Application of the infinite matrix theory to the solvability of sequence spaces inclusion equations with operators
UDC 517.9 Given any sequence $a = (a_n)_n \geq 1$ of positive real numbers and any set $E$ of complex sequences, we write $E_a$ for the set of all sequences $y=(y_{n})_{n \geq 1}$ such that y/a = $y/a=(y_{n}/a_{n})_{n\geq 1}\in E.$ In particular, $c_{a}$ denotes the set of all sequences $y$ such...
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Institute of Mathematics, NAS of Ukraine
2019
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Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507286530686976 |
|---|---|
| author | de, Malafosse B. де, Малафоссе Б. |
| author_facet | de, Malafosse B. де, Малафоссе Б. |
| author_sort | de, Malafosse B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:57:29Z |
| description | UDC 517.9
Given any sequence $a = (a_n)_n \geq 1$ of positive real numbers and any set $E$ of complex sequences, we write $E_a$ for the set of all sequences $y=(y_{n})_{n \geq 1}$ such that y/a = $y/a=(y_{n}/a_{n})_{n\geq 1}\in E.$
In particular, $c_{a}$ denotes the set of all sequences $y$ such that $y/a$ converges.
We deal with sequence spaces inclusion equations (SSIE) of the form $F\subset E_{a}+F_{x}'$ with $e\in F$
and explicitly find the solutions of these SSIE when $a=(r^{n})_{n\geq 1},$ $F$ is either $c$ or $s_{1},$ and $E,$ $F'$ are any sets $c_{0},$ $c,$ $s_{1},$ $\ell_{p},$ $w_{0},$ and $w_{\infty }.$ Then we determine the sets of all positive sequences satisfying each of the SSIE $c\subset D_{r}\ast (c_{0})_{\Delta }+c_{x}$ and $c\subset D_{r}\ast (s_{1})_{\Delta}+c_{x},$ where $\Delta $ is the operator of the first difference defined by $\Delta _{n}y=y_{n}-y_{n-1}$ for all $n\geq 1$ with $y_{0}=0.$
Then we solve the SSIE $c\subset D_{r}\ast E_{C_{1}}+s_{x}^{(c)}$ with $E\in \{ c,s_{1}\} $
and $s_{1}\subset D_{r}\ast(s_{1}) _{C_{1}}+s_{x},$ where $C_{1}$, is the Cesaro operator defined by $(C_{1}) _{n}y=n^{-1}
\displaystyle \sum \nolimits_{k=1}^{n}y_{k}$ for all $y$.
We also deal with the solvability of the sequence
spaces equations (SSE) associated with the previous SSIE and defined as $D_{r}\ast E_{C_{1}}+s_{x}^{(c)}=c$ with $E\in \{c_{0},c, s_{1}\} $ and $D_{r}\ast E_{C_{1}}+s_{x}=s_{1}$ with $E\in \{ c,s_{1}\}.$ |
| first_indexed | 2026-03-24T02:06:54Z |
| format | Article |
| fulltext |
UDC 517.9
B. de Malafosse (Univ. Le Havre, France)
APPLICATION OF THE INFINITE MATRIX THEORY
TO THE SOLVABILITY OF SEQUENCE
SPACES INCLUSION EQUATIONS WITH OPERATORS
ЗАСТОСУВАННЯ ТЕОРIЇ НЕСКIНЧЕННИХ МАТРИЦЬ
ДО РОЗВ’ЯЗАННЯ ВIДНОШЕНЬ ВКЛЮЧЕННЯ
ДЛЯ ПРОСТОРIВ ПОСЛIДОВНОСТЕЙ З ОПЕРАТОРАМИ
Given any sequence a = (an)n\geq 1 of positive real numbers and any set E of complex sequences, we write Ea for the
set of all sequences y = (yn)n\geq 1 such that y/a = (yn/an)n\geq 1 \in E. In particular, ca denotes the set of all sequences y
such that y/a converges. We deal with sequence spaces inclusion equations (SSIE) of the form F \subset Ea +F \prime
x with e \in F
and explicitly find the solutions of these SSIE when a = (rn)n\geq 1, F is either c or s1, and E, F \prime are any sets c0, c, s1,
\ell p, w0, and w\infty . Then we determine the sets of all positive sequences satisfying each of the SSIE c \subset Dr \ast (c0)\Delta + cx
and c \subset Dr \ast (s1)\Delta + cx, where \Delta is the operator of the first difference defined by \Delta ny = yn - yn - 1 for all n \geq 1
with y0 = 0. Then we solve the SSIE c \subset Dr \ast EC1 + s
(c)
x with E \in \{ c, s1\} and s1 \subset Dr \ast (s1)C1 + sx, where C1
is the Cesàro operator defined by (C1)ny = n - 1
\sum n
k=1
yk for all y. We also deal with the solvability of the sequence
spaces equations (SSE) associated with the previous SSIE and defined as Dr \ast EC1 + s
(c)
x = c with E \in \{ c0, c, s1\} and
Dr \ast EC1 + sx = s1 with E \in \{ c, s1\} .
Для заданої послiдовностi додатних дiйсних чисел a = (an)n\geq 1 i будь-якої множини комплексних послiдовностей
E вираз Ea позначає множину всiх послiдовностей y = (yn)n\geq 1 таких, що y/a = (yn/an)n\geq 1 \in E. Зокрема, ca
позначає множину всiх послiдовностей y таких, що y/a збiгається. Розглянуто вiдношення включення для просторiв
послiдовностей (ВВПП) вигляду F \subset Ea + F \prime
x з e \in F, а також знайдено явнi розв’язки цих ВВПП у випадку,
коли a = (rn)n\geq 1, F — це c або s1, а E i F \prime — будь-якi з множин c0, c, s1, \ell p, w0 i w\infty . Крiм того, визначено
множини всiх додатних послiдовностей, що задовольняють кожне з ВВПП c \subset Dr \ast (c0)\Delta +cx i c \subset Dr \ast (s1)\Delta +cx,
де \Delta — оператор першої рiзницi, визначений як \Delta ny = yn - yn - 1 для всiх n \geq 1 з y0 = 0. Також розв’язано
ВВПП c \subset Dr \ast EC1 + s
(c)
x , де E \in \{ c, s1\} , s1 \subset Dr \ast (s1)C1 + sx, а C1 — оператор Чезаро, визначений як
сума (C1)ny = n - 1
\sum n
k=1
yk для всiх y. Крiм того, розглянуто питання про iснування розв’язкiв рiвнянь для
просторiв послiдовностей (РПП), пов’язаних iз попереднiми ВВПП i визначених таким чином: Dr \ast EC1 + s
(c)
x = c
з E \in \{ c0, c, s1\} i Dr \ast EC1 + sx = s1 з E \in \{ c, s1\} .
1. Introduction. We write \omega for the set of all complex sequences y = (yn)n\geq 1, \ell \infty , c and
c0 for the sets of all bounded, convergent and null sequences, respectively, also \ell p =
\Bigl\{
y \in \omega :\sum \infty
n=1
| yn| p < \infty
\Bigr\}
for 1 \leq p < \infty . If y, z \in \omega , then we write yz = (ynzn)n\geq 1. Let U =
\bigl\{
y \in \omega :
yn \not = 0
\bigr\}
and U+ =
\bigl\{
y \in \omega : yn > 0
\bigr\}
. We write z/u = (zn/un)n\geq 1 for all z \in \omega and all u \in U, in
particular 1/u = e/u, where e is the sequence with en = 1 for all n. Finally, if a \in U+ and E is
any subset of \omega , then we put Ea = (1/a) - 1\ast E =
\bigl\{
y \in \omega : y/a \in E
\bigr\}
. Let E and F be subsets of \omega .
