Vol. 13, No. 2 (2018), pp 101-109 DOI: 10.7508/ijmsi.2018.13.009
On
I
-Statistical Convergence
Shyamal Debnath∗,a, Debjani Rakshitb
aDepartment of Mathematics, Tripura University, Agartala-799022, India. bDepartment of FST, ICFAI University, Tripura, Kamalghat, West
Tripura-799210, India.
E-mail: [email protected]
E-mail: [email protected]
Abstract. In this paper we investigate the notion of I-statistical con-vergence and introduceI-st limit points andI-st cluster points of real number sequence and also studied some of its basic properties.
Keywords: I-limit point,I-cluster point, I-statistically Convergent. 2000 Mathematics subject classification: 40A35, 40D25.
1. Introduction
In 1951 Fast [6] and Steinhaus [18] introduced the concept of statistical conver-gence independently and established a relation with summability. Later on it was further investigated from sequence space point of view by Fridy [8], Salat [19] and many others. Some applications of statistical convergence in number theory and mathematical analysis can be found in [1, 2, 13, 14, 21].
The notion ofI-convergence is a generalization of the statistical convergence which was introduced by Kostyrko et al. [12]. They used the notion of an ideal I of subsets of the setN to define such a concept. For an extensive view of this article we refer [4, 11, 20].
The idea ofI-convergence was further extended toI-statistical convergence by Savas and Das [16]. Later on more investigation in this direction was done
∗Corresponding Author
Received 04 February 2016; Accepted 27 September 2017 c
2018 Academic Center for Education, Culture and Research TMU
101
by Savas and Das [17], Debnath and Debnath [3], Mursaleen et.al [15], Et et al. [5] and many others [9, 10, 22, 23] . In [16], Savas and Das introduced the I-statistical convergence andI-λ-statistical convergence and the relation between them. Also they studied these concept in the notion of [V, λ]- summability method.
In the present paper we return to the view ofI-statistical convergence as a sequential limit concept and we extend this concept in a natural way to define a I-statistical analogue of the set of limit points and cluster points of a real number sequence.
2. Definitions and Preliminaries
Definition 2.1. [8] IfK is a subset of the positive integersN, then Kn de-notes the set {k∈K:k≤n}. The natural density of K is given by D(K) = limn→∞|Kn|
n .
Definition 2.2. [8] A sequence (xn) is said to be statistically convergent tox0
if for each ε >0, the set A(ε) = {k∈N:d(xk, x0)≥ε} has natural density
zero. x0 is called the statistical limit of the sequence (xn) and we write
st-limn→∞xn=x0.
Definition 2.3. [7] If xk(j)
be a subsequence of a sequence x= (xn) and density ofK={k(j) :j∈N}is zero then xk(j)
is called a thin subsequence. Otherwise xk(j)
is called a non-thin subsequence ofx.
x0 is said to be a statistical limit point of a sequence (xn), if there exist a
non-thin subsequence of (xn) which conveges tox0.
LetΛx denotes the set of all statistical limit points of the sequence (xn). Definition 2.4. [7]x0is said to be a statistical cluster point of a sequencex=
(xn), provided that for eachε >0 the density of the set{k∈N:d(xk, x0)< ε}
is not equal to 0.
LetΓxdenotes the set of all statistical cluster points of the sequence (xn). Definition 2.5. [12] LetX is a non-empty set. A family of subsetsI ⊂P(X) is called an ideal onX if and only if
(i)∅ ∈I;
(ii) for eachA, B∈I impliesA∪B∈I; (iii) for eachA∈I andB⊂AimpliesB∈I.
Definition 2.6. [12] LetXis a non-empty set. A family of subsetsF ⊂P(X) is called a filter onX if and only if
(i)∅∈ F/ ;
(ii) for eachA, B∈ F impliesA∩B ∈ F; (iii) for eachA∈ F andB⊃AimpliesB∈ F.
An ideal I is called non-trivial if I 6= ∅ and X /∈ I. The filter F =
F(I) = {X−A:A∈I} is called the filter associated with the ideal I. A non-trivial ideal I ⊂ P(X) is called an admissible ideal in X if and only if
I ⊃ {{x}:x∈X}
Definition 2.7. [12] LetI ⊂P(N) be a non-trivial ideal onN. A sequence (xn) is said to be I-convergent to x0 if for each ε > 0, the set A(ε) =
{k∈N :d(xk, x0)≥ε} belongs to I. x0 is called theI-limit of the sequence
(xn) and we write I-limn→∞xn=x0.
Definition 2.8. [12] x0 is said to be I-limit point of a sequence x = (xn)
provided that there is a subsetK={k1< k2< ...} ⊂N such thatK /∈I and
lim xki =x0.
