggr] pk
\leq
\leq T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk)
\rho
\biggr) \biggr] pk
+ T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk+1)
\rho
\biggr) \biggr] pk
\leq
(by using inequality (2.1))
\leq T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk)
\rho
\biggr) \biggr] pk
+ T
\infty \sum
k=1
an(k+1)
\biggl[
Mk+1
\biggl(
q(\Delta l
vxk+1)
\rho
\biggr) \biggr] pk
.
As X is a normal space, so
\biggl( \sum \infty
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l+1
v xk)
\rho
\biggr) \biggr] pk\biggr)
\in X, i.e., x \in X[E,A,\Delta l+1
v ,M, p].
Consequently, X[E,A,\Delta l
v,M, p] \subseteq X[E,A,\Delta l+1
v ,M, p].
Now, for strictness of inclusion, let us consider the following example.
Let E = \BbbC , A = [ank] such that ank = 1 for n = k, and 0 otherwise, pk = 1 for all k,
Mk(x) = x for all k, vk =
1
k
for any k and xk = kl+1 for any k. Then \Delta l+1
v xk = (0, 0, . . .), which
means x \in c0[E,A,\Delta l+1
v ,M, p]. But \Delta l
vxk = ( - 1)ll!, which implies that x /\in c0[E,A,\Delta l
v,M, p].
Theorem 7. The sequence space X[E,A,\Delta m
v ,M, p] is a normal space if m = 0.
Proof. Let x \in X[E,A,\Delta 0
v,M, p], i.e.,
\Biggl( \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta 0
vxk)
\rho
\biggr) \biggr] pk\Biggr)
\in X. Again, let (\lambda k)
be a sequence of scalars such that | \lambda k| \leq 1 for all k \in \BbbN . Then by non-decreasing property of Orlicz
function, we have
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta 0
v(\lambda kxk))
\rho
\biggr) \biggr] pk
\leq
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta 0
vxk)
\rho
\biggr) \biggr] pk
for all n \in \BbbN .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
494 A. K. VERMA, S. KUMAR
As X is a normal space, so
\biggl( \sum \infty
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta 0
v(\lambda kxk))
\rho
\biggr) \biggr] pk\biggr)
\in X and result follows.
Theorem 8. Let X1 and X2 be two normal sequence spaces with X1 \subseteq X2. Then
X1[E,A,\Delta m
v ,M, p] \subseteq X2[E,A,\Delta m
v ,M, p].
Proof. Inclusion follows by the definition of X[E,A,\Delta m
v ,M, p].
Theorem 9. Let A = [ank] be non-negative infinite matrix such that ank \leq an(k+1) for all
n, k \in \BbbN and m \geq 1. Suppose (Mk) is non-decreasing sequence of Orlicz functions, i.e., Mk(x) \leq
\leq Mk+1(x) for all x \geq 0. Then
X[E,A,\Delta l
v,M, p] \subset X[E,A,\Delta l+1
v ,M, p] for any l \in \{ 1, 2, . . . ,m - 1\} .
Proof. Let x \in X[E,A,\Delta l
v,M, p]. Then\Biggl( \infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk)
\rho
\biggr) \biggr] pk\Biggr)
\in X for some \rho > 0.
Since seminorm q is subadditive and each Mk is non-decreasing convex function, so we have
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l+1
v xk)
2\rho
\biggr) \biggr] pk
=
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk - \Delta l
vxk+1)
2\rho
\biggr) \biggr] pk
\leq
\leq T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk)
\rho
\biggr) \biggr] pk
+ T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk+1)
\rho
\biggr) \biggr] pk
\Rightarrow
(by using inequality (2.1))
\Rightarrow
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l+1
v xk)
2\rho
\biggr) \biggr] pk
\leq T
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l
vxk)
\rho
\biggr) \biggr] pk
+
+T
\infty \sum
k=1
an(k+1)
\biggl[
Mk+1
\biggl(
q(\Delta l
vxk+1)
\rho
\biggr) \biggr] pk
.
As X is a normal space, so
\biggl( \sum \infty
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta l+1
v xk)
\rho
\biggr) \biggr] pk\biggr)
\in X, i.e., x \in X[E,A,\Delta l+1
v ,
M, p]. Consequently, X[E,A,\Delta l
v,M, p] \subseteq X[E,A,\Delta l+1
v ,M, p].
Now, for strictness of inclusion, let us consider the following example.
Let E = \BbbC , A = [ank] such that ank = 1 for n = k, and 0 otherwise, pk = 1 for all k,
Mk(x) = x for all k, vk =
1
k
for any k and xk = kl+1 for any k. Then \Delta l+1
v xk = (0, 0, . . .), which
means x \in c0[E,A,\Delta l+1
v ,M, p]. But \Delta l
vxk = ( - 1)ll!, which implies that x /\in c0[E,A,\Delta l
v,M, p].
