MATEMATIQKI VESNIK
originalni nauqni rad
research paper
66, 1 (2014), 91–100
March 2014
SOME SPACES OF DOUBLE DIFFERENCE
SEQUENCES OF FUZZY NUMBERS
Kuldip Raj and Sunil K. Sharma
Abstract. In this paper, we introduce some spaces of double difference sequences of fuzzy
numbers defined by a sequence of modulus functions F = (fk,l ). We also make an effort to study
some topological properties and prove some inclusion relations between these spaces.
1. Introduction and preliminaries
Fuzzy set theory, compared to other mathematical theories, is perhaps the most
easily adaptable theory to practice. The main reason is that a fuzzy set has the
property of relativity, variability and inexactness in the definition of its elements.
Instead of defining an entity in calculus by assuming that its role is exactly known,
we can use fuzzy sets to define the same entity by allowing possible deviations and
inexactness in its role. This representation suits well the uncertainties encountered
in practical life, which make fuzzy sets a valuable mathematical tool. The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [35]
and subsequently several authors have discussed various aspects of the theory and
applications of fuzzy sets such as fuzzy topological spaces, similarity relations and
fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming.
Matloka [16] introduced bounded and convergent sequences of fuzzy numbers and
studied some of their properties. For more details about sequence spaces of fuzzy
numbers (see [4], [8], [10], [15], [20], [21], [30], [31]) and references therein.
The initial works on double sequences is found in Bromwich [7]. Later on, it
was studied by Hardy [11], Moricz [17], Moricz and Rhoades [18], Tripathy ([32],
[33]), Başarır and Sonalcan [5] and many others. Hardy [11] introduced the notion
of regular convergence for double sequences. Quite recently, Zeltser [36] in her
Ph.D thesis has essentially studied both the theory of topological double sequence
spaces and the theory of summability of double sequences. Mursaleen and Edely
[22] have recently introduced the statistical convergence and Cauchy convergence for
double sequences and given the relation between statistical convergent and strongly
2010 AMS Subject Classification: 40A05, 40D25
Keywords and phrases: Fuzzy numbers; modulus function; double sequence spaces.
91
92
K. Raj, S.K. Sharma
Cesaro summable double sequences. Subsequently, Mursaleen [19] and Mursaleen
and Edely [23] have defined the almost strong regularity of matrices for double
sequences and applied these matrices to establish a core theorem and introduced
the M -core for double sequences and determined those four dimensional matrices
transforming every bounded double sequence x = (xk,l ) into one whose core is a
subset of the M -core of x. More recently, Altay and Başar [1] have defined the
spaces BS, BS(t), CSp , CSbp , CSr and BV of double sequences consisting of all
double series whose sequence of partial sums are in the spaces Mu , Mu (t), Cp ,
Cbp , Cr and Lu , respectively and also examined some properties of these sequence
spaces and determined the α-duals of the spaces BS, BV, CSbp and the β(v)duals of the spaces CSbp and CSr of double series. Now, recently Başar and Sever
[6] have introduced the Banach space Lq of double sequences corresponding to
the well known space `q of single sequences and examined some properties of the
space Lq . By the convergence of a double sequence we mean the convergence in
the Pringsheim sense i.e. a double sequence x = (xk,l ) has Pringsheim limit L
(denoted by P − lim x = L) provided that given ² > 0 there exists n ∈ N such
that |xk,l − L| < ² whenever k, l > n see [25]. We shall write more briefly as P convergent. The double sequence x = (xk,l ) is bounded if there exists a positive
number M such that |xk,l | < M for all k and l.
A fuzzy number is a function on the real axis, i.e., a mapping X : Rn → [0, 1]
which satisfies the following four conditions:
(1) X is normal, i.e., there exist an x0 ∈ Rn such that X(x0 ) = 1;
(2) X is fuzzy convex, i.e., for x, y ∈ Rn and 0 ≤ λ ≤ 1, X(λx + (1 − λ)y) ≥
min[X(x), X(y)];
(3) X is upper semi-continuous;
(4) the closure of {x ∈ Rn : X(x) > 0}, denoted by [X]0 , is compact.