LetI(Λx) denotes the set of allI-limit points of the sequence x.
Definition 2.9. [12] x0 is said to be I-cluster point of a sequence x= (xn)
provided that for eachε >0 the set{k∈N :d(xk, x0)< ε}∈/ I.
LetI(Γx) denotes the set of allI-cluster points of the sequencex.
Definition 2.10. [16] A sequencex= (xn) is said to beI-statistically conver-gent tox0if for everyε >0 and everyδ >0 ,
n∈N : 1
n| {k≤n:d(xn, x0)≥ε} | ≥δ ∈I.
x0is calledI-statistical limit of the sequence (xn) and we write,I-st lim xn =
x0.
Throughout the paper we considerI as an admissible ideal. 3. Main Results
Theorem 3.1. If (xn) be a sequence such that I-st lim xn = x0, then x0
determined uniquely.
Proof. If possible let the sequence (xn) be I-statistically convergent to two different numbersx0 andy0
i.e, for anyε >0,δ >0 we have, A1=
n∈N : 1n| {k≤n:d(xk, x0)≥ε} |< δ ∈ F(I)
andA2=n∈N : 1n| {k≤n:d(xk, y0)≥ε} |< δ ∈ F(I)
Therefore,A1∩A26=∅,sinceA1∩A2∈ F(I).
Letm∈A1∩A2 and takeε= d(x03,y0)>0
so, m1| {k≤m:d(xk, x0)≥ε} |< δ
and 1
m| {k≤m:d(xk, y0)≥ε} |< δ
i.e, for maximum k≤m will satisfyd(xk, x0)< εand d(xk, y0)< εfor a
very smallδ >0. Thus, we must have
{k≤m:d(xk, x0)< ε} ∩ {k≤m:d(xk, y0)< ε} 6= ∅ a contradiction, as
the neighbourhood ofx0 andy0are disjoint.
Hence the theorem is proved.
Theorem 3.2. For any sequence(xn),st-limxn =x0impliesI-st lim xn =x0.
Proof. Letst-limxn=x0.
Then for each ε > 0, the set A(ε) = {k≤n:d(xk, x0)≥ε} has natural
density zero.
i.e,limn→∞n1| {k≤n:d(xk, x0)≥ε} |= 0
So for everyε >0 andδ >0,
n∈N : 1
n| {k≤n:d(xk, x0)≥ε} | ≥δ is a finite set and therefore be-longs toI , asI is an admissible ideal.
HenceI-st lim xn=x0.
But the converse is not true.
Example 3.3. LetI=ζbe the class ofA⊂N that intersect a finite number
of4j’s whereN =∪∞j=14j and4i∩ 4j=∅fori6=j.
Letxn =n1 and so limn→∞d(xn,0) = 0. Putn=d(xn,0) forn∈N. Now define a sequence (yn) byyn=xj ifn∈ 4j
Letη >0. Chooseν∈N such thatν < η. Then A(η) ={n:d(yn,0)≥η} ⊂ 41∪...∪ 4ν ∈ζ. Now,{k≤n:d(yk,0)≥η} ⊆ {n∈N :d(yn,0)≥η}
i.e, n1| {k≤n:d(yk,0)≥η} | ≤ | {n∈N:d(yn,0)≥η} | so for anyδ >0,
n∈N : 1n| {k≤n:d(yk,0)≥η} | ≥δ ⊆ {n∈N :d(yn,0)≥η} ∈ζ. Therefore (yn) isζ-statistically convergent to 0.
But (yn) is not a statistically convergent.
Theorem 3.4. For any sequence(xn),I-limxn=x0impliesI-st lim xn =x0.
Proof. The proof is obvious. But the converse is not true.
Example 3.5. If we takeI =If the sequence (xn),
wherexn=
(
0, n=k2, k∈N
1, otherwise
isI-statistically convergent to 1. But (xn) is notI-convergent.
Theorem 3.6. If each subsequence of (xn) is I-statistically convergent to ξ
then(xn)is alsoI-statistically convergent toξ.
Proof. Suppose (xn) is not I-statistically convergent to ξ, then there exists ε >0 andδ >0 such that
A =
n∈N: n1| {k≤n:d(xk, ξ)≥ε} | ≥δ ∈/ I. Since I is admissible ideal soAmust be an infinite set.
Let A = {n1< n2< ... < nm< ...}. Let ym = xnm for m ∈ N. Then
(ym)m∈N is a subsequence of (xn) which is not I-statistically convergent to ξ,
a contradiction. Hence the theorem is proved.