In [1], the sets sa, s
0
a and s
(c)
a were defined for positive sequences a by (1/a) - 1 \ast E and E = \ell \infty ,
c0, c, respectively. In [2], the sum Ea + Fb and the product Ea \ast Fb were defined, where E, F are
any of the symbols s, s0 or s(c). Then in [5] the solvability was determined of sequences spaces
inclusion equations Gb \subset Ea+Fx, where E, F, G \in
\bigl\{
s0, s(c), s
\bigr\}
and some applications were given
to sequence spaces inclusions with operators. Recall that the spaces w\infty and w0 of strongly bounded
and summable sequences are the sets of all y such that
\Bigl(
n - 1
\sum n
k=1
| yk|
\Bigr)
n
is bounded and tends to
zero, respectively. These spaces were studied by Maddox [21] and Malkowsky, Rakočević [20]. In
c\bigcirc B. DE MALAFOSSE, 2019
1040 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
APPLICATION OF THE INFINITE MATRIX THEORY TO THE SOLVABILITY OF SEQUENCE . . . 1041
[9, 12], were given some properties of well known operators definedby the sets Wa = (1/a) - 1 \ast w\infty
and W 0
a = (1/a) - 1 \ast w0. In this paper, we deal with special sequence spaces inclusion equations
(SSIE) (resp. sequence spaces equations (SSE)), which are determined by an inclusion (resp. identity),
for which each term is a sum or a sum of products of sets of the form (Ea)T and
\bigl(
\itE f(x)
\bigr)
T
, where
f maps U+ to itself, E is any linear space of sequences and T is a triangle. Some results on SSE
and SSIE were stated in [3 – 6, 8, 13, 14, 16, 17]. In [5], we dealt with the SSIE with operators
Ea + (Fx)\Delta \subset s
(c)
x , where E and F are any of the sets c0, c or s1. Then we gave a resolution
of the next inclusion equations with operator s
(c)
x +
\bigl(
s0b
\bigr)
\Delta
\subset sb and s0x +
\bigl(
s0b
\bigr)
\Delta
\subset s
(c)
b . Note that
the SSIE s
(c)
x + (s0b)\Delta \subset sb means yn/xn \rightarrow l and (zn - zn - 1)/bn \rightarrow 0 n \rightarrow \infty , together imply
| yn+ zn| \leq Kbn for all y, z \in \omega and for some scalars l and K with K > 0. In [13], we determined
the set of all positive sequences x for which the SSIE
\bigl(
s
(c)
x
\bigr)
B(r,s)
\subset
\bigl(
s
(c)
x
\bigr)
B(r\prime ,s\prime )
holds, where r,
r\prime , s\prime , and s are real numbers, and B(r, s) is the generalized operator of the first difference defined
by (B(r, s)y)n = ryn + syn - 1 for all n \geq 2 and (B(r, s)y)1 = ry1. In this way we determined the
set of all positive sequences x for which
\bigl(
ryn + syn - 1
\bigr)
/xn \rightarrow l implies
\bigl(
r\prime yn + s\prime yn - 1
\bigr)
/xn \rightarrow l
(n \rightarrow \infty ) for all y and for some scalar l. In the paper [8], we used the sets of analytic and entire
sequences denoted by \bfLambda and \bfGamma and defined by \mathrm{s}\mathrm{u}\mathrm{p}n\geq 1
\bigl(
| yn| 1/n
\bigr)
< \infty and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\bigl(
| yn| 1/n
\bigr)
= 0,
respectively. Then we dealt with a class of SSE with operators of the form ET + Fx = Fb, where T
is either \Delta or \Sigma and E is any of the sets c0, c, \ell \infty , \ell p (p \geq 1), w0, \bfGamma or \bfLambda and F = c, \ell \infty or
\bfLambda . In [11], we solved the SSE defined by (Ea)\Delta + s
(c)
x = s
(c)
b , where E is either c0 or \ell p, and the
SSE (Ea)\Delta + s0x = s0b , where E is either c or \ell \infty . Finally, in [10], we dealt with the SSIE defined
by F \subset Ea + F \prime
x, where a is positive sequence and E, F, and F \prime are linear subspaces of \omega and we
solved the SSE Er + (\ell p)x = (\ell p)b.
In this paper, we deal with the SSIE of the form F \subset Ea + F \prime
x, where E, F, and F \prime are linear
spaces of sequences a is a positive sequence with e \in F. We obtain a solvability of these SSIE for
a = (rn)n\geq 1. Throughout this paper we consider the SSIE F \subset Ea + F \prime
x as a perturbed inclusion
equation of the elementary inclusion equation F \subset F \prime
x. In this way it is interesting to determine
the set of all positive sequences a for which the elementary and the perturbed inclusions equations
have the same solutions. Then writing Dr for the diagonal matrix with (Dr)nn = rn, we study the
solvability of the SSIE using the operator of the first difference \Delta , defined by c \subset Dr \ast E\Delta + cx
with E = c0 or s1. Then we consider the SSIE c \subset Dr \ast EC1 + s
(c)
x with E = c0, c or s1 and
s1 \subset Dr \ast (s1)C1
+ sx with E = c or s1, where C1 is the Cesàro operator defined by (C1)ny =
=
\Bigl( \sum n
k=1
yk
\Bigr)
/n.
This paper is organized as follows. In Section 2, we recall some well-known results on sequence
spaces and matrix transformations. In Section 3, we recall some results on the multipliers and on
some characterizations of matrix transformations. In Section 4, we give some general results on the
SSIE F \subset Ea + F \prime
x, where E, F, and F \prime are linear spaces of sequences and e \in F. In Section 5,
we study the solvability of the SSIE of the form F \subset Ea + F \prime
x, where a = (rn)n\geq 1. In Section 6,
we deal with the SSIE of the form F \subset Ea + Fx and we explicitly calculate the solutions of the
SSIE of the form F \subset Ea+Fx, where a = (rn)n\geq 1. Finally, in Section 7, we study some SSIE with
operators of the form F \subset (ET )r + Fx, where T is a either \Delta or C1, and we solve the SSE of the
form (EC1)r + Fx = F.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1042 B. DE MALAFOSSE
2. Preliminaries and notations. An FK space is a complete linear metric space, for which
convergence implies coordinatewise convergence. A BK space is a Banach space of sequences that
is an FK space. A BK space E is said to have AK if for every sequence y = (yk)k\geq 1 \in E, then
y = \mathrm{l}\mathrm{i}\mathrm{m}p\rightarrow \infty
\sum p
k=1
yke
(k), where e(k) = (0, . . . , 0, 1, 0, . . .), 1 being in the k th position.
Let \BbbR be the set of all real numbers. For any given infinite matrix A = (\bfa nk)n,k\geq 1 we define the
operators An = (\bfa nk)k\geq 1 for any integer n \geq 1, by Any =
\sum \infty
k=1
\bfa nkyk, where y = (yk)k\geq 1, and
the series are assumed convergent for all n. So we are led to the study of the operator A defined by
Ay = (Any)n\geq 1 mapping between sequence spaces. When A maps E into F, where E and F are
subsets of \omega , we write A \in (E,F ) (cf. [21, 22]). It is well known that if E has AK, then the set
\scrB (E) of all bounded linear operators L mapping in E, with norm \| L\| = \mathrm{s}\mathrm{u}\mathrm{p}y \not =0
\bigl(
\| L(y)\| E/\| y\| E
\bigr)
satisfies the identity \scrB (E) = (E,E). We write \ell p for the set of all p-absolutely convergent series
with p \geq 1, that is, \ell p =
\Bigl\{
y \in \omega :
\sum \infty
k=1
| yk| p < \infty
\Bigr\}
. For any subset F of \omega , we write FA =
\bigl\{
y \in
\in \omega : Ay \in F
\bigr\}
for the matrix domain of A in F. Then for any given sequence u = (un)n\geq 1 \in \omega
we define the diagonal matrix Du by [Du]nn = un for all n. It is interesting to rewrite the set Eu
using a diagonal matrix. Let E be any subset of \omega and u \in U+, then we have Eu = Du \ast E =
\bigl\{
y =
= (yn)n \in \omega : y/u \in E
\bigr\}
. We use the sets s0a, s
(c)
a , sa, and (\ell p)a defined as follows (cf. [1]). For
given a \in U+ and p \geq 1 we put Da\ast c0 = s0a, Da\ast c = s
(c)
a , Da\ast \ell \infty = sa, and Da\ast \ell p = (\ell p)a. We
will frequently write ca instead of s(c)a to simplify. Each of the spaces Da\ast E, where E \in \{ c0, c, \ell \infty \}
is a BK space normed by \| y\| sa = \mathrm{s}\mathrm{u}\mathrm{p}n (| yn| /an) and s0a has AK. The set \ell p, p \geq 1, normed by
\| y\| \ell p =
\Bigl( \sum \infty
k=1
| yk| p
\Bigr) 1/p
is a BK space with AK. If a = (Rn)n\geq 1 with R > 0, we write sR,
s0R, s
(c)
R (or cR), and (\ell p)R for the sets sa, s
0
a, s
(c)
a , and (\ell p)a, respectively. We also write DR for
D(rn)n\geq 1
. When R = 1, we obtain s1 = \ell \infty , s01 = c0, and s
(c)
1 = c. Recall that S1 = (s1, s1) is a
Banach algebra and (c0, s1) = (c, s1) = (s1, s1) = S1. We have A \in S1 if and only if
\mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl( \infty \sum
k=1
| \bfa nk|
\Biggr)
< \infty . (1)
We also use the characterizations of the classes (c0, c0), (c0, c), (c, c0), (c, c), (s1, c), and
(\ell p, F ), where F = c0, c or \ell \infty . In this way we state the next well-known results.