Let C(Rn ) = {A ⊂ Rn : A is compact and convex }. The spaces C(Rn ) has a
linear structure induced by the operations
A + B = {a + b, a ∈ A, b ∈ B}
and
λA = {λa : a ∈ A}
for A, B ∈ C(R ) and λ ∈ R. The Hausdorff distance between A and B of C(Rn )
is defined as
n
d(A, B) = max{sup inf ka − bk, sup inf ka − bk}
b∈B a∈A
a∈A b∈B
n
where k.k denotes the usual Euclidean norm in R . It is well known that (C(Rn ), d)
is a complete (non separable) metric space. For 0 < α ≤ 1, the α-level set
X α = {x ∈ Rn : X(x) ≥ α}
is a nonempty compact convex, subset of Rn , as the support X 0 = limα→0+ X α . By
L(Rn ) we denote the set of all fuzzy numbers. Define a map d¯ : L(Rn )×L(Rn ) → Rn
by
¯
d(X,
Y ) = sup d(X α , Y α ).
α∈[0,1]
Some spaces of double difference sequences
93
¯ For X, Y ∈
It is showed that L(Rn ) is a complete metric space with the metric d.
n
α
α
L(R ), define X ≤ Y if and only if X ≤ Y for any α ∈ [0, 1].
We denote by w(f ) the set of all double sequences X = (Xk,l ) of fuzzy numbers.
A sequence X = (Xk,l ) of fuzzy numbers is said to be bounded if the set {Xk,l :
k, l ∈ N} of fuzzy numbers is bounded. We denote by c∞ (f ) the set of all bounded
sequences of fuzzy numbers. The sequence X = (Xk,l ) of fuzzy numbers is said
to be convergent to the fuzzy number X0 , written as limk,l Xk,l = X0 , if for every
¯ k,l , X0 ) < ² for every
² > 0 there exists a positive integer k0 = k0 (²), such that d(X
k > k0 . By c(f ) we denote the set of all convergent sequences of fuzzy numbers. It
is clear that c(f ) ⊂ c∞ (f ) ⊂ w(f ). A metric on L(Rn ) is said to be a translation
invariant if d(X + Z, Y + Z) = d(X, Y ) for X, Y, Z ∈ L(Rn ).
Let X be a linear metric space. A function p : X → R is called a paranorm, if
(1) p(x) ≥ 0 for all x ∈ X,
(2) p(−x) = p(x) for all x ∈ X,
(3) p(x + y) ≤ p(x) + p(y) for all x, y ∈ X,
(4) if (λn ) is a sequence of scalars with λn → λ as n → ∞ and (xn ) is a sequence
of vectors with p(xn − x) → 0 as n → ∞, then p(λn xn − λx) → 0 as n → ∞.
A paranorm p for which p(x) = 0 implies x = 0 is called a total paranorm and
the pair (X, p) is called a total paranormed space. It is well known that the metric
of any linear metric space is given by some total paranorm (see [34], Theorem
10.4.2, pp. 183).
A modulus function is a function f : [0, ∞) → [0, ∞) such that
(1) f (x) = 0 if and only if x = 0,
(2) f (x + y) ≤ f (x) + f (y) for all x ≥ 0, y ≥ 0,
(3) f is increasing
(4) f is continuous from right at 0.
It follows that f must be continuous everywhere on [0, ∞). The modulus
x
,
function may be bounded or unbounded. For example, if we take f (x) = x+1
p
then f (x) is bounded. If f (x) = x , 0 < p < 1, then the modulus function f (x) is
unbounded. Subsequentially, modulus function has been discussed in ([2, 3, 13, 14,
24, 26–29]) and references therein.
