R E S E A R C H
Open Access
Statistical summability through a lacunary
sequence in locally solid Riesz spaces
Syed Abdul Mohiuddine
*and Mohammed A Alghamdi
*Correspondence:
[email protected] Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
Abstract
In this paper, we introduce the concepts of lacunary statistical
τ
-convergence, lacunary statisticallyτ
-bounded and lacunary statisticallyτ
-Cauchy in the framework of locally solid Riesz spaces. We also define a new type of convergence, that is, S∗(τ)-convergence in this setup and prove some interesting results related to these notions.MSC: 40A35; 40G15; 46A40
Keywords: density; statistical convergence; lacunary sequence; locally solid Riesz
space
1 Introduction and preliminaries
In Fast [] presented the following definition of statistical convergence for sequences of real numbers. We shall denote byNthe set of all natural numbers. LetK ⊆Nand Kn={k≤n:k∈K}. Then thenatural densityofKis defined byδ(K) =limnn–|Kn|if the limit exists, where the vertical bars indicate the number of elements in the enclosed set. The sequencex= (xk) is said to bestatistically convergenttoLif for everyε> , the set Kε:={k∈N:|xk–L| ≥ε}has natural density zero (cf.Fast []),i.e., for eachε> ,
lim n
nj≤n:|xj–L| ≥ε= .
In this case, we writeL=st-limx. Note that every convergent sequence is statistically convergent, but not conversely. For example, suppose that the sequencex= (xn) is defined as
x= (xn) =
⎧ ⎨ ⎩
√
n, ifnis a square,
, otherwise.
It is clear that the sequencex= (xn) is statistically convergent to , but it is not convergent. In Fridy [] presented the notion of a statistically Cauchy sequence and proved that it is equivalent to statistical convergence. Active research on this topic was started after the papers of Fridy. Mursaleen and Edely [] extended these concepts from single sequences to double sequences by using two dimensional analogue of natural density. In the recent past, Mursaleen and Mohiuddine [, ] defined these notions for double sequences in locally solid Riesz spaces as well as in intuitionistic fuzzy normed spaces and proved some
interesting results. Subsequently, the statistical convergence for sequences of real numbers in several spaces has been extensively investigated by a number of authors, and there are many interesting results concerning this concept. For more details related to this concept, we refer to [–] and references therein.
Now, we recall some basic definitions and notions related to the concept of locally solid Riesz spaces. LetXbe a real vector space and≤be a partial order on this space. ThenX is said to be anordered vector spaceif it satisfies the following properties:
(i) ifx,y∈Xandy≤x, theny+z≤x+zfor eachz∈X. (ii) ifx,y∈Xandy≤x, thenλy≤λxfor eachλ≥.
If in additionXis a lattice with respect to the partial order≤, thenXis said to be aRiesz space(or avector lattice) [].
For an elementxof a Riesz spaceX, the positive part ofxis defined byx+=x∨θ = sup{x,θ}, the negative part ofxbyx–= (–x)∨θand the absolute value ofxby|x|=x∨(–x),
whereθ is the zero element ofX.
A subsetSof a Riesz spaceXis said to besolidify∈Sand|x| ≤ |y|impliesx∈S. Atopological vector space(X,τ) is a vector spaceX, which has a (linear) topologyτ, such that the algebraic operations of addition and scalar multiplication inXare continuous. Continuity of addition means that the functionf :X×X→Xdefined byf(x,y) =x+y is continuous onX×X, and continuity of scalar multiplication means that the function f :C×X→Xdefined byf(λ,x) =λxis continuous onC×X.
Every linear topologyτ on a vector spaceXhas a baseN for the neighborhoods ofθ
satisfying the following properties:
(C) EachY∈N is abalanced set, that is,λx∈Yholds for allx∈Yand everyλ∈Rwith |λ| ≤.
(C) EachY∈N is anabsorbing set, that is, for everyx∈X, there existsλ> such that λx∈Y.
(C) For eachY∈N, there exists someE∈N withE+E⊆Y.
A linear topologyτ on a Riesz spaceXis said to belocally solid[] ifτ has a base at zero consisting of solid sets. Alocally solid Riesz space(X,τ) is a Riesz space equipped with a locally solid topologyτ.