Lemma 1 ([20, p. 160], Theorem 1.36, [21]). (i) A \in (c0, c0) if and only if (1) holds and
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \bfa nk = 0 for all k.
(ii) A \in (c0, c) if and only if (1) holds and
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bfa nk = lk for all k and for some scalar lk. (2)
(iii) A \in (c, c0) if and only if (1) holds and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum \infty
k=1
\bfa nk = 0 and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \bfa nk = 0 for
all k.
(iv) A \in (c, c) if and only if (1) holds and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum \infty
k=1
\bfa nk = l for some scalar l.
(v) A \in (s1, c) if and only if (2) holds and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\sum \infty
k=1
| \bfa nk| =
\sum \infty
k=1
| lk| .
Characterization of (\ell p, F ), where F = c0, c or \ell \infty . For this, we let q = p/(p - 1) for p > 1.
By using the notations of [20], we define \scrM (\ell p, \ell \infty ) = \mathrm{s}\mathrm{u}\mathrm{p}n
\bigl(
| \bfa nk|
\bigr)
if p = 1 and \scrM (\ell p, \ell \infty ) =
= \mathrm{s}\mathrm{u}\mathrm{p}n
\Bigl( \sum \infty
k=1
| \bfa nk| q
\Bigr)
if p > 1.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
APPLICATION OF THE INFINITE MATRIX THEORY TO THE SOLVABILITY OF SEQUENCE . . . 1043
Lemma 2 ([20, p. 161], Theorem 1.37). Let p \geq 1. Then we have:
(i) A \in (\ell p, \ell \infty ) if and only if
\scrM (\ell p, \ell \infty ) < \infty . (3)
(ii) A \in (\ell p, c0) if and only if the condition in (3) holds and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \bfa nk = 0 for all k.
(iii) A \in (\ell p, c) if and only if the conditions in (3) and (2) hold.
We also use the well known properties, stated as follows.
Lemma 3. Let a, b \in U+ and let E, F \subset \omega be any linear spaces. We have A \in (Ea, Fb) if
and only if D1/bADa \in (E,F ).
Lemma 4 ([3, p. 45], Lemma 9). Let T \prime and T \prime \prime be any given triangles and let E, F \subset \omega .
Then for any given operator T represented by a triangle we have T \in (ET \prime , FT \prime \prime ) if and only if
T \prime \prime TT
\prime - 1 \in (E,F ).
3. Some results on matrix transformations and on the multipliers of special sets. 3.1. On
the triangles \bfitC (\bfitlambda ) and \bfDelta (\bfitlambda ) and the sets \bfitW \bfita and \bfitW 0
\bfita . The infinite matrix T = (tnk)n,k\geq 1 is
said to be a triangle if tnk = 0 for k > n and tnn \not = 0 for all n. For \lambda \in U the infinite matrices
C(\lambda ) and \Delta (\lambda ) are triangles. We have [C (\lambda )]nk = 1/\lambda n for k \leq n, and the nonzero entries of
\Delta (\lambda ) are determined by
\bigl[
\Delta (\lambda )
\bigr]
nn
= \lambda n for all n, and
\bigl[
\Delta (\lambda )
\bigr]
n,n - 1
= - \lambda n - 1 for all n \geq 2. It can
be shown that the matrix \Delta (\lambda ) is the inverse of C(\lambda ), that is, C(\lambda )
\bigl(
\Delta (\lambda )y
\bigr)
= \Delta (\lambda )
\bigl(
C(\lambda )y
\bigr)
= y
for all y \in \omega . If \lambda = e we obtain the well known operator of the first difference represented by
\Delta (E) = \Delta . Then we have \Delta ny = yn - yn - 1 for all n \geq 1 with the convention y0 = 0. It is usually
written \Sigma = C(E), and then we may write C(\lambda ) = D1/\lambda \Sigma . Note that \Delta = \Sigma - 1. The Cesàro
operator is defined by C1 = C
\bigl(
(n)n\geq 1
\bigr)
. We use the sets of sequences that are a-strongly bounded
and a-strongly convergent to zero sequences defined for a \in U+ by
Wa =
\Biggl\{
y \in \omega : \| y\| Wa = \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl(
n - 1
n\sum
k=1
| yk| /ak
\Biggr)
< \infty
\Biggr\}
and
W 0
a =
\Biggl\{
y \in \omega : \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\Biggl(
n - 1
n\sum
k=1
| yk| /ak
\Biggr)
= 0
\Biggr\}
(cf. [7, 9, 12, 15]). It can easily be seen that Wa =
\bigl\{
y \in \omega : C1D1/a| y| \in s1
\bigr\}
. If a = (rn)n\geq 1 the
sets Wa and W 0
a are denoted by Wr and W 0
r . For r = 1, we obtain the well-known sets
w\infty =
\Biggl\{
y \in \omega : \| y\| w\infty = \mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl(
n - 1
n\sum
k=1
| yk|
\Biggr)
< \infty
\Biggr\}
and
w0 =
\Biggl\{
y \in \omega : \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\Biggl(
n - 1
n\sum
k=1
| yk|
\Biggr)
= 0
\Biggr\}
called the spaces of sequences that are strongly bounded and strongly summable to zero sequences
by the Cesàro method (cf. [18]).
3.2. On the multipliers of some sets. First we need to recall some well known results. Let y
and z be sequences and let E and F be two subsets of \omega , then we write M(E,F ) =
\bigl\{
y \in \omega :
yz \in F for all z \in E
\bigr\}
, the set M(E,F ) is called the multiplierspace of E and F. In the following
we will use the next well known results.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1044 B. DE MALAFOSSE
Lemma 5. Let E, \widetilde E, F and \widetilde F be arbitrary subsets of \omega . Then: (i) M(E,F ) \subset M( \widetilde E,F ) for
all \widetilde E \subset E; (ii) M(E,F ) \subset M(E, \widetilde F ) for all F \subset \widetilde F .
The \alpha -dual of a set of sequences E is defined as E\alpha = M (E, \ell 1) and the \beta -dual of E is defined
as E\beta = M(E, cs), where cs = c\Sigma is the set of all convergent series.
Lemma 6. Let a, b \in U+ and let E and F be two subsets of \omega . Then DaE \subset DbF if and only
if a/b \in M(E,F ).
In the following we use the results stated below.
Lemma 7. Let p \geq 1. Then we have:
(i) a) M(c, c0) = M(\ell \infty , c) = M(\ell \infty , c0) = c0 and M(c, c) = c,
b) M(E, \ell \infty ) = M(c0, F ) = \ell \infty for E, F = c0, c or \ell \infty ,
c) M(c0, \ell p) = M(c, \ell p) = M(\ell \infty , \ell p) = \ell p,
d) M(\ell p, F ) = \ell \infty for F \in
\bigl\{
c0, c, s1, \ell p
\bigr\}
;
(ii) a) M(w0, F ) = M(w\infty , \ell \infty ) = s(1/n)n\geq 1
for F = c0, c or \ell \infty ,
b) M(w\infty , c0) = s0(1/n)n\geq 1
,
c) M(\ell 1, w\infty ) = s(n)n\geq 1
and M(\ell 1, w0) = s0(n)n\geq 1
,
d) M(E,w0) = w0 for E = s1 or c,
e) M(E,w\infty ) = w\infty for E = c0, s1 or c.