The notion of difference sequence spaces was introduced by Kızmaz [12], who
studied the difference sequence spaces l∞ (∆), c(∆) and c0 (∆). The notion was
further generalized by Et and Çolak [9] by introducing the spaces l∞ (∆m ), c(∆m )
and c0 (∆m ). Let w denote the set of all real or complex sequences x = (xk ). Let
m, n be non-negative integers, then for Z = l∞ , c, c0 . We have sequence spaces
m
Z(∆m
n ) = {x = (xk ) ∈ w : (∆n xk ) ∈ Z},
m−1
where ∆m
xk − ∆m−1
xk+1 ) and ∆0 xk = xk for all k ∈ N, which is
n xk = (∆n
n
equivalent to the following binomial representation
µ ¶
m
X
v m
∆m
x
=
(−1)
xk+nv .
n k
v
v=0
94
K. Raj, S.K. Sharma
Let X = (Xk,l ) be a double sequence of fuzzy numbers, F = (fk,l ) be a sequence of
modulus functions and p = (pk,l ) be a bounded sequence of positive real numbers.
In this paper we define the following classes of double difference sequences of fuzzy
numbers:
©
£ ¡ ¡ m
¢¢¤pk,l
ª
−s
cf [F, ∆m
fk,l d¯ ∆n Xk,l , 0̄
=0 ,
n , p, s]0 = X = (Xk,l ) ∈ w(f ) : lim(kl)
k,l
©
£ ¡ ¡
¢¢¤pk,l
f
m
c [F, ∆n , p, s]1 = X = (Xk,l ) ∈ w(f ) : lim(kl)−s fk,l d¯ ∆m
= 0,
n Xk,l , X0
k,l
ª
for some X0 and s ≥ 0
and
©
£ ¡ ¡ m
¢¢¤pk,l
ª
−s
cf [F, ∆m
fk,l d¯ ∆n Xk,l , 0̄
<∞ ,
n , p, s]∞ = X = (Xk,l ) ∈ w(f ) : sup(kl)
k,l
If we take F (x) = x, we get the following classes of sequences
©
¡ ¡ m
¢¢p
ª
−s ¯
cf [∆m
d ∆n Xk,l , 0̄ k,l = 0 ,
n , p, s]0 = X = (Xk,l ) ∈ w(f ) : lim(kl)
k,l
©
¡ ¡ m
¢¢p
−s ¯
cf [∆m
d ∆n Xk,l , X0 k,l = 0,
n , p, s]1 = X = (Xk,l ) ∈ w(f ) : lim(kl)
k,l
ª
for some X0 and s ≥ 0
and
¢¢p
ª
©
¡ ¡ m
−s ¯
d ∆n Xk,l , 0̄ k,l < ∞ .
cf [∆m
n , p, s]∞ = X = (Xk,l ) ∈ w(f ) : sup(kl)
k,l
If we take p = (pk,l ) = 1, we get
©
£ ¡ ¡ m
¢¢¤
ª
−s
cf [F, ∆m
fk,l d¯ ∆n Xk,l , 0̄
=0 ,
n , s]0 = X = (Xk,l ) ∈ w(f ) : lim(kl)
k,l
©
£ ¡ ¡ m
¢¢¤
−s
cf [F, ∆m
fk,l d¯ ∆n Xk,l , X0
= 0,
n , s]1 = X = (Xk,l ) ∈ w(f ) : lim(kl)
k,l
ª
for some X0 and s ≥ 0
and
©
£ ¡ ¡ m
¢¢¤
ª
−s
cf [F, ∆m
fk,l d¯ ∆n Xk,l , 0̄
<∞ .
