Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 507950, 11 pages
http://dx.doi.org/10.1155/2013/507950
Research Article
Some Spaces of Double Sequences Obtained through
Invariant Mean and Related Concepts
S. A. Mohiuddine and Abdullah Alotaibi
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203,
Jeddah 21589, Saudi Arabia
Correspondence should be addressed to S. A. Mohiuddine; mohiuddine@gmail.com
Received 29 November 2012; Accepted 11 March 2013
Academic Editor: Elena Braverman
Copyright © 2013 S. A. Mohiuddine and A. Alotaibi. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We introduce some double sequences spaces involving the notions of invariant mean (or 𝜎-mean) and 𝜎-convergence for double
sequences while the idea of 𝜎-convergence for double sequences was introduced by Çakan et al. 2006, by using the notion of
invariant mean. We determine here some inclusion relations and topological results for these new double sequence spaces.
1. Preliminaries, Background, and Notation
In 1900, Pringsheim [1] presented the following notion of
convergence for double sequences. A double sequence 𝑥 =
(𝑥𝑗𝑘 ) is said to converge to the limit 𝐿 in Pringsheim’s sense
(shortly, 𝑝-convergent to 𝐿) if for every 𝜀 > 0 there exists an
integer 𝑁 such that |𝑥𝑗𝑘 − 𝐿| < 𝜀 whenever 𝑗, 𝑘 > 𝑁. In this
case, 𝐿 is called the 𝑝-limit of 𝑥.
A double sequence 𝑥 = (𝑥𝑗𝑘 ) of real or complex numbers
is said to be bounded if ‖𝑥‖∞ = sup𝑗,𝑘 |𝑥𝑗𝑘 | < ∞. The space of
all bounded double sequences is denoted by M𝑢 .
If 𝑥 ∈ M𝑢 and is 𝑝-convergent to 𝐿, then 𝑥 is said to be
boundedly 𝑝-convergent to 𝐿 (shortly, 𝑏𝑝-convergent to 𝐿). In
this case, 𝐿 is called the 𝑏𝑝-limit of 𝑥. The assumption of 𝑝convergent was made because a double sequence on which
𝑝-convergent is not necessarily bounded.
In general, for any notion of convergence ], the space of
all ]-convergent double sequences will be denoted by C] , the
space of all ]-convergent to 0 double sequences by C]0 and
the limit of a ]-convergent double sequence 𝑥 by ]-lim𝑗,𝑘 𝑥𝑗𝑘 ,
where ] ∈ {𝑝, 𝑏𝑝}.
Let Ω denote the vector space of all double sequences with
the vector space operations defined coordinatewise. Vector
subspaces of Ω are called double sequence spaces. In addition
to the above-mentioned double sequence spaces, we consider
the double sequence space as
}
{
󵄨 󵄨
L𝑢 := {𝑥 ∈ Ω | ‖𝑥‖1 := ∑ 󵄨󵄨󵄨󵄨𝑥𝑗𝑘 󵄨󵄨󵄨󵄨 < ∞}
𝑗,𝑘
}
{
(1)
of all absolutely summable double sequences.
All considered double sequence spaces are supposed to
contain
Φ := span {ejk | 𝑗, 𝑘 ∈ N} ,
(2)
where
1;
jk
eil = {
0;
if (𝑗, 𝑘) = (𝑖, ℓ) ,
otherwise.
(3)
We denote the pointwise sums ∑𝑗,𝑘 ejk , ∑𝑗 ejk (𝑘 ∈ N),
and ∑𝑘 ejk (𝑗 ∈ N) by e, ek , and ej , respectively.
Let 𝜎 be a one-to-one mapping from the set N of natural
numbers into itself. A continuous linear functional 𝜑 on
the space ℓ∞ of bounded single sequences is said to be an
invariant mean or a 𝜎-mean if and only if (i) 𝜑(𝑥) ≥ 0 when
2
Abstract and Applied Analysis
the sequence 𝑥 = (𝑥𝑘 ) has 𝑥𝑘 ≥ 0 for all 𝑘, (ii) 𝜑(𝑒) = 1, where
𝑒 = (1, 1, 1, . . .), and (iii) 𝜑(𝑥) = 𝜑((𝑥𝜎(𝑘) )) for all 𝑥 ∈ ℓ∞ .
Throughout this paper we consider the mapping 𝜎 which
has no finite orbits, that is, 𝜎𝑝 (𝑘) ≠ 𝑘 for all integer 𝑘 ≥ 0
and 𝑝 ≥ 1, where 𝜎𝑝 (𝑘) denotes the 𝑝th iterate of 𝜎 at 𝑘.
Note that a 𝜎-mean extends the limit functional on the space
𝑐 of convergent single sequences in the sense that 𝜑(𝑥) =
lim 𝑥 for all 𝑥 ∈ 𝑐, (see [2]). Consequently, 𝑐 ⊂ 𝑉𝜎 the set
of bounded sequences all of whose 𝜎-means are equal. We
say that a sequence 𝑥 = (𝑥𝑘 ) is 𝜎-convergent if and only if
𝑥 ∈ 𝑉𝜎 . Using this concept, Schaefer [3] defined and characterized 𝜎-conservative, 𝜎-regular, and 𝜎-coercive matrices
for single sequences. If 𝜎 is translation then 𝑉𝜎 is reduced to
the set 𝑓 of almost convergent sequences [4]. Recently, Mohiuddine [5] has obtained an application of almost convergence
for single sequences in approximation theorems and proved
some related results.
In 2006, Çakan et al. [6] presented the following definition of 𝜎-convergence for double sequences and further
studied by Mursaleen and Mohiuddine [7–9]. A double sequence 𝑥 = (𝑥𝑗𝑘 ) of real numbers is said to be 𝜎-convergent to
a number 𝐿 if and only if 𝑥 ∈ V𝜎 , where
For more details of spaces for single and double sequences
and related concepts, we refer to [14–31] and references
therein.
In this paper, we define and study some new spaces
involving the idea of invariant mean and 𝜎-convergence
for double sequences and establish a relation between these
spaces. Further, we extend above spaces to more general
spaces by considering the double sequences 𝛼 = (𝛼𝑗𝑘 ) such
that 𝛼𝑗𝑘 > 0 for all 𝑗, 𝑘 and sup𝑗,𝑘 𝛼𝑗𝑘 = 𝐻 < ∞ and prove
some topological results.
2. The Double Sequence Spaces
We construct the following spaces involving the idea of
invariant mean and 𝜎-convergence for double sequences:
1
(𝑚 + 1) (𝑛 + 1)
W𝜎 = {𝑥 = (𝑥𝑗𝑘 ) :
𝑚
𝑛
× ∑ ∑ 𝜁𝑝𝑞𝑠𝑡 (𝑥 − ℓ𝐸) 󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞,
𝑝=0 𝑞=0
uniformly in 𝑠, 𝑡, for some ℓ} ,
V𝜎 = {𝑥 ∈ M𝑢 : lim 𝜁𝑝𝑞𝑠𝑡 (𝑥) = 𝐿 uniformly in 𝑠, 𝑡;
𝑝,𝑞 → ∞
[W𝜎 ] = {𝑥 = (𝑥𝑗𝑘 ) :
𝐿 = 𝜎-lim𝑥} ,
𝑝
𝑚 𝑛
󵄨
󵄨
× ∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥 − ℓ𝐸)󵄨󵄨󵄨󵄨 󳨀→ 0 as 𝑚, 𝑛 󳨀→ ∞,
𝑞
1
𝜁𝑝𝑞𝑠𝑡 (𝑥) =
∑ ∑𝑥 𝑗 𝑘 ,
(𝑝 + 1) (𝑞 + 1) 𝑗=0 𝑘=0 𝜎 (𝑠),𝜎 (𝑡)
𝑝=0 𝑞=0
(4)
while here the limit means 𝑏𝑝-limit. Let us denote by V𝜎 the
space of 𝜎-convergent double sequences 𝑥 = (𝑥𝑗𝑘 ). For 𝜎(𝑛) =
𝑛 + 1, the set V𝜎 is reduced to the set F of almost convergent
double sequences [10]. Note that C𝑏𝑝 ⊂ V𝜎 ⊂ M𝑢 .