Proof. Statements (i) a), (i) b) and (i) d) with F \in
\bigl\{
c0, c, \ell \infty
\bigr\}
follow from [19, p. 648],
(Lemma 3.1), Lemma 2 and [20, p. 157] (Example 1.28). The case M(\ell p, \ell p) = \ell \infty is immediate.
Then statements (ii) a) with F = c0 or c, (ii) b), (ii) c), (ii) d) follow from [14, p. 598] (Lemma 4.2).
It remains to successively show the identity M(c0, \ell p) = \ell p in (i) c), statement (ii) a) with F = \ell \infty
and the identity M(c0, w\infty ) = w\infty in (ii) e). We show M(c0, \ell p) = \ell p.
Case p = 1. We have M(c0, \ell 1) = c\alpha 0 = \ell 1 (cf. [20], Theorem 1.29).
Case p > 1. By [22, p. 124] (Theorem 8.3.9) with X = c0 and Z = \ell q, we have M(c0, \ell p) =
= M(c0, \ell
\beta
q ) = M(\ell q, c
\beta
0 ) = M(\ell q, \ell 1) = \ell \alpha q . Then by [20] (Theorem 1.29) we have \ell \alpha q = \ell \beta q = \ell p.
We conclude M(c0, \ell p) = \ell p. The identity M(\ell \infty , \ell p) = \ell p follows from [21], since (c0, \ell p) =
= (\ell \infty , \ell p). We conclude by Lemma 5 that \ell p = M(\ell \infty , \ell p) \subset M(c, \ell p) \subset M(c0, \ell p) = \ell p,
which shows (i) c). For (ii) a) it is enough to notice that by [20, p. 219] (Theorem 3.58), we
have (w0, s1) = (w\infty , s1). Then, by [14, p. 598] (Lemma 4.2), we obtain M(w0, F ) = s(1/n)n\geq 1
for F \in \{ c0, c, s1\} . It remains to show M(c0, w\infty ) = w\infty in the statement (ii) e). By [20, p. 218]
(Lemma 3.56), the set \scrM = w\beta
\infty is a BK space with AK, and is \beta -perfect, that is, w\beta \beta
\infty = w\infty . Again
by [22, p. 124] (Theorem 8.3.9) with X = c0 and Z = \scrM , we obtain M(c0, w\infty ) = M(c0, w
\beta \beta
\infty ) =
= M
\bigl(
\scrM , c\beta 0
\bigr)
. But we have c\beta 0 = \ell 1. We conclude M(c0, w\infty ) = \scrM \beta = w\infty .
3.3. The equivalence relation \bfitR \bfscrE . We need to recall some results on the equivalence relation
R\scrE which is defined using the multiplier of sequence spaces. For b \in U+ and for any subset \scrE
of \omega , we denote by cl\scrE (b) the equivalence class for the equivalence relation R\scrE defined by xR\scrE y
if \scrE x = \scrE y for x, y \in U+. It can easily be seen that cl\scrE (b) is the set of all x \in U+ such that
x/b \in M(\scrE , \scrE ) and b/x \in M(\scrE , \scrE ) (cf. [16]). Then we have cl\scrE (b) = clM(\scrE ,\scrE )(b). For instance,
clc(b) is the set of all x \in U+ such that s(c)x = s
(c)
b . This is the set of allsequences x \in U+ such that
xn \thicksim Cbn (n \rightarrow \infty ) for some C > 0. In [16], we denote by cl\infty (b) the class cl\ell \infty (b). Recall that
cl\infty (b) is the set of all x \in U+ such that K1 \leq xn/bn \leq K2 for all n and for some K1, K2 > 0.
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APPLICATION OF THE INFINITE MATRIX THEORY TO THE SOLVABILITY OF SEQUENCE . . . 1045
4. On the SSIE \bfitF \subset \bfitE \bfita + \bfitF \prime
\bfitx with \bfite \in \bfitF and \bfitF \prime \subset \bfitM (\bfitF , \bfitF \prime ). Here we are interested in
the study of the set of all positive sequences x that satisfy the inclusion F \subset Ea + F \prime
x, where E, F,
and F \prime are linear spaces of sequences and a is a positive sequence. We may consider this problem as
a perturbation problem. If we know the set M(F, F \prime ), then the solutions of the elementary inclusion
F \prime
x \supset F are determined by 1/x \in M(F, F \prime ). Now the question is: let \scrE be a linear space of
sequences. What are the solutions of the perturbed inclusion F \prime
x + \scrE \supset F ? An additionnal question
may be the following one: what are the conditions on \scrE under which the solutions of the elementary
and the perturbed inclusions are the same? The solutions of the perturbed inclusion F \subset Ea + F \prime
x,
where E, F, and F \prime are linear spaces of sequences cannot be obtained in the general case. So are
led to deal with the case when a = (rn)n\geq 1, r > 0, for which most of these SSIE can be totally
solved. In the following we write \scrI a(E,F, F \prime ) =
\bigl\{
x \in U+ : F \subset Ea + F \prime
x
\bigr\}
, where E, F, and
F \prime are linear spaces of sequences and a \in U+. For any set \chi of sequences we let \chi =
\bigl\{
x \in U+ :
1/x \in \chi
\bigr\}
.
4.1. General case. The next theorem is the main result and is used throuhgout this paper. We
use the set \Phi =
\bigl\{
c0, c, s1, \ell p, w0, w\infty
\bigr\}
with p \geq 1. By c(1) we define the set of all sequences
\alpha \in U+ that satisfy \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \alpha n = 1. Then we consider the condition
G \subset G1/\alpha for all \alpha \in c(1) (4)
for any given linear space G of sequences. Notice that condition (4) is satisfied for all G \in \Phi .
Theorem 1. Let a \in U+ and let E, F, F \prime be linear spaces of sequences. Assume: a) e \in
\in F, b) F \prime \subset M(F, F \prime ), c) F \prime satisfies condition (4). Then we have: (i) a \in M(E, c0) implies
\scrI a(E,F, F \prime ) = F \prime , (ii) 1/a \in M(F,E) implies \scrI a(E,F, F \prime ) = U+.
Proof. (i) Let x \in \scrI a(E,F, F \prime ). Then there are \xi \in E and f \prime \in F \prime such that 1 = an\xi n + xnf
\prime
n,
hence,
1 - an\xi n
xn
= f \prime
n for all n.
Since a \in M(E, c0) we have 1 - an\xi n \rightarrow 1 (n \rightarrow \infty ) and
1
xn
=
1
1 - an\xi n
f \prime
n for all n.
By the condition in c) we conclude x \in F \prime . Conversely, the condition x \in F \prime implies 1/x \in F \prime , and
the condition in b) implies 1/x \in M (F, F \prime ) . We conclude F \subset F \prime
x and x \in \scrI a(E,F, F \prime ). So we
have shown (i). Statement (ii) follows from the equivalence of 1/a \in M(F,E) and F \subset Ea. This
concludes the proof.
We immediately deduce the following.
Corollary 1. Let E, F, F \prime be linear spaces of sequences. Assume: a) e \in F, b) F \prime \subset M(F, F \prime )
and c) E \subset c0. Then the next statements are equivalent: (i) F \subset E + F \prime
x, (ii) F \subset F \prime
x, (iii) x \in F \prime .
In some cases, where E = cs or \ell 1 and F \prime = \ell 1, we obtain the next results using the \alpha - and
\beta -duals.
Corollary 2. Let a \in U+ and let F and F \prime be linear spaces of sequences. Assume a), b), c)
in Theorem 1 hold. Then the set \scrI a(cs, F, F \prime ) of all positive sequences x such that F \subset csa +
+ F \prime
x, satisfies the next properties: (i) a \in s1 implies \scrI a(cs, F, F \prime ) = F \prime , (ii) 1/a \in F \beta implies
\scrI a(cs, F, F \prime ) = U+.