But the converse is not true. We can easily show this from example 3.5.
Theorem 3.7. Let (xn) and(yn)be two sequences then (i) I -st lim xn=x0 andc∈R implies I -st lim cxn=cx0.
(ii) I -st lim xn = x0 and I -st lim yn = y0 implies I -st lim(xn +yn) =
x0+y0.
Proof. (i) Ifc= 0, we have nothing to prove. So we assume thatc6= 0.
Now, n1| {k≤n:d(cxk, cx0)≥ε} |=n1| {k≤n:|c|d(xk, x0)≥ε} |
≤ 1
n|
n
k≤n:d(xk, x0)≥ |c|ε o
|< δ
Therefore,n∈N : 1n| {k≤n:d(cxk, cx0)≥ε} |< δ ∈ F(I) .
i.e,I-st lim cxn=cx0.
(ii) We haveA1=
n∈N : 1n|
k≤n:d(xk, x0)≥ε2 |< δ2 ∈ F(I)
andA2=n∈N : 1n|k≤n:d(yk, y0)≥ ε2 |< δ2 ∈ F(I).
SinceA1∩A26=∅, therefore for alln∈A1∩A2 we have, 1
n| {k≤n:d(xk+yk, x0+y0)≥ε} |
≤ 1
n|
k≤n:d(xk, x0)≥2ε |+1n|
k≤n:d(yk, y0)≥ ε2 |< δ.
i.e,
n∈N : n1| {k≤n:d(xk+yk, x0+y0)≥ε}< δ ∈ F(I).
HenceI-st lim(xn+yn) = (x0+y0).
Definition 3.8. A sequence x = (xn)n∈N of elements of X is said to be I∗-statistical convergent to ξ ∈ X if and only if there exists a set M =
{m1< m2< ... < mk < ...} ∈ F(I), such thatst-lim d(xmk, ξ) = 0.
Theorem 3.9. If I∗-st limn→∞xn =ξthen I -st limn→∞xn=ξ.
Proof. LetI∗-st limn→∞xn =ξ. By assumption there exist a setH ∈I such that forM =N\H ={m1< m2< ... < mk < ...}we have st-lim xmk=ξ
i.e,limn→∞n1| {mk ≤n:d(xmk, ξ)≥ε} |= 0
so for anyδ >0,n∈N : n1| {mk≤n:d(xmk, ξ)≥ε} | ≥δ ∈I sinceI is an admissible ideal.
Now,A(ε, δ) =
n∈N : n1| {k≤n:d(xk, ξ)≥ε} | ≥δ
⊂H∪
n∈N: n1| {mk≤n:d(xmk, ξ)≥ε} | ≥δ ∈I
i.e,I-st limn→∞xn =ξ.
But the converse may not be true.
From example 3.3. we haveζ-st limn→∞yn= 0.
Suppose thatζ∗-st limn→∞yn = 0. Then there exist a setH ∈ζ such that for M = N\H = {m1< m2< ... < mk< ...} we have st-lim ymk = 0. By
definition of ζ there exist a p ∈ N such that H ⊂ 41∪...∪ 4p. But then
4p+1⊂M, so for infinitely manymk ∈ 4p+1,
D{mk∈ 4p+1:d(ymk,0)≥η}= 2
−(p+1)>0 for 0< η < 1
p+1
i.e,D{mk ∈ 4p+1:d(ymk,0)≥η} 6= 0, which is a contradictsst-lim ymk =
0.
Henceζ∗-st limn→∞yn6= 0.
Definition 3.10. An element x0 is said to be an I-statistical limit point
of a sequence x = (xn) provided that for each ε > 0 there is a set M =
{m1< m2< ...} ⊂N such that M /∈I andst-lim xmk =x0.
I-S(Λx) denotes the set of allI-statistical limit points of the sequence (xn). Theorem 3.11. If (xn) be a sequence such that I -st lim xn = x0 then I
-S(Λx) ={x0}.
Proof. Since (xn) isI-statistically convergent tox0, so for eachε >0 andδ >0
the set, A=
n∈N : 1
n| {k≤n:d(xk, x0)≥ε} | ≥δ ∈I, whereI is an admisible ideal.
SupposeI-S(Λx) containsy0 different fromx0. i.e,y0∈I-S(Λx).
So there exist aM ⊂N such that M /∈I andst-lim Xmk=y0.
Let B =
n∈M : 1
n| {k≤n:d(xk, y0)≥ε} | ≥δ . So B is a finite set and thereforeB∈I and soBc =
n∈M : n1| {k≤n:d(xk, y0)≥ε} |< δ ∈
F(I).