The rest of the paper is organized as follows. In Section , first we recall the notion of lacunary sequences and define the concepts of lacunary statisticallyτ-convergent and la-cunary statisticallyτ-bounded and prove some interesting results. Section is devoted to introduce concept of lacunary statisticallyτ-Cauchy and to proving that a lacunary sta-tisticallyτ-convergent sequence is lacunary statisticallyτ-Cauchy. Also, we defineSθ∗(τ )-convergent and prove that it is equivalent to lacunary statisticallyτ-convergent for a first countable space.
2 Lacunary statistical
τ
-convergenceBy a lacunary sequence, we mean an increasing integer sequenceθ= (kr) such thatk=
andhr:=kr–kr–→ ∞asr→ ∞. Throughout this paper, the intervals determined byθ
will be denoted byIr:= (kr–,kr], and the ratiokr/kr–will be abbreviated byqr.
LetK⊆N. The number
δθ(K) =lim r
hr
is said to be theθ-densityofK, provided the limit exists.
In Fridy and Orhan [] defined the concept of lacunary statistical convergence as follows.
Letθ be a lacunary sequence. Then a sequencex= (xk) is said to beSθ-convergent to the numberLif for every> , the setK() hasθ-density zero, where
K() :=k∈N:|xk–L| ≥
.
In this case, we writeSθ-limx=Lorxk→L(Sθ).
Remarks . It is well known that every convergent sequence is lacunary statistically con-vergent, but the converse is not true. For example, let the sequencex= (xk) be defined by
xk=
⎧ ⎨ ⎩
k; forkr– [
√
hr] + ≤k≤kr,r∈N, ; otherwise.
Thenxis lacunary statistically convergent to , but it is not convergent.
We shall assume throughout this paper that the symbol Nsol will denote any base at zero consisting of solid sets and satisfying the conditions (C), (C) and (C) in a locally
solid topology. For our convenience, here and in what follows, we shall write an LSR-space instead of a locally solid Riesz space.
Definition . Let (X,τ) be an LSR-space andθbe a lacunary sequence. Then a sequence x= (xj) inXis said to belacunary statisticallyτ-convergent(or Sθ(τ)-convergent) to the elementξ ∈X if for everyτ-neighborhoodU of zero,δθ(KU) = , where KU={j∈N: xj–ξ∈/U},i.e.,
lim r→∞
hr
{j∈Ir:xj–ξ∈/U}= .
In this case, we writeSθ(τ)-limx=ξ orxj Sθ(τ)
–→ξ.
Definition . Let (X,τ) be an LSR-space andθ be a lacunary sequence. We say that a sequencex= (xj) inXislacunary statisticallyτ-bounded(or Sθ(τ)-bounded) if for every
τ-neighborhoodUof zero there exists someλ> such that the setMU={j∈N:λxj∈/U} hasθ-density zero (shortly,δθ(MU) = ),i.e.,
lim r→∞
hr
{j∈Ir:λxj∈/U}= .
Theorem . Let(X,τ)be a Hausdorff LSR-space andθbe a lacunary sequence.Suppose that x= (xj)and y= (yk)are two sequences in X.Then the following hold:
(i) IfSθ(τ)-limjxj=ξandSθ(τ)-limjxj=ξ,thenξ=ξ. (ii) IfSθ(τ)-limjxj=ξ,thenSθ(τ)-limjαxj=αξ,α∈R.
Proof (i) Suppose thatSθ(τ)-limjxj=ξandSθ(τ)-limjxj=ξ. LetUbe anyτ
-neighbor-hood of zero. Then there existsY∈Nsolsuch thatY⊆U. Choose anyE∈Nsolsuch that E+E⊆Y. We define the following sets:
K={j∈N:xj–ξ∈E},
K={j∈N:xj–ξ∈E}.
SinceSθ(τ)-limjxj=ξandSθ(τ)-limjxj=ξ, we haveδθ(K) =δθ(K) = . Thusδθ(K∩
K) = , and in particularK∩K=∅. Now, letj∈K∩K. Then
ξ–ξ=ξ–xj+xj–ξ∈E+E⊆Y⊆U.
Hence, for everyτ-neighborhoodUof zero, we haveξ–ξ∈U. Since (X,τ) is Hausdorff,
the intersection of allτ-neighborhoodsUof zero is the singleton set{}. Thus, we get
ξ–ξ=,i.e.ξ=ξ.