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1046 B. DE MALAFOSSE
Proof. It is enough to show M(cs, c0) = s1. We have a \in M(cs, c0) if and only if Da\Delta \in
\in (c, c0), and Da\Delta is the infinite matrix whose the nonzero entries are [Da\Delta ]nn = - [Da\Delta ]n,n - 1 =
= an for all n \geq 2, and [Da\Delta ]1,1 = a1. By the characterization of (c, c0) recalled in Lemma 1 we
conclude M(cs, c0) = s1.
Corollary 3. Let a \in U+ and let F and F \prime be linear spaces of sequences. Assume a), b), c)
in Theorem 1 hold. Then the set \scrI a(\ell 1, F, F \prime ) of all positive sequences x such that F \subset (\ell 1)a +
+ F \prime
x, satisfies the next properties: (i) a \in s1 implies \scrI a(\ell 1, F, F \prime ) = F \prime , (ii) 1/a \in F\alpha implies
\scrI a(\ell 1, F, F \prime ) = U+.
Proof. This result follows from the identity M(\ell 1, c0) = s1 which is a direct consequence of
Lemma 2.
Corollary 4. Let a \in U+ and let E and F be linear spaces of sequences. Assume e \in F and
\ell 1 \subset F\alpha . Then the set \scrI a(E,F, \ell 1) of all positive sequences x such that F \subset Ea + (\ell 1)x satisfies:
(i) and (ii) in Theorem 1 with F \prime = \ell 1.
4.2. On SSIE of the form \bfitF \subset \bfitE \bfita + \bfitF \prime
\bfitx , where \bfitE , \bfitF , and \bfitF \prime are any of the sets \bfitc 0, \bfitc , \bfits 1,
\ell \bfitp , \bfitw 0, or \bfitw \infty . In this part we use the set \Omega =
\bigl(
\{ s1\} \times (\Phi \diagdown \{ c\} )
\bigr)
\cup
\bigl(
\{ c\} \times \Phi
\bigr)
with p \geq 1, and
we deal with the perturbed inclusions of the form F \subset Ea + F \prime
x, where E = c0, s1, \ell p, w0, or w\infty
and (F, F \prime ) \in \Omega . As a direct consequence of Lemma 7 we obtain.
Lemma 8. We have (F, F \prime ) \in \Omega \Rightarrow F \prime \subset M(F, F \prime ).
As a direct consequence of Corollary 1 and Lemma 8 we get the following result.
Proposition 1. Let E \subset c0 be a linear space of sequences and let (F, F \prime ) \in \Omega . Then the next
statements are equivalent, where
(i) F \subset E + F \prime
x, (ii) F \subset F \prime
x, (iii) x \in F \prime .
Proposition 2. Let a \in U+ and (F, F \prime ) \in \Omega . We have:
(i) \scrI a(c0, F, F \prime ) = F \prime if a \in s1, and \scrI a(c0, F, F \prime ) = U+ if 1/a \in c0,
(ii) \scrI a(s1, F, F \prime ) = F \prime if a \in c0, and \scrI a(s1, F, F \prime ) = U+ if 1/a \in s1,
(iii) \scrI a(\ell p, F, F \prime ) = F \prime if a \in s1, and \scrI a(\ell p, F, F \prime ) = U+ if 1/a \in \ell p for p \geq 1,
(iv) \scrI a(w0, F, F
\prime ) = F \prime if a \in s(1/n)n\geq 1
, and \scrI a(w0, F, F
\prime ) = U+ if 1/a \in w0,
(v) \scrI a(w\infty , F, F \prime ) = F \prime if a \in s0(1/n)n\geq 1
, and \scrI a(w\infty , F, F \prime ) = U+ if 1/a \in w\infty .
Proof. The proof is a direct consequence of Theorem 1 and Lemma 7. Indeed, we successively
have M(E, c0) = s1 for E = c0 or \ell p, M(E, c0) = c0 for E = c or s1, M(w0, c0) = s(1/n)n\geq 1
and M(w\infty , c0) = s0(1/n)n\geq 1
. Then we have M(F,E) = M(s1, E) = M(c, E) for E \in \Phi \diagdown \{ c\} and
M(s1, c0) = c0, M(s1, s1) = s1, M(s1, \ell p) = \ell p, M(s1, w0) = w0, and M(s1, w\infty ) = w\infty .
In the case when E = c we obtain the following result.
Proposition 3. Let a \in U+ and F \prime \in \Phi . We have:
(i) \scrI a(c, c, F \prime ) = F \prime if a \in c0, and \scrI a(c, c, F \prime ) = U+ if 1/a \in c,
(ii) \scrI a(c, s1, F \prime ) = F \prime if a \in c0, and \scrI a(c, s1, F \prime ) = U+ if 1/a \in c0.
Proof. The proof follows from Theorem 1 and Lemma 7. Here we have M(E, c0) = M(c, c0) =
= c0 and M(F,E) = M(F, c) = c0 for F = s1, and M(F, c) = c for F = c.
5. Solvability of the SSIE of the form \bfitF \subset \bfitE \bfitr + \bfitF \prime
\bfitx , where \bfitE and \bfitF \prime are any of the
sets \bfitc 0, \bfitc , \bfits 1, \ell \bfitp (\bfitp \geq \bfone ), \bfitw 0, or \bfitw \infty . For a = (rn)n\geq 1, we write \scrI r(E,F, F \prime ) for the set
\scrI a(E,F, F \prime ). Then we solve the perturbed inclusions F \subset Er + F \prime
x, where F is either c or s1 and
E \in \Phi \diagdown \{ w0\} , F \prime \in \Phi . It can easily beseen that in most of the cases the set \scrI r(E,F, F \prime ) may be
determined by
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APPLICATION OF THE INFINITE MATRIX THEORY TO THE SOLVABILITY OF SEQUENCE . . . 1047
\scrI r(E,F, F \prime ) =
\left\{ F \prime if r < 1,
U+ if r \geq 1
(5)
or
\scrI r(E,F, F \prime ) =
\left\{ F \prime if r \leq 1,
U+ if r > 1.
(6)
As a direct consequence of Propositions 2 and 3 we obtain the following result.
Proposition 4. Let a \in U+ and (F, F \prime ) \in \Omega . We have:
(i) The sets \scrI r(s1, F, F \prime ), \scrI r(c, c, F \prime ), and \scrI r(w\infty , F, F \prime ) are determined by (5).
(ii) The sets \scrI r(c0, F, F \prime ) and \scrI r(\ell p, F, F \prime ) for p \geq 1 are determined by (6).
Rewriting Proposition 4 we obtain.
Corollary 5. Let r > 0. Then we have:
(i) Let F \prime \in \Phi . Then:
a) The solutions of the SSIE c \subset Er + F \prime
x with E = c, s1 or w\infty are determined by (5).
b) The solutions of the SSIE c \subset Er + F \prime
x with E = c0 or \ell p (p \geq 1) are determined by (6).
(ii) Let F \prime \in \Phi \diagdown \{ c\} . Then we have:
a) The solutions of the SSIE s1 \subset Er + F \prime
x with E = s1 or w\infty are determined by (5).
b) The solutions of the SSIE s1 \subset Er + F \prime
x with E = c0 or \ell p (p \geq 1) are determined by (6).
Remark 1. The set \scrI r(w0, c, F
\prime ) of all the solutions of the SSIE c \subset W 0
r + F \prime
x, where F \prime \in \Phi ,
is determined for all r \not = 1. We obtain \scrI r(w0, c, F
\prime ) = F \prime for r < 1, and \scrI r(w0, c, F
\prime ) = U+ if
r > 1.
6. Application to the SSIE of the form \bfitF \subset \bfitE \bfita + \bfitF \bfitx with \bfite \in \bfitF . In the following we
write \scrI a(E,F ) = \scrI a(E,F, F ) =
\bigl\{
x \in U+ : F \subset Ea+Fx
\bigr\}
. In this part we give results on the SSIE
F \subset Ea + Fx and we explicitly calculate the solutions of special SSIE of the form F \subset Er + Fx.
6.1. Some general results on the SSIE of the form \bfitF \subset \bfitE \bfita +\bfitF \bfitx . From Theorem 1 we obtain
the next corollary.
