AIYUB, MOHAMMAD

English

Research Publication Journals Repository

ADV  MATH
SCI  JOURNAL
Advances in Mathematics: Scientic Journal 2 (2013), no.1, 18
ISSN 1857-8365, UDC: 515.175:517.521
STRONGLY ALMOST SUMMABLE DIFFERENCE SEQUENCES AND
STATISTICAL CONVERGENCE
M. AIYUB
Abstract. The idea of dierence sequence space was introduced by Kizmaz [12] and
was generalized by Et and Çolak [6]. In this paper, we introduce and examine some
properties of three sequence spaces dened by using a modulus function and give
various properties and inclusion relation on these spaces.
1. Introduction and Preliminaries
Let ! be the set of all sequences of real numbers and `1; c and c0 be respectively the
Banach spaces of bounded, convergent and null sequences x = (xk) with the usual norm
kxk = sup jxkj, where k 2 N = f1; 2; 3:::::g, the positive integers.
A sequence x 2 `1 is said to be almost convergent [14] if all Banach limits of x coincide.
Lorentz [14] dened:
c^ =
n
x : lim
n
1
n
nX
k=1
xk+m exists, uniformly in m
o
:
Several authors including Lorentz [14], Duran [2] and King [11], have studied almost
convergent sequences. Maddox [16, 17] has dened x to be strongly almost convergent to
a number L if
lim
n
1
n
nX
k=1
xk+m   L = 0 uniformly in m:
By

c^

we denote the spaces of all strongly almost convergent sequences. It is easy to see
that c 

c^

 c^  `1:
The space of strongly almost convergent sequences was generalized by Nanda [20, 21].
2010 Mathematics Subject Classication. 40A05, 46A45.
Key words and phrases. Dierence sequence, modulus function, statistical convergent.
1
2 M. AIYUB
Let p = (pk) be a sequence of strictly positive numbers. Nanda [20] dened:

c^; p

=

x = (xk) : lim
n
1
n
nX
k=1
xk+m   Lpk = 0; uniformly in m

;

c^; p

0
=

x = (xk) : lim
n
1
n
nX
k=1
xk+mpk = 0; uniformly in m

;

c^; p

1
=

x = (xk) : sup
m;n
1
n
nX
k=1
xk+mpk <1

:
Let  = (k) be a nondecreasing sequence of positive numbers tending to 1 such that
n+1  n + 1; 1 = 1:
The generalized de la Vallée-poussin mean is dened by
tn(x) =
1
n
X
k2In
xk;
where In = [n  n + 1; n] for n = 1; 2; ::::
A sequence x = (xk) is said to be (V; ) summable to a number L (see [13]), if tn(x)! L
as n!1: If n = n, then (V; ) summability and strongly (V; ) summability are reduced
to (C; 1) summability and [C; 1] summability, respectively.
The idea of dierence sequence spaces was introduced by Kizmaz in [12]. In 1981,
Kizmaz dened the sequence spaces:
X(4) =

x = (xk) : 4x 2 X

for X = `1; c and c0, where 4x =
 
xk   xk+1

:
Then Et and Çolak [6] generalized the above sequence spaces as below:
X(4r) =

x = (xk) : 4
rx 2 X

for X = `1; c and c0, where r 2 N, 4
0x = (xk), 4x = (xk   xk+1); 4
rx = (4r 1xk  
4r 1xk+1); and so that
4rx =
rX
v=0
( 1)v

r
v

xk+v:
Recently, Et and Basarir [5] extended the above sequence spaces to the sequence spaces
X(4r) for X = `1(p), c(p), c0(p),

c^; p

,

c^; p

0
and

c^; p

1
:
We recall that a modulus f is a function from [0;1)! [0;1) such that
(i) f(x) = 0 if and only if x = 0;
(ii) f(x+ y)  f(x) + f(y) for all x  0; y  0;
(iii) f is increasing,
(iv) f is continuous from right at 0.
It follows that f must be a continuous everywhere on [0;1). The modulus function
may be bounded or unbounded. Ruckle [23] and Maddox [15] used a modulus function
f to construct some sequence spaces. Subsequently modulus function has been discussed
in [3, 4, 19, 22, 26].
STRONGLY ALMOST SUMMABLE DIFFERENCE ... 3
Further, let X;Y  !: Then we shall write ([27]):
M(X;Y ) =
\
x2X
x 1  Y =

a 2 ! : ax 2 Y for all x 2 X

:
The set X = M(X; `1) is called the Köthe-Toeplitz dual space or - dual of X. Let X
be a sequence space. Then X is called:
(i) Solid (or normal) if (kxk) 2 X, whenever, (xk) 2 X for all sequences (k) of
scalars with jkj  1 for all k 2 N:
(ii) Symmetric if (xk) 2 X implies
 
x(k)

