MATEMATIQKI VESNIK
UDK 517.521
originalni nauqni rad
research paper
60 (2008), 95–100
ON QUASI ALMOST LACUNARY STRONG CONVERGENCE
DIFFERENCE SEQUENCE SPACES DEFINED BY
A SEQUENCE OF MODULI
Vakeel A. Khan and Q. M. Danish Lohani
Abstract. The idea of difference sequence sets, X(4) = { x = (xk ) : 4x ∈ X }, where
X = l∞ , c or c0 was introduced by Kizmaz [3], and then this subject has been studied and
generalized by various mathematicians. In this article we define quasi almost 4m -Lacunary
strongly P-convergent sequences defined by sequence of moduli and give inclusion relations on
these sequence spaces.
1. Preliminaries
The difference sequence space X(4) was introduced by Kizmaz [3] as follows
X(4) = { x = (xk ) ∈ ω : (4xk ) ∈ X }
for X = l∞ , c or c0 ,
where 4xk = (xk − xk+1 ) for all k ∈ N.
The difference sequence spaces were generalized by Et and Colak [1] as follows
X(4m ) = { x = (xk ) ∈ ω : 4m x = (4m xk ) ∈ X }
m
m−1
for X = l∞ , c or c0 ,
m−1
where 4 xk = (4
xk − 4
xk+1 ).
A sequence of positive integers θ = (kr ) is called “lacunary” if k0 = 0, 0 <
kr < kr+1 and hr = kr − kr−1 → ∞ as r → ∞. The intervals determined by θ
kr
. The space of lacunary strongly
will be denoted by Ir = (kr−1 , kr ) and qr = kr−1
convergent sequence Lθ was defined by Freedman et al. [2] as:
1 P
Lθ = { x = (xk ) : lim
|xk − L| = 0 for some L }.
r hr k∈I
r
The double lacunary sequence was defined by E. Savas and R. F. Patterson
[11] as follows: The double sequence θr,s = {(kr , ls )} is called double lacunary if
there exists two increasing sequences of integers such that
k0 = 0, hr = kr − kr−1 → ∞
as r → ∞
AMS Subject Classification: 40C05, 42B15.
Keywords and phrases: Difference sequence; lacunary sequence; sequence of moduli; double
sequence.
96
V. A. Khan, Q. M. D. Lohani
and
Let kr,s = kr ls , hr,s
l0 = 0, h¯s = ls − ls−1 → ∞ as s → ∞.
= hr h¯s and θr,s is determined by
Ir, s = { (k, l) : kr−1 < k ≤ kr and ls−1 < l ≤ ls },
qr =
kr
kr−1 ,
q¯s =
ls
ls−1
and qr,s = qr q¯s .
Definition 1.1. A function f : [0, ∞) → [0, ∞) is called modular if
1. f (t) = 0 if and only if t = 0,
2. f (t + u) ≤ f (t) + f (u) for all t, u ≥ 0,
3. f is increasing, and
4. f is continuous from the right at 0.
Let X be a sequence space. Then the sequence space X(f ) is defined as
X(f ) = { x = (xk ) : (f (|xk |)) ∈ X }
for a modulus f ([6],[8],[10]). Kolk [4], [5] gave an extension of X(f ) by considering
a sequence of moduli F = (fk ) i.e.
X(F ) = { x = (xk ) : (fk (|xk |)) ∈ X }.
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 [9]. We shall denote it briefly as “P-convergent”.
Recently Moricz and Rhoades [7] defined almost P-convergent sequences as
follows: A double sequence x = (xk,l ) of real numbers is called almost P -convergent
to a limit L if
P
P n+q−1
1 m+p−1
P - lim sup
|xk,l − L| = 0.
p,q→∞ m,n≥0 pq k=m
l=n
In this paper we introduce the following definition.
A double sequence x = (xk,l ) of elements of the real vector space w (the
space of bounded sequences) in a real normed space X is said to be quasi almost
P-convergent to a limit L if
°
°
m+p−1 n+q−1
°
°
P
°P - lim sup 1 P
° = 0.