Theorem 10. The sequence space X[E,A,\Delta m
v ,M, p] is a normal space if m = 0.
Proof. Let x \in X[E,A,\Delta 0
v,M, p], i.e.,
\biggl( \sum \infty
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta 0
vxk)
\rho
\biggr) \biggr] pk\biggr)
\in X. Again, let
(\lambda k) be a sequence of scalars such that | \lambda k| \leq 1 for all k \in \BbbN . Then by non-decreasing property of
Orlicz function, we have
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta 0
v(\lambda kxk))
\rho
\biggr) \biggr] pk
\leq
\infty \sum
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta 0
vxk)
\rho
\biggr) \biggr] pk
for all n \in \BbbN .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
GENERALIZED VECTOR-VALUED PARANORMED SEQUENCE SPACES . . . 495
As X is a normal space, so
\biggl( \sum \infty
k=1
ank
\biggl[
Mk
\biggl(
q(\Delta 0
v(\lambda kxk))
\rho
\biggr) \biggr] pk\biggr)
\in X and result follows.
Acknowledgements. Author would like to thank funding agency University Grant Commission
(UGC) of the Government of India for providing financial support during the research work, in the
form of CSIR-UGC NET-JRF.
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ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 4
|
| id | umjimathkievua-article-6549 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:28:43Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8d/a3e5ff001a955ef5775ae76d9807a08d.pdf |
| spelling | umjimathkievua-article-65492022-07-06T16:22:31Z Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions Verma, A. K. Kumar, S. Verma, A. K. Kumar, S. функцiї Oрлiча Orlicz function Paranormed space Difference sequence space Normal space UDC 517.9 We introduced a class of generalized vector-valued paranormed sequence space $X[E,A,\Delta_v^m,M,p]$ by using a sequence of Orlicz functions $M=(M_k),$ a non-negative infinite matrix $A=[a_{nk}],$ generalized difference operator $\Delta_v^m$ and bounded sequence of positive real numbers $p_k$ with $\inf_k p_k&gt;0.$&nbsp;Properties related to this space are studied under certain conditions.&nbsp;Some inclusion relations are obtained and a result related to subspace with Orlicz functions satisfying $\Delta_2$-condition has also been proved. УДК 517.9Узагальненi векторнозначнi паранормовнi просторовi послiдовностi, що визначаються послiдовнiстю функцiй Oрлiча Введено клас узагальнених векторнозначних паранормовних послiдовностей простору $X[E,A,\Delta_v^m,M,p]$ на базi послiдовностi функцiй Орлiча $M=(M_k)$, невiд’ємної нескiнченної матрицi $A=[a_{nk}]$, узагальненого рiзницевого оператора $\Delta_v^m$ та обмеженої послiдовностi додатних дiйсних чисел $p_k$ з $\inf_k p_k &gt; 0$. Властивостi, пов’язанi з цим простором, вивчаються за наявностi деяких умов. Отримано деякi спiввiдношення включення та доведено результати, якi вiдносяться до пiдпростору з функцiями Орлiча, що задовольняють $\Delta_2$-умову. Institute of Mathematics, NAS of Ukraine 2022-05-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6549 10.37863/umzh.v74i4.6549 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 4 (2022); 486 - 495 Український математичний журнал; Том 74 № 4 (2022); 486 - 495 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6549/9218 Copyright (c) 2022 SUDHANSHU KUMAR |
| spellingShingle | Verma, A. K. Kumar, S. Verma, A. K. Kumar, S. Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions |
| title | Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions |
| title_alt | Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions |
| title_full | Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions |
| title_fullStr | Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions |
| title_full_unstemmed | Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions |
| title_short | Generalized vector-valued paranormed sequence spaces defined by a sequence of Orlicz functions |
| title_sort | generalized vector-valued paranormed sequence spaces defined by a sequence of orlicz functions |
| topic_facet | функцiї Oрлiча Orlicz function Paranormed space Difference sequence space Normal space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6549 |
| work_keys_str_mv | AT vermaak generalizedvectorvaluedparanormedsequencespacesdefinedbyasequenceoforliczfunctions AT kumars generalizedvectorvaluedparanormedsequencespacesdefinedbyasequenceoforliczfunctions AT vermaak generalizedvectorvaluedparanormedsequencespacesdefinedbyasequenceoforliczfunctions AT kumars generalizedvectorvaluedparanormedsequencespacesdefinedbyasequenceoforliczfunctions |