n , s]∞ = X = (Xk,l ) ∈ w(f ) : sup(kl)
k,l
If we take s = 0, F (x) = x and p = (pk,l ) = 1, for all k, l then, we obtain new
difference sequence spaces of fuzzy numbers :
©
¡ ¡ m
¢¢
ª
¯
cf [∆m
=0 ,
n ]0 = X = (Xk,l ) ∈ w(f ) : lim d ∆n Xk,l , 0̄
k,l
©
¡ ¡ m
¢¢
ª
cf [∆m
]
=
X
=
(X
)
∈
w(f
)
:
lim
d¯ ∆n Xk,l , X0 = 0, for some X0
k,l
n 1
k,l
and
©
¡ ¡ m
¢¢
ª
¯
cf [∆m
<∞ .
n ]∞ = X = (Xk,l ) ∈ w(f ) : sup d ∆n Xk,l , 0̄
k,l
For a sequence of modulus functions F = (fkl ), we give the following conditions:
(1) sup fk,l (t) < ∞ for all t > 0,
95
Some spaces of double difference sequences
(2) limt→∞ sup fk,l (t) = 0 uniformly in k, l ≥ 1.
The following inequality will be used throughout the paper. Let p = (pk,l ) be
a double sequence of positive real numbers with 0 < pk,l ≤ supk,l pk,l = H and
let K = max{1, 2H−1 }. Then for the factorable sequences {ak,l } and {bk,l } in the
complex plane, we have
|ak,l + bk,l |pk,l ≤ K(|ak,l |pk,l + |bk,l |pk,l ).
(1.1)
The main purpose of this paper is to introduce some spaces of double difference
sequences of fuzzy numbers defined by a sequence of modulus functions and make
an effort to study some topological properties and prove inclusion relations between
the above defined sequence spaces.
2. Main results
Theorem 2.1. If d¯ is a translation invariant metric, then the classes of
f
m
f
m
sequences cf [F, ∆m
n , p, s]0 , c [F, ∆n , p, s]1 and c [F, ∆n , p, s]∞ are closed under
the operations of addition and scalar multiplication.
Proof. Since d¯ is a translation invariant metric, it implies that
¡
¢
¡
¢
¡
¢
m
¯ m
¯ m
d¯ ∆m
n Xk,l + ∆n Yk,l , X0 + Y0 ≤ d ∆n Xk,l , X0 + d ∆n Yk,l , Y0
and
¡
¢
¡
¢
¯ m
d¯ ∆m
n λXk,l , λX0 ≤ |λ|d ∆n Xk,l , X0 ,
f
(2.1)
(2.2)
[F, ∆m
n , p, s]1 .
Suppose
where λ is a scalar and |λ| > 1. We shall prove only for c
,
p,
s]
.
Then
by
using
inequality
(1.1),
we
X = (Xk,l ) and Y = (Yk,l ) ∈ cf [F, ∆m
1
n
have
¢¢¤pk,l
£ ¡ ¡
m
(kl)−s fk,l d¯ ∆m
n Xk,l + ∆n Yk,l , X0 + Y0
¢¢¤pk,l
¢
¡
£ ¡ ¡
¯ m
≤ (kl)−s fk,l d¯ ∆m
n Xk,l , X0 + d ∆n Yk,l , Y0
£ ¡ ¡
¢¢
¡ ¡
¢¢¤pk,l
≤ (kl)−s fk,l d¯ ∆m
+ fk,l d¯ ∆m
n Xk,l , X0
n Yk,l , Y0
£ ¡ ¡
¢¢¤pk,l
£ ¡ ¡
¢¢¤pk,l
≤ (kl)−s K H fk,l d¯ ∆m Xk,l , X0
+ (kl)−s K H fk,l d¯ ∆m Yk,l , Y0
n
f
n
[F, ∆m
n , p, s]1 .
f
[F, ∆m
n , p, s]1 .
