Generalized I-Convergent Difference Double Sequence
Spaces Defined By A Moduli Sequence
Naveen Kumar Srivastava Department of Mathematics St. Andrew’s College, Gorakhpur, U.P. srivastava.drnaveenkumar@gmail.com
Abstract:-In this article we introduce the sequence space I
0
c
(F,
n) and
I(F,
n) for the sequence of F = (fk) and given some inclusion relations.Keywords:-Ideal Filter, Sequence of Moduli, Difference Sequence Space, F-Convergent Sequence Space.
I. INTRODUCTION
Let ,
,c
0 be the set of all sequences of complex numbers, the linear spaces of bounded, convergent and null sequences x = (xk) with complex terms, respectively, normed by||x||
= where K
N = 1, 2, 3 ….. The idea of difference sequence spaces was introduced by H. Kizmaz [10]. In 1981, Kizmaz defined the sequence spaces as follows;
(
) = {x = (xk)
: (
x
k)
}, c(
) = {x = (xk)
: (
x
k)
c},0
c
(
) = {x = (xk)
: (
x
k)
c
0}, Where
x= (xk – xk+1) and
0x= (xk), There are Banach spaces with the norm
||
x
||
= |x1| +||
x||
. Later Colak and Et [2] defined the sequence spaces:
(
n) = {x = (xk)
: (
nxk)
},c(
n) = {x = (xk)
(
nxk)
c},0
c
(
n) = {x = (xk)
: (
nxk)
c
0}, Wheren
N,
0x = (xk),
x = (xk – xk+1),
nx = (
nxk) = (1 n
xk
n1xk+1)}
nxk = n
0 v
(1)n
v
n
xk+v,
And so that these are Banach space with the norm
||
x
||
= n
1 i
|xi| +
||
nx
||
.The idea of modulus was defined by Nakano [15] in 1953. A function
f : [0,
)[0,
) is called a modulus if(i) f(t) = 0 if and only if t = 0, (ii)f(t+u)
f(t)+f(u), for all t,u
, (iii)f is increasing and(iv)f is continuous from the right at 0.
Let X be a sequence spaces. Then the sequence spaces X(f) is defined as
X(f) = {x = (xk) : (f(|xk|))
X}For a modulus f. Maddox and Ruckle [14,16]
Kolak [11, 12] gave an extension of X(f) by considering a sequence of moduli F = (fk) that is
After then Gaur and Mursaleen [9] defined the following sequence spaces
(F,
) = {x = (
xk)
(F)},0
c
(F,
) = {x = (xk) : (
xk)
c
0(F)}, For a sequence of moduli F = (fk).We defined the following sequence spaces :
2
(F,
nm) = {x = (xi,j) : (
nmxi,j)
2(F)},2 0
c
(F,
nm) = {x = (xi,j) : (
nmxi,j)
c
02(F)}, Wheren m
xij =
n
0 v m
0 u
(1)u+v
v
n
u
n
xi+mu, k+mv
for a sequence of moduli F = (fij) we will give the necessary and sufficient conditions for the inclusion relations between X(
nm) and sufficient conditions for the inclusion relations between X(
nm) and Y(F,
nm), where X,Y =
2 orc
20. Sequences of moduli have been studied by C.A. Bektas and R. Colak [1] and many other authors.The notion of statical convergence was introduced by H. Fast [6]. Later on it was studied by J.A. Fridy [7,8] from the sequence space point view and linked with the summability theory.
The notion of I-convergence is a generalization of the statical convergence. It was studied at initial stage by Kostyrko, Salat and Wilezynski [13]. Later on it was studied by Salat [19], Salat, Tripathy and Ziman [20], Demric [3].
Let N be a non empty set. Then a family of sets I
2N (power set of N) is said to be an ideal if I is additive i.e. (A,B)
I
(A
B)
I and i.e. A
I, B
A
B
I. A non empty family of sets £(I) 2N is said to be filter on N if and only if£(I) for A, £(I) we have (
) £(I) and each A£(I) and implies B £(I).
An ideal I 2N is called non trivial it I 2N. A non trivial
is called admissible if {(x) : x } I. A non trivial
ideal is maximal ideal is maximal it there cannot exist any non-trivial ideal J containing I as a subset. For each ideal I, there exist a filter £(I) corresponding to I, i.e. £(I) = {
K
c}, whereK
c = N – K.Definition 1.1: A sequence (xij) is said to be I-convergent to a number L if for every > 0. {i,j : |xijL| } I. In this case we write I –
j i
lim
xij = L.Definition 1.2: A sequence (xij) is said to be I-null if L = 0. In this case we write I
j i
lim
xij = 0.Definition 1.3: A sequence (xij) is said to be I-cauchy if for every > 0, there exist a number m = m() such that {i,j N : |xij xm,n| } I.
Definition 1.4: A sequence (xij) is said to be I-bounded if there exist M > 0 such that {K N : |xij| M}.
We need the following Lemmas.
Lemma 1.5 : The condition supij fij(t) < , t > 0 hold if and only if there is a point t0 > 0 such that supij fij(t0) < (see [1, 9]).
Lemma 1.6: The condition infij fij(t) > 0 hold if and only if there exist is a point t0 > 0 such that infij fij(t0) > 0 (see [1, 9]).
