CC BY
16DOI: 10.15393/j3.art.2022.11930
UDC 517.521
C. CHOUDHURY, S. DEBNATH
ON X^-SUPREMUM, X^-INFIMUM AND RELATED
RESULTS
Abstract. In this paper, we introduce the concept of X^-supremum, X^-infimum, X^ —limit superior, and X^—limit inferior and investigate a few implication relationships between them.
Key words: ideal, filter, X^-supremum, X^-infimum, XK-limit superior, X^-limit inferior
2020 Mathematical Subject Classification: 40A35, 40A05
1. Introduction and background. In 1951, Fast [9] and Steinhaus [19] introduced the concept of statistical convergence via the natural density. The natural density of a set A C N is defined as
n cardiA n{1, 2 ,...,k}) d(A) = lim----,
k^x k
provided that the limit exists. A sequence x = (xk) is called statistically convergent to a real number x0, if for any e > 0, the set {k E N: Xk E (x0 — £,x0 + e)} has zero natural density. Now, since every finite subsets of N have zero natural density, statistical convergence has appeared to be one of the generalizations of the usual convergence. Apart from Fast [9] and Steinhaus [19], a lot of investigation and generalizations in this direction has been carried out by Fridy [10], [11], Salat [17], Tripathy [20] and many others [1], [3], [14].
In an attempt to extend the notion of statistical convergence, X and X*—convergence of sequences was introduced in 2001 by Kostyrko et. al. [13] in the metric space setting, where X represents an ideal in N. A sequence % = (xk) is called X-convergent to a real number x0 if for any e > 0, the set {k E N: Xk E (xo — £,xo + e)} E X. Interestingly, X-convergence was appeared not only as a generalization of statistical convergence, but, also,
© Petrozavodsk State University, 2022
some other important known notions of convergences, such as logarithmic statistical convergence, uniform statistical convergence etc., turned out to be the particular cases of Z-convergence. For more details on Z-convergence and its several generalizations, [8], [12], [16], [18] can be addressed, where many more references can be found.
On the other hand, the notion of Z*-convergence was further extended in 2011 to Z^-convergence by M. Macaj and M. Sleziak [15]. It should be mentioned that Z*-convergence of a sequence x = (xk) was defined in terms of usual convergence of the subsequence (xmk), where M = {m\ < m2 < ... < m-k < • • • } is an element of the associated filter T(Z). But in the case of Z^-convergence, that usual convergence was replaced by ^-convergence, where K is another ideal. The involvement of two ideals at the same point of time makes this concept more complicated and more interesting. Over the last few years, the study of Z^-convergence of sequences has got much attention from researchers and the research carried out so far shows a strong analogy in the behavior of Z^-convergence of sequences. The relation between Z and Z^-convergence can be found in works by Das et. al. [4] and Macaj and Sleziak [15]. In [5], [6], Das et. al. introduced and investigated ZK-convergence of sequence of function and Z^-Cauchy functions. For more details on -convergence, see [7], where many more references can be found.
When studying some new notion of convergence of sequences, several closely related concepts occur quite naturally, such as cluster points, supre-mum, infimum, limit superior, limit inferior, etc. In this paper, our aim is to introduce ZK-analogue of the above concepts and investigate some fundamental properties.
2. Definitions and preliminaries.
Definition 1. [13] Let X be a non-empty set. A family of subsets Z C P (X) is called an ideal in X if
(i) for every A,B e Z we have A U B e Z;
(ii) for every A e Z and B C A we have B e Z.
An ideal Z is called non-trivial if Z = 0 and X e Z. A non-trivial ideal Z C P(X) is called an admissible ideal in X if and only if Z D {{^} : x e X}. Some standard examples of ideals are given below:
(i) The set Zf of all finite subsets of N is an admissible ideal in N.
(ii) The set Z^ of all subsets of natural numbers having natural density 0 is an admissible ideal in N.
(iii) The set Zc = [A C N: a 1 < is an admissible ideal in N.
oo
(iv) Suppose N = (J Dp be a decomposition of N (for i = j, Di H Dj = 0).
p=i
Then the set Z of all subsets of N, which intersects finitely many Dp's, forms an ideal in N.
More important examples can be found in [12].
