R E S E A R C H
Open Access
On generalized double statistical
convergence in a random
-normed space
Ekrem Savas
**Correspondence:
[email protected]; [email protected] Department of Mathematics, Istanbul Commerce University, Uskudar, Istanbul, Turkey
Abstract
Recently, the concept of statistical convergence has been studied in 2-normed and random 2-normed spaces by various authors. In this paper, we shall introduce the concept of
λ-double statistical convergence and
λ-double statistical Cauchy in a
random 2-normed space. We also shall prove some new results.MSC: 40A05; 40B50; 46A19; 46A45
Keywords: statistical convergence;
λ-double statistical convergence;
t-norm; 2-norm; random 2-normed space1 Introduction
The probabilistic metric space was introduced by Menger [] which is an interesting and an important generalization of the notion of a metric space. The theory of probabilistic normed (or metric) space was initiated and developed in [–]; further it was extended to random/probabilistic -normed spaces by Goleţ [] using the concept of -norm which is defined by Gähler (see [, ]); and Gürdal and Pehlivan [] studied statistical conver-gence in -normed spaces. Also statistical converconver-gence in -Banach spaces was studied by Gürdal and Pehlivan in []. Moreover, recently some new sequence spaces have been studied by Savas [–] by using -normed spaces.
In order to extend the notion of convergence of sequences, statistical convergence of sequences was introduced by Fast [] and Schoenberg [] independently. A lot of devel-opments have been made in this areas after the works of ˘Salát [] and Fridy []. Over the years and under different names, statistical convergence has been discussed in the the-ory of Fourier analysis, ergodic thethe-ory and number thethe-ory. Recently, Mursaleen [] stud-iedλ-statistical convergence as a generalization of the statistical convergence, and in [] he considered the concept of statistical convergence of sequences in random -normed spaces. Quite recently, Bipan and Savas [] defined lacunary statistical convergence in a random -normed space, and also Savas [] studiedλ-statistical convergence in a ran-dom -normed space.
The notion of statistical convergence depends on the density of subsets ofN, the set of natural numbers. LetKbe a subset ofN. Then the asymptotic density ofKdenoted by δ(K) is defined as
δ(K) = lim n→∞
n{k≤n:k∈K},
where the vertical bars denote the cardinality of the enclosed set.
A single sequencex= (xk) is said to bestatistically convergenttoif for everyε> , the
setK(ε) ={k≤n:|xk–| ≥ε}has asymptotic density zero,i.e.,
lim n→∞
nk≤n:|xk–| ≥ε= .
In this case we writeS–limx=orxk→(S) (see [, ]).
2 Definitions and preliminaries
We begin by recalling some notations and definitions which will be used in this paper.
Definition A function f : R→R+
is called a distribution function if it is a non-decreasing and left continuous withinft∈Rf(t) = andsupt∈Rf(t) = . ByD+, we denote the set of all distribution functions such thatf() = . Ifa∈R+, thenHa∈D+, where
Ha(t) =
⎧ ⎨ ⎩
, ift>a; , ift≤a.
It is obvious thatH≥f for allf ∈D+.
A t-normis a continuous mapping∗: [, ]×[, ]→[, ] such that ([, ],∗) is an Abelian monoid with unit one andc∗d≥a∗bifc≥aandd≥bfor alla,b,c,d∈[, ]. A triangle functionτ is a binary operation onD+, which is commutative, associative and τ(f,H) =f for everyf ∈D+.
In [], Gähler introduced the following concept of a -normed space.
Definition LetXbe a real vector space of dimensiond> (dmay be infinite). A real-valued function·,·fromXintoRsatisfying the following conditions:
() x,x= if and only ifx,xare linearly dependent, () x,xis invariant under permutation,
() αx,x=|α|x,x, for anyα∈R, () x+x,x ≤ x,x+x,x
is called a -norm onXand the pair (X,·,·) is called a -normed space.