2 X whenever (k) is a permutation of N:
(iii) Perfect if X = X:
(iv) Sequence algebra if x:y 2 X, whenever x; y 2 X: It is well known that if X is
perfect then X is normal [10].
The following inequality will be used throughout this paper:
(1.1)
ak + bkpk  C jakjpk + jbkjpk ;
where ak; bk 2 C; 0  pk  supk pk = H, C = max
 
1; 2H 1

; (see [18]).
2. Main Results
In this section we prove some results involving the sequence space

V^ ; ; f; p

0
(4r; E),
V^ ; ; f; p

1
(4r; E) and

V^ ; ; f; p

1
(4r; E):
Denition 2.1. Let E be Banach space. We dene !(E) to be the vector space of
all E-valued sequences that is !(E) =

x = (xk) : xk 2 E

. Let f be a modulus
function and p = (pk) be any sequence of strictly positive real numbers.
We dene the following sequence sets:
V^ ; ; f; p

1
(4r; E) =
=

x 2 !(E) : lim
n
1
n
X
k2In

f
 4rxk+m   L
pk
= 0; uniformly in m for some L > 0

;

V^ ; ; f; p

0
(4r; E) =

x 2 !(E) : lim
n
1
n
X
k2In

f
 4rxk+m
pk
= 0;uniformly in m

;
and 
V^ ; ; f; p

1
(4r; E) =

x 2 !(E) : sup
n;m
1
n
X
k2In

f
 4rxk+m
pk
 1

:
If x 2

V^ ; ; f; p

1
(4r; E) then we shall write xk ! L

V^ ; ; f; p

1
(4r; E) and L will
be called E- strongly almost dierence limit of x with respect to the modulus f:
Through this paper, Z will denote any one of the notions 0; 1 or 1:
In this case f(x) = x and pk = 1 for all k 2 N, we shall write

V^ ; 

z
(4r; E) and
V^ ; ; f

z
(4r; E) instead of

V^ ; ; f; p

z
(4r; E). If x 2

V^ ; 

1
(4r; E) then we say that
x is 4r;E strongly almost convergent to L.
4 M. AIYUB
The proofs of the following two theorems are obtained by using the known standard
techniques, therefore we give them without proofs (see for detail [3, 22]).
Theorem 2.1. Let p = (pk) be a bounded. Then the spaces

V^ ; ; f; p

z
(4r; E) are
linear spaces over the set of complex numbers C:
Theorem 2.2. Let p = (pk) be a bounded and f be modulus function, then
V^ ; ; f; p

0
(4r; E) 

V^ ; ; f; p

1
(4r; E 

V^ ; ; f; p

1
(4r; E):
Theorem 2.3. If r  1, then the inclusion

V^ ; ; f; p

z
(4r 1; E) 

V^ ; ; f; p

z
(4r; E)
is strict. In general

V^ ; ; f; p

z
(4i; E) 

V^ ; ; f; p

z
(4r; E) for all i = 1; 2; 3:::r   1
and the inclusion is strict.
Proof. We give the proof for Z = 1 only. It can be proved in similar way for Z = 0; 1:
Let x 2

V^ ; ; f; p

1
(4r 1; E). Then we have:
sup
m;n
1
n
X
k2In
f
4r 1xk+m

<1 :
By denition of f , we have:
1
n
X
k2In
f
4rxk+m


1
n
X
k2In
f
4r 1xk+m

+
1
n
X
k2In
f
4r 1xk+m+1

<1 :
Thus, 
V^ ; ; f; p

1
(4r 1; E) 

V^ ; ; f; p

1
(4r; E) :
Proceeding in this way, we have:
[V^ ; ; f; p]1(4
iE)  [V^ ; ; f; p]1(4
r; E) ;
for all i = 1; 2; 3::::r   1: Let n = n for all n 2 N: Then the sequence x = (k
r), for
example, belongs to

