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1URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 99-111
DOI: 10.15826/umj.2024.1.009
STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES IN NEUTROSOPHIC 2-NORMED SPACES
Rahul Mondal
Department of Mathematics, Vivekananda Satavarshiki Mahavidyalaya,
Vidyasagar University, Manikpara, Jhargram - 721513, West Bengal, India
imondalrahul@gmail.com
Nesar Hossain
Department of Mathematics, University of Burdwan, Burdwan - 713104, West Bengal, India
nesarhossain24@gmail.com
Abstract: In this paper, we have studied the notion of statistical convergence for double sequences in neutrosophic 2-normed spaces. Also, we have defined statistically Cauchy double sequences and statistically completeness for double sequences and investigated some interesting results in connection with neutrosophic 2-normed space.
Keywords: Neutrosophic 2-normed space, Double natural density, Statistically double convergent sequence, Statistically double Cauchy sequence.
1. Introduction
In 1951, Fast [12] and Steinhaus [29] independently extended the concept of usual convergence of real sequences to statistical convergence of real sequences based on the natural density of a set. Later on, this idea has been studied in different directions and various spaces by many authors such as [8-10, 13, 14, 25, 26, 28, 31, 35], and many others.
After the introduction of the fuzzy set theory by Zadeh [37], there has been an extensive effort to find applications and fuzzy analogs of the classical theories and it is being applied in various branches of engineering and science [4, 15, 17, 19, 24]. Later on, the notion of the fuzzy set theory was developed effectively and generalized into new notions as its extensions like intuitionistic fuzzy set [1], interval-valued fuzzy set [36], interval-valued intuitionistic fuzzy set [2], and vague fuzzy set [3]. As a generalization of a crisp set, fuzzy set, intuitionistic fuzzy set, and Pythagorean fuzzy set, Smarandache [32] studied the concept of neutrosophic set. Later, Bera and Mahapatra introduced the notion of neutrosophic soft linear space [5] and neutrosophic soft normed linear space [6]. Recently, Kirisci and §imsek [21] defined neutrosophic normed space and, in this space, many summability methods such as statistical convergence [21], statistical convergence of double sequences [18], ideal convergence [22], lacunary statistical convergence [23], deferred statistical convergence [11] etc.
Mursaleen and Edely [26] defined and studied statistical convergence and statistically Cauchy double sequences in R. Sarabadan and Talebi [35] studied the notion of statistical convergence of double sequences in 2-normed spaces. Granados and Dhital [18] discussed statistical convergence and statistical Cauchy property for double sequences in neutrosophic normed spaces. Recently,
Murtaza et al. [27] introduced neutrosophic 2-normed space and studied statistical convergence for single sequences. In the present paper, we study statistical convergence and statistically Cauchy double sequences in neutrosophic 2-normed spaces and prove some associated results in the line of investigations of them with respect to neutrosophic 2-norm.
2. Preliminaries
Throughout the paper, N and R indicate the set of natural numbers and the set of reals, respectively; |A| denotes the cardinality of the set A. First, we recall some basic definitions and notations.
Definition 1 [26]. Let K C NxN be a two-dimensional set of positive integers, and let K(m,n) be the number of (j, k) in K such that j < m and k < n. Then, the two-dimensional analog of natural density can be defined as follows.
The lower asymptotic density of the set K C N x N is defined as
62( X) = liminf^l
— m,n mn
In case the sequence (K(m,n)/(mn)) has a limit in Pringsheim's sense, we say that K has double natural density defined as
m,n mn
Example 1. [26] Let Then,
K = {(i2,j2): i,j € N}.
¿2(3C) = lim v ' ; < lim ^—= 0; m,n mn m,n mn
i.e., the set K has double natural density zero, while the set {(i, 2j) : i, j € N} has double natural density 1/2.
Note that, setting m = n, we obtain the two-dimensional natural density due to Christopher [7].
Definition 2 [26]. A real double sequence {lmn} is said to be statistically convergent to a number £ if the set
{(m,n),m < i,n < j : |lmn - £| > 4 has double natural density zero for all e > 0.
Definition 3 [16]. Let Z be a real vector space of dimension d, where 2 < d < to. A 2-norm on Z is a function ||., .|| : Z x Z ^ R which satisfies the following conditions:
(1) ||x,y|| =0 if and only if x and y are linearly dependent in Z;
(2) ||x, y|| = ||y, x|| for all x and y in Z;
(3) ||ax, y|| = |a| ||x, y|| for all a in R and for all x and y in Z;
(4) ||x + y, z|| < ||x, z|| + ||y, for all x, y, and z in Z.