Maddox [11] has defined the concepts of strong almost
convergence and 𝑀-convergence for single sequences and
established inclusion relation between strong almost convergence, 𝑀-convergence, and almost convergence for single
sequence. Başarir [12] extended the notion of strong almost
convergence from single sequences to double sequences and
proved some interesting results involving this idea and the
notion of almost convergence for double sequences. In the
recent past, Mursaleen and Mohiuddine [13] presented the
notions of absolute and strong 𝜎-convergence for double
sequences. A bounded double sequence 𝑥 = (𝑥𝑗𝑘 ) is said to
be strongly 𝜎-convergent if there exists a number ℓ such that
𝑝
𝑞
1
󵄨
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝑥𝜎𝑗 (𝑠),𝜎𝑘 (𝑡) − ℓ󵄨󵄨󵄨󵄨 󳨀→ 0
(𝑝 + 1) (𝑞 + 1) 𝑗=0 𝑘=0
1
(𝑚 + 1) (𝑛 + 1)
uniformly in 𝑠, 𝑡, for some ℓ} ,
W󸀠𝜎 = {𝑥 = (𝑥𝑗𝑘 ) :
∞ ∞
󵄨
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑝=0 𝑞=0
converges uniformly in 𝑠, 𝑡} ,
W󸀠󸀠𝜎 = {𝑥 = (𝑥𝑗𝑘 ) :
∞ ∞
󵄨
sup ∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡
𝑠,𝑡
𝑝=0 𝑞=0
(5)
as 𝑝, 𝑞 󳨀→ ∞ uniformly in 𝑠, 𝑡,
while here the limit means 𝑏𝑝-limit. In this case, we write
[V𝜎 ]-lim 𝑥 = ℓ. Let us denote by [V𝜎 ] the set of all strongly
𝜎-convergent sequences 𝑥 = (𝑥𝑗𝑘 ). If 𝜎 is translation then
[V𝜎 ] is reduced to the set [F] of strong almost convergence
double sequences due to Başarir [12].
󵄨
+ 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨 < ∞} ,
(6)
where 𝐸 = (𝑒𝑗𝑘 ) with 𝑒𝑗𝑘 = 1 for all 𝑗, 𝑘;
𝜉𝑚𝑛𝑠𝑡 = 𝜉𝑚𝑛𝑠𝑡 (𝑥) =
𝑚 𝑛
1
∑ ∑ 𝜁𝑝𝑞𝑠𝑡 (𝑥) ,
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
Abstract and Applied Analysis
3
(L2) (1/𝑢V) ∑𝑢𝑚=1 ∑V𝑛=1 |Ω†𝑚𝑛𝑠𝑡 − ℓ| → 0 as 𝑢, V → ∞
(uniformly in 𝑠, 𝑡);
𝜉0,0,𝑠,𝑡 (𝑥) = 𝜁0,0,𝑠,𝑡 = 𝑥𝑠,𝑡 ,
𝜉−1,0,𝑠,𝑡 (𝑥) = 𝜁−1,0,𝑠,𝑡 (𝑥) = 𝑥𝑠−1,𝑡 ,
(L3) (1/𝑢V) ∑𝑢𝑚=1 ∑V𝑛=1 |Ω‡𝑚𝑛𝑠𝑡 − ℓ| → 0 as 𝑢, V → ∞
(uniformly in 𝑠, 𝑡);
𝜉0,−1,𝑠,𝑡 (𝑥) = 𝜁0,−1,𝑠,𝑡 (𝑥) = 𝑥𝑠,𝑡−1 ,
𝜉−1,−1,𝑠,𝑡 (𝑥) = 𝜁−1,−1,𝑠,𝑡 (𝑥) = 𝑥𝑠−1,𝑡−1 .
(7)
(L4) (1/𝑢V) ∑𝑢𝑚=1 ∑V𝑛=1 |𝜁𝑚𝑛𝑠𝑡 + 𝜉𝑚𝑛𝑠𝑡 − Ω†𝑚𝑛𝑠𝑡 − Ω‡𝑚𝑛𝑠𝑡 | →
0 as 𝑢, V → ∞ (uniformly in 𝑠, 𝑡),
where
Remark 1. If [W𝜎 ]-lim 𝑥 = ℓ, that is,
𝑚 𝑛
1
󵄨
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 − ℓ󵄨󵄨󵄨󵄨 󳨀→ 0
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
Ω†𝑚𝑛𝑠𝑡 =
(8)
Ω‡𝑚𝑛𝑠𝑡
as 𝑚, 𝑛 → ∞, uniformly in 𝑠, 𝑡; then
󵄨󵄨
𝑝
𝑚 𝑛 󵄨󵄨󵄨
󵄨󵄨
1
󵄨󵄨 1
∑ ∑ 󵄨󵄨󵄨
∑𝜁𝑗𝑞𝑠𝑡 − ℓ󵄨󵄨󵄨󵄨 󳨀→ 0,
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0 󵄨󵄨 𝑝 + 1 𝑗=0
󵄨󵄨
󵄨
󵄨
󵄨
󵄨󵄨
𝑞
𝑚 𝑛 󵄨󵄨
󵄨󵄨
1
󵄨󵄨 1
∑ ∑ 󵄨󵄨
∑ 𝜁𝑝𝑘𝑠𝑡 − ℓ󵄨󵄨󵄨 󳨀→ 0.
󵄨
󵄨󵄨
+
1)
+
1)
𝑞
+
1
(𝑚
(𝑛
𝑝=0 𝑞=0 󵄨󵄨
𝑘=0
󵄨
𝑚
𝑖=1
𝑖=1
∑V𝑖 (𝑢𝑖 ∓ 𝑢𝑖+1 ) = ∑𝑢𝑖 (V𝑖 ∓ V𝑖−1 ) ∓ 𝑢𝑚+1 V𝑚 .
(9)
(10)
(13)
1 𝑢 V 󵄨󵄨
󵄨
+ 𝜉𝑚𝑛𝑠𝑡 − Ω†𝑚𝑛𝑠𝑡 − Ω‡𝑚𝑛𝑠𝑡 󵄨󵄨󵄨󵄨
∑ ∑ 󵄨󵄨𝜁
𝑢V 𝑚=1 𝑛=1 󵄨 𝑚𝑛𝑠𝑡
=
1 𝑢 V 󵄨󵄨
󵄨
− ℓ + 𝜉𝑚𝑛𝑠𝑡 − ℓ − Ω†𝑚𝑛𝑠𝑡 + ℓ − Ω‡𝑚𝑛𝑠𝑡 + ℓ󵄨󵄨󵄨󵄨
∑ ∑ 󵄨󵄨𝜁
𝑢V 𝑚=1 𝑛=1 󵄨 𝑚𝑛𝑠𝑡
≤
𝑢 V
1 𝑢 V 󵄨󵄨
󵄨
󵄨 1
󵄨
∑ ∑ 󵄨󵄨𝜁𝑚𝑛𝑠𝑡 − ℓ󵄨󵄨󵄨 +
∑ ∑ 󵄨󵄨󵄨𝜉𝑚𝑛𝑠𝑡 − ℓ󵄨󵄨󵄨
𝑢V 𝑚=1 𝑛=1
𝑢V 𝑚=1 𝑛=1
+
We get Abel’s transformation for double series
𝑝
𝑛
1
=
∑ 𝜁𝑚𝑞𝑠𝑡 .
(𝑛 + 1) 𝑞=0
Proof. Suppose that [W𝜎 ]-lim 𝑥 = ℓ. Thus, we have W𝜎 lim 𝑥 = ℓ, that is, (L1) holds. We see that conditions (L2) and
(L3) follows from the Remark 1. Write
We remark that by using Abel’s transformation for single
series
𝑚
𝑚
1
∑ 𝜁𝑝𝑛𝑠𝑡 ,
(𝑚 + 1) 𝑝=0
𝑞
∑ ∑ V𝑗𝑘 (𝑢𝑗𝑘 − 𝑢𝑗+1,𝑘 − 𝑢𝑗,𝑘+1 + 𝑢𝑗+1,𝑘+1 )
1 𝑢 V 󵄨󵄨 †
󵄨 1 𝑢 V 󵄨󵄨 ‡
󵄨
− ℓ󵄨󵄨󵄨󵄨
∑ ∑ 󵄨󵄨󵄨Ω𝑚𝑛𝑠𝑡 − ℓ󵄨󵄨󵄨󵄨 +
∑ ∑ 󵄨󵄨Ω
𝑢V 𝑚=1 𝑛=1
𝑢V 𝑚=1 𝑛=1 󵄨 𝑚𝑛𝑠𝑡
= Σ1 + Σ2 + Σ3 + Σ4
𝑗=1 𝑘=1
(say) .
(14)
𝑝
𝑞
𝑝
= ∑ ∑ 𝑢𝑗𝑘 (Δ 11 V𝑗𝑘 ) − ∑ 𝑢𝑗,𝑞+1 (Δ 10 V𝑗𝑞 )
𝑗=1 𝑘=1
(11)
𝑗=1
𝑞
− ∑ 𝑢𝑝+1,𝑘 (Δ 01 V𝑝𝑘 ) + 𝑢𝑝+1,𝑞+1 V𝑝𝑞 ,
𝑘=1
where
Δ 10 V𝑗𝑞 = V𝑗𝑞 − V𝑗−1,𝑞 ,
Δ 01 V𝑝𝑘 = V𝑝𝑘 − V𝑝,𝑘−1
Δ 11 V𝑗𝑘 = V𝑗𝑘 − V𝑗−1,𝑘 − V𝑗,𝑘−1 + V𝑗−1,𝑘−1 .
