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13URAL MATHEMATICAL JOURNAL, Vol. 7, No. 1, 2021, pp. 16-25
DOI: 10.15826/umj.2021.1.002
SOME REMARKS ON ROUGH STATISTICAL A-CONVERGENCE OF ORDER a
Reena Antal
Department of Mathematics, Chandigarh University, NH-95, Chandigarh-Ludhiana Highway, Mohali, Punjab 140413, India
reena.antal@gmail.com
Meenakshi Chawla
Department of Mathematics, Chandigarh University, NH-95, Chandigarh-Ludhiana Highway, Mohali, Punjab 140413, India chawlameenakshi7@gmail.com
Vijay Kumar
Department of Mathematics, Panipat Institute of Engineering and Technology, 70, Milestone GT Road, Samalkha, Panipat, 132102 Haryana, India vjy_kaiishik@yahoo.com
Abstract: The main purpose of this work is to define Rough Statistical A-Convergence of order a (0 < a < 1) in normed linear spaces. We have proved some basic properties and also provided some examples to show that this method of convergence is more generalized than the rough statistical convergence. Further, we have shown the results related to statistically A-bounded sets of order a and sets of rough statistically A-convergent sequences of order a.
Keywords: Statistical convergence, Rough statistical convergence, Rough statistical limit points.
1. Introduction
In 1951, Fast [5] presented a new idea of convergence named as statistical convergence that is more generalized than the usual convergence for the sequences.
Definition 1 [5]. A sequence x = {xm} of numbers is said to be statistically convergent to £ if for every e > 0 we have lim |M(x,e)|/n = 0, where |M(x,e)| represents the order of the enclosed
set M(x, e) = {m < n : |xm — £| > e}.
This idea has interesting applications in the field of Fourier Analysis [1], Measure Theory [16], Approximation Theory [7] etc. It has been studied by many researchers for various types of sequences in different setups like locally convex spaces [10], probabilistic normed spaces [8], random normed spaces [3], intuitionistic fuzzy normed spaces [9] etc.
An interesting generalization of usual convergence named as rough convergence was introduced by Phu [19] for the sequences in finite dimensional normed linear spaces and later on introduced on infinite dimensional normed linear spaces [20]. He mainly worked on rough limits, roughness degree, rough continuity of linear operators and also introduced rough Cauchy sequences.
Definition 2 [19]. A sequence x = {xm} in a normed linear space (X, || ■ ||) is said to be rough convergent to £ € X if for every e > 0 there exists a non-negative number r and m0 € N such that ||xm — £|| < r + e, for all m > m0.
Aytar [2] extended the rough convergence to rough statistical convergence like usual convergence is extended to statistical convergence with the help of natural density.
Definition 3 [2]. A sequence x = {xm} in a normed linear space (X, || ■ ||) is said to be rough statistically convergent to £ € X if for every e > 0 there exists a non-negative number r such that
lim — |{m < n : \\xm — £11 > r + e}| = 0,
n^X n
where £ is known as r-St-limit of sequence x = {xm}.
Aytar [2] also defined the rough statistical bounded sequence along with the set of rough statistical limit points of a sequence. Further, some criterion associated with the convexity and closeness of the set of rough statistical limit points of a sequence was investigated.
Inspired by the work of Aytar [2], Maity [12] presented the concept of rough statistical convergence of order a (0 < a < 1) in normed linear spaces and explained some important results for the set of rough statistical limit points of order a. The idea of pointwise rough statistical convergence and rough statistical Cauchy sequences for real valued functions was introduced in [11]. The concept of rough convergence has been defined for double sequences by Malik and Maity in [13] and after that the authors extended this idea in [14] and defined rough statistical convergence for double sequences in normed linear spaces.
This idea has motivated many authors to use the concepts of ideals also. Pal et al. [18] introduced rough /-convergence with the help of ideals of N. Later, Malik et al. in [15] extended this concept of rough /-convergence to rough /-statistical convergence and described some topo-logical properties of the set of all rough /-statistical limits of sequences in normed linear spaces. A lot of work has been done on rough convergence and its generalizations. More investigations and applications of rough convergence can be revealed as it is an active area of research.
In this paper, we are introducing the concept of rough statistical A-convergence of order a (0 < a < 1) in the normed linear spaces.