Hence X + Y ∈ c
Let X = (Xk,l ) ∈ c
For λ ∈ R there
exists an integer K such that |λ| ≤ K. Then by equation (2.2) and the sequence
of modulus functions F = (fk,l ) for all k, l ∈ N, we have
£ ¡ ¡
¢¢¤pk,l
£ ¡ ¡
¢¢¤pk,l
(kl)−s fk,l d¯ λ∆m
≤ (kl)−s fk,l d¯ |λ|(∆m
n Xk,l , λX0
n Xk,l , X0 )
£ ¡ ¡
¢¢¤pk,l
≤ K H (kl)−s fk,l d¯ ∆m Xk,l , X0
.
n
f
This implies that λX ∈ c
[F, ∆m
n , p, s]1 .
Similarly, we can prove for other cases.
Theorem 2.2. Let F = (fk,l ) be a sequence of modulus functions. Then
f
m
f
m
cf [F, ∆m
n , p, s]0 ⊂ c [F, ∆n , p, s]1 ⊂ c [F, ∆n , p, s]∞ .
96
K. Raj, S.K. Sharma
f
m
Proof.
Clearly cf [F, ∆m
Let X
n , p, s]0 ⊂ c [F, ∆n , p, s]1 .
m
∈ c [F, ∆n , p, s]1 . Then, there is some fuzzy number X0 , such that
£ ¡ ¡
¢¢¤pk,l
lim(kl)−s fk,l d¯ ∆m Xk,l , X0
= 0, s ≥ 0.
f
= (Xk,l )
n
k,l
Now, by inequality (1.1), we have
£ ¡ ¡
¢¢¤pk,l
(kl)−s fk,l d¯ ∆m
n Xk,l , 0̄
¢¢¤pk,l
¢¢¤pk,l
£ ¡ ¡
£ ¡ ¡
.
+ K.(kl)−s fk,l d¯ X0 , 0̄
≤ K.(kl)−s fk,l d¯ ∆m
n Xk,l , X0
This implies that X = (Xk,l ) ∈ cf [F, ∆m
n , p, s]∞ .
Theorem 2.3. Let F = (fk,l ) be a sequence of modulus functions. Then,
cf [F, ∆m
n , p, s]1 is a complete metric space with the metric defined by
m,n
¢¢¤pk,l ¢ p 1
£ ¡ ¡
¡
P
m
k,l .
g(X, Y ) =
d(Xk,l , Yk,l ) + supk,l (kl)−s fk,l d ∆m
n Xk,l , ∆n Yk,l
k,l=1,1
t
t
Proof. Let (X t ) be a Cauchy sequence in cf [F, ∆m
n , p, s]1 , where X = (Xk,l )
for every k, l, t ∈ N. Then
m,n
P
t
v
d(Xk,l
, Xk,l
)
g(X t , X v ) =
k,l=1,1
¡
£ ¡ ¡
¢¢¤pk,l ¢ p 1
t
m v
k,l → 0
+ sup (kl)−s fk,l d ∆m
n Xk,l , ∆n Xk,l
k,l
as t, v → ∞, for each k, l ∈ N. Hence,
m,n
¢
¡ ¡
¢¢
P ¡ t
v
t
m v
d Xk,l , Xk,l
→ 0 and fk,l d ∆m
→ 0 as t, v → ∞,
n Xk,l , ∆n Xk,l
k,l=1,1
for each k, l ∈ N. As (fk,l ) is a sequence of modulus functions for all k, l ∈ N, then
¡
¢
t
m v
d ∆m
n Xk,l , ∆n Xk,l → 0 as t, v → ∞, for each k, l ∈ N.