(ii) LetUbe an arbitraryτ-neighborhood of zero andSθ(τ)-limjxj=ξ. Then there exists Y∈Nsolsuch thatY⊆Uand also
lim r→∞
hr
{j∈Ir:xj–ξ∈Y}= .
SinceYis balanced,xj–ξ∈Yimpliesα(xj–ξ)∈Yfor everyα∈Rwith|α| ≤. Hence,
{j∈N:xj–ξ∈Y} ⊆ {j∈N:αxj–αξ∈Y}
⊆ {j∈N:αxj–αξ∈U}.
Thus, we obtain
lim r→∞
hr
{j∈Ir:αxj–αξ∈U}= ,
for eachτ-neighborhoodUof zero. Now, let|α|> and [|α|] be the smallest integer greater than or equal to|α|. There existsE∈Nsolsuch that [|α|]E⊆Y. SinceSθ(τ)-limjxj=ξ, the set
K={j∈N:xj–ξ∈E}
hasθ-density zero. Therefore,
|αξ–αxj|=|α||ξ–xj| ≤
|α| |ξ–xj| ∈
|α|E⊆Y⊆U.
Since the setY is solid, we haveαξ–αxj∈Y. This implies thatαξ–αxj∈U. Thus,
lim r→∞
hr
{j∈Ir:αxj–αξ∈U}= ,
(iii) LetUbe an arbitraryτ-neighborhood of zero. Then there existsY∈Nsolsuch that Y⊆U. ChooseEinNsolsuch thatE+E⊆Y. SinceSθ(τ)-limjxj=ξandSθ(τ)-limj,kyjk=
η, we haveδθ(H) = =δθ(H), where
H={j∈N:xj–ξ∈E}, H={j∈N:yj–η∈E}.
LetH=H∩H. Hence, we haveδθ(H) = and
(xj+yj) – (ξ+y) = (xj–x) + (yj–η)∈E+E⊆Y⊆U.
Therefore,
lim r→∞
hr
j∈Ir: (xj+yj) – (ξ+η)∈U= .
SinceUis arbitrary, we haveSθ(τ)-limj(xj+yj) =ξ+η.
Theorem . Let(X,τ)be an LSR-space andθ be a lacunary sequence.If a sequence x= (xj)is lacunary statisticallyτ-convergent,then it is lacunary statisticallyτ-bounded.
Proof Supposex= (xj) is lacunary statisticallyτ-convergent to the pointξ∈Xand letU be an arbitraryτ-neighborhood of zero. Then there existsY∈Nsolsuch thatY⊆U. Let us chooseE∈Nsolsuch thatE+E⊆Y. SinceSθ(τ)-limj→∞xj=ξ, the set
K={j∈N:xj–ξ∈/E}
hasθ-density zero. SinceEis absorbing, there existsλ> such thatλξ∈E. Letαbe such thatα≤ andα≤λ. SinceEis solid and|αξ| ≤ |λξ|, we haveαξ∈E. SinceEis balanced, xj–ξ∈Eimpliesα(xj–ξ)∈E. Then we have
αxj=α(xj–ξ) +αξ∈E+E⊆Y⊆U,
for eachj∈N\K. Thus,
lim r→∞
hr
{j∈Ir:αxj∈/U}= .
Hence, (xj) is lacunary statisticallyτ-bounded.
Theorem . Let(X,τ)be an LSR-space andθbe a lacunary sequence.If(xj), (yj)and(zj)
are three sequences such that
(i) xj≤yj≤zj,for allj∈N,
Proof Let U be an arbitraryτ-neighborhood of zero, there existsY ∈Nsol such that Y⊆U. ChooseE∈Nsol such thatE+E⊆Y. From the condition (ii), we haveδθ(A) = =δθ(B), where
A={j∈N:xj–ξ∈E}, B={j∈N:zj–ξ∈E}.
Also, we getδθ(A∩B) = , and from (i), we have
xj–ξ≤yj–ξ≤zj–ξ
for allj∈N. This implies that for allj∈A∩B, we get
|yj–ξ| ≤ |xj–ξ|+|zj–ξ| ∈E+E⊆Y.