V^ ; ; f; p

1
(4r; E); but doesn't belong to

V^ ; ; f; p

1
(4r 1; E)
for f(x) = x
 
if x = (kr); then 4rxk = ( 1)
rr! and 4r 1xk = ( 1)
r+1r!(k + (r 1)2 ) for
all k 2 N

. 
Similarly, as in the previous theorems for the cases [V^ ; ; f; p]0(4
r; E) and [V^ ; ; f; p]1(4
rE)
we have:
Proposition 2.1. Let f be a sequence of modulus functions. Then:
V^ ; ; f; p

1
(4r 1; E) 

V^ ; ; f; p

0
(4r; E):
Theorem 2.4. Let f1 and f2 be modulus functions. Then we have:
(i)

V^ ; ; f1; p

z
(4r; E) 

V^ ; ; f1of2; p

z
(4r; E);
(ii)

V^ ; ; f1; p

z
(4r; E) \

V^ ; ; f2; p

z
(4r; E) 

V^ ; ; f1 + f2; p

z
(4r; E):
The following results are consequence of Theorem 2.4.
Proposition 2.2. Let f be a modulus functions. Then:
V^ ; ; p

z
(4r; E) 

V^ ; ; f; p

z
(4r; E):
STRONGLY ALMOST SUMMABLE DIFFERENCE ... 5
Theorem 2.5. The sequence spaces

V^ ; ; f; p

0
(4r; E) ,

V^ ; ; f; p

1
(4r; E) and
V^ ; ; f; p

1
(4r; E) are not solid for r  1:
Proof. Let pk = 1 for all k, f(x) = x and n = n for all n 2 N: Then (xk) = (k
r) 2
[V^ ; ; f; p]1(4
r; E) but (kxk) =2 [V^ ; ; f; p]1(4
r; E), when k = ( 1)
k for all k 2 N.
Hence [V^ ; ; f; p]1(4
r; E) is not solid. The other cases can be proved by considering
similar examples. 
From the above theorem we may give the following corollary.
Corollary 2.1. The sequence spaces

V^ ; ; f; p

0
(4r; E) ,

V^ ; ; f; p

1
(4r; E) and
V^ ; ; f; p

1
(4r; E) are not perfect for r  1:
Theorem 2.6. The sequence spaces

V^ ; ; f; p

1
(4r; E) and

V^ ; ; f; p

1
(4r; E) are
not symmetric for r  1:
Proof. Let (pk) = 1 for all k, f(x) = x and n = n for all n 2 N: Then xk = (k
r) 2
V^ ; ; f; p

1
(4r; E). Let (yk) be an arrangement of (xk), which is dened by
(yk) =

x1; x2; x4; x3; x9; x5; x16; x6; x25; x7; x36; x8; x49; x10::::::

:
Then (yk) =2

V^ ; ; f; p

1
(4r; E): 
Remark 2.1. The space [V^ ; ; f; p]0(4
r; E) is not symmetric for r  2:
Theorem 2.7. The sequence spaces

V^ ; ; f; p

z
(4r; E): are not sequence of algebras.
Proof. Let pk = 1 for all k 2 N:, f(x) = x and n = n for all n 2 N: Then x = (k
r 2),
y = (kr 2) 2

V^ ; ; f; p

z
(4r; E), but x; y 2

V^ ; ; f; p

z
(4r; E): 
3. Statistical Convergent
The notion of statistical convergence was introduced by Fast [7] and studied by various
authors [1, 9, 24, 25]. In this section we dene 4r;E almost statistically convergent
sequences and give some inclusion relations between s^(4r; E) and

V^ ; ; f; p

1
(4r; E):
Denition 3.1. A sequence x = (xk) is said to be 4
r
;E-almost statistically conver-
gent to the number L if for every  > 0,
lim
n
1
n
k 2 In : 4rxk+m   L  	 = 0 uniformly in m:
In this case we write s^(4r; E)  limx = L, or xk ! Ls^(4
r
; E):
When n = n and L = 0 we shall write s^(4
r; E) instead of s^(4r; E):
The proof of the following theorem is easily obtained by using the same technique as
in Theorem 2 in Sava³ [25], therefore we give it without proof.
Theorem 3.1. Let  = (n) be the same as in section 1, then:
(i) If xk ! L

V^ ; 