(12)
In the recent past, Altay and Başar [32] also presented
another form of Abel’s transformation for double series.
By our assumption, that is, [W𝜎 ]-lim 𝑥 = ℓ, Σ1 → 0 as
𝑢, V → ∞ uniformly in 𝑠, 𝑡. The condition (L1) implies
that 𝜉𝑚𝑛𝑠𝑡 tends to zero as 𝑚, 𝑛 tending to ∞ uniformly in
𝑠, 𝑡; therefore Σ2 → 0 as 𝑢, V → ∞ uniformly in 𝑠, 𝑡 and
Σ3 , Σ4 → 0 as 𝑢, V → ∞ uniformly in 𝑠, 𝑡 by the conclusion (L2) and (L3), respectively. Thus, (14) tends to zero
as 𝑢, V → ∞ uniformly in 𝑠, 𝑡, that is, (L4) holds.
Conversely, let (L1)–(L4) hold. Then,
1 𝑢 V 󵄨󵄨
󵄨
− ℓ󵄨󵄨󵄨
∑ ∑ 󵄨𝜁
𝑢V 𝑚=1 𝑛=1 󵄨 𝑚𝑛𝑠𝑡
≤
1 𝑢 V 󵄨󵄨
󵄨
+ 𝜉𝑚𝑛𝑠𝑡 − Ω†𝑚𝑛𝑠𝑡 − Ω‡𝑚𝑛𝑠𝑡 󵄨󵄨󵄨󵄨
∑ ∑ 󵄨󵄨𝜁
𝑢V 𝑚=1 𝑛=1 󵄨 𝑚𝑛𝑠𝑡
3. Inclusion Relations
+
1 𝑢 V 󵄨󵄨
󵄨
− ℓ󵄨󵄨󵄨
∑ ∑ 󵄨𝜉
𝑢V 𝑚=1 𝑛=1 󵄨 𝑚𝑛𝑠𝑡
In the following theorem, we establish a relationship between
spaces defined in Section 2. Before proceeding further, first
we prove the following lemmas which will be used to prove
our inclusion relations.
+
1 𝑢 V 󵄨󵄨 †
󵄨
− ℓ󵄨󵄨󵄨󵄨
∑ ∑ 󵄨󵄨Ω
𝑢V 𝑚=1 𝑛=1 󵄨 𝑚𝑛𝑠𝑡
+
1 𝑢 V 󵄨󵄨 ‡
󵄨
− ℓ󵄨󵄨󵄨󵄨
∑ ∑ 󵄨󵄨Ω
𝑢V 𝑚=1 𝑛=1 󵄨 𝑚𝑛𝑠𝑡
Lemma 2. Consider that [W𝜎 ]-lim 𝑥 = ℓ if and only if
(L1) W𝜎 -lim 𝑥 = ℓ;
→ 0 (𝑢, V → ∞) uniformly in 𝑠, 𝑡.
(15)
4
Abstract and Applied Analysis
Lemma 3. One has
𝜁𝑚𝑛𝑠𝑡 + 𝜉𝑚𝑛𝑠𝑡 − Ω†𝑚𝑛𝑠𝑡 − Ω‡𝑚𝑛𝑠𝑡
= 𝑚𝑛 [𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡 ] .
=
𝑚
1
1 𝑛−1
{(𝑚 + 1) 𝜁𝑚𝑞𝑠𝑡 − ∑ 𝜁𝑝𝑞𝑠𝑡 }]
∑[
𝑛 𝑞=0 𝑚 (𝑚 + 1)
𝑝=0
=
𝑚 𝑛−1
1
1 𝑛−1
∑ 𝜁𝑚𝑞𝑠𝑡 −
∑ ∑ 𝜁𝑝𝑞𝑠𝑡
𝑚𝑛 𝑞=0
𝑚 (𝑚 + 1) 𝑛 𝑝=0 𝑞=0
=
1 𝑛−1
1
∑ 𝜁𝑚𝑞𝑠𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 .
𝑚𝑛 𝑞=0
𝑚
(16)
Proof. Since
𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡
=[
𝑚 𝑛
𝑚−1 𝑛
1
1
∑ ∑ 𝜁𝑝𝑞𝑠𝑡 −
∑ ∑𝜁 ]
𝑚 (𝑛 + 1) 𝑝=0 𝑞=0 𝑝𝑞𝑠𝑡
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
(19)
Substituting (18) and (19) in (17), we get
𝑚 𝑛−1
1
1 𝑚−1 𝑛−1
−[
∑ ∑ 𝜁𝑝𝑞𝑠𝑡 −
∑ ∑𝜁 ] .
𝑚𝑛 𝑝=0 𝑞=0 𝑝𝑞𝑠𝑡
(𝑚 + 1) 𝑛 𝑝=0 𝑞=0
𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡
(17)
First, we solve the expression in the first bracket
[
𝑛
1
1 𝑛−1
1
∑ 𝜁𝑚𝑞𝑠𝑡 −
∑ 𝜁𝑚𝑞𝑠𝑡 − 𝜉𝑚𝑛𝑠𝑡
𝑚 (𝑛 + 1) 𝑞=0
𝑚𝑛 𝑞=0
𝑚
=
𝑚 𝑛
𝑚−1 𝑛
1
1
∑ ∑ 𝜁𝑝𝑞𝑠𝑡 −
∑ ∑𝜁 ]
𝑚 (𝑛 + 1) 𝑝=0 𝑞=0 𝑝𝑞𝑠𝑡
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
+
1
=
𝑚 (𝑚 + 1) (𝑛 + 1)
=
𝑛
𝑚
𝑚−1
𝑞=0
𝑝=0
𝑝=0
=
𝑛
=
𝑚−1
1
∑ [𝑚𝜁𝑚𝑞𝑠𝑡 − ∑ 𝜁𝑝𝑞𝑠𝑡 ]
𝑚 (𝑚 + 1) (𝑛 + 1) 𝑞=0
𝑝=0
=
𝑛
𝑚
1
1
∑ [𝜁𝑚𝑞𝑠𝑡 −
∑ 𝜁𝑝𝑞𝑠𝑡 ]
𝑚 (𝑛 + 1) 𝑞=0
(𝑚 + 1) 𝑝=0
𝑛
𝑚 𝑛
1
1
∑ 𝜁𝑚𝑞𝑠𝑡 −
∑ ∑ 𝜁𝑝𝑞𝑠𝑡
𝑚 (𝑛 + 1) 𝑞=0
𝑚 (𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
=
𝑛
1
1
∑ 𝜁𝑚𝑞𝑠𝑡 − 𝜉𝑚𝑛𝑠𝑡 .
𝑚 (𝑛 + 1) 𝑞=0
𝑚
=
(18)
𝑚 𝑛−1
1
1 𝑚−1 𝑛−1
∑ ∑ 𝜁𝑝𝑞𝑠𝑡 −
∑ ∑𝜁
𝑚𝑛 𝑝=0 𝑞=0 𝑝𝑞𝑠𝑡
(𝑚 + 1) 𝑛 𝑝=0 𝑞=0
𝑚
𝑚−1
1 𝑛−1
1
= ∑[
{𝑚 ∑ 𝜁𝑝𝑞𝑠𝑡 − (𝑚 + 1) ∑ 𝜁𝑝𝑞𝑠𝑡 }]
𝑛 𝑞=0 𝑚 (𝑚 + 1)
𝑝=0
𝑝=0
=
𝑚−1
1 𝑛−1
1
{𝑚𝜁𝑚𝑞𝑠𝑡 − ∑ 𝜁𝑝𝑞𝑠𝑡 }]
∑[
𝑛 𝑞=0 𝑚 (𝑚 + 1)
𝑝=0
1
− 𝜉𝑚,𝑛−1,𝑠,𝑡 )
(𝜉
𝑚 𝑚𝑛𝑠𝑡
1
− 𝜉𝑚,𝑛−1,𝑠,𝑡 )
(𝜉
𝑚 𝑚𝑛𝑠𝑡
𝑛
1
1
[𝜁𝑚𝑛𝑠𝑡 −
∑ 𝜁𝑚𝑞𝑠𝑡 ]
𝑚𝑛
(𝑛 + 1) 𝑞=0
−
Now, the expression in the second bracket
(20)
𝑛
1
[(𝑛 + 1) 𝜁𝑚𝑛𝑠𝑡 − ∑ 𝜁𝑚𝑞𝑠𝑡 ]
𝑚𝑛 (𝑛 + 1)
𝑞=0
−
=
1
− 𝜉𝑚,𝑛−1,𝑠,𝑡 )
(𝜉
𝑚 𝑚𝑛𝑠𝑡
𝑛−1
1
[𝑛𝜁𝑚𝑛𝑠𝑡 − ∑ 𝜁𝑚𝑞𝑠𝑡 ]
𝑚𝑛 (𝑛 + 1)
𝑞=0
−
𝑛
𝑚
1
=
∑ [(𝑚 + 1) 𝜁𝑚𝑞𝑠𝑡 − ∑ 𝜁𝑝𝑞𝑠𝑡 ]
𝑚 (𝑚 + 1) (𝑛 + 1) 𝑞=0
𝑝=0
=
𝑛
𝑛−1
1
[𝑛 ∑ 𝜁𝑚𝑞𝑠𝑡 − (𝑛 + 1) ∑ 𝜁𝑚𝑞𝑠𝑡 ]
𝑚𝑛 (𝑛 + 1) 𝑞=0
𝑞=0
−
× [ ∑ (𝑚 ∑ 𝜁𝑝𝑞𝑠𝑡 − (𝑚 + 1) ∑ 𝜁𝑝𝑞𝑠𝑡 )]
1
𝜉
𝑚 𝑚,𝑛−1,𝑠,𝑡
1
− 𝜉𝑚,𝑛−1,𝑠,𝑡 ) .