Definition 2. [13] Let X be a non-empty set. A family of subsets T C P (X) is called a filter in X if
(i) for each A,B eT we have A H B e T;
(ii) for each A eT and B D A we have B eT.
The filter T = T(Z) = [X - A: A e Z} is called the filter associated with the ideal Z.
Remark 1. If Z and K are two ideals in N, then the set Z V £ = [A U B: A eZ ,B e £} forms an ideal in N. Further, if IVK is non-trivial, then the dual filter ofIV^ is denoted and defined by T(ZV£) = [M H N: M e T(Z),N e T(£)}.
Definition 3. [13] Let Z C P(N) be a non-trivial ideal in N. A real-valued sequence x = (x^) is said to be Z-convergent to x0 if the set [k e N: | Xk — x0 e} belongs to Z for each £ > 0. In this case, x0 is called the Z-limit of the sequence (xk) and is written as Z — lim x = x0.
Definition 4. [13] Let Z be an admissible ideal in N. A real-valued sequence x = (xk) is said to be Z* — convergent to x0, if there exists a set M = [m1 < m2 < ... < m-k < ...} in the associated filter T(Z), such that lim Xk = x0. In this case, x0 is called the I *-limit of the sequence
k£M
(xk) and is written as Z* — lim x = x0.
Definition 5. [15] Let Z and K be two ideals in N. A real-valued sequence x = (xk) is said to be ZK-convergent to x0 if there exists M e T (Z), such that the sequence y = (yk) defined by
xk, k e M, xo, k e M
is IC-convergent to x0. In this case, x0 is called the ZK-limit of the sequence (xk) and is written as ZK — lim x = x0.
When K = Zf, then Z^-convergence concept coincides with Z*-convergence [13].
Definition 6. [2] A real number I is said to be an X lower bound of the real-valued sequence x = (xk), if
{kE N: xk ^ 1}eT(X) (or {k E N: xk <1}eX).
The set of all X lower bound of the sequence x = (xk) is denoted by Lx(x).
Theorem 1. [2] If I E R is a usual lower bound of the real-valued sequence x = (xk), then I is also an X lower bound of the sequence x.
Definition 7. [2] A real number t is said to be the X-infimum of the real-valued sequence x = (xk) if t is the supremum of Lx(x). In other words,
X — inf x = sup Lx (z).
Definition 8. [2] A real number u is said to be an X upper bound of the real-valued sequence x = (xk), if
{k E N: xk ^ u} E T(X) (or {k E N: xk > u} E X).
The set of all X upper bounds of the sequence x = (xk) is denoted by Ux (x).
Definition 9. [2] A real number s is said to be the X-supremum of the real-valued sequence x = (xk) if s is the infimum of Ux(x). In other words,
X — sup x = inf Ux (x).
Theorem 2. [2] Let x = (xk) be any real-valued sequence. Then
inf x ^ X — inf x ^ X — sup x ^ sup x.
Definition 10. [2] (a) Let x = (xk) be a real-valued sequence. Then the X— limit inferior is denoted and defined by
X — lim inf x = X — sup v,
where v = (vk) is the sequence defined by vk = X — inf {xn,xn+\,...}.
(b) Let x = (xk) be a real-valued sequence. Then the X—limit superior is denoted and defined by
X — lim sup x = X — inf w, where w = (wk) is the sequence defined by wk = X — sup{xra, xn+l,...}.
Definition 11. [13] A real number 7 is said to be an X-cluster point of a sequence x = (xk) if for any £ > 0
{kE N: lxk — 71 < e} E X
holds.
3. Main results. Throughout the article, X, K, and X V K denotes the non-trivial admissible ideal in N.
Definition 12. A real number I is said to be an X* lower bound of the real-valued sequence x = (xk), if there exists M = {m\ < m2 < ... < mk < ...} E T(X), such that I is an usual lower bound of the subsequence (xmk).
The set of all X* lower bounds of the sequence x = (xk) is denoted by Lx * (x).
Definition 13. A real number t is said to be the X*-infimum of the real-valued sequence x = (xk), if t is the supremum of Lx* (x). In other words,
X* — inf x = sup Lx* (x).
Definition 14. A real number u is said to be an X* upper bound of the real-valued sequence x = (xk), if there exists M = {m\ < m2 < • • • < mk < • • •} E T(X), such that u is an usual upper bound of the subsequence (xmk).