1
(4r; E)) xk then Ls^
 
4r; E

;
(ii) If x 2 `1
 
4r; E

and xk ! Ls^
 
4r; E

, then xk ! L

V^ ; 

1
(4r; E);
(iii) s^
 
4r; E

\ `1(4
r; E) =

V^ ; 

1
(4r; E) \ `1(4
r; E):
6 M. AIYUB
Theorem 3.2. s^
 
4r; E

 s^
 
4r; E

if and only if lim infn
n
n
> 0:
Proof. The suciency part of this proof can be obtained using the same technique as the
suciency part of proof of Theorem 3 in Sava³ [25].
For necessity suppose that lim infn
n
n
= 0. As in ([8], p. 510) we can choose a
subsequence (n(j)) such that
n(j)
n(j) <
1
j
: We dene x = (xi) such that:
4rxi =

1 if i 2 In(j); j = 1; 2; 3:::::;
0 otherwise:
Then x 2

c^

(4r; E) and by [4, Theorem 3.1 (i)], x 2 s^
 
4r; E

. But x =2

V^ ; 

1
(4r; E)
and Theorem 3.1 (ii) implies that x =2 s^
 
4r; E

. This completes the proof. 
Theorem 3.3. Let f be a modulus function and supk pk = H. Then:
V^ ; ; f; p

1
(4r; E)  s^(4r; E):
Proof. Let x 2

V^ ; ; f; p

1
(4r; E) and  > 0 be given. Let 1 denote the sum over
k  n such that
4rxk+m   L   and 2 denote the sum of over k  n such that4rxk+m   L < . Then:
1
n
X
k2In

f
 4rk+m   L
pk
=
1
n
X
1

f
 4rk+m   Lpk + 1n
X
2

f
 4rk+m   L
pk

1
n
X
1

f
 4rk+m   Lpk  1n
X
1

f
 

pk

1
n
X
1
min
 
f
 

inf pk
;

f
 

H!

k 2 In : 4rxk+m   L  	min

f()
inf pk
;

f()
H
:
Hence x 2 s^(4r; E): 
Theorem 3.4. Let f be a bounded and 0 < h = infk pk  pk  supk pk = H < 1.
Then: s^(4r; E) 

V^ ; ; f; p

1
(4r; E):
Proof. Suppose that f is bounded. Let  > 0 and 1 and 2 be denoted in the previous
theorem. Since f is bounded there exists an integer K such that f(x) < K for all x  0:
Then:
1
n
X
k2In

f
 4rk+m   L
pk
=
1
n
X
1

f
 4rk+m   L
pk
+
1
n
X
2

f
 4rk+m   L
pk

1
n
X
1
max
 
Kh;KH

+
1
n
X
2

f()
pk
 max
 
Kh;KH
 1
n
k 2 In : 4rk+m   L  	+
+max
 
f()h; f()H

STRONGLY ALMOST SUMMABLE DIFFERENCE ... 7
Hence x 2

V^ ; ; f; p

1
(4r; E): 
Theorem 3.5. Let 0 < h = infk pk  pk  supk pk = H <1. Then
s^(4r; E) =

V^ ; ; f; p

1
(4r; E) ;
if and only if f is bounded.
Proof. Let f is bounded bounded. By Theorem 3.4 and Theorem 3.5 we have s^
 
4r; E

=
V^ ; ; f; p

1
(4r; E): Conversely, suppose that f is unbounded. Then there exists a posi-
tive sequence (tk) with f(tk) = k
2, for k = 1; 2; 3::::. If we choose
4rxi =

tk i = k
2; i = 1; 2; 3::::::
0 otherwise:
;
then we have:
1
n
k 2 In : 4rk+m  	 
p
n 1
n
for all n;m
and so that x 2 s^(4r; E) but x =2

V^ ; ; f; p

1
(4r; E) for E = C: This contradict to
s^
 
4r; E

=

V^ ; ; f; p

1
(4r; E


Acknowledgment
The author thanks the referee(s) for their valuable suggestions that improved the
presentation of the paper.
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Department of Mathematics
University of Bahrain
P.O. Box-32038
Kingdom of Bahrain
E-mail address: maiyub2002@yahoo.com


STRONGLY ALMOST SUMMABLE DIFFERENCE SEQUENCES AND STATISTICAL CONVERGENCE

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STRONGLY ALMOST SUMMABLE DIFFERENCE SEQUENCES AND STATISTICAL CONVERGENCE

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