2. Main results
In order to study the basic concept of rough statistical A-convergence, we first consider a
sequence A = {Aj} of real numbers such that 0 < Ao < Ai <......< Aj < ... and Aj — to as j — to.
The concept of A-convergence for real sequences have been defined by Mursaleen[17] as given below: a sequence x = {xm} of real numbers is A-convergent to a number L if Axm — L as m — to where
1 m
A.rm — (Aj - Xj-i)xj.
A m ■ „
3 =0
Here, without loss of generality we take all the terms with negative subscripts equal to zero.
Using this concept, we are defining the notion of the rough A-convergence and rough statistical A-convergence as follows:
Definition 4. A sequence x = {xm} in a normed linear space (X, || ■ ||) is said to be rough A-convergent to £ € X if for every e > 0 there exist a non-negative number r and m0 € N such that ||Axm — £|| < r + e, for all m > m0.
Definition 5. A sequence x = {xm} in a normed linear space (X, || ■ ||) is said to be rough statistically K-convergent to £ if for every e > 0 there exists some non-negative number r such that
lim — lim. < n : ||Aa:m — £11 > r + e}| = 0,
n^X n
where £ is known as r-St\-limit of sequence x = {xm}.
Remark 1. For the case r = 0, the notion of rough statistical K-convergence agrees with the statistical K-convergence.
Qolak [4] has given an interesting idea related to the statistical convergence of order a (0 < a < 1) with the help of a-density. Motivated by his idea, now we are defining a rough statistical K-convergence of order a (0 < a < 1) as follows:
Definition 6. A sequence x = {xm} in a normed linear space (X, || ■ ||) is said to be rough statistically K-convergent of order a (0 < a < 1) to the number £ € X if for every e > 0 there exists some non-negative number r such that
lim —|{m < n : ||Aa:m — £11 > r + e}| = 0,
n^x na
where £ is known as r-St\-limit of sequence x = {xm}. It is denoted by
r-St% xm ^ £-
The set of all the rough statistically K-convergent sequences of order a(0 < a < 1) is denoted by rSt\ for fixed r.
In general, the r-St^-limit of a sequence may be not unique. So we consider r-St^-limit set of a sequence x = {xm} as
r-Sfi
r-St%-LTX = {£ : xm-^ £}.
The sequence x = {xm} is said to be r-St^-convergent such that r-St^-LTX = For unbounded sequence the rough limit set is always empty.
But in case of rough statistical K-convergence of order a, we have r-St^-LTX = 0 even though sequence may be unbounded. For this we have given the next example.
Example 1. Let X = R. Then, define a sequence
Kxm —
Take a = 1, then
(-1)m, m = n2, m, otherwise.
r-StaK-LTx =
0, r < 1,
[1 - r, r - 1] , otherwise
and r-K-LTX = 0 for all r > 0. Thus, this sequence is divergent in ordinary sense as it is unbounded. Also, the sequence is not statistically K-convergent for any r.
With the help of statistically cluster points defined by Fridy [6], we are giving the following definition as follows:
Definition 7. A point £ is said to be rough statistically K-cluster point of order a (0 < a < 1) of a sequence x = {xm} in a normed linear space (X, || ■ ||) if for every e > 0 there exists some non-negative number r such that
lim —|{m < n : ||Aa:m — £11 > r + e}| / 0.
n^x na
Definition 8. A sequence x = {xm} is said to be statistically A-bounded if there exists a real number M0 > 0 such that
lim -|{m < n : ||Aa;m|| > M0}| = 0.
n^x n
Definition 9. A sequence x = {xm} is said to be statistically A-bounded of order a (0 < a < 1) if there exists a real number M0 > 0 such that
lim — |{m < n : ||Aa;m|| > M0} \ = 0.
n^x na
In view of above definitions, we obtained the following interesting results on rough statistical A-convergence.
Theorem 1. Every rough A-convergent sequence is also rough statistically A-convergent of order a (0 < a < 1), but converse may be not true.
Proof. Let the sequence x = {xm} be rough A-convergent in a normed linear space (X, || ■ ||). Then, for every e > 0 and some r > 0 there exists a real number M0 > 0 such that ||Axm—£|| > r+e for all m > Mo.