The set of all X* upper bounds of the sequence x = (xk) is denoted by Ux* (x).
Definition 15. A real number s is said to be the X*-supremum of the real-valued sequence x = (xk), if s is the infimum of Ux* (x). In other words,
X* — sup x = inf Ux* (x). Theorem 3. For any real-valued sequence x = (xk),
X* — inf x ^ X — inf x ^ X — sup x ^ X* — sup x.
Proof. We first prove that the inclusions Lx* (x) C Lx(x) and Ux* (x) C Ux(x) hold for the sequence x. Let I E Lx* (x). Then, by definition, there exists M = {m\ < m2 < ... < mk < ...} E T(X), such that I is an usual lower bound of (xmk). In other words, xmk ^ I for any k E N. But then the inclusion
{kE N: xk ^ 1} = M U{kE N \ M : xk ^ 1} D M
holds and, subsequently, [k e N: xk ^ /} e T(Z). Hence, I e Lx(x) and the first inclusion is established. Applying a similar technique, the second inclusion can be obtained. Now the inclusions Lx* (x) C Lx(x) and Ux* (x) C Ux(x) and Theorem 2 altogether give
Z* — inf x ^ Z — inf x ^ Z — sup x ^ Z* — sup x.
□
Definition 16. A real number I is said to be an Z^ lower bound of the real-valued sequence x = (xk), if there exists M = [m1 < m2 < • • • < mk < • • • } e T(Z), such that I is an K lower bound of the sequence y = (yk) defined by
xk k e M, I k /M.
The set of all ZK lower bounds of the sequence x = (xk) is denoted by Lx^ (x).
Definition 17. A real number t is said to be the Z^-infimum of the real-valued sequence x = (xk), if t is the supremum of Lxk (x). In other words,
Z^ — inf x := sup Lxk (x).
Definition 18. A real number u is said to be an upper bound of the real-valued sequence x = (xk), if there exists M = [m1 < m2 < • • • < mk < • • • } e T(Z), such that u is an K upper bound of the sequence y = (yk) defined by
xk k e M, u k/M.
The set of all ZK upper bounds of the sequence x = (xk) is denoted by Ux>c (x).
Definition 19. A real number s is said to be the Z^-supremum of the real-valued sequence x = (xk) if s is the infimum of Uxk (x). In other words,
Z^ — sup x := inf Uxic (x).
Theorem 4. (i) Let x = (xk) be a real-valued sequence, such that I e Lxk (x). If I' < I, then I' e Lxk (x);
(ii) Let x = (xk) be a real-valued sequence, such that u E Uxk (x). If u' > u, then u' E Uxk (x).
Proof. (i) Let I E Lxk (x). Then there exists M = [m\ < m2 < ... < mk < ...} E T(X), such that I E L^(y), where y = (yk) is the sequence defined by
xk, k E M, I, k/M.
This implies {k E N: yk ^ 1} E T(K). Now, since I' < I by the assumption, the inclusion
{k E N: yk > I'} D{kE N: yk > 1}
holds, and, consequently, {k E N: yk ^ I'} E T (K), i.e., I' E Lxk (x). This completes the proof.
(ii) The proof can be obtained by applying a similar technique. □
Corollary 1. (i) Let x = (xk) be a real-valued sequence, such that I E Lx* (x). If I' < I, then I' E Lx* (x);
(ii) Let x = (xk) be a real-valued sequence, such that u E Ux* (x). If u' > u, then u' E Ux* (x) .
Theorem 5. For any real-valued sequence x = (xk),
X* — inf x ^ XK — inf x ^ XK — sup x ^ X* — sup x. Proof. To prove the theorem, we prove the following three inequalities:
X* — inf x ^ XK — inf x, (1)
XK — inf x ^ XK — sup x, (2)
and
XK — sup x ^ X* — sup x. (3)
To prove (1), let I E Lx* (x). Then, by definition, there exists M = {m\ < m2 < ... < mk < ...} E T(X), such that I is an usual lower bound of (xmk). In other words, I E L(y), where y = (yk) is defined by
xk, k E M, I, k/M.
Therefore, by Theorem 1, we have I G L^(y). This implies I G Lxk (x). Hence, we have Lx* (x) C Lxk (x) and, consequently, (1) holds.