SinceYis solid, we haveyj–ξ∈Y⊆U. Thus,
lim r→∞
hr
{j∈Ir:yj–ξ∈U}= ,
for eachτ-neighborhoodUof zero. Hence,Sθ(τ)-limjyj=ξ.
3 Lacunary statistically
τ
-Cauchy andS∗θ(τ
)-convergenceIn the present section, first we define the concept of lacunary statisticallyτ-Cauchy in locally solid Riesz spaces as follows.
Definition . Let (X,τ) be an LSR-space andθ be a lacunary sequence. A sequencex= (xj) inXislacunary statisticallyτ-Cauchyif for everyτ-neighborhoodUof zero there existsp∈Nsuch that
lim r→∞
hr
{j∈Ir:xj–xp∈/U}= .
Theorem . Let(X,τ)be an LSR-space andθbe a lacunary sequence.If a sequence x= (xj)is lacunary statisticallyτ-convergent,then it is lacunary statisticallyτ-Cauchy.
Proof Suppose thatSθ(τ)-limj→∞xj=ξ. LetU be an arbitraryτ-neighborhood of zero, there existsY∈Nsolsuch thatY⊆U. ChooseE∈Nsolsuch thatE+E⊆Y. Then
lim r→∞
hr
{j∈Ir:xj–ξ∈/E}= .
Also, we have
xj–xp=xj–ξ+ξ–xp∈E+E⊆Y⊆U,
Therefore, the set
{j∈N:xj–xp∈/U} ⊆K.
For everyτ-neighborhoodUof zero, there existsN∈Nsuch that for allj,p≥N,
lim r
hr{
j∈Ir:xj–xp∈/U}= .
Hence, (xj) is lacunary statisticallyτ-Cauchy.
Now, we define another type of convergence in locally solid Riesz spaces.
Definition . Letθ be a lacunary sequence. A sequence (xj) in an LSR-space (X,τ) is said to beS∗θ(τ)-convergenttoξ∈Xif there exists an index setK={jn} ⊆N,n= , , . . . , withδθ(K) = such thatlimn→∞xjn=ξ. In this case, we writeξ=S∗θ(τ)-limx.
Theorem . Letθ be a lacunary sequence.A sequence x= (xj)is lacunary statistically
τ-convergent to a numberξif it is S∗θ(τ)-convergent toξin a locally solid Riesz space(X,τ).
Proof LetUbe an arbitraryτ-neighborhood ofξ. Sincex= (xj) isSθ∗(τ)-convergent toξ, there is an index setK={jn} ⊆N,n= , , . . . , withδθ(K) = andj=j(U), such thatj≥j
andj∈Kimplyxj–ξ∈U. Then
KU={j∈N:xj–ξ∈/U} ⊆N–{jN+,jN+, . . .}.
Therefore,
δθ(KU)≤ – = .
Hence,xis lacunary statisticallyτ-convergent toξ.
Note that the converse holds for a first countable space.
Recall that a first countable space is a topological space satisfying the ‘first axiom of countability’. Specifically, a spaceXis said to be first countable if each point has a countable neighborhood basis (local base). That is, for each pointx inXthere exists a sequence U,U, . . . of open neighborhoods ofxsuch that for any open neighborhoodVofxthere
exists an integeriwithUicontained inV.
Theorem . Let(X,τ)be a first countable LSR-space andθ be a lacunary sequence.If
a sequence x= (xj)is lacunary statisticallyτ-convergent to a numberξ,then it is S∗θ(τ) -convergent toξ.
Proof Letxbe lacunary statisticallyτ-convergent to a numberξ. Fix a countable local baseU⊃U⊃U⊃ · · ·atξ. For everyi∈N, put
and
Mi={j∈N:xj–ξ∈Ui} (i= , , , . . .). Thenδθ(Ki) = and
() M⊃M⊃ · · · ⊃Mi⊃Mi+⊃ · · ·
and
() δλ(Mi) = ,i= , , , . . ..
Now, we have to show that forj∈Mi, (xjn) is convergent toξ. Suppose that (xjn) is not convergent toξ. Therefore,xjn–ξ∈/Uifor infinitely many terms. Let
Mr={j∈N:xjn–ξ∈Ur} (r>i). Then
() δθ(Mr) = ,
and by (),Mi⊂Mr. Hence,δθ(Mi) = , which contradicts (). Therefore, (xjn) is
conver-gent toξ. Hence, by Definition .,xisSθ∗(τ)-convergent toξ.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. Both the authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments.