Example 2. [34] Let Z = R2. Define ||-, on R2 by ||x,y|| = |xiy2 — x2yi|, where x = (xi,x2) and y = (y1,y2) € R2. Then, (Z, ||-, is a 2-normed space.
Definition 4 [35]. A double sequence {lmn} in a 2-normed space (Z, ||., .y) is called statistically convergent to £ € Z if, for all e > 0 and all nonzero z € Z, the set
{(m,n) € N x N : ||lmn - z| > e} has double natural density zero ; i.e.,
lim 1 {(m, n),m < i, n < j : \\lmn — > e} | =0.
Definition 5 [35]. A double sequence {lmn} in a 2-normed space (Z, ||., .||) is called a statistically Cauchy double sequence if, for all e > 0 and all z € Z, there exist no, mo € N such that, for all m,p > n0 and n, q > m0, the set
{(m,n), m < i,n < j : ||lmn - lpq,z|| > e} has double natural density zero.
Definition 6 [30]. A binary operation □ : [0,1] x [0,1] ^ [0,1] is called a continuous t-norm if the following conditions hold :
(1) □ is associative and commutative ;
(2) □ is continuous ;
(3) x □ 1 = x for all x € [0,1];
(4) x □ y < z □ w whenever x < z and y < w for all x, y, z, w € [0,1].
Definition 7 [30]. A binary operation * : [0,1] x [0,1] ^ [0,1] is called a continuous t-conorm if the following conditions are satisfied :
(1) * is associative and commutative ;
(2) * is continuous ;
(3) x * 0 = x for all x € [0,1];
(4) x * y < z * w whenever x < z and y < w for all x, y, z, w € [0,1].
Example 3. [20] Here are examples of t-norms:
(1) x □ y = min{x,y};
(2) x □ y = x.y;
(3) x □ y = max{x + y — 1, 0}. This t-norm is known as Lukasiewicz t-norm.
Example 4. [20] Here are examples of t-conorms:
(1) x * y = max{x,y};
(2) x * y = x + y — x.y;
(3) x * y = min{x + y, 1}. This is known as the Lukasiewicz t-conorm.
Lemma 1 [33]. If □ is a continuous t-norm, * is a continuous t-conorm, and r € (0,1) for 1 < i < 7, then the following statements hold:
(1) if r1 > r2, then there are r3, r4 € (0,1) such that r1 □ > r2 and r1 > r2 * r4;
(2) if r5 € (0,1), then there are r6, r7 € (0,1) such that r6 □ r6 > r5 and r5 > r7 * r7.
Now we recall the notion of neutrosophic 2-normed space.
Definition 8 [27]. Let Y be a vector space, and let
N2 = {< (e, f), 8(e, f), tf(e, f),^(e, f) >: (e, f) € Y x Y} be a 2-normed space such that
N2 : Y x Y x R+ ^ [0,1].
Suppose that □ and * are continuous t-norm and t-conorm, respectively. Then, the quadruple Z = (Y, N2, □, *) is called a neutrosophic 2-normed space (N2-NS) if the following conditions hold for all e, f, g € Z, n, Z > 0, and ft = 0:
(1) 0 < 6(e, f; n) < 1, 0 < tf (e, f; n) < 1, and 0 < ^(e, f; n) < 1 for every n > 0;
(2) 8(e,f; n) + tf(e, f; n) + ^(e, f; n) < 3;
(3) 6(e, f; n) = 1 iff e and f are linearly dependent;
(4) 6(fte, f; n) = 6(e, f; n/|ft|) for all ft = 0;
(5) 8(e,f; n) □ 6(e,g; Z) < ©(e,f + g; n + Z);
(6) 6(e, f; ■) : (0, to) ^ [0,1] is a continuous nonincreasing function;
(7) lim^rc 6(e,f; n) = 1;
(8) 8(e,f; n) = ©(f,e; n);
(9) tf (e, f; n) = 0 iff e and f are linearly dependent;
(10) tf (fte, f; n) = tf (e, f; n/|ft|) for all ft = 0;
(11) tf(e,f; n) * tf(e,g; Z) > tf(e, f + g; n + Z);
(12) tf (e, f; ■) : (0, to) ^ [0,1] is a continuous nonincreasing function;
(13) lim^rc, tf (e, f; n) = 0;
(14) tf(e,f; n) = tf(f,e; n);
(15) ^(e, f; n) =0 iff e and f are linearly dependent;
(16) ^(fte, f; n) = ^(e, f; n/|ft|) for each ft = 0;
(17) ^(e,f; n) * ^(e,g; Z) > ^(e, f + g; n + Z);
(18) ^(e, f; ■) : (0, to) ^ [0,1] is a continuous nonincreasing function;
(19) lim^rc ^(e, f; n) = 0;
(20) ^(e,f; n) = ^(f,e; n);
(21) If n < 0, 8(e, f; n) = 0, tf (e, f; n) = 1, and ^(e, f; n) = 1. In this case, N2 = is called neutrosophic 2-norm on Y.