(x
−
L)
k,l
° p,q→∞ m,n≥0 pq
°
k=m
l=n
X
Let us denote the above set of sequences as t̄2 .
For a sequence F = (fk ) of moduli, we define the following sequence spaces:
[Lθr,s , 4m , F, P ] = { x = (xk,l ) : P - lim
r,s
1
hr,s
P
[fk (k4m xk+m,l+n −Lk)]pk,l = 0,
(k,l)∈Ir,s
uniformly in m and n for some L }.
On difference sequence spaces defined by a sequence of moduli
1
hr,s
[Lθr,s , 4m , F, P ]0 = { x = (xk,l ) : P - lim
r,s
P
97
[fk (k4m xk+m,l+n k)]pk,l = 0,
(k,l)∈Ir,s
uniformly in m and n for some l }.
We shall denote [Lθr,s , 4m , F, P ] and [Lθr,s , 4m , F, P ]0 as [Lθr,s , 4m , F ] and
[Lθr,s , 4m , F ]0 , respectively when pk,l = 1 for all k and l. If x is in [Lθr,s , 4m , F ],
we shall say that x is quasi almost lacunary strongly P-convergent with respect to
the sequence of moduli F = (fk ). Also note that if F (x) = x, pk,l = 1 for all k and
l then [Lθr,s , 4m , F, P ] = [Lθr,s , 4m ] and [Lθr,s , 4m , F, P ]0 = [L0θr,s , 4m ] which are
defined as follows:
[Lθr,s , 4m ] = { x = (xk,l ) : P - lim
r,s
1
hr,s
P
k4m xk+m,l+n − Lk = 0,
(k,l)∈Ir,s
uniformly in m and n for some L }.
and
[L0θr,s , 4m ] = { x = (xk,l ) : P - lim
r,s
1
hr,s
P
k4m xk+m,l+n k = 0,
(k,l)∈Ir,s
uniformly in m and n }.
Again note that if pk,l = 1 for all k and l then [Lθr,s , 4m , F, P ] = [Lθr,s , 4m , F ]
and [Lθr,s , 4m , F, P ]0 = [Lθr,s , 4m , F ]0 .
We define
1
hr,s
[Lθr,s , 4m , F ] = { x = (xk,l ) : P - lim
r,s
P
[fk (k4m xk+m,l+n − Lk)] = 0,
(k,l)∈Ir,s
uniformly in m and n for some L },
and
[Lθr,s , 4m , F ]0 = { x = (xk,l ) : P - lim
r,s
1
hr,s
P
[fk (k4m xk+m,l+n k)] = 0,
(k,l)∈Ir,s
uniformly in m and n }.
Now we extend quasi almost convergent double sequences to a sequence of
moduli as follows: Let F = (fk ) be a sequence of moduli and P = (pk,l ) be
any factorable sequence of strictly positive real numbers, we define the following
sequence space:
[t̄2 , 4m , F, P ] = {x = (xk,l ) : P - lim
pq
1
p, q
p,q
P
[fk (k4m xk+m,l+n − Lk)]pk,l = 0,
k,l=1,1
uniformly in m and n for some L }.
If we take F (x) = x, pk,l = 1 for all k and l, then [t̄2 , 4m , F, P ] = [t̄2 , 4m ].
98
V. A. Khan, Q. M. D. Lohani
2. Main results
Theorem 1. Let θr,s = {kr , ls } be a double lacunary sequence with lim inf r qr >
1, and lim inf s q¯s > 1. Then for any sequence of moduli F = (fk ), [t̄2 , 4m , F, P ] ⊂
[Lθr,s , 4m , F, P ].