Received: 19 May 2012 Accepted: 21 September 2012 Published: 9 October 2012 References
1. Fast, H: Sur la convergence statistique. Colloq. Math.2, 241-244 (1951) 2. Fridy, JA: On statistical convergence. Analysis5, 301-313 (1985)
3. Mursaleen, M, Edely, OHH: Statistical convergence of double sequences. J. Math. Anal. Appl.288, 223-231 (2003) 4. Mohiuddine, SA, Alotaibi, A, Mursaleen, M: Statistical convergence of double sequences in locally solid Riesz spaces.
Abstr. Appl. Anal.2012, Article ID 719729 (2012)
5. Mursaleen, M, Mohiuddine, SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals41, 2414-2421 (2009)
6. Albayrak, H, Pehlivan, S: Statistical convergence and statistical continuity on locally solid Riesz spaces. Topol. Appl. 159, 1887-1893 (2012)
7. Çakalli, H: Lacunary statistical convergence in topological groups. Indian J. Pure Appl. Math.26(2), 113-119 (1995) 8. Çakalli, H: On Statistical Convergence in topological groups. Pure Appl. Math. Sci.43, 27-31 (1996)
9. Çakalli, H, Khan, MK: Summability in topological spaces. Appl. Math. Lett.24, 348-352 (2011)
10. Çakalli, H, Sava¸s, E: Statistical convergence of double sequence in topological groups. J. Comput. Anal. Appl.12(2), 421-426 (2010)
11. Maddox, IJ: Statistical convergence in a locally convex space. Math. Proc. Camb. Philos. Soc.104, 141-145 (1988) 12. Di Maio, G, Koˇcinac, LDR: Statistical convergence in topology. Topol. Appl.156, 28-45 (2008)
13. Mohiuddine, SA, Aiyub, M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inf. Sci.6(3), 581-585 (2012)
14. Mohuiddine, SA, Alotaibi, A, Alsulami, SM: Ideal convergence of double sequences in random 2-normed spaces. Adv. Differ. Equ.2012, 149 (2012)
15. Mohiuddine, SA, Danish Lohani, QM: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solitons Fractals42, 1731-1737 (2009)
16. Mohiuddine, SA, ¸Savas, E: Lacunary statistically convergent double sequences in probabilistic normed spaces. Ann. Univ. Ferrara (2012). doi:10.1007/s11565-012-0157-5
17. Mohiuddine, SA, ¸Sevli, H, Cancan, M: Statistical convergence in fuzzy 2-normed space. J. Comput. Anal. Appl.12(4), 787-798 (2010)
18. Mursaleen, M, Çakan, C, Mohiuddine, SA, Sava¸s, E: Generalized statistical convergence and statistical core of double sequences. Acta Math. Sin. Engl. Ser.26, 2131-2144 (2010)
20. Mursaleen, M, Mohiuddine, SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math.233, 142-149 (2009)
21. Mursaleen, M, Mohiuddine, SA: On ideal convergence of double sequences in probabilistic normed spaces. Math. Rep. (Bucur.)12(64)(4), 359-371 (2010)
22. Mursaleen, M, Mohiuddine, SA: On ideal convergence in probabilistic normed spaces. Math. Slovaca62, 49-62 (2012) 23. Mursaleen, M, Mohiuddine, SA, Edely, OHH: On the ideal convergence of double sequences in intuitionistic fuzzy
normed spaces. Comput. Math. Appl.59, 603-611 (2010)
24. Sava¸s, E, Mohiuddine, SA:λ¯-statistically convergent double sequences in probabilistic normed spaces. Math. Slovaca 62(1), 99-108 (2012)
25. Zaanen, AC: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin (1997) 26. Roberts, GT: Topologies in vector lattices. Math. Proc. Camb. Philos. Soc.48, 533-546 (1952) 27. Fridy, JA, Orhan, C: Lacunary statistical convergence. Pac. J. Math.160, 43-51 (1993)
doi:10.1186/1029-242X-2012-225