The set {m < n : ||Axm — £|| > r + e} has finitely many terms. Thus,
lim — I lm < n : ||Aa;m — £11 > r + e} I =0.
n^x na
Hence, the sequence x = {xm} is rough statistically A-convergent of order a (0 < a < 1). But the contrary part is not true which can be justified by the next example.
Example 2. Consider the normed space (R, || ■ ||) under the usual norm. Define a sequence
Ax = i 1, m is a square, m \ 0, otherwise.
For e > 0 and some r > 0 we have
M(r, e) = {m < n : ||Axm — £|| > r + e}; £ = 0 = {m < n : ||Axm|| > r + e > 0} = {m < n : ||Axm|| = 1} = {m < n : m is a square}.
Thus,
lim —\M(r,e)\< lim = 0.
n^x na n^x na
Therefore, x = {xm} is rough statistically A-convergent of order a to 0 for a > 1/2.
□
In the next theorem we discuss the algebraic characterization of rough statistically A-convergent sequences of order a (0 < a < 1).
Theorem 2. Let x = {xm} and y = {ym} be two sequences in a normed linear space (X, || ■ ||) and a (0 < a < 1) be given. Then for some non-negative number r the following holds:
r-StA r-StA
(1) if xm-> x0 and k € N then kxm-> kxo;
xo and ym-— yo then (xm + ym)-— (xo + yo).
(2) if xm -
P r o o f. (1) If k = 0 then there is nothing to prove. r-stt
If k = 0. Since xm-> xo then for given e > 0 and some r > 0, we have the set
f r + e t 1
M(r,e) = <m <n : \\Axm - x0\\ >—-—} with lim —\M(r, e)| = 0. I k J n^x na
Let m € Mc(r, e). Then
||Akxm — kxo|| = |k| ||Axm — xo|| < |k|
r + e
~w
This implies that
i. e.
lim —
n^x na
lim —
n^x na
( r + e i
| m<n: || A kxm - kx0\\ < f
( r + e i
| m<n: || A kxm - kx0\\ > —-— j
< r + e.
= 1,
0.
Therefore, kxm (2) Since xm
> kxo.
A r-St%
— xo and ym-> yo then for given e > 0 and some r > 0, we have sets
r + ei ,. 1
f r + e ] 1
Mx(r,e) = <m<n:\\Axm — Xo\\>—-—f with lim —\Mx(r,e) 1=0, I 2 J n^x na
( r + e T 1
My(r, e) = \m <n : \\Aym - y0\\ > —— \ with lim — \My(r, e)| = 0.
I 2 J n^x na
Let m € M£(r, e) n M£(r, e). Then
r + e r + e
||A(a;m + ym) ~ (x0 +y0)|| < ||Aa;m -a;0|| + ||Aym -y0\\ < —--1--— = r + e.
This implies that
i. e.
lim — \{m < n : \\A(xm + ym) ~ (x0 + yo)\\ < r + e}| = 1,
n^x na
lim —\{m<n: \\A(xm + ym) - (x0 + yo) || > r + e} = 0.
n^x na
Therefore, (xm + ym)-— (xo + yo).
□
Theorem 3. Let 0 < a < ft < 1 then rSt^ — rSt^ where rSt^ and rSt^ represent the sets of all rough statistically A-convergent of order a and ft respectively.
Proof. Let x = {xm} be a sequence in a normed linear space (X, || ■ ||). If 0 < a < ft < 1 then for every e > 0 and some r > 0 with the limit point £, we have
| {m <n: ||Aa;m - £|| > r + e}| < < n : ||Aa;m - £|| > r + e}|.
Therefore, we get rSt^ — rSt^.
□
r-sta
Theorem 4. A sequence x = {xm} in a normed linear space (X, || ■ ||) is statistically K-bounded of order a(0 < a < 1) if and only if r-St^-LTX = 0, for some non-negative number r.
Proof. Let the sequence x = {xm} is statistically K-bounded of order a (0 < a < 1), then there exists a real number M0 > 0 such that
lim — \{m < n : ||Aa;m|| > M0} \ = 0.
n^x na
Let M = {m € N : ||Kxm|| > M0}. Define r0 = sup {||Kxm|| : m € Mc}. As
0 € ro-St%-LTX ^ ro-St%-LTX = 0.