(𝜉
𝑚 𝑚𝑛𝑠𝑡
We know that
𝜉𝑚𝑛𝑠𝑡 =
=
𝜉𝑚−1,𝑛,𝑠,𝑡 =
𝑚 𝑛
1
∑ ∑ 𝜁𝑝𝑞𝑠𝑡
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
𝑚−1 𝑛
𝑛
1
[ ∑ ∑ 𝜁𝑝𝑞𝑠𝑡 + ∑ 𝜁𝑚𝑞𝑠𝑡 ] , (21)
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
𝑞=0
𝑚−1 𝑛
1
∑ ∑𝜁 .
𝑚 (𝑛 + 1) 𝑝=0 𝑞=0 𝑝𝑞𝑠𝑡
Abstract and Applied Analysis
5
From (21), we have
Also, we have
(𝑚 + 1) 𝜉𝑚𝑛𝑠𝑡 − 𝑚𝜉𝑚−1,𝑛,𝑠,𝑡 =
𝑛
1
∑𝜁 .
(𝑛 + 1) 𝑞=0 𝑚𝑞𝑠𝑡
(22)
Thus, (20) becomes
󵄨󵄨
󵄨󵄨 𝑚 𝑛
󵄨󵄨
1
󵄨󵄨󵄨
󵄨󵄨 ∑ ∑ 𝜁𝑝𝑞𝑠𝑡 (𝑥 − ℓ)󵄨󵄨󵄨
󵄨󵄨
󵄨
(𝑚 + 1) (𝑛 + 1) 󵄨󵄨𝑝=0 𝑞=0
󵄨󵄨
󵄨
𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡
=
1
− (𝑚 + 1) 𝜉𝑚𝑛𝑠𝑡 + 𝑚𝜉𝑚−1,𝑛,𝑠,𝑡 ]
[𝜁
𝑚𝑛 𝑚𝑛𝑠𝑡
−
=
1
− 𝜉𝑚,𝑛−1,𝑠,𝑡 )
(𝜉
𝑚 𝑚𝑛𝑠𝑡
≤
𝑚 𝑛
1
󵄨
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥 − ℓ)󵄨󵄨󵄨󵄨
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
≤
𝑚 𝑛
1
∑ ∑ 𝜁𝑝𝑞𝑠𝑡 (|𝑥 − ℓ|) .
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
(23)
Hence, [V𝜎 ] ⊂ [W𝜎 ] ⊂ W𝜎 and
1
− 𝜉𝑚𝑛𝑠𝑡 − 𝑚 (𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 )
[𝜁
𝑚𝑛 𝑚𝑛𝑠𝑡
[W𝜎 ]-lim 𝑥 = W𝜎 -lim 𝑥 = ℓ.
−𝑛 (𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 )] .
𝑛
1
∑ 𝜁𝑚𝑞𝑠𝑡 − 𝜉𝑚𝑛𝑠𝑡 .
(𝑛 + 1) 𝑞=0
(24)
Similarly, we can write
𝑛 (𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 ) =
𝑚
1
∑ 𝜁𝑝𝑛𝑠𝑡 − 𝜉𝑚𝑛𝑠𝑡 .
(𝑚 + 1) 𝑞=0
∞ ∞
󵄨
󵄨
I𝑚𝑛𝑠𝑡 = ∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑝=𝑚 𝑞=𝑛
(32)
󳨀→ 0
(25)
as 𝑚, 𝑛 → ∞, uniformly in 𝑠, 𝑡; and
Using (24) and (25) in (23), we get
𝜉𝑝𝑞𝑠𝑡 󳨀→ ℓ
𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡
=
(31)
(ii) We have to show that W󸀠𝜎 ⊂ [W𝜎 ]. If 𝑥 ∈ W󸀠𝜎 , then we
have
Also (22) can be written as
𝑚 (𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 ) =
(30)
𝑚
𝑛
1
1
1
[𝜁𝑚𝑛𝑠𝑡 +𝜉𝑚𝑛𝑠𝑡 −
∑ 𝜁𝑝𝑛𝑠𝑡 −
∑ 𝜁𝑚𝑞𝑠𝑡 ] .
𝑚𝑛
(𝑚 + 1) 𝑝=0
(𝑛 + 1) 𝑞=0
(26)
(say) as 𝑝, 𝑞 󳨀→ ∞, uniformly in 𝑠, 𝑡,
(33)
that is, W𝜎 -lim 𝑥 = ℓ.
In order to prove that 𝑥 ∈ [W𝜎 ], it is enough to show that
condition (L4) of Lemma 2 holds. Now,
I𝑚𝑛𝑠𝑡 − I𝑚,𝑛+1,𝑠,𝑡 − I𝑚+1,𝑛,𝑠,𝑡 + I𝑚+1,𝑛+1,𝑠,𝑡
This implies that
∞ ∞
󵄨
󵄨
= ∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑝=𝑚 𝑞=𝑛
𝜁𝑚𝑛𝑠𝑡 + 𝜉𝑚𝑛𝑠𝑡 − Ω†𝑚𝑛𝑠𝑡 − Ω‡𝑚𝑛𝑠𝑡
= 𝑚𝑛 [𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡 ] .
(27)
Theorem 4. One has the following inclusions and the limit is
preserved in each case:
(i) [V𝜎 ] ⊂ [W𝜎 ] ⊂ W𝜎 ,
(ii) W󸀠𝜎 ⊂ [W𝜎 ] if the conditions (L2) and (L3) of
Lemma 2 hold,
(iii) W󸀠𝜎 ⊂ W󸀠󸀠𝜎 .
Proof. (i) Let 𝑥 ∈ [V𝜎 ] with [V𝜎 ]-lim 𝑥 = ℓ, say. Then,
𝜁𝑝𝑞𝑠𝑡 (|𝑥 − ℓ|) 󳨀→ 0
This implies that
as 𝑝, 𝑞 󳨀→ ∞, uniformly in 𝑠, 𝑡.
𝑞=𝑛+1
∞
∞
󵄨
󵄨
− ∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑞=𝑛
𝑝=𝑚+1
∞
+ ∑
∞
󵄨
󵄨
∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑝=𝑚+1 𝑞=𝑛+1
∞
∞
∞
𝑝=𝑚
𝑞=𝑛
𝑞=𝑛+1
= ∑ [∑ − ∑ ]
as 𝑝, 𝑞 󳨀→ ∞, uniformly in 𝑠, 𝑡. (28)
𝑚 𝑛
1
∑ ∑ 𝜁𝑝𝑞𝑠𝑡 (|𝑥 − ℓ|) 󳨀→ 0
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
∞
∞
󵄨
󵄨
− ∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑝=𝑚
(29)
󵄨
󵄨
× 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
∞
∞
∞
𝑝=𝑚+1
𝑞=𝑛
𝑞=𝑛+1
− ∑ [∑ − ∑ ]
󵄨
󵄨
× 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
6
Abstract and Applied Analysis
∞
󵄨
󵄨
= ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑛𝑠𝑡 − 𝜉𝑝−1,𝑛,𝑠,𝑡 − 𝜉𝑝,𝑛−1,𝑠,𝑡 + 𝜉𝑝−1,𝑛−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑝=𝑚
Hence, it is left to show that for fixed 𝑝, 𝑞
󵄨
󵄨󵄨
󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨 ≤ 𝐾,
󵄨
󵄨
∞
󵄨
󵄨
− ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑛𝑠𝑡 − 𝜉𝑝−1,𝑛,𝑠,𝑡 − 𝜉𝑝,𝑛−1,𝑠,𝑡 + 𝜉𝑝−1,𝑛−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
From (40), we have
𝑝=𝑚+1
∞
󵄨󵄨
󵄨
󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨 < 1,
󵄨
󵄨
∞
󵄨
󵄨
= [ ∑ − ∑ ] 󵄨󵄨󵄨󵄨𝜉𝑝𝑛𝑠𝑡 − 𝜉𝑝−1,𝑛,𝑠,𝑡 − 𝜉𝑝,𝑛−1,𝑠,𝑡 + 𝜉𝑝−1,𝑛−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑝=𝑚
𝑝=𝑚+1
󵄨
󵄨
= 󵄨󵄨󵄨𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡 󵄨󵄨󵄨 .