Corollary 6. Let a \in U+ and let E, F be two linear spaces of sequences. Assume: a) e \in F, b)
F \subset M(F, F ) and c) F satisfies condition (4). Then we have:
(i) a \in M(E, c0) implies \scrI a(E,F ) = F ,
(ii) 1/a \in M(F,E) implies \scrI a(E,F ) = U+.
Now we deal with the SSIE F \subset Ea+Fx, where F is either c or s1 and E \in \Phi . By Corollary 6
and Lemma 7 we obtain the following result.
Corollary 7. Let a \in U+. We have:
(i) a) \scrI a(c0, c) = c if a \in s1, and \scrI a(c0, c) = U+ if 1/a \in c0,
b) \scrI a(c, c) = c if a \in c0, and \scrI a(c, c) = U+ if 1/a \in c,
c) \scrI a(s1, c) = c if a \in c0, and \scrI a(s1, c) = U+ if 1/a \in s1,
d) \scrI a(\ell p, c) = c if a \in s1, and \scrI a(\ell p, c) = U+ if 1/a \in \ell p for p \geq 1,
e) \scrI a(w0, c) = c if a \in s(1/n)n\geq 1
, and \scrI a(w0, c) = U+ if 1/a \in w0,
f) \scrI a(w\infty , c) = c if a \in s0(1/n)n\geq 1
, and \scrI a(w\infty , c) = U+ if 1/a \in w\infty ;
(ii) a) \scrI a(c0, s1) = s1 if a \in s1, and \scrI a(c0, s1) = U+ if 1/a \in c0,
b) \scrI a(c, s1) = s1 if a \in c0, and \scrI a(c, s1) = U+ if 1/a \in c0,
c) \scrI a(s1, s1) = s1 if a \in c0, and \scrI a(s1, s1) = U+ if 1/a \in s1,
d) \scrI a(\ell p, s1) = s1 if a \in s1, and \scrI a(\ell p, s1) = U+ if 1/a \in \ell p for p \geq 1,
e) \scrI a(w0, s1) = s1 if a \in s(1/n)n\geq 1
, and \scrI a(w0, s1) = U+ if 1/a \in w0,
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1048 B. DE MALAFOSSE
f) \scrI a(w\infty , s1) = s1 if a \in s0(1/n)n\geq 1
, and \scrI a(w\infty , s1) = U+ if 1/a \in w\infty .
6.2. Solvability of SSIE of the form \bfitF \subset \bfitE \bfitr + \bfitF \bfitx . In this part we consider the case when
a = (rn)n\geq 1 for r > 0. We write Er for E(rn)n\geq 1
, and the set of all positive sequences x that satisfy
F \subset Er + Fx (7)
is denoted by \scrI r(E,F ). Here we explicitly calculate the solutions of the SSIE defined by (7),
where F is either c or s1 and E \in \Phi . We consider the conditions
(rn)n\geq 1 \in M(E, c0) for all 0 < r < 1, (8)
(rn)n\geq 1 \in M(E, c0) for all 0 < r \leq 1, (9)
(r - n)n\geq 1 \in M(F,E) for all r \geq 1, (10)
(r - n)n\geq 1 \in M(F,E) for all r > 1. (11)
We will use the next result which is a direct consequence of Corollary 6.
Corollary 8. Let r > 0 and let E and F be linear spaces of sequences. We assume that F
satisfies the conditions a), b) and c) in Corollary 6. Then we have:
(i) Assume that the conditions in (8) and (10) hold. Then the solutions of the SSIE defined by (7)
are determined by (5) with F \prime = F.
(ii) Assume that the conditions in (9) and (11) hold. Then the solutions of the SSIE defined by (7)
are determined by (6) with F \prime = F.
As a direct consequence of the preceding we obtain the following result.
Corollary 9. Let r > 0. Then we have:
(i) a) The solutions of the SSIE c \subset Er + cx for E = c, s1 or w\infty are determined by (5) with
F \prime = c.
b) The solutions of the SSIE c \subset Er + cx with E = c0 or \ell p (p \geq 1) are determined by (6) with
F \prime = c.
(ii) a) The solutions of the SSIE s1 \subset Er + sx with E = s1 or w\infty are determined by (5) with
F \prime = s1.
b) The solutions of the SSIE s1 \subset Er + sx with E = c0 or \ell p (p \geq 1) are determined by (6)
with F \prime = s1.
7. On some SSIE and SSE with operators. In this part we consider the SSIE associated
with the operator \Delta , defined by c \subset Dr \ast (c0)\Delta + cx, c \subset Dr \ast c\Delta + cx, c \subset Dr \ast (s1)\Delta + cx,
s1 \subset sr + sx and s1 \subset Dr \ast (s1)\Delta + sx for r > 0. Then we consider the SSIE c \subset Dr \ast EC1 + s
(c)
x
with E \in \{ c, s1\} and s1 \subset Dr \ast (s1)C1 + sx, where C1 is the Cesàro operator. Then we solve
the SSE Dr \ast EC1 + s
(c)
x = c with E \in \{ c0, c, s1\} , and Dr \ast (s1)C1 + sx = s1. Notice that since
Da \ast ET = ETD1/a
, where TD1/a is a triangle, for any linear space E of sequences and any triangle
T the previous inclusions and identities can be considered as SSIE and SSE. More precisely, the
previous SSE can be considered as the perturbed equations of the equations Fx = F with F = c,
or s1.
7.1. On the SSIE of the form \bfitF \subset \bfitD \bfitr \ast \bfitE \Delta + \bfitF \bfitx . In the next result among other things we
deal with the SSIE c \subset Dr \ast (c0)\Delta + cx which is associated with the next statement. The condition
yn \rightarrow l (n \rightarrow \infty ) implies that there are u, v \in \omega such that y = u+v and unr
- n - un - 1r
- (n - 1) \rightarrow 0
and vn/xn \rightarrow l\prime (n \rightarrow \infty ) for some scalars l, l\prime and for all y. The corresponding set of sequences
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APPLICATION OF THE INFINITE MATRIX THEORY TO THE SOLVABILITY OF SEQUENCE . . . 1049
is denoted by \scrI r((c0)\Delta , c). In a similar way \scrI r
\bigl(
(s1)\Delta , c
\bigr)
is the set of all sequences that satisfy the
SSIE c \subset Dr \ast (s1)\Delta + cx. We obtain the following result.
Corollary 10. Let r > 0. We have:
(i) \scrI r(E, c) = \scrI r(s1, c) for E = c, (c0)\Delta , c\Delta or (s1)\Delta , and \scrI r(s1, c) is determined by (5) with
F \prime = c.
(ii) \scrI r((s1)\Delta , s1) = \scrI r(s1, s1) is determined by (5) with F \prime = s1.
Proof. (i) The identity \scrI r(s1, c) = \scrI r(c, c) is a direct consequence of Corollary 8, and \scrI r(s1, c)
is determined by (5) with F = c. Indeed, we have M(E, c0) = M(c, c0) = M(s1, c0) = c0. Then
we get M(E, c) = c for E = c, and M(E, s1) = s1 for E = s1. Now we deal with \scrI ((c0)\Delta , c).
We have (Rn)n\geq 1 \in M((c0)\Delta , c0) if and only if DR\Sigma \in (c0, c0). The operator DR\Sigma is the triangle
defined by (DR\Sigma )nk = Rn for k \leq n, for all n. Then from the characterization of (c0, c0) the
condition DR\Sigma \in (c0, c0) is equivalent to nRn = O(1) (n \rightarrow \infty ), and R < 1. Then the condition
(R - n)n\geq 1 \in M(c, (c0)\Delta ) implies \Delta D1/R \in (c, c0). The nonzero entries of \Delta D1/R are given by
[\Delta D1/R]nn = R - n and [\Delta D1/R]n,n - 1 = - R - n+1 for all n \geq 1, and from the characterization of
(c, c0) we conclude R \geq 1. By similar arguments we obtain \scrI r((s1)\Delta , c) = \scrI r(c\Delta , c) = \scrI r(s1, c).
The proof of (ii) is similar and is left to the reader.