A trivial example of a -normed space isX=R, equipped with the Euclidean -norm x,xE= the area of the parallelogram spanned by the vectorsx,xwhich may be given explicitly by the formula
x,xE=det(xij)=abs
detxi,xj
,
wherexi= (xi,xi)∈Rfor eachi= , .
Recently, Goleţ [] used the idea of a -normed space to define a random -normed space.
(PN) F(x,y;t) =H(t)ifxandyare linearly dependent, whereF(x,y;t)denotes the value ofF(x,y)att∈R,
(PN) F(x,y;t)=H(t)ifxandyare linearly independent, (PN) F(x,y;t) =F(y,x;t), for allx,y∈X,
(PN) F(αx,y;t) =F(x,y;|αt|), for everyt> ,α= andx,y∈X,
(PN) F(x+y,z;t)≥τ(F(x,z;t),F(y,z;t)), wheneverx,y,z∈X. If (PN) is replaced by
(PN) F(x+y,z;t+t)≥F(x,z;t)∗F(y,z;t), for allx,y,z∈Xandt,t∈R+; then (X,F,∗) is called arandom -normedspace (for short, RNS).
Remark Every -normed space (X,·,·) can be made a random -normed space in a natural way by settingF(x,y;t) =H(t–x,y) for everyx,y∈X,t> anda∗b=min{a,b},
a,b∈[, ].
Example Let (X,·,·) be a -normed space withx,z=xz–xz,x= (x,x),z= (z,z) anda∗b=ab,a,b∈[, ]. For allx∈X,t> and nonzeroz∈X, consider
F(x,z;t) =
⎧ ⎨ ⎩
t
t+x,z, ift> ;
, ift≤.
Then (X,F,∗) is a random -normed space.
Definition A sequencex= (xk,l) in a random -normed space (X,F,∗) is said to be
double convergent(orF-convergent) to∈Xwith respect toFif for eachε> ,η∈(, ), there exists a positive integernsuch thatF(xk,l–,z;ε) > –η, wheneverk,l≥nand for nonzeroz∈X. In this case we writeF–limk,lxk,l=, andis called theF-limit of
x= (xk,l).
Definition A sequencex= (xk,l) in a random -normed space (X,F,∗) is said to be
double Cauchywith respect to F if for each ε> ,η∈(, ) there existN=N(ε) and
M=M(ε) such thatF(xk,l–xp,q,z;ε) > –η, wheneverk,p≥N andl,q≥Mand for
nonzeroz∈X.
Definition A sequencex= (xk,l) in a random -normed space (X,F,∗) is said to be
double statistically convergentorSRN-convergentto some∈Xwith respect toFif for
eachε> ,η∈(, ) and for nonzeroz∈Xsuch that
δ(k,l)∈N×N:F(xk,l–,z;ε)≤ –η = .
In other words, we can write the sequence (xk,l)double statistically convergestoin
random -normed space (X,F,∗) if
lim m,n→∞
or equivalently,
δk,l∈N:F(xk,l–,z;ε) > –η = ,
i.e.,
S– lim
k,l→∞F(xk,l–,z;ε) = .
In this case we writeSRN –limx=, andis called theSRN-limit ofx. LetSRN(X)
denote the set of all double statistically convergent sequences in a random -normed space (X,F,∗).
In this article, we studyλ-double statistical convergence in a random -normed space which is a new and interesting idea. We show that some properties ofλ-double statistical convergence of real numbers also hold for sequences in random -normed spaces. We es-tablish some relations related to double statistically convergent andλ-double statistically convergent sequences in random -normed spaces.
3
λ
-double statistical convergence in a random 2-normed spaceRecently, the concept ofλ-double statistical convergence has been introduced and studied in [] and []. In this section, we defineλ-double statistically convergent sequence in a random -normed space (X,F,∗). Also we get some basic properties of this notion in a random -normed space.
Definition Letλ= (λn) andμ= (μn) be two non-decreasing sequences of positive real
numbers such that each is tending to∞and
λn+≤λn+ , λ=
and
μn+≤μn+ , μ= .