Thus we have
¡ t
¢
¡
¢
v
t
m v
d Xk+m,l+n
, Xk+m,l+n
≤ d ∆m
n Xk,l , ∆n Xk,l → 0 as t, v → ∞,
for each k, l ∈ N. Therefore,
t
1
2
3
(Xk,l
)t = (Xk,l
, Xk,l
, Xk,l
,···)
is a Cauchy sequence in L(Rn ). As L(Rn ) is complete, it is convergent, say
t
limt Xk,l
= Xk,l , for every k, l ∈ N. Since (X t ) is a Cauchy sequence for each
² > 0 there exists n0 = n0 (²) such that g(X t , X v ) < ² for all t, v ≥ n0 . So, we have
m,n
m,n
¢
¢
P ¡ t
P ¡ t
v
lim
d Xk,l , Xk,l
=
d Xk,l , Xk,l < ²
v k,l=1,1
k,l=1,1
and
£ ¡ ¡
£ ¡ ¡
¢¢¤pk,l
¢¢¤pk,l
t
m v
t
m
lim(kl)−s fk,l d ∆m
= (kl)−s fk,l d ∆m
n Xk,l , ∆n Xk,l
n Xk,l , ∆n Xk,l
v
< ²H , for all t ≥ n0 .
Some spaces of double difference sequences
97
This implies that g(X t , X) < 2² for all t ≥ n0 , this means X t → X as t → ∞,
where X = (Xk,l ). As
£ ¡ ¡
¢¢¤pk,l
©
£ ¡ ¡
¢¢¤pk,l
n0
(kl)−s fk,l d ∆m
≤ 2pk,l (kl)−s ) fk,l d ∆m
n Xk,l , X0
n Xk,l , X0
£ ¡ ¡
¢¢¤pk,l ª
n0
m
+ (kl)−s ) fk,l d ∆m
n Xk,l , ∆n Xk,l
→ 0 as k, l → ∞,
f
m
we obtain X = (Xk,l ) ∈ cf [X, F, ∆m
n , p, s]1 . Therefore, c [X, F, ∆n , p, s]1 is a
complete metric space.
Theorem 2.4. Let inf pkl = h > 0. Then we have the following:
f
m
(i) Xk,l → X0 (cf [F, ∆m
n ]1 ), implies that Xk,l → X0 (c [F, ∆n , p, s]1 );
f
m
(ii) Xk,l → X0 (cf [∆m
n , p, s]1 ), implies that Xk,l → X0 (c [F, ∆n , p, s]1 );
(iii) β = limt→∞
fk,l (t)
t
f
m
> 0, implies cf [∆m
n , p, s] = c [F, ∆n , p, s].
Proof. (i) Suppose that (Xk,l ) → X0 (cf [F, ∆m
n ]1 ) (as k, l → ∞). Since (fk,l )
is a modulus function for all k, l, then
£ ¡ ¡
¢¢¤
£
¡ ¡
¢¢¤
lim fk,l d¯ ∆m
= fk,l lim d¯ ∆m
= 0.
n Xk,l , X0
n Xk,l , X0
k,l
k,l
£ ¡ ¡
¢¢¤h
Since inf pk,l = h > 0, then limk,l fk,l d¯ ∆m
= 0. So, for 0 < ² < 1,
n Xk,l , X0
there exists k0 such that for all k > k0 ,
£ ¡ ¡ m
¢¢¤h
fk,l d¯ ∆n Xk,l , X0
< ² < 1,
and as pk,l ≥ h for all k, l,
¢¢¤h
¢¢¤pk,l £ ¡ ¡ m
£ ¡ ¡ m
< ² < 1,
≤ fk,l d¯ ∆n Xk,l , X0
fk,l d¯ ∆n Xk,l , X0
£ ¡ ¡ m
¢¢¤pk,l
then, we obtain limk,l fk,l d¯ ∆n Xk,l , X0
= 0. Since (kl)−s is bounded, we
write
¢¢¤pk,l
£ ¡ ¡
= 0.
lim(kl)−s fk,l d¯ ∆m
n Xk,l , X0
k,l
Therefore, X = (Xk,l ) ∈ cf [F, ∆m
n , p, s]1 .
