Acta Scientiarum http://www.uem.br/acta ISSN printed: 1806-2563 ISSN on-line: 1807-8664 Doi: 10.4025/actascitechnol.v35i1.15523
Some Generalized Lacunary statistically difference double seminormed sequence spaces defined by Orlicz function Bipan Hazarika1* and Ayhan Esi2 1 Department of Mathematics, Rajiv Gandhi University,Rono Hills, Doimukh-791 112, Arunachal Pradesh, India. 2Department of Mathematics, Science and Art Faculty, Adiyaman University, 02040, Adiyaman, Turkey. *Author for correspondence. E-mail:
[email protected]
ABSTRACT. In this article, we have introduced the idea of statistically convergent generalized difference lacunary double sequence spaces
[ w2 ( M , Δ n , p, q)]θ , [ w02 ( M , Δ n , p, q)]θ
and defined over a semi
norm space (X, q). Also we have study some basic properties and obtained some inclusion relations between them. Keywords: statistical convergent, P-convergent, difference sequence, lacunary sequence, Orlicz function.
Espaços sequenciais semi-normais duplos com diferença estatística lacunar generalizada definidos pela função de Orlicz RESUMO. Este trabalho apresenta a ideia de espaços sequenciais duplos com diferença lacunar generalizada e convergente
[ w2 ( M , Δ n , p, q)]θ e [ w02 ( M , Δ n , p, q)]θ
definidos em um espaço
semi normalizado. Além disso, foram também estudadas algumas propriedades básicas e obtidas algums relações de inclusão entre elas. Palavras-chave: estatística convergente, P-convergente, sequência de diferença, sequência lacunar, função de orlicz.
χE
Introduction
exists, where
The concept involving statistical convergence plays a vital role not only in pure mathematics but also in other branches of mathematics especially in information theory, computer science and biological science. Let ∞ ,c and c0 be the Banach spaces of
of E. A sequence ( xk ) ∈ w is said to be statistically
bounded, convergent and null sequences x = ( xk ) with the usual norm || x ||= sup | xk | . In order to k
extend the notion of convergence of sequences, statistical convergence was introduced by Fast (1951) and Schoenberg (1959) independently. Later on it was further investigated by Fridy (1985), Mursaleen and Mohiuddine (2009a and b), Mohiuddine and Danish Lohani (2009), Mohiuddine et al. (2010), Šalát (1980), Tripathy (2003), Tripathy and Sen (2001) and many others. The idea depends on the notion of density (natural or asymptotic) of subsets of N. A subset E of N is said to have natural density δ (E ) if 1 n χ E (k ) n →∞ n k =1
δ ( E ) = lim
Acta Scientiarum. Technology
denotes the characteristic function
convergent to a number if for every
ε >0
the set
{k ∈ :| xk − |≥ ε } has natural density zero. Kizmaz (1981) introduced the difference sequence spaces as follows:
notion
of
X (Δ) = { x = ( xk ) ∈ w : (Δxk ) ∈ X } for X= ∞ ,c
and co. Later on, the notion was
generalized by Et and Çolak (1995) as follows:
X (Δ m ) = { x = ( xk ) ∈ w : (Δ m xk ) ∈ X } for X= ∞ ,c and co, where
Δ m x = (Δ m xk ) = (Δ m −1 xk − Δ m −1 xk +1 ), Δ 0 x = x and also this generalized difference notion has the following binomial representation: Maringá, v. 35, n. 1, p. 131-138, Jan.-Mar., 2013
132
Hazarika and Esi
m m Δ m xk = (−1)i xk +i for all k ∈ . i =0 i
Subsequently, difference sequence spaces were studied by Esi (2009a and b), Esi and Tripathy (2008), Tripathy et al. (2005) and many others. An Orlicz function M is a function M :[0, ∞) → [0, ∞) , which is continuous, convex, nondecreasing function such that M(0) = 0, M(x) > 0, for x > 0 and M(x) → ∞ as x → ∞. If convexity of Orlicz function is replaced by M(x + y) ≤ M(x) + M(y) then this function is called the modulus function and characterized by Ruckle (1973). An Orlicz function M is said to satisfy ∆2 – condition for all values
if there exists K > 0 such that
M (2u ) ≤ KM (u ), u ≥ 0 .