(34)
󵄨
󵄨
= 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨 .
By Lemma 3, we have
𝜁𝑝𝑞𝑠𝑡 + 𝜉𝑝𝑞𝑠𝑡 − Ω†𝑝𝑞𝑠𝑡 − Ω‡𝑝𝑞𝑠𝑡
= 𝑝𝑞 [𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 ] .
(42)
for every fixed 𝑝 > 𝑝0 , 𝑞 > 𝑞0 and for all 𝑠, 𝑡. Since
𝑚 (𝑚 + 1) 𝑛 (𝑛 + 1) (𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡 )
𝑚
Replacing 𝑚 and 𝑛 by 𝑝 and 𝑞, respectively, we have
I𝑝𝑞𝑠𝑡 − I𝑝,𝑞+1,𝑠,𝑡 − I𝑝+1,𝑞,𝑠,𝑡 + I𝑝+1,𝑞+1,𝑠,𝑡
∀𝑠, 𝑡. (41)
𝑛
= ∑ ∑ 𝑝𝑞 (𝜁𝑝𝑞𝑠𝑡 − 𝜁𝑝−1,𝑞,𝑠,𝑡 − 𝜁𝑝,𝑞−1,𝑠,𝑡 + 𝜁𝑝−1,𝑞−1,𝑠,𝑡 ) .
𝑝=1 𝑞=1
(43)
(35)
Accordingly,
𝑚𝑛 [ (𝑚 + 1) (𝑛 + 1)
(36)
× (𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡 )
− (𝑚 − 1) (𝑛 + 1)
So that we have
1 𝑚 𝑛 󵄨󵄨
󵄨
∑ ∑ 󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 + 𝜉𝑝𝑞𝑠𝑡 − Ω†𝑝𝑞𝑠𝑡 − Ω‡𝑝𝑞𝑠𝑡 󵄨󵄨󵄨󵄨
𝑚𝑛 𝑝=1 𝑞=1
1 𝑚 𝑛
=
∑ ∑ 𝑝𝑞 [I𝑝𝑞𝑠𝑡 − I𝑝,𝑞+1,𝑠,𝑡
𝑚𝑛 𝑝=1 𝑞=1
× (𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚−2,𝑛,𝑠,𝑡 − 𝜉𝑚−1,𝑛−1,𝑠,𝑡 + 𝜉𝑚−2,𝑛−1,𝑠,𝑡 )
− (𝑚 + 1) (𝑛 − 1)
(37)
+ (𝑚 − 1) (𝑛 − 1)
−I𝑝+1,𝑞,𝑠,𝑡 + I𝑝+1,𝑞+1,𝑠,𝑡 ] .
By using Abel’s transformation for double series in the right
hand side of above equation, we have
1 𝑚 𝑛 󵄨󵄨
󵄨
∑ ∑ 󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 + 𝜉𝑝𝑞𝑠𝑡 − Ω†𝑝𝑞𝑠𝑡 − Ω‡𝑝𝑞𝑠𝑡 󵄨󵄨󵄨󵄨
𝑚𝑛 𝑝=1 𝑞=1
𝑚 𝑛
𝑛
1
=
[ ∑ ∑ I𝑝𝑞𝑠𝑡 − 𝑚 ∑ I𝑚+1,𝑞,𝑠,𝑡
𝑚𝑛 𝑝=1 𝑞=1
𝑞=1
× (𝜉𝑚,𝑛−1,𝑠,𝑡 − 𝜉𝑚−1,𝑛−1,𝑠,𝑡 − 𝜉𝑚,𝑛−2,𝑠,𝑡 + 𝜉𝑚−1,𝑛−2,𝑠,𝑡 )
× (𝜉𝑚−1,𝑛−1,𝑠,𝑡 − 𝜉𝑚−2,𝑛−1,𝑠,𝑡 − 𝜉𝑚−1,𝑛−2,𝑠,𝑡 + 𝜉𝑚−2,𝑛−2,𝑠,𝑡 ) ]
𝑛
𝑚
𝑞=1
𝑝=1
= ∑ 𝑞 [ ∑ 𝑝 (𝜁𝑝𝑞𝑠𝑡 − 𝜁𝑝−1,𝑞,𝑠,𝑡 − 𝜁𝑝,𝑞−1,𝑠,𝑡 + 𝜁𝑝−1,𝑞−1,𝑠,𝑡 )
𝑚−1
− ∑ 𝑝 (𝜁𝑝𝑞𝑠𝑡 − 𝜁𝑝−1,𝑞,𝑠,𝑡 − 𝜁𝑝,𝑞−1,𝑠,𝑡 + 𝜁𝑝−1,𝑞−1,𝑠,𝑡 )]
(38)
𝑚
𝑝=1
𝑛−1
𝑚
𝑞=1
𝑝=1
− ∑ 𝑞 [ ∑ 𝑝 (𝜁𝑝𝑞𝑠𝑡 − 𝜁𝑝−1,𝑞,𝑠,𝑡 − 𝜁𝑝,𝑞−1,𝑠,𝑡 + 𝜁𝑝−1,𝑞−1,𝑠,𝑡 )
−𝑛 ∑ I𝑝,𝑛+1,𝑠,𝑡 + 𝑚𝑛I𝑚+1,𝑛+1,𝑠,𝑡 ]
𝑝=1
𝑚−1
→ 0 as 𝑚, 𝑛 → ∞, uniformly in 𝑠, 𝑡 (by (32)). Hence, by
Lemma 2, 𝑥 ∈ [W𝜎 ].
(iii) Let 𝑥 ∈ W󸀠𝜎 , and we have to show that
− ∑ 𝑝 (𝜁𝑝𝑞𝑠𝑡 − 𝜁𝑝−1,𝑞,𝑠,𝑡
𝑝=1
∞ ∞
󵄨
󵄨
sup ∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨 ≤ 𝐾,
𝑠,𝑡
− 𝜁𝑝,𝑞−1,𝑠,𝑡 + 𝜁𝑝−1,𝑞−1,𝑠,𝑡 ) ]
𝑝=0 𝑞=0
(39)
where 𝐾 is an absolute constant. Since 𝑥 ∈ W󸀠𝜎 , there exist
integers 𝑝0 , 𝑞0 such that
󵄨
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨 < 1, ∀𝑠, 𝑡.
𝑝>𝑝0 𝑞>𝑞0
(40)
𝑛
= ∑ 𝑞 [𝑚 (𝜁𝑚𝑞𝑠𝑡 − 𝜁𝑚−1,𝑞,𝑠,𝑡 − 𝜁𝑚,𝑞−1,𝑠,𝑡 + 𝜁𝑚−1,𝑞−1,𝑠,𝑡 )]
𝑞=1
𝑛−1
− ∑ 𝑞 [𝑚 (𝜁𝑚𝑞𝑠𝑡 − 𝜁𝑚−1,𝑞,𝑠,𝑡 − 𝜁𝑚,𝑞−1,𝑠,𝑡 + 𝜁𝑚−1,𝑞−1,𝑠,𝑡 )] .