To prove (2), assume the contrary. Then there exist some I' G Lxk (x) and u' G Uxk (x), such that u' < I'. But then, by Theorem 4, I' G Uxk (x), which is a contradiction.
The proof of (3) is analogous to that of (1), so omitted. Combining (1), (2), and (3) we obtain the desired inequality. □
Corollary 2. For any real-valued sequence x = (xk):
inf x ^ K — inf x ^ X^ — inf x ^ X^ — sup x ^ K — sup x ^ sup x.
Proof. We omit the proof as it can be easily obtained by combining Theorem 2, Theorem 5, and considering M = N from T(X). □
Theorem 6. Let XVK be a non-trivial ideal in N and x = (xk) be a real-valued sequence. Then:
(i) if x is a monotonic increasing sequence, then XK — inf x = X* — sup x;
(ii) if x is monotonic decreasing, then X^ — sup x = X* — inf x.
Proof. (i) We divide the entire proof into considering two cases. Case-I: X* — sup x < ro
Suppose X* — sup x = s. Then there exists some M = [m\ < m2 < ... < m-k < ...} G T(X), such that xmk ^ s holds for all k G N. This implies
M C{kG N: xk ^ s}. (4)
Also, for any e > 0, there exists k0 G N, such that xmko > s — e. We claim that s G Lxic (x). Otherwise, if s G Lxk (x), then there exists N = {n\ < n2 < ... <nk <...}, such that s G L^(y), where y = (yk) is the sequence defined by
Xk, k G N, s, kG N.
In other words, {k G N: Xk ^ s} G T(^). Now, as the inclusion
{k G N: xk > s} C {k G N: xk > s}
holds, we have {k G N: Xk ^ s} G T(£). Consequently, from (4), we have N \ M G T(^). Now M G T(X) and N \ M G Ttogether yield
M n (N \ M) G T(X VK), i. e., 0 G T(X V K), which is a contradiction. This proves our claim. Now, let e > 0 be arbitrary. Then we have
{kGM : xk < s — e} Ç{kG M : xk < xmkQ} Ç {1, 2,...,mko}.
Since K is admissible, so {1,2,...,mko} G K and, as a consequence, {kGM : xk < s — e} G K,, i. e., s — e G Lxk (x). Therefore, by Theorem 4, we obtain Lxk (x) = (—œ, s — ej. Hence, XK — inf x = sup Lxk (x) = s. Case-II: X* — sup x = œ
If X* — sup x = œ, then, for any l G R, there exists M = {m\ < m2 < ... < mk < ...} G T(X), such that xmko ^ I for some k0 G N. Now, since x is monotonic increasing, xmko ^ xk for all k ^ mko. Thus, for all k ^ mko, we have xk ^ I. Eventually, {k G N : xk < 1} Ç {1, 2,...,mko} G K, i.e., I G Lfc(x), which further implies that l G Lxk(x). Now, since I is arbitrary, Lxk (x) = (—œ, œ). Hence, XK — inf x = sup Lxk (x) = œ. (ii) The proof is analogous to that of (i), so omitted. □
Theorem 7. Let x = (xk) be a real-valued sequence and t G R be fixed. Then XK — inf x = t if and only if for every £ > 0 there exists M G T(X), such that
{k E M: xk <t — £} EK and {k E M: xk ^ t + £} E T (K).
Proof. Suppose XK — inf x = t. Then, for any I E Lxk (x), I ^ t and for any e > 0, there exists I' E Lxk (x), such that t — £ < I'. So, by Theorem 4(i), t — £ E Lxk (x). This implies that there exists a set M E T(X), such that {k E N: yk < t — £} E K, where y = (yk) is defined as
xk, k E M, t — £, k/M.
Therefore, {k E M : xk <t — £} E K holds.
Now, to prove {k E M: xk ^ t + £} E T(K), we assume the contrary. Then there exists some £0 > 0, such that for any M E T(X) {k E M: xk ^ t + e0} E T(K). In particular, if we take M = N, then we have t + £0 E LK(x) and, consequently, t + £0 E Lxk (x), which is a contradiction to the fact that t = sup Lxk (x).