(ii) Let X = (Xk,l ) ∈ cf [∆m
n , p, s]1 , so that
¢¢pk,l
¡
¡
Sk,l = (kl)−s d¯ ∆m
→ 0 as k, l → ∞.
n Xk,l , X0
Suppose ² > 0 and choose δ with 0 < δ < 1, such that fk,l (t) < ² for 0 ≤ t ≤ δ and
for all k, l. Now, we take
©
¡
¢
A1 = (k, l) ∈ N × N : d¯ ∆m
n Xk,l , X0 ≤ δ}
and
©
¡
¢
A2 = (k, l) ∈ N × N : d¯ ∆m
n Xk,l , X0 > δ}.
¡
¢
For d¯ ∆m
n Xk,l , X0 > δ,
¡
¢
¡
¢ −1
£ ¡
¢ −1 ¤
¯ m
d¯ ∆m
< 1 + |d¯ ∆m
| ,
n Xk,l , X0 < d ∆n Xk,l , X0 δ
n Xk,l , X0 δ
98
K. Raj, S.K. Sharma
where (k, l) ∈ A2 and [|t|] ¡denotes the ¢integer part of t. By using properties of
modulus function and for d¯ ∆m
n Xk,l , X0 > δ, we have
¡ ¡ m
¢¢ ¡
£ ¡
¢ −1 ¤¢
fk,l d¯ ∆n Xk,l , X0 ≤ 1 + |d¯ ∆m
| fk,l (1)
n Xk,l , X0 δ
¡ m
¢ −1
¯
≤ 2fk,l (1)d ∆n Xk,l , X0 δ .
¡ m
¢
¡ ¡ m
¢¢
For d¯ ∆n Xk,l , X0 ≤ δ, fk,l d¯ ∆n Xk,l , X0 < ², where (k, l) ∈ A1 . Hence
¢¢¤pk,l
£ ¡ ¡
¢¢¤pk,l
£ ¡ ¡
n0
|(k, l) ∈ A1
(kl)−s fk,l d¯ ∆m
= (kl)−s ) fk,l d¯ ∆m
n Xk,l , X0
n Xk,l , X0
£ ¡ ¡ m n0 m
¢¢¤pk,l
−s
¯
+ (kl) fk,l d ∆ X , ∆ Xk,l
|(k, l) ∈ A2 .
n
Then
k,l
n
£ ¡ ¡
¢¢¤pk,l
£
¤H
(kl)−s fk,l d¯ ∆m
≤ (kl)−s ²H + 2fk,l (1)δ −1 Sk,l → 0
n Xk,l , X0
as k, l → ∞. Thus X = (Xk,l ) ∈ cf [F, ∆m
n , p, s]1 .
f
m
(iii) In (ii), it was shown that cf [∆m
n , p, s]1 ⊂ c [F, ∆n , p, s]1 . We have to only
f
m
f
m
show that c [∆n , p, s]1 ⊃ c [F, ∆n , p, s]. For any modulus function, the existence
of positive limit given with β in Maddox [13, Proposition 1]. Now β > 0 and let
X = (Xk,l ) ∈ cf [F, ∆m
n , p, s]1 . Since β > 0, for every t > 0, we have fk,l (t) ≥ βt for
all k, l ∈ N. From this inequality, it is seen that X = (Xk,l ) ∈ cf [∆m
n , p, s]1 . This
completes the proof.
Theorem 2.5. Let F = (fk,l ) and G = (gk,l ) be two sequences of modulus
functions and s1 , s2 ≥ 0. Then we have the following:
f
m
f
m
(i) cf [F, ∆m
n , p, s]1 ∩ c [G, ∆n , p, s]1 ⊂ c [F ∩ G, ∆n , p, s]1 ;
f
m
(ii) s1 ≤ s2 implies cf [F, ∆m
n , p, s1 ]1 ⊂ c [F, ∆n , p, s2 ]1 .