𝑞=1
(44)
Abstract and Applied Analysis
7
𝑚−1 𝑛
𝑚 𝑛−1
𝑚−1 𝑛−1 }
{𝑚 𝑛
= { ∑ ∑ 𝑗𝑘 − ∑ ∑ 𝑗𝑘 − ∑ ∑ 𝑗𝑘 + ∑ ∑ 𝑗𝑘}
𝑗=1 𝑘=1
𝑗=1 𝑘=1
𝑗=1 𝑘=1
{𝑗=1 𝑘=1
}
This implies that
(𝑚 + 1) (𝑛 + 1) (𝜉𝑚𝑛𝑠𝑡 − 𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚,𝑛−1,𝑠,𝑡 + 𝜉𝑚−1,𝑛−1,𝑠,𝑡 )
𝑑𝑗󸀠
− (𝑚 − 1) (𝑛 + 1)
× ( ∑ ∑ 𝑎𝑢V )
𝑢=𝑑𝑗 V=ℎ𝑘
× (𝜉𝑚−1,𝑛,𝑠,𝑡 − 𝜉𝑚−2,𝑛,𝑠,𝑡 − 𝜉𝑚−1,𝑛−1,𝑠,𝑡 + 𝜉𝑚−2,𝑛−1,𝑠,𝑡 )
𝑚
𝑚−1
𝑛−1
𝑚
𝑚−1
}
{𝑛
= { ∑ 𝑘 [ ∑𝑗 − ∑ 𝑗] − ∑ 𝑘 [∑𝑗 + ∑ 𝑗]}
𝑗=1
𝑗=1
] 𝑘=1 [𝑗=1
]}
{𝑘=1 [𝑗=1
− (𝑚 + 1) (𝑛 − 1)
× (𝜉𝑚,𝑛−1,𝑠,𝑡 − 𝜉𝑚−1,𝑛−1,𝑠,𝑡 − 𝜉𝑚,𝑛−2,𝑠,𝑡 + 𝜉𝑚−1,𝑛−2,𝑠,𝑡 )
𝑑𝑗󸀠
+ (𝑚 − 1) (𝑛 − 1)
ℎ𝑘󸀠
× ( ∑ ∑ 𝑎𝑢V )
× (𝜉𝑚−1,𝑛−1,𝑠,𝑡 − 𝜉𝑚−2,𝑛−1,𝑠,𝑡 − 𝜉𝑚−1,𝑛−2,𝑠,𝑡 + 𝜉𝑚−2,𝑛−2,𝑠,𝑡 )
𝑢=𝑑𝑗 V=ℎ𝑘
= [𝜁𝑚𝑛𝑠𝑡 − 𝜁𝑚−1,𝑛,𝑠,𝑡 − 𝜁𝑚,𝑛−1,𝑠,𝑡 + 𝜁𝑚−1,𝑛−1,𝑠,𝑡 ] .
(45)
𝑛
𝑛−1
𝑘=1
𝑘=1
󸀠
𝑑𝑚
󵄨󵄨
󵄨
󵄨󵄨𝜁𝑚𝑛𝑠𝑡 − 𝜁𝑚−1,𝑛,𝑠,𝑡 − 𝜁𝑚,𝑛−1,𝑠,𝑡 + 𝜁𝑚−1,𝑛−1,𝑠,𝑡 󵄨󵄨󵄨 ≤ 𝐾 (𝑚, 𝑛)
ℎ𝑛󸀠
𝑢=𝑑𝑚 V=ℎ𝑛
(46)
(50)
Thus, we have
󸀠
𝑑𝑚
ℎ𝑛󸀠
∑ ∑ 𝑎𝑢V = (𝑚 + 1) (𝑛 + 1) 𝜙𝑚,𝑛,𝑠,𝑡
𝑢=𝑑𝑚 V=ℎ𝑛
𝑗, 𝑘 = 1, 2, . . .
+ (𝑚 − 1) (𝑛 − 1) 𝜙𝑚−1,𝑛−1,𝑠,𝑡 .
and 𝜎 be monotonically increasing. For simplicity in notation,
we denote
𝜙𝑚𝑛𝑠𝑡 = 𝜁𝑚𝑛𝑠𝑡 (𝑎) − 𝜁𝑚−1,𝑛,𝑠,𝑡 (𝑎) − 𝜁𝑚,𝑛−1,𝑠,𝑡 (𝑎)
+ 𝜁𝑚−1,𝑛−1,𝑠,𝑡 (𝑎) .
(48)
𝑑𝑗󸀠
ℎ󸀠
𝑘
𝑗−1
for all 𝑚, 𝑛 ≥ 1 and 𝜙0,0,𝑠,𝑡 (𝑎) = 𝑎𝑠𝑡 with 𝑑𝑗 = 𝜎
𝜎 (𝑠), ℎ𝑘 = 𝜎
(𝑡) + 1,
𝑘
(𝑠) + 1,
𝑑𝑗󸀠
= 𝜎 (𝑡). Further, we calculate
𝑚 (𝑚 + 1) 𝑛 (𝑛 + 1) 𝜙𝑚𝑛𝑠𝑡 − 𝑚 (𝑚 − 1) 𝑛 (𝑛 + 1) 𝜙𝑚−1,𝑛,𝑠,𝑡
− 𝑚 (𝑚 + 1) 𝑛 (𝑛 − 1) 𝜙𝑚,𝑛−1,𝑠,𝑡
+ 𝑚 (𝑚 − 1) 𝑛 (𝑛 − 1) 𝜙𝑚−1,𝑛−1,𝑠,𝑡
󵄨󵄨 𝑑󸀠 ℎ󸀠
󵄨󵄨
󵄨󵄨 𝑚 𝑛
󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨 ∑ ∑ 𝑎𝑢V 󵄨󵄨󵄨 ≤ 𝐾 (𝑚, 𝑛) ,
󵄨󵄨𝑢=𝑑 V=ℎ
󵄨
󵄨󵄨 𝑚 𝑛 󵄨󵄨󵄨
∀𝑠, 𝑡.
(52)
(53)
where 𝐾 is independent of 𝑢, V. By (49), we have
𝑚 𝑛
𝑘
1
=
∑ ∑ 𝑗𝑘 ( ∑ ∑ 𝑎𝑢V ) (49)
𝑚 (𝑚 + 1) 𝑛 (𝑛 + 1) 𝑗=1 𝑘=1
𝑢=𝑑 V=ℎ
𝑗
Hence, it follows from (46) that for each fixed 𝑚 > 𝑝0 , 𝑛 > 𝑞0 ,
Hence, it follows from (52) that
󵄨󵄨󵄨𝑎𝑢V 󵄨󵄨󵄨 ≤ 𝐾 ∀𝑢, V,
󵄨 󵄨
Again from the definition of 𝜁𝑚𝑛𝑠𝑡 , it is easy to obtain
ℎ𝑘󸀠
(51)
− (𝑚 + 1) (𝑛 − 1) 𝜙𝑚,𝑛−1,𝑠,𝑡
(47)
𝑠=0 𝑡=0
𝑘−1
𝑢=𝑑𝑚 V=ℎ𝑘
− (𝑚 − 1) (𝑛 + 1) 𝜙𝑚−1,𝑛,𝑠,𝑡
𝑘
𝑥𝑗𝑘 = ∑ ∑𝑎𝑠𝑡 ,
𝑗
ℎ𝑘󸀠
= 𝑚𝑛 ( ∑ ∑ 𝑎𝑢V ) .
for every fixed 𝑚 > 𝑝0 , 𝑛 > 𝑞0 and for all 𝑠, 𝑡, where 𝐾(𝑚, 𝑛)
is a constant depending upon 𝑚, 𝑛.
Now, for any given infinite double series ∑𝑠 ∑𝑡 𝑎𝑠𝑡 denoted
as “𝑎”, let us write
𝜙𝑚𝑛𝑠𝑡
󸀠
𝑑𝑚
= {𝑚 ∑ 𝑘 − 𝑚 ∑ 𝑘} ( ∑ ∑ 𝑎𝑢V )
Using (42) and (45), we have
𝑗
ℎ𝑘󸀠
=
󵄨󵄨
󵄨
󵄨󵄨𝜁𝑚𝑛𝑠𝑡 − 𝜁𝑚−1,𝑛,𝑠,𝑡 − 𝜁𝑚,𝑛−1,𝑠,𝑡 + 𝜁𝑚−1,𝑛−1,𝑠,𝑡 󵄨󵄨󵄨
≤ 𝐾,
∀𝑚, 𝑛, 𝑠, 𝑡.
Also from (43) and (54), we have
󵄨󵄨
󵄨
󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡 − 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨 ≤ 𝐾,
󵄨
󵄨
(54)
∀𝑝, 𝑞, 𝑠, 𝑡.
(55)
4. Topological Results
Here, we extend the spaces [W𝜎 ], W󸀠𝜎 , W󸀠󸀠𝜎 to more general
spaces, respectively, denoted by [W𝜎 (𝛼)], W󸀠𝜎 (𝛼), W󸀠󸀠𝜎 (𝛼),
8
Abstract and Applied Analysis
where 𝛼 = (𝛼𝑗𝑘 ) is a double sequence of positive real numbers
for all 𝑗, 𝑘 and sup𝑗,𝑘 𝛼𝑗𝑘 = 𝐻 < ∞. First, we recall the notion
of paranorm as follows.