To prove the converse part, assume that Ve > 0 there exists M E T(X): {k EM: xk < t — £} E K. and {k E M: xk ^ t + £} E T (K). Then we
have t — e G Lxk (x) and t + £ G Lxk (x). Therefore, Lxk (x) = (—ro,t — e). Now, since £ is arbitrary, we have
X^ — inf x = sup Lxic (x) = t.
This completes the proof. □
Corollary 3. Let x = (xk) be a real-valued sequence and s G E be fixed. Then X^ — sup x = s if and only if for every £ > 0 there exists M G T(X), such that
{k G M: xk > s + £} G K and {k G M: xfc ^ s — e} G T(£).
Proof. The proof is similar to that of Theorem 7, so it is omitted here. □
Theorem 8. For a real-valued sequence x = (xk), X^ — limx = x0 if and only if XK — inf x = x0 = XK — sup x.
Proof. Let XK — lim X - X§. Then there exists a set M = {m\ < m2 <
... < mk < ...} G T (X), such that for any £ > 0, {k G M: lxk — xo| ^ £} G This implies {fc G M: > x0 + e} G i.e., xo + e G f/XK(x) and {fc G M: < x0 — e} G i.e., x0 — £ G Lxk(x). By Theorem 4, we have Uxk (x) = (x0, ro) and Lxk (x) = (—ro,x0), which further gives
X^ — inf x = sup Lxk (x) = x0 = inf UXic (x) = X^ — sup x.
For the converse part, let X^ — infx = x0 = X^ — sup x. Then sup Lxk (x) = x0 = inf f/XK (x). Then, by definition of the usual supre-mum and infimum, for any £ > 0 there exists I G Lxk (x) and u G Uxk (x), such that x0 — £ < I and x0+£ > u. Now, I G Lxk (x) and u G Uxk (x) imply the existence of two sets M', M" G T(X), such that {k G M': xk < /} G K and {k G M": xk > u} G Let M denote the set M' n M''. Then M G T(X) and, since x0 — £< I and x0 + £ > u hold, by the hereditary property of K we have:
{k G M: xk ^ x0 — £} C {k G M: xfc < /} C {fc g M': xfc < /} G £,
and
{k G M: xfc ^ x0 + e} C {fc g M: xk > u} C {fc G M": xfc > «} G
Consequently,
{k G M: lxk—x0l ^ e} = {k G M: xk ^ x0—e}U{k G M: xk ^ x0+e} G K. Hence, XK — limx = x0. □
Corollary 4. Let x = (x^) be a real-valued sequence, such that K — inf x = x0 = K — sup x. Then, XK — limx = x0.
Proof. The proof follows directly from Corollary 2 and Theorem 8, so omitted. □
Theorem 9. Let x = (xk) and y = (yk) be two real-valued sequences, such that there exists a set M G T(X) satisfying {k G M: xk = } G K. Then
XK — inf x = X^ — inf y and X^ — sup x = X^ — sup y.
Proof. We only prove the first part, i.e., X^ — infx = X^ — infy. The proof of the second part can be obtained by applying a similar technique.
Let the given conditions hold and suppose I G Lxk (x) be arbitrary. Then, by definition, there exists N G T(X), such that {k G N: xk < 1} G K,. Consequently,
{k G M n N: yk <1} = = {k G M n N: xk = yfc,yk <l}U{kGM n N: xfc = yfc,yk < 1} C
C {fc G M: xfc = yk} U {fc G N: xfc < /} G
From the above inclusion, it is clear that {fc G M n N: yk < 1} G K,. Since M n N G T(X), we have I G Lxk (y). This proves that Lxk (x) C Lxk (y). Similarly, one can establish LXic (y) C LXic (x). Hence, LXic (x) = LXic (y) holds and, finally, sup Lxk (x) = sup Lxk (y), i.e., XK — inf x = XK — inf y. □
Theorem 10. Let x = (xk) and y = (yk) be two real-valued sequences. Then:
(i) XK — inf(x + y) = XK — inf x + XK — inf y;
(ii) XK — sup(x + y) = XK — sup x + XK — sup y.
Proof. (i) Let X^—inf x = and X^ — inf y = . Then, by Theorem 7, for any e > 0 there exists M,N G T(X), such that {k G M: xfc < — §} G 1C and {k G ^: yk <ty — |} G^. Now, as the inclusion
[keM n N: xk + yk< (tx + ty) - e] C
C {keM: xk <tx - U {ke N: yk < ty -
holds and M n N e T(X), [k e M n N: xk + yk < (tx + ty) - e] e K and, consequently, XK - inf(x + y) = tx + ty = X^ - inf x + X^ - inf y. This completes the proof.