Remark 1.1. An Orlicz function satisfies the inequality M(λx) ≤ λM(x) for all λ with 0 < λ < 1. Lindenstrauss and Tzafriri (1971) used the idea of Orlicz function to construct the sequence space ∞ | x | lM = ( xk ) : M k < ∞, for some r > 0 r k =1 which is a Banach space normed by
∞ | x | || ( xk ) ||= inf r > 0 : M k ≤ 1 . r k =1
Pringsheim sense that is, a double sequence x = (xk,l) has Pringsheim limit L (denoted by P − lim x = L ) provided that given ε > 0 there exists N∈ such that | xk ,l − L |< ε for k,l > N (PRINGSHEIM, 1900). And we called it as "P convergent". We shall denote the space of all P convergent sequences by c2. The double sequence x = (xk,l) is bounded if and only if there exists a positive number M such that | xk ,l |< M for all k and l. We shall denote the space of all bounded double sequences by l∞2 . The zero single sequence will be denoted by θ = (θ, θ, θ,…) and the zero double sequence will be denoted by θ2 = (θ). The notion of asymtotic density for subsets of × was introduced by Tripathy (2003). A subset E of × is said to have density ρ (E) if
ρ ( E ) = lim
p , q →∞
1 χ E (n, k ) pq n≤ p k ≤ q
exists. The notion of statistically convergent double sequences was introduced by Mursaleen and Edely (2003) and Tripathy (2003) independently. A double sequence (xk,l) is said to be statistically convergent to in Pringsheim’s sense if for every ε > 0, ρ({(k, l) ∈ × : | xk ,l − L |≥ ε })= 0. The double sequence
The space lM is closely related to the space l p , which
is
an
Orlicz
sequence
space
with
θ r ,s = {(kr , ls )}
is called
double lacunary sequence if there exist two increasing sequence of integers such that
p
M ( x) =| x | , for 1 ≤ p < ∞. In a later stage different Orlicz sequence spaces were introduced and studied by Tripathy and Mahanta (2004), Esi (1999, 2009a and b, 2010), Esi and Et (2000), Parashar and Choudhary (1994) and many others. The following well-known inequality will be used throughout the article. Let p = (pk) be any sequence of positive real numbers with
0 < pk ≤ sup pk = G, D = max{1, 2G −1} then k
pk
ko = 0, hr = kr − kr −1 → ∞, as r → ∞ and
l0 = 0, hs = ls − ls −1 → ∞ as s → ∞ . Notations: kr , s = kr ls , hr , s = hr hs and
θ r ,s
is
determined by
I r , s = {( k , l ) : kr −1 < k ≤ kr and ls −1 < l ≤ ls } pk
pk
| ak + bk | ≤ D(| ak | + | bk | ) qr =
for all k ∈ and ak, bk ∈ . Also | a | pk ≤ max{1,| a |G } for all a ∈ . Let w2 denote the set of all double sequences of complex numbers. By the convergence of a double sequence we mean the convergence in the Acta Scientiarum. Technology
l kr , qs = s and qr , s = qr qs . kr −1 ls −1
The set of all double lacunary sequences is denoted by Nθr ,s and defined by Savas and Patterson (2006) as follows: Maringá, v. 35, n. 1, p. 131-138, Jan.-Mar., 2013
Statistically convergent generalized difference lacunary double sequence spaces
Nθr ,s = x = ( xk ,l ) : P − lim (hr , s ) −1 | xk ,l − L |= 0, for some L r ,s ( k ,l )∈I r ,s
New types of double sequence spaces
In this presentation our goal is to extend a few results known in the literature from ordinary (single) difference sequences to difference double sequences. Some studies on double sequence spaces can be found in Gökhan and Çolak (2004, 2005 2006). Let M be an Orlicz function and p = ( pk ,l ) be a factorable double sequence of strictly positive real numbers and θ r , s be a lacunary sequence. Let X be a seminormed space over the complex field with the seminorm q. We now define the following new statistically convergent generalized difference lacunary double sequence spaces:
133
and q ( x) =| x | then we obtain the double sequence spaces [ w2 ]θ ,[ w02 ]θ and [ w∞2 ]θ . (iii) If M(x)=x, n = 0 for all k , l ∈ and q ( x) =| x | then we obtain the double sequence spaces [ w2 ( p )]θ ,[ w02 ( p )]θ and [ w∞2 ( p )]θ defined as follows:
and
(iv) If n = 0 and q ( x) =| x | then we obtain the double sequence spaces [ w2 ( M , p )]θ ,[ w02 ( M , p )]θ and
[ w∞2 ( M , p )]θ defined as follows:
and
and where: Δ u x = (Δ u xk ,l ) = (Δ u −1 xk ,l − Δ u −1 xk ,l +1 − Δ u −1 xk +1,l + Δ u −1 xk +1,l +1 )
(Δ1 xk ,l ) = (Δxk ,l )
= ( xk ,l − xk ,l +1 − xk +1,l + xk +1,l +1 ), Δ 0 xk ,l = xk ,l and also this generalized difference double notion has the following binomial representation:
Δ n xk ,l
n m = (−1)i + j xk +i ,l + j i =0 j =0 i j n
n
(v) If n=1 and double 2
q( x) =| x | then we obtain the sequence 2 0
[ w ( M , Δ, p)]θ ,[ w ( M , Δ, p )]θ
spaces and
2 ∞
[ w ( M , Δ, p)]θ defined as follows:
Some double sequence spaces are obtained by specializing θ r , s M, p, q and n. Here are some examples: (i) If =0,
θ r , s = {(kr , ls )} = {(2r , 2s )} , M(x)=x, n
pk ,l = 1 for all k , l ∈ and q( x) =| x | then
we obtain the double sequence spaces and
and
[ w2 ],[ w02 ]
[ w∞2 ] .