A paranorm is a function 𝑔 : 𝑋 → R defined on a linear
space 𝑋 such that for all 𝑥, 𝑦, 𝑧 ∈ 𝑋
Theorem 5. Let 𝛼 = (𝛼𝑝𝑞 ) be a bounded double sequence
of strictly positive real numbers. Then, [W𝜎 (𝛼)] is a complete
linear topological space paranormed by
𝑔 (𝑥) = sup (
𝑚,𝑛,𝑠,𝑡
(P1) 𝑔(𝑥) = 0 if 𝑥 = 𝜃
,
where 𝑀 = max(1, sup𝑝,𝑞 𝛼𝑝𝑞 ). If 𝛼 ≥ 1 then [W𝜎,𝛼 ] is a
Banach space and [W𝜎,𝛼 ] is a 𝛼-normed space if 0 < 𝛼 < 1.
(P3) 𝑔(𝑥 + 𝑦) ≤ 𝑔(𝑥) + 𝑔(𝑦)
(P4) If (𝛼𝑛 ) is a sequence of scalars with 𝛼𝑛 → 𝛼0 (𝑛 →
∞) and 𝑥𝑛 , 𝑎 ∈ 𝑋 with 𝑥𝑛 → 𝑎 (𝑛 → ∞) in the
sense that 𝑔(𝑥𝑛 − 𝑎) → 0 (𝑛 → ∞), then 𝛼𝑛 𝑥𝑛 →
𝛼0 𝑎 (𝑛 → ∞), in the sense that 𝑔(𝛼𝑛 𝑥𝑛 − 𝛼0 𝑎) →
0 (𝑛 → ∞).
A paranorm 𝑔 for which 𝑔(𝑥) = 0 implies 𝑥 = 𝜃 is called
a total paranorm on 𝑋, and the pair (𝑋, 𝑔) is called a total
paranormed space. Note that each seminorm (norm) 𝑝 on 𝑋
is a paranorm (total) but converse needs not be true.
A paranormed space (𝑋, 𝑔) is a topological linear space
with the topology given by the paranorm 𝑔.
Now, we define the following spaces:
𝑚
1/𝑀
(57)
(P2) 𝑔(−𝑥) = 𝑔(𝑥)
[W𝜎 (𝛼)] = {𝑥 = (𝑥𝑗𝑘 ) :
𝑚 𝑛
1
󵄨𝛼𝑝𝑞
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥)󵄨󵄨󵄨󵄨 )
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
Proof. Let (𝑥𝑝𝑞 ) and (𝑦𝑝𝑞 ) be two double sequences. Then,
󵄨󵄨
󵄨𝛼
󵄨 󵄨𝛼
󵄨 󵄨𝛼
󵄨󵄨𝑥𝑝𝑞 + 𝑦𝑝𝑞 󵄨󵄨󵄨 𝑝𝑞 ≤ 𝐾 (󵄨󵄨󵄨𝑥𝑝𝑞 󵄨󵄨󵄨 𝑝𝑞 + 󵄨󵄨󵄨𝑦𝑝𝑞 󵄨󵄨󵄨 𝑝𝑞 ) ,
󵄨
󵄨
󵄨 󵄨
󵄨 󵄨
where 𝐾 = max(1, 2𝐻−1 ) and 𝐻 = sup𝑝,𝑞 𝛼𝑝𝑞 . Since
|𝜆|𝛼𝑝𝑞 ≤ max (1, |𝜆|𝐻) ,
(59)
therefore
𝑚 𝑛
1
󵄨
󵄨𝛼𝑝𝑞
∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝜆𝑥 + 𝜇𝑦)󵄨󵄨󵄨󵄨
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
≤ 𝐾𝐾1
1
(𝑚 + 1) (𝑛 + 1)
𝑚 𝑛
1
󵄨𝛼𝑝𝑞
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥)󵄨󵄨󵄨󵄨
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
+ 𝐾𝐾2
𝑛
󵄨𝛼𝑝𝑞
󵄨
× ∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥 − ℓ𝐸)󵄨󵄨󵄨󵄨 󳨀→ 0,
𝑝=0 𝑞=0
(58)
(60)
𝑚 𝑛
1
󵄨
󵄨𝛼𝑝𝑞
∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑦)󵄨󵄨󵄨󵄨 ,
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
where 𝐾1 = max(1, |𝜆|𝐻) and 𝐾2 = max(1, |𝜇|𝐻). From (60),
we have that if 𝑥, 𝑦 ∈ [W𝜎 (𝛼)], then 𝜆𝑥+𝜇𝑦 ∈ [W𝜎 (𝛼)]. Thus,
[W𝜎 (𝛼)] is a linear space. Without loss of generality, we can
take
as 𝑚, 𝑛 󳨀→ ∞, uniformly in 𝑠, 𝑡} ,
[W𝜎 (𝛼)] -lim 𝑥 = 0.
W󸀠𝜎 (𝛼) = {𝑥 = (𝑥𝑗𝑘 ) :
(61)
Clearly, 𝑔(0) = 0, 𝑔(−𝑥) = 𝑔(𝑥). From (58) and
Minkowski’s inequality, we have
∞ ∞
󵄨
󵄨𝛼𝑝𝑞
∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 −𝜉𝑝−1,𝑞,𝑠,𝑡 −𝜉𝑝,𝑞−1,𝑠,𝑡 +𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
1/𝑀
𝑝=0 𝑞=0
𝑚 𝑛
󵄨
󵄨𝛼𝑝𝑞
( ∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥 + 𝑦)󵄨󵄨󵄨󵄨 )
𝑝=0 𝑞=0
converges uniformly in 𝑠, 𝑡} ,
𝑚
𝑛
󵄨𝛼𝑝𝑞 /𝑀 󵄨󵄨
󵄨
󵄨𝛼𝑝𝑞 /𝑀
≤ [ ∑ ∑ (󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥)󵄨󵄨󵄨󵄨
+ 󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑦)󵄨󵄨󵄨󵄨
)]
W󸀠󸀠𝜎 (𝛼) = {𝑥 = (𝑥𝑗𝑘 ) :
1/𝑀
𝑝=0 𝑞=0
1/𝑀
∞
𝑚 𝑛
󵄨𝛼𝑝𝑞
󵄨
≤ ( ∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥)󵄨󵄨󵄨󵄨 )
∞
󵄨
󵄨𝛼𝑝𝑞
sup ∑ ∑󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 −𝜉𝑝−1,𝑞,𝑠,𝑡 −𝜉𝑝,𝑞−1,𝑠,𝑡 +𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨󵄨
𝑠,𝑡
𝑝=0 𝑞=0
𝑝=0 𝑞=0
1/𝑀
𝑚 𝑛
󵄨
󵄨𝛼𝑝𝑞
+ ( ∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑦)󵄨󵄨󵄨󵄨 )
< ∞} .
.
𝑝=0 𝑞=0
(62)
(56)
We remark that if 𝛼 = (𝛼𝑗𝑘 ) is a constant sequence, then
we write [W𝜎,𝛼 ] in place of [W𝜎 (𝛼)].
Hence,
𝑔 (𝑥 + 𝑦) ≤ 𝑔 (𝑥) + 𝑔 (𝑦) .
(63)
Abstract and Applied Analysis
9
Since 𝛼 = (𝛼𝑚𝑛 ) is bounded away from zero, there exists a
constant 𝛿 > 0 such that 𝛼𝑚𝑛 ≥ 𝛿 for all 𝑚, 𝑛. Now for |𝜆| ≤
1, |𝜆|𝛼𝑠𝑡 ≤ |𝜆|𝛿 and so
𝑔 (𝜆𝑥) ≤ |𝜆|𝛿/𝑀𝑔 (𝑥) ,
(64)
that is, the scaler multiplication is continuous. Hence, 𝑔 is a
paranorm on [W𝜎 (𝛼)].
Let (𝑥𝑏 ) be a Cauchy sequence in [W𝜎 (𝛼)], that is,
𝑑
𝑏
𝑔 (𝑥 − 𝑥 ) 󳨀→ 0 as 𝑏, 𝑑 󳨀→ ∞.
which is proved in the next theorem (i.e., Theorem 7). Using
the standard technique as in the previous theorem, we can
prove that ℎ is subadditive.