Lemma 1.7: Let K £(I) and . If MI, the M
KI (see [20]).
Lemma 1.8: If I 2N and . If M I then M
K I (see [13]).II. MAIN RESULT
In this article we introduce the following classes of sequence cpaces.
I 0 2
c
(F,
nm) = {(xi,j) : I – j ilim
fi,j(|
nmxi,j|)= 0} I,I 2
(F,
nm) = {(xi,j) : I – supi,j fi,j(|
nmxi,j|) < } I.(a)
2
I(
nm) 2
I(F,
nm),(b)
2c
0I(
nm) 2c
I0(F,
nm),(c)
Supi,j fi,j(t) < , (t > 0).Proof: (a) Implies (b) is obvious.
(b) implies (c). Let 2
c
I0(
nm) 2c
0I(F,
nm). Suppose that (c) is not true. Then by Lemma (1.5)j , i
sup
fi,j(t) = , for all t > 0,And therefore there is a sequence (ki) of positive integers such that
i k
f
i
1
> i, for each i = 1, 2, 3 ….. (1)
Define x = (xi,j) as follows
xij =
.
otherewise
0
...;
3
,
2
,
1
i
k
j
,
i
if
i
1
i
Then x 2
c
I0(
nm) but by (1), x
I(F,
nm) which contradicts (b). Hence (c) must hold, (c) implies (a). Let (c) be satisfied and x 2
I(F,
nm). If we suppose that x 2
I (F,
nm) thenj , i
sup
fi,j(|
nmxi,j|) = for x 2
IIf we take t = |
nmx| then j , isup
fi,j(t) = which contradicts (c).Hence 2
I(
nm) 2
I(F,
nm).Theorem 2.2: For a sequence F = fi,j is a sequence of moduli, the following statements are equivalent :
(a)
2c
I0(F,
nm) 2c
0I(
nm),(b)
2c
I0(F,
nm) 2
I(
nm),(c)
Infi,j fi,j(t) > 0, (t > 0).Proof: (a) implies (b) is obvious.
(b) implies (c). Let 2
c
0I(F,
nm) 2
I(
nm). Suppose that (c) is not true. Then by Lemma (1.6)Infi,j fi,j(t) = 0, (t > 0)
And therefore there is a sequence (ki) of positive integers such that
i k
f
(i2) <i
1
for each I = 1, 2, 3 ….. (2)
Define x = (xi,j) as follows
xi,j =
otherwise
0
...;
3
,
2
,
1
i
k
k
if
,
i
2 iBy (2) x 2
c
0I(F,
nm) but x
I(
n) which contradicts (b). Hence (c) must hold. (c) implies (a). Let (c) be satisfied and x 2c
I0(F,
nm) that isI – j i
lim
fi,j(|
nmxi,j|) = 0Suppose that x 2
c
I0(
nm). Then for some number 0 > 0 and positive integer k0 we have |
nmxi,j| c0 for ki,j > k0. Therefore fij(c0) fij(|
nmxi,j|) for I,j > k0 and hence j i
lim
fi,j(c0) > 0,
Which contradicts our assumption that x 2
c
0I(
nm).Thus 2
c
I0(F,
nm) 2c
0I(
nm).Theorem 2.3: The inclusion 2
I(F,
nm) 2c
I0(
nm) holds in and only if j i
lim
fi,j(t) = for t > 0 ……… (3)such that
j i
lim
fij(t) = for t > 0 does not hold. Then there is a number t0>0 and a sequence (ki) of positive integer such thati k
f
(t0) M < . ……. (4)Define the sequence x = (xij) by
xij =
otherwise
0
...;
3
,
2
,
1
i
ki
j
,
i
if
,
t
0
Thus x 2
I(F,
nm) by (4).But x 2
c
0I(
nm), so that (3) must hold.If
I(F,
n) c
I0(
n).Conversely, let (3) hold. If x 2
I(F,
nm),then fi,j(|
nmxij|) M < , for i,j = 1, 2, 3 ….. Suppose that x c
I0(
n).Then for some number c0 > 0 and positive integer k0 we have |
n
xij| < c0 for i,j k0.Therefore fi,j(c0) fi,j(|
nxij|) M for i,j k0, which contradicts (3).Hence x 2
c
I0(
nm).Theorem 2.4: The inclusion 2
I(
nm) 2c
I0(F,
nm) holds if and only if j i
lim
fi,j(t) = 0, for t > 0 …….. (5)Proof: Suppose that 2
I(
nm) 2c
I0(F,
nm) but (5) does not holdThen
j i
lim
fij(t0) = i 0, for some t0 > 0 ……….(6)Define the sequence x = (xij) by
xij = t0 u v
n i
0 v n i
0 u
)
1
(
v
j
v
i
1
u
i
n
i
v
i
n
for i,j = 1, 2, 3 ….. Then x 2
c
0I(F,
nm) by (6). Hence (5) must hold, conversely, let x 2
I(
nm) and suppose that (5) holds.Then |
nmxij|M< for K=1, 2, 3, …..There for fij(|
nmxij|)fij(M) for i,j = 1, 2, 3 ….. andij
lim
fij(|
nmxij|) j i
lim
fij(M) = 0 by (5).Hence x2
c
0I(F,
nm).REFERENCES
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