7.2. On the SSIE of the form \bfitF \subset \bfitD \bfitr \ast \bfitE \bfitC 1 + \bfitF \bfitx , where \bfitC 1 is the Cesàro operator. In
this part we consider the SSIE with the Cesàro operator C1 of the form c \subset Dr \ast EC1 + s
(c)
x with
E \in \{ c, s1\} and of the form s1 \subset Dr \ast EC1 + sx with E \in \{ c, s1\} . We obtain the following result.
Proposition 5. Let r > 0. Then we have:
(i) The solutions of the SSIE defined by c \subset Dr \ast EC1 + s
(c)
x with E \in \{ c, s1\} , are determined
by (5) with F \prime = c.
(ii) The solutions of the SSIE s1 \subset Dr \ast (s1)C1 + sx are determined by (5) with F \prime = s1.
Proof. (i). Case E = c. Let R > 0. We have (rn)n\geq 1 \in M(cC1 , c0) if and only if DRC
- 1
1 \in
\in (c, c0). It can easily be seen that the entries of the matrix C - 1
1 are defined by
\bigl[
C - 1
1
\bigr]
nn
= n,
[C - 1
1 ]n,n - 1 = - (n - 1) for all n \geq 2 and [C - 1
1 ]1,1 = 1. Then DRC
- 1
1 is the triangle whose the
nonzero entries are given by
\bigl[
DRC
- 1
1
\bigr]
nn
= nRn,
\bigl[
DRC
- 1
1
\bigr]
n,n - 1
= - (n - 1)Rn for all n \geq 2
and
\bigl[
DRC
- 1
1
\bigr]
1,1
= R. From the characterization of (c, c0) this means \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\Bigl\{
Rn
\bigl[
n - (n -
- 1)
\bigr] \Bigr\}
= \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Rn = 0, and (2n - 1)Rn \leq K for some K > 0 and for all n. We conclude
(Rn)n\geq 1 \in M(cC1 , c0) if and only if R < 1. Then we have (r - n)n\geq 1 \in M(c, cC1) if and only if
C1D1/R \in (c, c). But C1D1/R is the triangle defined by [C1D1/R]nk = n - 1R - k for k \leq n, for
all n. So the condition C1D1/R \in (c, c) is equivalent to n - 1
\sum n
k=1
R - k \rightarrow L (n \rightarrow \infty ) for some
scalar L, and R \geq 1. We conclude by Corollary 8.
Case E = s1. We have (rn)n\geq 1 \in M
\bigl(
(s1)C1 , c0
\bigr)
if and only if DRC
- 1
1 \in (s1, c0), and from
the characterization of (s1, c0), that is, \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\bigl[
(2n - 1)Rn
\bigr]
= 0 and R < 1. Then we have
(r - n)n\geq 1 \in M
\bigl(
c, (s1)C1
\bigr)
if and only if C1D1/R \in (c, s1), that is, \mathrm{s}\mathrm{u}\mathrm{p}n
\Bigl(
n - 1
\sum n
k=1
R - k
\Bigr)
< \infty
and R \geq 1. Again we conclude by Corollary 8.
(ii). As we have just seen above we have (rn)n\geq 1 \in M
\bigl(
(s1)C1 , c0
\bigr)
if and only if R < 1. Then
we have (r - n)n\geq 1 \in M
\bigl(
s1, (s1)C1
\bigr)
if and only if C1D1/R \in (s1, s1), that is,
\mathrm{s}\mathrm{u}\mathrm{p}
n
\Biggl(
n - 1
n\sum
k=1
R - k
\Biggr)
< \infty
and we conclude R \geq 1.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
1050 B. DE MALAFOSSE
Corollary 11. The solutions of the SSIE c \subset Dr \ast (c0)C1 + s
(c)
x are determined by
\scrI r
\bigl(
(c0)C1 , c
\bigr)
=
\left\{ c if r < 1,
U+ if r > 1.
Proof. We apply Theorem 1. We have (rn)n\geq 1 \in M ((c0)C1 , c0) if and only if DRC
- 1
1 \in
\in (c0, c0). We conclude DRC
- 1
1 \in (c0, c0) if and only if
\bigl(
(2n - 1)Rn
\bigr)
n\geq 1
\in \ell \infty and R < 1. Then
we get (r - n)n\geq 1 \in M
\bigl(
c, (c0)C1
\bigr)
if and only if C1D1/R \in (c, c0), that is, n - 1
\sum n
k=1
R - k \rightarrow 0
(n \rightarrow \infty ) and R > 1.
7.3. Application to the solvability of the SSE with operator of the form \bfitD \bfitr \ast \bfitE \bfitC 1 + \bfitF \bfitx = \bfitF .
In this subsection the deal with the solvability of each of the SSE Dr \ast EC1 + s
(c)
x = c, where
E \in
\bigl\{
c0, c, s1
\bigr\}
and Dr \ast EC1 + sx = s1 with E \in \{ c, s1\} . For instance, the SSE defined by
Dr \ast (s1)C1 + cx = c is associated with the next statement. The condition yn \rightarrow l (n \rightarrow \infty )
holds if and only if there are u, v \in \omega such that y = u + v and \mathrm{s}\mathrm{u}\mathrm{p}n
\Bigl\{
n - 1
\bigm| \bigm| \bigm| \sum n
k=1
uk/r
k
\bigm| \bigm| \bigm| \Bigr\} <
< \infty and vn/xn \rightarrow l\prime (n \rightarrow \infty ) for some scalars l, l\prime and for all y. Here we also use the SSIE
defined by Dr \ast (c0)C1
+ cx \subset c which is associated with the next statement. The conditions
n - 1
\Bigl( \sum n
k=1
uk/r
k
\Bigr)
\rightarrow 0 and vn/xn \rightarrow l together imply un + vn \rightarrow l\prime (n \rightarrow \infty ) for all u, v \in \omega
and for some scalars l, l\prime . Let E and F be two linear spaces of sequences. We write \scrI \prime
a(E,F ) =
=
\bigl\{
x \in U+ : Ea + Fx \subset F
\bigr\}
. Notice that since E and F are linear spaces of sequences, we have
x \in \scrI \prime
a(E,F ) if and only if and Ea \subset F and Fx \subset F. This means that x \in \scrI \prime
a(E,F ) if and only
if a \in M(E,F ) and x \in M(F, F ). Then we have \scrS (E,F ) = \scrI a(E,F ) \cap \scrI \prime
a(E,F ) =
\bigl\{
x \in U+ :
Ea + Fx = F
\bigr\}
, see [16].
From Proposition 5 and Corollary 11 we obtain the next results on the SSE Dr \ast EC1 + s
(c)
x = c
with E \in \{ c0, c, s1\} , and Dr \ast EC1 + sx = s1 with E \in \{ c, s1\} , where we write G+ = G\cap U+ for
any set G of sequences.
Proposition 6. Let r > 0. Then we have:
(i) The solutions of the SSIE Dr \ast EC1 + s
(c)
x \subset c with E \in \{ c0, c, s1\} , and Dr \ast EC1 + sx \subset s1
with E \in \{ c, s1\} are determined by
\scrI \prime
r(EC1 , c) =
\left\{ c+ if r < 1,
\varnothing if r \geq 1,
for E \in \{ c0, c, s1\} , (12)
and
\scrI \prime
r(EC1 , s1) =
\left\{ s+1 if r < 1,
\varnothing if r \geq 1,
for E \in \{ c, s1\} .
(ii) The solutions of the perturbed equations Dr \ast EC1 + s
(c)
x = c with E \in \{ c0, c, s1\} , and
Dr \ast EC1 + sx = s1 with E \in \{ c, s1\} are determined by
\scrS r(EC1 , c) =
\left\{ clc(E) if r < 1,
\varnothing if r \geq 1,
for E \in \{ c0, c, s1\} ,
and
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
APPLICATION OF THE INFINITE MATRIX THEORY TO THE SOLVABILITY OF SEQUENCE . . . 1051
\scrS r(EC1 , s1) =
\left\{ cl\infty (E) if r < 1,
\varnothing if r \geq 1,
for E \in \{ c, s1\} .