Conversely, suppose that r-St^-LTX = 0 for some r > 0. Then, for each e > 0 there exists £ € X such that £ € r-St%-LTX. Then
lim — I {m < n : ||Aa;m — £11 > r + e} I =0.
n^x na
Hence, the sequence x = {xm} is statistically K-bounded of order a. □
Theorem 5. If x' = {xmk} is a non-thin subsequence of a sequence x = {xm} then
r-St% -LTx C r-St% -LTX,.
Proof. The proof of above results is obvious, so we are omitting it. □
Theorem 6. Let x = {xm} be a sequence in a normed linear space (X, || ■ ||). Then, the rough statistical limit set of order a (0 < a < 1) is convex, i.e., r-St^-LTX is convex.
Proof. Let £i,£2 € r-St^-LTX and e > 0 be given. For the convexity of the set r-St^-LTX, we have to show that [(1 — ^)£i + ft£2] € r-St^-LTX for some ft € (0,1). Now, we define
Mi(r, e) = jm € N : \\Axm - 6|| >
M2(r, e) = {m € N : \\Kxm - £2|| > As £1,£2 € r-St%-LTX, we have
lim J-|Mi(r,e)|= lim ~^\M2(r, e)| = 0.
n^x na n^x na
Let m € Mf(r, e) n M£(r, e). Then
||Axm — [(1 — ft)£l + ft£2] || = 11(1 — ft)(Kxm — £l) + ft(Kxm — £2) ||
< (1 — ft) ||Kxm — £l || + ft|Kxm — £2|
< r + e.
Since
we get
1. e.
lim — \M?{r,e) nMS(r,e)\ = 1,
n^x na
lim 4-| {m<n: ||Aa:m - [(1 - ft)6 + ft£2] || >r + e}| = 0,
[(1 - ft)6 + ft6] € r-Si^LT*.
Hence, r-Si^-LTX 1s a convex set. □
Theorem 7. A sequence x = {xm} in a normed linear space (X, || ■ ||) is rough statistically K-convergent of order a (0 < a < 1) to £ € X for some non-negative number r if and only if there exists a sequence y = {ym} in X which is rough statistically K-convergent of order a to £ and ||Kxm — Kym|| < r for all m € N.
Proof. Necessity. Let xm-> £. Then, for each e > 0 and some r > 0 we have
1
lim
n^x na
{m < n : ||Axm - C|| > r + e} = 0. (2.1)
Now, we define the sequence as
C, ||Axm - £||< r,
Aym = <( A , C - Axm ,,
I Axm + r-rT---77, otherwise.
||Axm -
Then, we have
0, ||Axm - £|| < r,
Aym"e = <! ||ATl-eii(l|A'Ym~ell~r)' otherwise
such that ||Axm - Aym|| < r for all m € N. Further,
0, ||Axm - £|| < r,
||Aym 1 ||Axm - C||- r, otherwise.
Hence, by the definition of Aym and (2.1), we have
1
lim —
n^x na
{m < n : ||Aym - C|| > r + e}
which prove that the sequence y = {ym} is rough statistically K-convergent of order a to £.
Sufficiency. Since the sequence y = {ym} is rough statistically K-convergent of order a (0 < a < 1) to £ then for e > 0 we have
lim —
n^x na
0.
{m < n : ||Aym - C|| > r + e}
Now for some r > 0 and sequence x = |xm} with ||Axm - Ay m|| < r, the following inclusion holds {m < n : ||Axm - £|| > r + e} C {m < n : ||Aym - £|| > r + e}.
Hence, we get
lim —lim < n : ||Aa:m - £11 > r + e}| = 0.
n^x na
□
Theorem 8. The set r-St^-LTX of rough statistical K-limit set of order a (0 < a < 1) is closed.
0
Proof. (i) If r-St^-LTX = 0, then we have to prove nothing.
(ii) If r-St^-LTX = 0. Then, take a sequence y = {ym} — r-St^-LTX such that Aym — y* for m — to. It is sufficient to show that y* € r-St^-LTx.
As Aym — y*, then for given e > 0 there exists me € N such that
ii a ii r + e
||Aym — y*|| < ——
for m > me.