(ii) The proof is similar to that of (i), so omitted. □
Definition 20. A real number 7 is said to be the X^-cluster point of a real-valued sequence x = (xk) if there exists M = [m\ < m2 <.. .< mk < ...] e T(X), such that 7 is a ^-cluster point of the subsequence (xmk).
The set of all X^-cluster points of a real-valued sequence x = (xk) is denoted by XK - (rx).
Theorem 11. Let x = (xk) be a real-valued sequence, such that XK -sup x and XK - inf x are finite. Then, XK - sup x e XK - (rx) and XK - inf xe XK - (rx).
Proof. Le^ X^ - sup x = inf ujk. (x) = s. Then, by definition of the usual infimum, for any e > 0 there exists t0 e (x), such that s ^ t0 < s + e. Consequently, there exists M e T (X), such that [keM: xk > t0] e K. Now, as the inclusion
[keM: xk ^ s + e] C[keM: xk > to]
holds,
[keM: xk ^ S + e]e1C. (5)
Again, s = inf uxk (x) gives s - £ e (x), which further implies
[keM: xk > s - e] e (6)
Now, since the following relation
[k e M: xk > s - e] = [keM: s - £ < xk < s + e]U[k e M: xk ^ s+e]
holds, from (5) and (6) we have [keM: s - £ < xk < s + e] e i.e.,
s e XK - (rx). This completes the proof of the first part.
Applying a similar technique, we can show that XK - inf x e XK - (rx). □
Definition 21. (a) Let x = (xk) be a real-valued sequence. Then X^-limit inferior is denoted and defined by
XK - lim inf x = XK - sup v,
where v = (vk) is the sequence defined by = XK — inf {xn,xn+\,...}.
n'k
(b) Let x = (xk) be a real-valued sequence. Then XK—limit superior is denoted and defined by
X^ — lim sup x = X^ — inf w,
where w = (wk) is the sequence defined by Wk = XK — sup{xra, xn+1,...}.
n'k
Corollary 5. Let x = (xk) be a real-valued sequence. Then:
(i) if Vk = XK — inf {xn, xn+\,...} for all n G N, then v = (vk) is a constant
n'k
sequence and Vk = XK — inf x;
(ii) if Wk = XK — sup{xn,xn+\,...} for all n G N, then w = (wk) is a
n'k
constant sequence and Wk = XK — sup x.
Proof. The proof of (i) and (ii) are easy, so omitted. □
Corollary 6. Let x = (xk) be a real-valued sequence. Then:
X^ — lim inf x = X^ — inf x and X^ — lim sup x = X^ — sup x.
Proof. The proof is omitted as it can be easily obtained from the Definition 21 and Corollary 5. □
Corollary 7. Let x = (xk) and y = (yk) be two real-valued sequences. Then:
(i) X^ — lim inf x ^ X^ — lim sup x;
(ii) lim inf x ^ K — lim inf x ^ X^ — lim inf x ^
^ XK — lim sup x ^ K — lim sup x ^ lim sup x;
(iii) X^ — lim inf (x + y) = X^ — lim inf x + X^ — lim inf y;
(iv) XK — lim sup(x + y) = XK — lim sup x + XK — lim sup y.
4. Conclusion. In this paper, we investigated the notions of -supremum, X^-infimum, X^-limit superior, and X^-limit inferior for a real-valued sequence x = (xk), and presented some interrelationships between these notions. Theorem 7 and Corollary 3 give the necessary and sufficient conditions for a real number to become X^-infimum and
XK -supremum, respectively, of a real-valued sequence. Theorem 8 gives the necessary and sufficient condition regarding the X^-convergence of a real-valued sequence. Theorem 11 proves the inclusion of the numbers - sup x and XK - inf x in the set XK - (rx). The obtained results may be helpful for future researchers to explore the notion of X^-convergence in more detail.
Acknowledgment. The authors thank the anonymous referees for their constructive suggestions that helped improving the quality of the paper. The first author is grateful to the University Grants Commission, India for their fellowship funding under the UGC-SRF scheme (F.No.16-6(DEC. 2018)/2019(NET/CSIR)) during the preparation of this paper.