Proof. (i) The inclusion Dr \ast (c0)C1 \subset c is equivalent to DrC
- 1
1 \in (c0, c), that is, DrC
- 1
1 \in S1
and (n+ n - 1)rn \leq K for all n and for some K > 0. So we have Dr \ast (c0)C1 \subset c if and only if
r < 1. Then the inclusion s
(c)
x \subset c is equivalent to x \in c. So the SSIE Dr \ast (c0)C1 + s
(c)
x \subset c is
equivalent to r < 1 and x \in c and the identity in (12) holds for E = c0. Case E = c. The inclusion
Dr \ast cC1 \subset c is equivalent to DrC
- 1
1 \in (c, c), that is,
\bigl[
n - (n - 1)
\bigr]
rn = rn \rightarrow L (n \rightarrow \infty ) for
some scalar L, and
\bigl[
n+(n - 1)
\bigr]
rn = O(1) (n \rightarrow \infty ). This implies r < 1. Using similar arguments
as those above we conclude (12) holds for E = c. The proof of the case E = s1 is similar and is
left to the reader.
(ii) is obtained from (i) and Proposition 5 (i) for \scrS r(EC1 , c) with E = c, or s1, and is obtained
from (i) and Corollary 11 for \scrS r
\bigl(
(c0)C1 , c
\bigr)
. Then the determination of the set \scrS r ((s1)C1 , s1) is
obtained from (i) and Proposition 5 (ii). It remains to determine the set \scrS r(cC1 , s1). For this we deal
with the solvability of the SSIE s1 \subset Dr \ast cC1 + sx. As we have seen in the proof of Proposition 5
we have (rn)n\geq 1 \in M(cC1 , c0) if and only if DrC
- 1
1 \in (c, c0), that is, r < 1. Then we have
(r - n)n\geq 1 \in M(s1, cC1) if and only if C1D1/r \in (s1, c). Since \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty [C1D1/R]nk = 0 for all
k \geq 1, by v) in Lemma 1 we have C1D1/r \in (s1, c) if and only if n - 1
\sum n
k=1
r - k \rightarrow 0 (n \rightarrow \infty )
and r > 1. So we have shown \scrI r(cC1 , s1) = s1 for r < 1 and \scrI r(cC1 , s1) = s1 for r > 1. We
conclude for the set \scrS r(cC1 , s1) using the determination of \scrI \prime
r(cC1 , s1) given in (i).
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Received 30.06.16,
after revision — 12.09.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 8
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| id | umjimathkievua-article-1496 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:54Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/21/83f20290944152929475cf3f178fc121.pdf |
| spelling | umjimathkievua-article-14962019-12-05T08:57:29Z Application of the infinite matrix theory to the solvability of sequence spaces inclusion equations with operators Застосування теорiї нескiнченних матриць до розв’язання вiдношень включення для просторiв послiдовностей з операторами de, Malafosse B. де, Малафоссе Б. UDC 517.9 Given any sequence $a = (a_n)_n \geq 1$ of positive real numbers and any set $E$ of complex sequences, we write $E_a$ for the set of all sequences $y=(y_{n})_{n \geq 1}$ such that y/a = $y/a=(y_{n}/a_{n})_{n\geq 1}\in E.$ In particular, $c_{a}$ denotes the set of all sequences $y$ such that $y/a$ converges. We deal with sequence spaces inclusion equations (SSIE) of the form $F\subset E_{a}+F_{x}'$ with $e\in F$ and explicitly find the solutions of these SSIE when $a=(r^{n})_{n\geq 1},$ $F$ is either $c$ or $s_{1},$ and $E,$ $F'$ are any sets $c_{0},$ $c,$ $s_{1},$ $\ell_{p},$ $w_{0},$ and $w_{\infty }.$ Then we determine the sets of all positive sequences satisfying each of the SSIE $c\subset D_{r}\ast (c_{0})_{\Delta }+c_{x}$ and $c\subset D_{r}\ast (s_{1})_{\Delta}+c_{x},$ where $\Delta $ is the operator of the first difference defined by $\Delta _{n}y=y_{n}-y_{n-1}$ for all $n\geq 1$ with $y_{0}=0.$ Then we solve the SSIE $c\subset D_{r}\ast E_{C_{1}}+s_{x}^{(c)}$ with $E\in \{ c,s_{1}\} $ and $s_{1}\subset D_{r}\ast(s_{1}) _{C_{1}}+s_{x},$ where $C_{1}$, is the Cesaro operator defined by $(C_{1}) _{n}y=n^{-1} \displaystyle \sum \nolimits_{k=1}^{n}y_{k}$ for all $y$. We also deal with the solvability of the sequence spaces equations (SSE) associated with the previous SSIE and defined as $D_{r}\ast E_{C_{1}}+s_{x}^{(c)}=c$ with $E\in \{c_{0},c, s_{1}\} $ and $D_{r}\ast E_{C_{1}}+s_{x}=s_{1}$ with $E\in \{ c,s_{1}\}.$ УДК 517.9 Для заданої послідовності додатних дійсних чисел $a=(a_{n})_{n\geq 1}$ і будь-якої множини комплексних послідовностей $E$ вираз $E_{a}$ позначає множину всіх послідовностей $y=(y_{n})_{n\geq 1}$ таких, що $y/a=(y_{n}/a_{n})_{n\geq 1}\in E.$ Зокрема, $c_{a}$ позначає множину всіх послідовностей $y$ таких, що $y/a$ збігається. Розглянуто відношення включення для просторів послідовностей (ВВПП) вигляду $F\subset E_{a}+F_{x}'$ з $e\in F,$ а також знайдено явні розв'язки цих ВВПП у випадку, коли $a=(r^{n})_{n\geq 1},$ $F$ --- це $c$ або $s_{1},$ а $E$ і $F'$ --- будь-які з множин $c_{0},$ $c,$ $s_{1},$ $\ell_{p},$ $w_{0}$ і $w_{\infty }.$ Крім того, визначено множини всіх додатних послідовностей, що задовольняють кожне з ВВПП $c\subset D_{r}\ast (c_{0})_{\Delta }+c_{x}$ і $c\subset D_{r}\ast (s_{1})_{\Delta}+c_{x},$ де $\Delta $ --- оператор першої різниці, визначений як $\Delta _{n}y=y_{n}-y_{n-1}$ для всіх $n\geq 1$ з $y_{0}=0.$ Також розв'язано ВВПП $c\subset D_{r}\ast E_{C_{1}}+s_{x}^{(c)},$ де $E\in \{ c,s_{1}\} ,$ $s_{1}\subset D_{r}\ast(s_{1}) _{C_{1}}+s_{x},$ а $C_{1}$ --- оператор Чезаро, визначений як сума $(C_{1}) _{n}y=n^{-1} \displaystyle \sum\nolimits_{k=1}^{n}y_{k}$ для всіх $y.$ Крім того, розглянуто питання про існування розв'язків рівнянь для просторів послідовностей (РПП), пов'язаних із попередніми ВВПП і визначених таким чином: $D_{r}\ast E_{C_{1}}+s_{x}^{(c)}=c$ з $E\in \{c_{0},c,s_{1}\} $ і $D_{r}\ast E_{C_{1}}+s_{x}=s_{1}$ з $E\in \{ c,s_{1}\}.$ Institute of Mathematics, NAS of Ukraine 2019-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1496 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 8 (2019); 1040-1052 Український математичний журнал; Том 71 № 8 (2019); 1040-1052 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1496/480 Copyright (c) 2019 de Malafosse B. |
| spellingShingle | de, Malafosse B. де, Малафоссе Б. Application of the infinite matrix theory to the solvability of sequence spaces inclusion equations with operators |
| title | Application of the infinite matrix theory to the solvability of sequence spaces
inclusion equations with operators |
| title_alt | Застосування теорiї нескiнченних матриць до розв’язання вiдношень включення для просторiв послiдовностей з операторами |
| title_full | Application of the infinite matrix theory to the solvability of sequence spaces
inclusion equations with operators |
| title_fullStr | Application of the infinite matrix theory to the solvability of sequence spaces
inclusion equations with operators |
| title_full_unstemmed | Application of the infinite matrix theory to the solvability of sequence spaces
inclusion equations with operators |
| title_short | Application of the infinite matrix theory to the solvability of sequence spaces
inclusion equations with operators |
| title_sort | application of the infinite matrix theory to the solvability of sequence spaces
inclusion equations with operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1496 |
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