LetK⊆N×N. The number
δ¯λ(K) =lim mn
¯ λmn
k∈In,l∈Jm: (k,l)∈K,
whereIn= [n–λn+ ,n],Jm= [m–μm+ ,m] andλ¯nm=λnμm, is said to be theλ-double
densityofK, provided the limit exists.
Definition A sequencex= (xk,l) is said to beλ-double statistically convergent orSλ¯
-convergent to the number if for everyε> , the setN(ε) hasλ-double density zero, where
N(ε) =k∈In,l∈Jm:|xk,l–| ≥ε
.
Now we defineλ-double statistical convergence in a random -normed space (see []).
Definition A sequencex= (xk,l) in a random -normed space (X,F,∗) is said to be
λ-double statistically convergentorSλ¯-convergentto∈Xwith respect toFif for every ε> ,η∈(, ) and for nonzeroz∈Xsuch that
δ¯λ
k∈In,l∈Jm:F(xk,l–,z;ε)≤ –η =
or equivalently,
δ¯λ
k∈In,l∈Jm:F(xk,l–,z;ε) > –η = ,
i.e.,
S¯λ– lim
k,l→∞F(xk,l–,z;ε) = .
In this case we writeSλ¯RN–limx=orxk,l→(Sλ¯RN) and
S¯λRN(X) =x= (xk,l) :∃∈R,Sλ¯RN–limx=
.
LetSRN ¯
λ (X) denote the set of allλ-double statistically convergent sequences in a random
-normed space (X,F,∗).
Ifλ¯mn=mnfor everyn,mthenλ-double statistically convergent sequences in a random
-normed space (X,F,∗) reduce to double statistically convergent sequences in a random -normed space (X,F,∗).
Definition immediately implies the following lemma.
Lemma Let(X,F,∗)be a random-normed space.If x= (xk,l)is a sequence in X,then
for everyε> ,η∈(, )and for nonzero z∈X,the following statements are equivalent: (i) SRλ¯N–limk,l→∞xk,l=;
(ii) δλ¯({k∈In,l∈Jm:F(xk,l–,z;ε)≤ –η}) = ;
(iii) δλ¯({k∈In,l∈Jm:F(xk,l–,z;ε) > –η}) = ;
(iv) Sλ¯–limk,l→∞F(xk,l–,z;ε) = .
Theorem Let(X,F,∗)be a random-normed space.If x= (xk,l)is a sequence in X such
that Sλ¯RN–limxk,l=exists,then it is unique.
Proof Suppose thatSRN ¯
λ –limk,l→∞xk,l=;S
RN ¯
λ –limk,l→∞xk,l=, where (=).
Letε> be given. Choosea> such that ( –a)∗( –a) > –ε. Then, for anyt> and for nonzeroz∈X, we define
K(a,t) =
k∈In,l∈Jm:F
xk,l–,z;
t
≤ –a
;
K(a,t) =
k∈In,l∈Jm:F
xk,l–,z;
t
≤ –a
Since Sλ¯RN –limk,l→∞xk,l = and Sλ¯RN – limk,l→∞xk,l = , we have Lemma δλ¯(K(a,t)) = andδλ¯(K(a,t)) = for allt> .
Now, letK(a,t) =K(a,t)∪K(a,t), then it is easy to observe thatδλ¯(K(a,t)) = . But we
haveδλ¯(Kc(r,t)) = .
Now, if (k,l)∈Kc(a,t), then we have F(–,z;t)≥F
xk,l–,z;
t
∗F
xk,l–,z;
t
> ( –a)∗( –a).
It follows that
F(–,z;t) > ( –ε).
Sinceε> was arbitrary, we getF(–,z;t) = for allt> and nonzeroz∈X. Hence =.
This completes the proof.
Next theorem gives the algebraic characterization ofλ-statistical convergence on ran-dom -normed spaces. We give it without proof.
Theorem Let(X,F,∗)be a random-normed space,and x= (xk,l)and y= (yk,l)be two
sequences in X.