Proof. We need to show that [t̄2 , 4m , F, P ]0 ⊂ [Lθr,s , 4m , F, P ]0 . The general
inclusion follows by linearity. Suppose lim inf r qr > 1, and lim inf s q¯s > 1; then
δ
δ
there exists δ > 0 such that qr > 1 + δ. This implies hkrr ≤ δ+1
and hlss ≤ δ+1
.
2
m
Then for x ∈ [t̄ , 4 , F, P ]0 , we can write for each m and n
Ar,s =
1
hr,s
P
[fk (k4m xk+m,l+n k)]pk,l =
(k,l)∈Ir,s
=
r−1 lP
s−1
1 kP
[fk (k4m xk+m,l+n k)]pk,l
hr,s k=1 l=1
=
1
hr,s
=
1
hr,s
kr
P
lP
s−1
kr P
ls
1 P
[fk (k4m xk+m,l+n k)]pk,l
hr,s k=1 l=1
[fk (k4m xk+m,l+n k)]pk,l
k=kr−1 +1 l=1
ls
P
kP
r−1
[fk (k4m xk+m,l+n k)]pk,l
l=ls−1 +1 k=1
µ
¶
kr P
ls
1 P
[fk (k4m xk+m,l+n k)]pk,l
kr ls k=l l=1
!
Ã
kP
r−1 lP
s−1
kr−1 ls−1
1
m
pk,l
[fk (k4 xk+m,l+n k)]
=
hr,s
kr−1 ls−1 k=l l=1
=
kr ks
hr,s
=
1
hr
s−1 lP
s−1
ls−1 1 lP
[fk (k4m xk+m,l+n k)]pk,l
k=kr−1 +1 hs ls−1 l=l l=1
=
1
hs
r−1 kP
r−1
kr−1 1 kP
[fk (k4m xk+m,l+n k)]pk,l .
l=ls−1 +1 hr kr−1 k=l k=1
kr
P
ls
P
Since x ∈ [t̄2 , 4m , F, P ] the last two terms tends to zero uniformly in m, n in the
Pringsheim sense, thus for each m and n
µ
¶
kr P
ls
kr ks
1 P
m
pk,l
Ar,s =
[fk (k4 xk+m,l+n k)]
hr,s kr ls k=1 l=1
=
kP
r−1 lP
s−1
kr−1 ls−1
1
[fk (k4m xk+m,l+n k)]pk,l ) + o(1).
(
hr,s
kr−1 ls−1 k=1 l=1
Since hrs = kr ls − kr−1 ls−1 we are granted for each m and n the following:
kr ls
1+δ
≤
hrs
δ
The terms
and
kr−1 ls−1
1
≤ .
hrs
δ
kr P
ls
1 P
[fk (k4m xk+m,l+n k)]pk,l
kr ls k=1 l=1
99
On difference sequence spaces defined by a sequence of moduli
and
kP
r−1 lP
s−1
1
[fk (k4m xk+m,l+n k)]pk,l
kr−1 ls−1 k=1 l=1
are both Pringsheim null sequences for all m and n. Thus Ars is Pringsheim.
Theorem 2. Let θr,s = {k, l} be a double lacunary sequence with lim supr qr <
∞, and lim sups q¯s < ∞. Then for any sequence of moduli F = (fk ),
[Lθr,s , 4m , F, P ] ⊂ [t̄2 , 4m , F, P ].
Proof. Since lim supr qr < ∞, and lim sups q¯s < ∞ there exists G > 0 such
that qr < G and q¯s < G for all r and s. Let x ∈ [Lθr,s , 4m , F, P ] and ² > 0. Also
there exist r0 > 0 and s0 > 0 such that for every i ≥ r0 and j ≥ s0 and m and n,
0
=
Di,j
1
hij
P
[fk (k4m xk+m,l+n k)]pk,l < ².