Now choose mo € N such that mo > me. Then we have
r + e
||Aymo -y*|| < —— ■ Again as y = {ym} — r-Si^-LTc, we have ymo € r-Si^-LTc. Clearly,
lim —
r + e
m < n : ||A.Tm - ymo || > —— \ = 0. (2.2)
Next we prove the inclusion
( r + e ]
| m < ??. : ||A.Tm - ymJ < —— j C {m < ??. : ||Aa:m - y*|| <r + e}. (2.3)
Let
k € <! m < ??. : ||Aa:m - ymJ < I => ||Aa;fc - ymo\\ <
Hence,
||Axfc - y*|| = ||Axfc - ymo + Aym - y* - Aym + ymo || < ||Axfc - ymo || + ||Aym - y*|| + ||Ayfc - ymo I Using equation (2.2) and Theorem 7 we get
r + e
l|Aj/fc Vmo 11 < —•
Thus,
,, r + e r + e r + e ||A:rfc - y*|| < —— + —— + —— = r + e.
This implies that
k € {m < n : ||Axm - y* || < r + e}.
Hence the inclusion (2.3) is proved. Thus,
( r + e i
{m < ??. : ||A.Tm - y*|| > r + e} C |m < ??, : ||Aa:m - ymo|| > —— j.
Now,
1 1f r + e 1
lim — |{m < n : ||Aa:m - y*|| > r + e}| < lim — m < ??. : ||Aa:m - ymo || > —— \ . (2.4)
Using equation (2.2), we obtained that the set on left side of (2.4) has density 0. Hence, we get
lim —Ilm < n : ||Aa:m — y*|| > r + e}| =0.
n^x na
□
Theorem 9. Let Ta^ be the set of all rough statistical A-cluster points of order a (0 < a < 1) for a sequence x = {xm} in the normed linear space (X, || ■ ||). Then for an arbitrary c € rAx and a positive real number r, we have ||£ — c|| < r for all £ € r-SiA-LTx.
Proof. We prove the result by contradiction. For given a (0 < a < 1), we take a point c € and £ € r-St^-LTX such that ||£ — c|| > r. By choosing e = (||£ — c|| — r)/3, we get the following inclusion
{m < n : ||Axm — £|| > r + e} 5 {m < n : ||Axm — c|| < e}. (2.5)
Since c € then
lim —\{m < n : ||Aa;m — c|| < e}| / 0.
n^x na
By (2.5), we get
lim —|{m < n : ||Aa:m — £11 < r + e}| / 0,
n^x na
which is a contradiction to £ € r-St^-LTX. □
Theorem 10. Let x = {xm} be a sequence in a .strictly convex normed linear space (X, || ■ ||). Let a and r be two positive real numbers. If any £o,£i € r-St^-LTX with ||£o — £i| = 2r, then x = {xm} is rough statistically A-convergent of order a (0 < a < 1) to (£0 + £1 )/2.
Proof. Let z € and £0,£1 € r-St^-LTx such that ||£0 — £1| = 2r. Then, we have
||£o — z|| < r and ||£i — z|| < r, (2.6)
and by triangle inequality, we get
||£o — £i|| < ||£o — z|| + ||£i — z|| ^ 2r < ||£o — z|| + ||£i — z||. (2.7)
We get from (2.6) and (2.7) Also
||£0 — z| = ||£1 — z| = r-^(6-eo) = ^[(z-£o) + (£i-z)], (2.8)
and using ||£o — £i| = 2r, we get (£1 — £o)/2 = r.
Now from equation (2.8) and from strict convexity of the normed linear space (X, || ■ ||), we have (z — £0) = (£1 — z) = (£1 — £0)/2 which implies that z = (£0 + £0/2. Thus, z is a unique statistical A-cluster point of sequence x = {xm}.
As £0,£1 € r-StA-LTX ^ r-SiA-LTx = Hence, by Theorem 4, the sequence x = {xm} is statistically A-bounded of order a.
Since z is the unique statistical A-cluster point to statistically A-bounded sequence x = {xm} of order a.
r-stx
This implies that xm-> z, where z = (£0 + £0/2. □
Acknowledgements
We express great sense of gratitude and deep respect to the referees of this paper for their valuable suggestions.
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