(ii) If M(x)=x, n = 0 , Acta Scientiarum. Technology
pk ,l = 1 for all k , l ∈
where
(Δxk ,l ) = (xk ,l − xk ,l +1 − xk +1,l + xk +1,l +1 ) . Maringá, v. 35, n. 1, p. 131-138, Jan.-Mar., 2013
134
Hazarika and Esi
Main Results
Theorem 3.1. Let p = ( pk ,l ) be bounded. The classes of [ w2 ( M , Δ n , p, q )]θ , [ w02 ( M , Δ n , p, q )]θ and [ w∞2 ( M , Δ n , p, q )]θ are linear spaces over the complex field . Proof. We give the proof for the space [ w∞2 ( M , Δ n , p, q )]θ and for the others spaces the proof can be obtained in a similar way. Let
x = ( xk ,l ), y = ( yk ,l ) ∈ [ w∞2 ( M , Δ n , p, q)]θ . Then
Proof. Since q is a seminorm, so we have
f (( xk ,l )) ≥ 0 for all x = ( xk ,l ) f (θ 2 ) = 0 and f ((λ xk ,l )) =| λ | f (( xk ,l )) for all scalars λ . Now,
x = ( xk ,l ) ,
let 2 0
n
y = ( yk ,l ) ∈ [ w ( M , Δ , p, q )]θ Then there exist
ρ1 > 0 , ρ 2 > 0
such that
we have
(3.1) Let
and
ρ = ρ1 + ρ 2
Then we have,
(3.2)
α, β ∈
be scalars and ρ = max{2 | α | ρ1 , 2 | β | ρ 2 } . Since M is nondecreasing convex function, we have Let
q (Δ n (α xk ,l + β yk ,l )) M ρ pk ,l
q(Δ n ( β yk ,l )) + M ρ
n ≤ D M q(Δ xk ,l )
pk ,l
q ( Δ n yk , l ) + M ρ
ρ
ρ
where D = max(1, 2
H −1
> 0 so we have,
pk ,l
n ≤ M q(Δ (α xk ,l ))
Since ρ1 , ρ 2
pk ,l
pk ,l
), H = sup pk ,l < ∞ .
=
k ,l
Now, from (3.1) and (3.2), we have
f (( xk ,l )) + f (( yk ,l )) .
Therefore f is a seminorm. Theorem 3.3. Let (X, q) be a complete seminormed space. Then the spaces
[ w2 ( M , Δ n , p, q)]θ Therefore α x + β y ∈ [ w∞2 ( M , Δ n , p, q )]θ .
Hence
[ w∞2 ( M , Δ n , p, q )]θ is a linear space. Theorem 3.2. The double sequence spaces and [ w2 ( M , Δ n , p, q )]θ [ w02 ( M , Δ n , p, q )]θ 2 n are seminormed spaces, [ w∞ ( M , Δ , p, q )]θ seminormed by Acta Scientiarum. Technology
2 ∞
[ w02 ( M , Δ n , p, q)]θ
n
[ w ( M , Δ , p, q)]θ
and
are complete seminormed
spaces seminormed by f. Proof. We prove the theorem for the space
[ w02 ( M , Δ n , p, q)]θ
The other cases can be
establish
similar
i
i k ,l
following
x = (x )
be
a
Cauchy
technique. sequence
Let in
Maringá, v. 35, n. 1, p. 131-138, Jan.-Mar., 2013
Statistically convergent generalized difference lacunary double sequence spaces
[ w02 ( M , Δ n , p, q)]θ . Let ε > 0 be given and for b bx x0 fixed such that M 0 ≥ 1 and 2 there exists m0 ∈ such that
((
))
f xki , l − xk , l < ε , for all i ≥ mo .