Again letA1=
n∈M : n1| {k≤n:d(xk, x0)≥ε} | ≥δ . SoA1⊂A∈I.
i.e,Ac
1∈ F(I). ThereforeAc1∩Bc6=∅,sinceAc1∩Bc∈ F(I)
Letp∈Ac1∩Bc and take ε=
d(x0,y0)
3 >0
so 1
p| {k≤p:d(xk, x0)≥ε} |< δ and 1p| {k≤p:d(xk, y0)≥ε} |< δ
i.e, for maximum k ≤pwill satisfy d(xk, x0)< ε and d(xk, y0) < ε for a
very smallδ >0. Thus we must have,
{k≤p:d(xk, x0)< ε} ∩ {k≤p:d(xk, y0)< ε} 6=∅a contradiction, as the
neighbourhood ofx0andy0are disjoint.
HenceI-S(Λx) ={x0}.
Definition 3.12. [15] An elementx0is said to be anI-statistical cluster point
of a sequencex= (xn) if for eachε >0 andδ >0
n∈N : 1n| {k≤n:d(xk, x0)≥ε} |< δ ∈/I.
I-S(Γx) denotes the set of all I-statistical cluster points of the sequence (xn).
Theorem 3.13. For any sequencex= (xn),I-S(Γx) is closed.
Proof. Lety0be a limit point of the setI-S(Γx) then for anyε >0,I-S(Γx)∩
B(y0, ε)6= 0, whereB(y0, ε) ={z∈R:d(z, y0)< ε}
Let z0 ∈ I-S(Γx)∩B(y0, ε) and choose ε1 > 0 such that B(z0, ε1) ⊆
B(y0, ε).
Then we have{k≤n:d(xk, z0)≥ε1} ⊇ {k≤n:d(xk, y0)≥ε}
⇒ 1
n| {k≤n:d(xk, z0)≥ε1} | ≥
1
n| {k≤n:d(xk, y0)≥ε} | Now for anyδ >0,
n∈N : 1n| {k≤n:d(xk, z0)≥ε1} |< δ
⊆
n∈N : n1| {k≤n:d(xk, y0)≥ε} |< δ
Sincez0∈I-S(Γx) therefore,
n∈N: 1
n| {k≤n:d(xk, y0)≥ε} |< δ ∈/I. i.e,y0∈I-S(Γx). Hence the theorem is proved.
Theorem 3.14. For any sequencex= (xn),I-S(Λx)⊆I -S(Γx).
Proof. Let x0 ∈ I-S(Λx). Then there exist a set M = {m1< m2< ...} ∈/ I
such that,st-lim xmk =x0⇒limk→∞1k| {mi≤k:d(xmi, x0)≥ε} |= 0.
Takeδ >0, so there existk0∈N such that forn > k0we have, 1
n| {mi≤n:d(xmi, x0)≥ε} |< δ.
LetA=
n∈N : 1n| {mi≤n:d(xmi, x0)≥ε} |< δ .
Also, A ⊃ M/{m1< m2< ... < mk0}. Since I is an admissible ideal and
M /∈I, thereforeA /∈I. So by definition ofI-statistical cluster point x0∈I
-S(Γx).
Hence the theorem is proved.
Theorem 3.15. Ifx= (xn) andy= (yn)be two sequences such that
{n∈N:xn 6=yn} ∈I , then
(i)I-S(Λx) =I -S(Λy)and (ii)I-S(Γx) =I -S(Γy). Proof. (i) Letx0∈I-S(Λx). So by definition there exist a set
K={k1< k2< k3<· · · } ofN such thatK /∈I andst-lim xkn=x0.
Since{n∈K:xn 6=yn} ⊂ {n∈N :xn6=yn} ∈I, thereforeK0 ={n∈K:xn=yn}∈/I andK0 ⊆K. So we havest-lim yk0
n=x0.
This shows thatx0∈I-S(Λy) and thereforeI-S(Λx)⊆I-S(Λy).
By symmetryI-S(Λy)⊆I-S(Λx) . HenceI-S(Λy) =I-S(Λx).
(ii) Letx0∈I-S(Γx). So by definition for eachε >0 the set,
A=n∈N : n1| {k≤n:d(xk, x0)≥ε} |< δ ∈/ I.
Let B =
n∈N : 1
n| {k≤n:d(yk, x0)≥ε}< δ . We have to prove that B /∈I.
SupposeB∈I. So,Bc=n∈N : n1| {k≤n:d(yk, x0)≥ε} ≥δ ∈ F(I).
By hypothesis the setC={n∈N :xn=yn} ∈ F(I).