References
[1] Altinok M., Kugukaslan M. A-statistical supremum-infimum and A-statistical convergence. Azerb. J. Math., 2014, vol. 4, no. 2, pp. 31-42.
[2] Altinok M., Kiigiikaslan M. Ideal limit superior-inferior. Gazi Univ. J. Sci., 2017, vol. 30, no. 1, pp. 401-411.
[3] Altinok M., Kiiciikaslan M., Kaya U. Statistical extension of bounded sequence space. Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 2021, vol. 70, no. 1, pp. 82-99.
DOI: https://doi.org/10.31801/cfsuasmas.736132
[4] Das P., Sengupta S., Glab S., Bienias M. Certain aspects of ideal convergence in topological spaces. Topology Appl., 2020, vol. 275, 107005. DOI: https://doi.org/10.1016/j.topol.2019.107005
[5] Das P., Sengupta S., Supina J. IK-convergence of sequence of functions. Math. Slovaca, 2019, vol. 69, no. 5, pp. 1137-1148.
DOI: https://doi .org/10.1515/ms-2017-0296
[6] Das P., Sleziak M., Toma V. IK-Cauchy functions. Topology Appl., 2014, vol. 173, pp. 9-27.
DOI: https://doi.org/10.1016Zj.topol.2014.05.008
[7] Debnath S., Choudhury C. On some properties of X^-convergence. Palest. J. Math., 2022, vol. 11, no. 2, pp. 129-135.
[8] Demirci K. I-limit superior and limit inferior. Math. Commun., 2001, vol. 6, no. 2, pp. 165-172.
[9] Fast H. Sur la convergence statistique. Colloq. Math., 1951, vol. 2, pp. 241-244. DOI: https://doi.org/10.4064/cm-2-3-4-241-244
[10] Fridy J. A. On statistical convergence. Analysis, 1985, vol. 5, no. 4, pp. 301-313. DOI: https://doi .org/10.1524/anly.1985.5.4.301
[11] Fridy J. A., Orhan C. Statistical limit superior and limit inferior. Proc. Amer. Math. Soc., 1997, vol. 125, no. 12, pp. 3625-3631.
DOI: https://doi.org/10.1090/S0002-9939-97-04000-8
[12] Kostyrko P., Macaj M., Salat T., Sleziak M. I-convergence and extremal I-limit points. Math. Slovaca, 2005, vol. 55, no. 4, pp. 443-464.
[13] Kostyrko P., Salat T., Wilczynski W. I-convergence. Real Anal. Exch., 2000-2001, vol. 26, no. 2, pp. 669-686.
DOI: https://doi.org/10.2307/44154069
[14] Kugukaslan M., Altinok M. Statistical supremum-infimum and statistical convergence. Aligarh Bull. Math., 2013, vol. 32, no. 1-2, pp. 39-54.
[15] Macaj M., Sleziak M. IK—convergence. Real Anal. Exch., 2010-2011, vol. 36, no. 1, pp. 177-194.
[16] Nabiev A., Pehlivan S., Gurdal M. On I —Cauchy sequences. Taiwanese J. Math., 2007, vol. 11, no. 2, pp. 569-576.
DOI: https://doi .org/10.11650/twjm/1500404709
[17] Salat T. On statistically convergent sequences of real numbers. Math. Slovaca, 1980, vol. 30, no. 2, pp. 139-150.
[18] Salat T., Tripathy B. C., Ziman M. On some properties of I—convergence. Tatra Mt. Math. Publ., 2004, vol. 28, no. 2, pp. 274-286.
[19] Steinhaus H. Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math., 1951, vol. 2, pp. 73-74.
[20] Tripathy B. C. On statistically convergent sequences. Bull. Calcutta Math. Soc., 1998, vol. 90, pp. 259-262.
Received June 04, 2022. In revised form, September 1, 2022. Accepted September 2, 2022. Published online September 13, 2022.
Department of Mathematics,
Tripura University (A Central University),
Suryamaninagar-799022, Agartala, India
Chiranjib Choudhury
E-mail: chiranjibchoudhury123@gmail.com Shyamal Debnath
E-mail: shyamalnitamath@gmail.com