> 0, choose
By definition of seminorm, we have
135
(x
Thus
i k ,l
− xk ,l ) ∈ [ w02 ( M , Δ n , p, q )]θ .
linearity of the space for all
By
[ w02 ( M , Δ n , p, q)]θ we have
i ≥ m0 ,
( xk ,l ) = ( xki ,l ) + ( xki ,l − xk ,l ) ∈ [ w02 ( M , Δ n , p, q)]θ
[ w02 ( M , Δ n , p, q)]θ is a complete semi-
Thus
normed space. Theorem 3.4. (a) If 0 < inf pk ,l ≤ pk ,l < 1
(3.3)
k ,l
This implies that n
(
n
) q(x
q xki ,1 − xkj,1 +
k =1
i 1, l
)
− x1j, l < ε
l =1
i
j
This shows that ( xk ,1 ) and ( x1,l )( k , l ≤ n) are Cauchy sequences in (X, q) Since (X, q) is complete, so there exists xk ,1 , x1,l ∈ X such that i k ,1
lim x i →∞
j 1,l
= xk ,1 and lim x = x1,l (k , l ≤ n) j →∞
Now from (3.3), we have
( ( )) ≤ 1 ≤ M bx (( ) ( )) 2
q Δn xki , l − xkj, l M f xki , l − x j k ,l (3.4)
o
, for all i, j ≥ mo .
This implies
( (
))
q Δn xki , l − xkj, l ≤
bxo ε ε . = , for all i, j ≥ mo 2 bxo 2
So ( Δ ( x ) ) is a Cauchy sequence in (X, q). n
i k ,l
Since (X, q) is complete, there exists xk,l ∈ X such n
i
that lim Δ ( xk ,l ) = xk ,l for all i
is continuous, so for
[ Z ( M , Δ , p, q)]θ ⊂ [ Z ( M , Δ u , q)]θ (b) If then 1 < pk ,l ≤ sup pk ,l < ∞
then
k , l ∈ Since M
i ≥ m0 , on taking limit as
j → ∞ we have from (3.4),
2
u
k ,l
2
u
[ Z ( M , Δ , q)]θ ⊂ [ Z ( M , Δ u , p, q)]θ 2
2
2
2 0
Acta Scientiarum. Technology
where
2 ∞
Z = w , w and w . Proof. The first part of the result follows from the inequality
q (Δ n xk ,l ) q (Δ n xk ,l ) M ≤ M ρ ρ
pk ,l
and the second part of the result follows from the inequality pk ,l
q (Δ n xk ,l ) M ρ
q ( Δ n xk ,l ) ≤M . ρ
Theorem 3.5. Let M 1 and M 2 be Orlicz
functions
β = lim t →∞
Δ 2 -condition.
satisfying
( M 2 (t )) ≥1 t
If then
[ Z 2 ( M 1 , Δ n , p, q )]θ = [ Z 2 ( M 1 M 2 , Δ n , p, q)]θ 2
2
2
2
where Z = w , w0 and w∞ . 2
2
Proof. We prove it for Z = w0 and the other
cases will follows on applying similar techniques. 2
n
Let x = ( xk ,l ) ∈ [ w0 ( M 1 , Δ , p, q )]θ then st 2 − lim(hr , s )
−1
On taking the infimum of such
2
ρ ′ s, we have
r,s
(
q Δn −1 x k ,l M1 ρ (k , l )∈l r ,s
)
p k ,l
=0
Maringá, v. 35, n. 1, p. 131-138, Jan.-Mar., 2013
136
Hazarika and Esi
Let 0 < ε < 1 and δ with 0 < δ < 1 such that M 2 (t ) < ε for 0 ≤ t < δ . Let
yk , l
q(Δ n xk ,l ) = M 1 ρ
implies x = ( xk ,l ) ∈ [ w02 ( M1 , Δ n , p, q)]θ . This implies
[ w02 ( M 1 , Δ n , p, q)]θ = [ w02 ( M 1 M 2 , Δ n , p, q)]θ Theorem 3.6. Let M, M 1 and M 2 be Orlicz functions, q, q1 and q2 be seminorms. Then (i) (ii) (iii)
and consider
[ M 2 ( yk ,l )]
pk ,l
= [ M 2 ( yk ,l )]
pk ,l
+ [ M 2 ( yk ,l )]
pk ,l
(3
.6)
q1
If
2
is
stronger
than
2
n
n
q2 ,
then
[ Z ( M , Δ , p, q1 )]θ ⊂ [ Z ( M , Δ , p, q2 )]θ , 2
2
2
2
where Z = w , w0 and w∞ .
where the first term is over yk ,l ≤ δ and the second is over yk ,l > δ From the first term in (3.6), using the Remark 1.1
[ M 2 ( yk ,l )]
pk ,l
2
2
Proof. (i) We establish it for only Z = w0 The
rest
of
the
cases
2 0
are
similar.
n
2 0
Let
n
x = ( xk ,l ) ∈ [ w ( M 1 , Δ , p, q)]θ ∩ [ w ( M 2 , Δ , p, q)]θ Then
H
< [ M 2 (2)] + [( yk ,l )]
pk ,l
(3.7)
On the other hand, we use the fact that
yk ,l