(a) IfSλ¯RN–limxk,l=andc(= ) ∈R,thenSλ¯RN–limcxk,l=c.
(b) IfSRN ¯
λ –limxk,l=andS RN ¯
λ –limyk,l=,thenS
RN ¯
λ –lim(xk,l+yk,l) =+. Theorem Let(X,F,∗)be a random-normed space.If x= (xk,l)is a sequence in X such
thatF–limxk,l=,then Sλ¯RN–limxk,l=.
Proof LetF–limxk,l=. Then for everyε> ,t> and nonzeroz∈X, there is a positive
integernandmsuch that F(xk–,z;t) > –ε
for allk≥n. Since the set
K(ε,t) =k∈In,l∈Jm:F(xk,l–,z;t)≤ –ε
has at most finitely many terms. Since every finite subset ofN×Nhasδλ¯-density zero,
finally we haveδλ¯(K(ε,t)) = . This shows thatSλ¯RN–limxk,l=. Remark The converse of the above theorem is not true in general. It follows from the following example.
Example LetX=R, with the -normx,z=|x
z–xz|,x= (x,x),z= (z,z) and
a∗b=abfor alla,b∈[, ]. LetF(x,y;t) = t+tx,y, for allx,z∈X,z= , andt> . We define a sequencex= (xk) by
xk,l=
⎧ ⎨ ⎩
(kl, ), ifn– [√λn] + ≤k≤nandm– [√μm] + ≤k≤m;
Now for every <ε< andt> , we write
Kn(ε,t) =
k∈In,l∈Jm:F(xk,l–,z;t)≤ –ε
.
Therefore, we get
δ¯λ
K(ε,t) = lim
nm→∞
[λ¯nm]
¯ λnm
= .
This shows thatSλ¯RN–limxk,l= , while it is obvious thatF–limxk,l= .
Theorem Let(X,F,∗)be a random-normed space.If x= (xk,l)is a sequence in X,then
SRN ¯
λ –limxk,l=if and only if there exists a subset K={(kn,ln) :k<k, . . . ;l<l, . . .} ⊆ N×Nsuch thatδλ¯(K) = andF–limn→∞xkn,ln=.
Proof Suppose first thatSλRN–limxk,l=. Then for anyt> ,a= , , , . . . and nonzero
z∈X, let
A(a,t) =
k∈In;l∈Jm:F(xk,l–,z;t) > –
a
and
K(a,t) =
k∈In;l∈Jm:F(xk,l–,z;t)≤ –
a
.
SinceSλ¯RN–limxk,l=, it follows that
δ¯λ
K(a,t) = .
Now, fort> anda= , , , . . . , we observe that
A(a,t)⊃A(a+ ,t)
and
δ¯λ
A(a,t) = . (.)
Now we have to show that for (k,l)∈A(a,t),F–limxk,l=. Suppose that for some
(k,l)∈A(a,t), (xk,l) is not convergent towith respect toF. Then there exist somes>
and a positive integerk,lsuch that
k∈In;l∈Jm:F(xk,l–,z;t)≤ –s
for allk≥kandl≥l. Let
A(s,t) =k∈In;l∈Jm:F(xk,l–,z;t) > –s
fork<kandl<land
s>
a, a= , , , . . . .
Then we have
δ¯λ
A(s,t) = .
Furthermore, A(a,t)⊂A(s,t) implies that δ¯λ(A(a,t)) = , which contradicts (.) as
δλ¯(A(a,t)) = . HenceF–limxk,l=.
Conversely, suppose that there exists a subset K={(kn,ln) :k <k, . . . ;l <l, . . .} ⊆
N×Nsuch thatδλ¯(K) = andF–limn,m→∞xkn,ln=. Then for everyε> ,t> and nonzeroz∈X, we can find a positive integernsuch that
F(xk,l,z;t) > –ε
for allk,l≥n. If we take
K(ε,t) =k∈In;l∈Jm:F(xk,l–,z;t)≤ –ε
,
then it is easy to see that
K(ε,t)⊂N×N–(kn+,ln+), (kn+,ln+), . . .