ThereforeBc∩C∈ F(I). Also it is clear that Bc∩C⊂Ac∈ F(I), i.e,A∈I, which is a contradiction.
HenceB /∈I and thus the result is proved.
Theorem 3.16. If g is a continuous function on X then it preserves I -statistical convergence inX.
Proof. LetI-st limn→∞xn=ξ.
Since g is continuous, then for eachε1 >0, there exist ε2 >0 such that if
x∈B(ξ, ε1) theng(x)∈B(g(ξ), ε2).
Also we have, C(ε1, δ) =
n∈N : 1
n| {k≤n:d(xk, ξ)≥ε1} |< δ ∈ F(I) Now,{k≤n:d(xk, ξ)≥ε1} ⊇ {k≤n:d(g(xk), g(ξ))≥ε2}
so, n1| {k≤n:d(xk, ξ)≥ε1} | ≥ n1| {k≤n:d(g(xk), g(ξ))≥ε2} |
forδ >0,n∈N: n1| {k≤n:d(xk, ξ)≥ε1} |< δ
⊆
n∈N : 1
n| {k≤n:d(g(xk), g(ξ))≥ε2} |< δ ∈ F(I) sinceC(ε1, δ)∈ F(I).
Hence the theorem is proved.
Acknowledgments
The authors are grateful to CSIR for their financial support to carry out this research work through the Project No. 25(0236)/14/EMR-II.
References
1. R. C. Buck, The measure theoretic approach to density, Amer. J. Math., 68, (1946), 560-580.
2. R. C. Buck, Generalized asymptotic density,Amer. J. Math.,75, (1953), 335-346. 3. S. Debnath, J. Debnath, On I-statistically convergent sequence spaces defined by
se-quences of Orlicz functions using matrix transformation,Proyecciones J. Math.,33(3), (2014), 277-285.
4. K. Dems, OnI-Cauchy sequences,Real Anal. Exchange,30(1), (2004/2005), 123-128. 5. M. Et, A. Alotaibi, S. A. Mohiuddine,On (4m, I)-statistical convergence of order α,
The Scientific world journal, vol 2014.
6. H. Fast, Sur la convergence statistique,Colloq. Math., 2, (1951), 241-244. 7. J. A. Fridy, Statistical limit points,Proc. Amer. Math. Soc.,4, (1993), 1187-1192. 8. J. A. Fridy, On statistical convergence,Analysis,5, (1985), 301-313.
9. M. Gurdal, H. Sari, Extremal A-statistical limit points via ideals, J. Egyptian Math. Soc.,22(1), (2014), 55-58.
10. M. Gurdal, M. O. Ozgur, A generalized statistical convergence via moduli,Electronic J. Math. Anal. & Appl.,3(2), (2015), 173-178.
11. P. Kostyrko, M. Macaj, T. Salat, M. Sleziak,I-convergence and extremalI-limit points, Math. Slov.,4, (2005), 443-464.
12. P. Kostyrko, T. Salat, W. Wilczynski, I-convergence, Real Anal. Exchange, 26(2), (2000/2001), 669-686.
13. M. Mamedov, S. Pehlivan, Statistical cluster points and turnpike theorem in non convex problems,J. Math. Anal. Appl., 256, (2001), 686-693.
14. D. S. Mitrinovic, J. Sandor, B. Crstici,Handbook of Number Theory, Kluwer Acad. Publ. Dordrecht-Boston-London, 1996.
15. M. Mursaleen, S. Debnath, D. Rakshit,I-statistical limit superior andI-statistical limit inferior,Filomat,31(7), (2017), 2103-2108.
16. E. Savas, P. Das, A generalized statistical convergence via ideals,App. Math. Lett.,24, (2011), 826-830.
17. E. Savas, P. Das, OnI-statistically pre-Cauchy sequences,Taiwanese J. Math.,18(1), (2014), 115-126.
18. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique,Colloq. Math.,
2, (1951), 73-74.
19. T. Salat, On statistically convergent sequences of real numbers,Math. Slov.,30, (1980), 139-150.
20. T. Salat, B. Tripathy, M. Ziman, On some properties ofI-convergence,Tatra. Mt. Math. Publ.,28, (2004), 279-286.
21. B. C. Tripathy, On statistically convergent and statistically bounded sequences, Bull. Malaysian. Math. Soc.,20, (1997), 31-33.
22. U. Yamanci, M. Gurdal, I -statistical convergence in 2-normed space, Arab J. Math. Sci.,20(1), (2014), 41-47.
23. U. Yamanci, M. Gurdal,I-statistically pre-Cauchy double sequences,Global J. Math. Anal.,2(4), (2014), 297-303.