(k,l)∈Ii,j
0
Let F 0 = max{Di,j
: 1 ≤ i ≤ r0 and 1 ≤ j ≤ s0 } and p and q be such that
kr−1 < p ≤ kr and ls−1 < q ≤ ls . Thus we obtain the following:
1
pq
p,q
P
[fk (k4m xk+m,l+n k)]pk,l
k,l=1,1
≤
1
kr−1 ls−1
1
≤
kr−1 ls−1
=
≤
≤
1
kr−1 ls−1
F0
kr−1 ls−1
kP
r ls
[fk (k4m xk+m,l+n k)]pk,l
k,l=1,1
r,s
P
t,u=1,1
rP
0 ,s0
t,u=1,1
rP
0 ,s0
t,u=1,1
Ã
!
P
[fk (k4m xk+m,l+n k)]pk,l
k,l∈It,u
0
+
ht,u Dt,u
ht,u +
1
kr−1 ls−1
1
kr−1 ls−1
P
(r0 <t≤r)∪(s0 <u≤s)
P
(r0 <t≤r)∪(s0 <u≤s)
F 0 kr0 ls0 r0 s0
1
0
+ ( sup Dt,u
)
kr−1 ls−1
kr−1 ls−1
t≥r0 ∪u≥s0
0
ht,u Dt,u
0
ht,u Dt,u
P
ht,u
(r0 <t≤r)∪(s0 <u≤s)
≤
P
F 0 kr0 ls0 r0 s0
1
+
²
ht,u
kr−1 ls−1
kr−1 ls−1 (r0 <t≤r)∪(s0 <u≤s)
≤
F 0 kr0 ls0 r0 s0
+ ²H 2 .
kr−1 ls−1
Since kr and ls both approach infinity as both p and q approach infinity, it follows
that
p,q
1 X
[fk (k4m xk+m,l+n k)]pk,l → 0, uniformly in m and n.
pq
k,l=1,1
Therefore x ∈ [t̄2 , 4m , F, P ].
100
V. A. Khan, Q. M. D. Lohani
As a consequence we obtain
Theorem 3. Let θr,s = {k, l} be a double lacunary sequence with lim inf rs qrs ≤
lim suprs qrs < ∞. Then for any sequence of moduli F = (fk ), [Lθr,s , 4m , F, P ] =
[t̄2 , 4m , F, P ].
Acknowledgement. The authors would like to record their gratitude to
the reviewer for his/her careful reading and making some useful corrections which
improved the presentation of the paper.
REFERENCES
[1] M. Et and Colak, On some generalized difference sequence spaces, Soochow J. Math. 21
(1995), 377–386.
[2] A. R. Freedman, J.J. Sember and M. Raphael, Some Cesaro-type summability spaces, Proc.
London Math. Soc. 37(3) (1978), 508–520.
[3] H. Kizmaz, On certain sequence spaces, Canadian Math. Bull., 24 (2) (1981), 169–176.
[4] E. Kolk, On strong boundedness and summmability with respect to a sequence of moduli,
Acta Comment. Univ. Tartu, 960 (1993), 41–50.
[5] E. Kolk, Inclusion theorems for some sequence spaces defined by a sequence of moduli, Acta
Comment. Univ. Tartu, 970 (1994), 65–72.
[6] I. J. Maddox, Sequence spaces defined by a modulus, Math. Camb. Phil. Soc., 100 (1986),
161–166.
[7] F. Moricz and B. E. Rhoades, Almost convergence of double sequences and strong regularity
of summability matrices, Math. Proc. Camb. Phil. Soc. 104 (1988), 283–293.
[8] S. Pehlivan and B. Fisher, Lacunary strong convergence with respect to a sequence of modulus
functions, Comment Math. Univ. Caroline 36 (1995), 69–76.
[9] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Annalen, 53
(1900), 289–321.
[10] W. H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J.
Math., 25 (1973), 973–975.
[11] E. Savas and R. F. Patterson, On some double almost lacunary sequences defined by Orlicz
function, FILOMAT, 19 (2005), 35–44.
(received 22.01.2007, in revised form 15.01.2008)
Department of Mathematics, A.M.U., Aligarh, INDIA
E-mail: vakhan@math.com