,
and finally,
δ¯λ
K(ε,t) ≤ – = .
ThusSλR¯N–limxk,l=. This completes the proof.
We now have
Definition A sequencex= (xk,l) in a random -normed space (X,F,∗) is said to be
λ-double statistically Cauchywith respect toFif for eachε> ,η∈(, ) and for nonzero
z∈X, there existN=N(ε) andM=M(ε) such that for allk,m>Nandl,n>M,
δ¯λ
k∈In;l∈Jm:F(xk,l–xMN,z;ε)≤ –η = ,
or equivalently,
δ¯λ
k∈In;l∈Jm:F(xk,l–xMN,z;ε) > –η = .
Theorem Let(X,F,∗)be a random-normed space.Then a sequence(xk,l)in X isλ
-double statistically convergent if and only if it isλ-double statistically Cauchy in random
Proof Let (xk,l) be a λ-double statistically convergent to with respect to random
-normed space,i.e.,SRN ¯
λ –limxk=. Letε> be given. Choosea> such that
( –a)∗( –a) > –ε. (.)
Fort> and for nonzeroz∈X, define
A(a,t) =
k∈In;l∈Jm:F
xk,l–,z;
t
≤ –a
.
Then
Ac(a,t) =
k∈In;l∈Jm:F
xk,l–,z;
t
> –a
.
SinceSRN ¯
λ –limxk,l=, it follows thatδλ¯(A(a,t)) = , and finally,δλ¯(A
c(a,t)) = .
Letp,q∈Ac(a,t). Then
F
xp,q–,z;
t
> –a. (.)
If we take
B(ε,t) =k∈In;l∈Jm:F(xk,l–xp,q,z;t)≤ –ε
,
then to prove the result it is sufficient to prove thatB(ε,t)⊆A(a,t). Let (k,l)∈B(ε,t)∩Ac(a,t), then for nonzeroz∈X, we have
F(xk,l–xp,q,z;t)≤ –ε and F
xk,l–,z;
t
> –a. (.)
Now, from (.), (.) and (.), we get
–ε≥F(xk,l–xp,q,z;t)≥F
xk,l–,z;
t
∗F
xp–,z;
t
> ( –a)∗( –a) > ( –ε),
which is not possible. ThusB(ε,t)⊂A(a,t). Sinceδλ¯(A(a,t)) = , it follows thatδλ¯(B(ε,t)) =
. This shows that (xk,l) isλ-double statistically Cauchy.
Conversely, suppose (xk,l) isλ-double statistically Cauchy but notλ-double statistically
convergent with respect toF. Then for eachε> ,t> and for nonzeroz∈X, there exist a positive integerN=N(ε) andM=M(ε) such that
A(ε,t) =k∈In;l∈Jm:F(xk,l–xNM,z;t)≤ –ε
.
Then
δ¯λ
and
δ¯λ
Ac(ε,t) = . (.)
Fort> , choosea> such that
( –a)∗( –a) > –ε (.)
is satisfied, and we take
B(a,t) =
k∈In;l∈Jm:F
xk,l–,z;
t
> –a
.
IfN,M∈B(a,t), thenF(xN,M–,z;t) > –a.
Since
F(xk,l–xNM,z;t)≥F
xk,l–,z;
t
∗F
xN,M–,z;
t
> ( –a)∗( –a) > –ε,
then we have
δ¯λ
xk,l:F(xk,l–xNM,z;t) > –ε = ,
i.e.,δλ¯(Ac(ε,t)) = , which contradicts (.) asδλ¯(Ac(ε,t)) = . Hence (xk,l) isλ-double
sta-tistically convergent.
This completes the proof.
Competing interests
The author declares that they have no competing interests.
Received: 12 March 2012 Accepted: 22 August 2012 Published: 25 September 2012 References
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