Now, we have to prove the continuity of scalar multiplication. Suppose that 𝑥 = (𝑥𝑗𝑘 ) ∈ W󸀠𝜎 (𝛼). Then, for 𝜖 > 0 there
exist integers 𝑀, 𝑁 > 0 such that
∞ ∞
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 (𝑥) − 𝜉𝑝−1,𝑞,𝑠,𝑡 (𝑥)
𝑝=𝑀 𝑞=𝑁
Since
𝑚 𝑛
𝑀
1
󵄨
󵄨𝛼𝑝𝑞
∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥𝑑 − 𝑥𝑏 )󵄨󵄨󵄨󵄨 ≤ [𝑔 (𝑥𝑑 − 𝑥𝑏 )] ,
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
(66)
If |𝜆| ≤ 1, then by (72) we have
∞
∞
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 (𝜆𝑥) − 𝜉𝑝−1,𝑞,𝑠,𝑡 (𝜆𝑥)
𝑝=𝑀 𝑞=𝑁
󵄨𝛼𝑝𝑞
− 𝜉𝑝,𝑞−1,𝑠,𝑡 (𝜆𝑥) + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 (𝜆𝑥 )󵄨󵄨󵄨󵄨
it follows that
󵄨󵄨
󵄨𝛼
󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥𝑑 − 𝑥𝑏 )󵄨󵄨󵄨 𝑝𝑞 󳨀→ 0 as 𝑏, 𝑑 󳨀→ ∞ ∀𝑝, 𝑞, 𝑠, 𝑡. (67)
󵄨
󵄨
(72)
󵄨𝛼𝑝𝑞
− 𝜉𝑝,𝑞−1,𝑠,𝑡 (𝑥) + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 (𝑥)󵄨󵄨󵄨󵄨 < 𝜖.
(65)
∞ ∞
󵄨
≤ ∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 (𝑥) − 𝜉𝑝−1,𝑞,𝑠,𝑡 (𝑥)
𝑝=𝑀 𝑞=𝑁
In particular,
󵄨󵄨
󵄨 󵄨 𝑑
𝑏 󵄨󵄨
󵄨󵄨𝜁0,0,𝑠,𝑡 (𝑥𝑑 − 𝑥𝑏 )󵄨󵄨󵄨 = 󵄨󵄨󵄨𝑥𝑠𝑡
󵄨
󵄨
󵄨 󵄨 − 𝑥𝑠𝑡 󵄨󵄨 󳨀→ 0
(68)
as 𝑏, 𝑑 󳨀→ ∞ for fixed 𝑠, 𝑡.
Hence, (𝑥𝑑 ) is a Cauchy sequence in C. Since C is complete,
there exists 𝑥 = (𝑥𝑗𝑘 ) ∈ C such that 𝑥𝑑 → 𝑥 coordinatewise
as 𝑑 → ∞. It follows from (66) that given 𝜖 > 0, there exists
𝑑0 such that
1/𝑀
𝑚 𝑛
1
󵄨
󵄨𝛼𝑝𝑞
(
∑ ∑ 󵄨󵄨󵄨󵄨𝜁𝑝𝑞𝑠𝑡 (𝑥𝑑 − 𝑥𝑏 )󵄨󵄨󵄨󵄨 )
(𝑚 + 1) (𝑛 + 1) 𝑝=0 𝑞=0
< 𝜖, (69)
for 𝑏, 𝑑 > 𝑑0 . Now taking 𝑏 → ∞ and sup𝑠,𝑡,𝑚,𝑛 in (69), we
have 𝑔(𝑥𝑑 − 𝑥) ≤ 𝜖 for 𝑑 > 𝑑0 . This proves that 𝑥𝑑 → 𝑥 and
𝑥 = (𝑥𝑗𝑘 ) ∈ [W𝜎 (𝛼)]. Hence [W𝜎 (𝛼)] is complete. If 𝛼 is a
constant then it is easy to prove the rest of the theorem.
Theorem 6. One has the following:
W󸀠𝜎 (𝛼) is a complete paranormed space, paranormed by
∞ ∞
󵄨
ℎ (𝑥) = sup ( ∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 − 𝜉𝑝−1,𝑞,𝑠,𝑡
𝑠,𝑡
𝑝=0 𝑞=0
󵄨󵄨𝛼𝑝𝑞
− 𝜉𝑝,𝑞−1,𝑠,𝑡 + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 󵄨󵄨󵄨
1/𝑀
(70)
)
which is defined on W󸀠𝜎 (𝛼). If 𝛼 ≥ 1 then W󸀠𝜎,𝛼 is a Banach
space and if 0 < 𝛼 < 1, W𝜎,𝛼 is 𝛼-normed space.
Proof. In order for the paranorm in (70) be defined, we
require that
W󸀠𝜎 (𝛼) ⊂ W󸀠󸀠𝜎 (𝛼)
󵄨𝛼𝑝𝑞
− 𝜉𝑝,𝑞−1,𝑠,𝑡 (𝑥) + 𝜉𝑝−1,𝑞−1,𝑠,𝑡 (𝑥)󵄨󵄨󵄨󵄨 < 𝜖.
(73)
(71)
Since for fixed 𝑀, 𝑁
𝑀−1 𝑁−1
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 (𝜆𝑥) − 𝜉𝑝−1,𝑞,𝑠,𝑡 (𝜆𝑥) − 𝜉𝑝,𝑞−1,𝑠,𝑡 (𝜆𝑥)
𝑝=0 𝑞=0
󵄨󵄨𝛼𝑝𝑞
+𝜉𝑝−1,𝑞−1,𝑠,𝑡 (𝜆𝑥)󵄨󵄨󵄨
(74)
󳨀→ 0
as 𝜆 → 0, it follows from (73) and (74) that for fixed 𝑥 =
(𝑥𝑗𝑘 ), ℎ(𝜆𝑥) → 0 as 𝜆 → 0. Also, since |𝜆|𝛼𝑝𝑞 ≤ max(1,
|𝜆|𝐻) implies that
1/𝑀
ℎ (𝜆𝑥) ≤ (sup |𝜆|𝛼𝑝𝑞 )
ℎ (𝑥) .
(75)
It follows that for fixed 𝜆, ℎ(𝜆𝑥) → 0 as 𝑥 → 0. This
proves the continuity of scalar multiplication. Hence, ℎ is a
paranorm. The proof of the completeness of W󸀠𝜎 (𝛼) can be
achieved by using the same technique as in Theorem 5.
Theorem 7. Suppose that 𝛼 = (𝛼𝑝𝑞 ) is bounded double
sequence of strictly positive real numbers. Then,
(i) W󸀠󸀠𝜎 (𝛼) is a complete linear space paranormed by the
function ℎ defined in (70). In particular, if 𝛼 is a
constant, W󸀠󸀠𝜎,𝛼 is a Banach space for 𝛼 ≥ 1 and 𝛼normed space for 0 < 𝛼 < 1,
(ii) W󸀠𝜎 (𝛼) is a closed subspace of W󸀠󸀠𝜎 (𝛼).
Proof. (i) Proceeding along the same lines as in Theorem 6,
except for the proof of continuity of scalar multiplication. If
𝑥 = (𝑥𝑗𝑘 ) ∈ W󸀠󸀠𝜎 (𝛼), we cannot assert (72) as in the case when
𝑥 = (𝑥𝑗𝑘 ) ∈ W󸀠𝜎 (𝛼). Since 𝛼 is bounded away from zero, there
10
Abstract and Applied Analysis
exists 𝛿 > 0 such that 𝛼𝑚𝑛 ≥ 𝛿 for all 𝑚, 𝑛. Hence, |𝜆| ≤ 1
implies |𝜆|𝛼𝑝𝑞 ≤ |𝜆|𝛿 . Since
ℎ (𝜆𝑥) ≤ |𝜆|𝛼𝑝𝑞 ℎ (𝑥) ,
(76)
continuity of scalar multiplication follows.
(ii) For this, first we have to show that
W󸀠𝜎 (𝛼) ⊂ W󸀠󸀠𝜎 (𝛼) .
(77)
Let 𝑥 = (𝑥𝑗𝑘 ) ∈ W󸀠𝜎 (𝛼). Then, there exist integers 𝑀, 𝑁 such
that
∞ ∞
󵄨
∑ ∑ 󵄨󵄨󵄨󵄨𝜉𝑝𝑞𝑠𝑡 (𝑥) − 𝜉𝑝−1,𝑞,𝑠,𝑡 (𝑥) − 𝜉𝑝,𝑞−1,𝑠,𝑡 (𝑥)
𝑝=𝑀 𝑞=𝑁
(78)
󵄨𝛼𝑝𝑞
+ 𝜉𝑝−1,𝑞−1,𝑠,𝑡 (𝑥)󵄨󵄨󵄨󵄨 < 1.
By an argument similar to Theorem 4(iii), we obtain (77).
Since W󸀠𝜎 (𝛼) and W󸀠󸀠𝜎 (𝛼) are complete with the same metric,
we have (ii).
Acknowledgement
This work was funded by the Deanship of Scientific Research
(DSR), King Abdulaziz University, Jeddah, under Grant no.
(130-068-D1433). The authors, therefore, acknowledge with
thanks DSR technical and financial support.
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