f
m
Proof. (i) Let X = (Xk,l ) ∈ cf [F, ∆m
n , p, s]1 ∩ c [G, ∆n , p, s]1 . From (1.1) we
have
£
¡ ¡
¢¢¤pk,l
(fk,l + gk,l ) d¯ ∆m
n Xk,l , X0
£ ¡ ¡
¢¢
¡ ¡
¢¢¤pk,l
= fk,l d¯ ∆m
+ gk,l d¯ ∆m
n Xk,l , X0
n Xk,l , X0
©£ ¡ ¡
¢¢
£ ¡ ¡
¢¢¤pk,l ª
≤ K fk,l d¯ ∆m Xk,l , X0 ]pk,l + gk,l d¯ ∆m Xk,l , X0
.
n
n
Since (kl)−s is bounded, write
£
¡ ¡
¢¢¤pk,l
£ ¡ ¡
¢¢¤pk,l
(kl)−s (fk,l + gk,l ) d¯ ∆m
≤ K(kl)−s fk,l d¯ ∆m
n Xk,l , X0
n Xk,l , X0
¢¢¤pk,l
£ ¡ ¡
+ K(kl)−s gk,l d¯ ∆m
n Xk,l , X0
Thus X = (Xk,l ) ∈ cf [F ∩ G, ∆m
n , p, s]1 . This completes the proof of (i).
(ii) Let s1 ≤ s2 . Then (kl)−s2 ≤ (kl)−s1 for all k, l ∈ N. As
£ ¡ ¡
¢¢¤pk,l
£ ¡ ¡
¢¢¤pk,l
(kl)−s2 fk,l d¯ ∆m
≤ (kl)−s1 fk,l d¯ ∆m
,
n Xk,l , X0
n Xk,l , X0
this implies that
f
m
cf [F, ∆m
n , p, s1 ]1 ⊂ c [F, ∆n , p, s2 ]1 .
Some spaces of double difference sequences
99
Theorem 2.6. Let F = (fk,l ) be a sequence of modulus functions. Then we
have the following:
(i) let 0 < inf k,l pk,l ≤ pk,l ≤ 1, then
f
m
cf [F, ∆m
n , p, s]1 ⊂ c [F, ∆n , s]1 ;
(ii) let 1 ≤ pk,l ≤ supk,l pk,l < ∞, then
f
m
cf [F, ∆m
n , s]1 ⊂ c [F, ∆n , p, s]1 ,
q
) be bounded, then
(iii) let 0 ≤ pk,l ≤ qk,l and ( pk,l
k,l
f
m
cf [F, ∆m
n , q, s]1 ⊂ c [F, ∆n , p, s]1 .
Proof. (i) Let X = (Xk,l ) ∈ cf [F, ∆m
n , p, s]1 , since 0 < inf k,l pk,l ≤ 1, we
obtain
£ ¡ ¡
¢¢¤
£ ¡ ¡
¢¢¤pk,l
(kl)−s fk,l d¯ ∆m
≤ (kl)−s fk,l d¯ ∆m
n Xk,l , X0
n Xk,l , X0
for all k, l and hence we obtain X = (Xk,l ) ∈ cf [F, ∆m
n , s]1 .
(ii) Let 1 ≤ pk,l ≤ supk,l pk,l < ∞ for each k, l and X = (Xk,l ) ∈ cf [F, ∆m
n , s]1 .
Then, for each 0 < ² < 1, there exists a positive integer k0 , such that
¢¢¤
£ ¡ ¡
(kl)−s fk,l d¯ ∆m
≤ ²,
n Xk,l , X0
for all k ≥ k0 . This implies that
¢¢¤
¢¢¤pk,l
£ ¡ ¡
£ ¡ ¡
.
≤ (kl)−s fk,l d¯ ∆m
(kl)−s fk,l d¯ ∆m
n Xk,l , X0
n Xk,l , X0
Thus X = (Xk,l ) ∈ cf [F, ∆m
n , p, s]1 .
(iii) The proof is easy so we omit it.
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(received 02.04.2012; available online 01.11.2012)
School of Mathematics Shri Mata Vaishno Devi University, Katra-182320, J & K (India)
E-mail: kuldipraj68@gmail.com, sunilksharma42@yahoo.co.in