Definition 9 [27]. Let {ln}ngN be a sequence in an N2-NS Z = (Y, N2, □, *). Choose e € (0,1) and n > 0. Then, {ln }neN is called convergent if there exist no € N and lo € Y such that
©(ln — lo, z; n) > 1 — e, tf(ln — lo,z; n) < e, ^(ln — lo,z; n) <e
for all n > no and z € Z; i.e.,
lim B(ln — lo, z; n) = 1, lim tf (ln — lo, z; n) = 0, lim ^(ln — lo, z; n) = 0.
n^-rc n^-rc n^-rc
In this case, we write
N2 — lim ln = lo or ln —^ lo
n^-rc
and lo is called an N2-limit of {ln}ngN.
Definition 10 [27]. Let {lk}keN be a sequence in an N2-NS Z = (Y, N2, □, *). Choose e € (0,1) and n > 0. Then, {lk}keN is called statistically convergent to £ if the natural density of the set
A(e, n) = {k < n : 0(lk — £, z; n) < 1 — e or tf (lk — £, z; n) > e and — £, z; n) > e}
is zero for all z € Z, i.e., ¿(A(e, n)) = 0.
Definition 11 [27]. Let {ln}neN be a sequence in an N2-NS Z = (Y, N2, *). Choose e € (0,1) and n > 0. Then, {ln}neN is called a Cauchy sequence if there exists m0 € N such that
0(l„ - n) > 1 - e, - n) < e, - ,z; n) <e
for all n, m > m0 and z € Z.
Definition 12 [27]. Let {lk}keN be a sequence in an N2-NS Z = (Y, N2, *), e > 0, and n > 0. Then, {lk}keN is called a statistical Cauchy sequence if there exists n0 € N such that
lim -1 {k < n : Q(lk - lno,z; r?) < 1 - t or §(lk - lno,z; r?) > t and ip(lk - lno,z; r?) > t} I = 0
n n 1 1
for every z € Z or, equivalently, the natural density of the set
A(e, n) = {k < n : B(lfc - ln0 ,z; n) < 1 - e or - ln0 ,z; n) > e and - ln0 ,z; n) > e} is zero; i.e., ¿(A(e, n)) = 0.
3. Main results
Throughout this section, Z and ¿2(A) stand for neutrosophic 2-normed space and double natural density of the set A respectively unless otherwise stated. First, We define the following:
Definition 13. A double sequence {lmn} in an N2-NS Z is said to be convergent to £ € Z with respect to N2 if, for all a € (0,1) and u > 0, there exists n0 € N such that
6(lmn - £, z; u) > 1 - a, - £, z; u) < a, ^(Un - £, z; u) < a
for all m, n > n0 and nonzero z € Z; i.e.,
lim B(lmn - £,z; u) = 1, lim - £,z; u) = 0, lim - £,z; u) = 0.
In this case, we write
N2 - lim lmn = £ or lmn £.
Definition 14. A double sequence {lmn} in an N2-NS Z is said to be statistically convergent to £ € Z with respect to N2 if, for all a € (0,1), u > 0, and nonzero z € Z,
£2({(m,n) € N x N : 0(lmn—£,z; u) < 1—a or £,z; u) > a and —£,z; u) > a}) = 0
or, equivalently,
lim -771 {'m < i, n < j : Q(lmn — £, 2;n) < 1—a or t>(lmn—£,, z; u) > a and £, z; u) > a\I =0.
ij
In this case, we write
St2(N2) - lim lmn = £ or lmn £
and £ is called an st2(N2)-limit of {lmn}.
Lemma 2. Let {lmn} be a double sequence in an N2-NS Z. Then, for all a € (0,1), u > 0, and nonzero z € Z, the following statements are equivalent:
(1) si2(N2) - lirnm;ra^ Imn = C;
(2) ¿2({(m,n) € N x N : B(1mra-£,z;u) < 1-a}) = ¿2({(m,n) € N x N : -C,z;u)>a}) = ¿2({(m, n) € N x N : ^(1mra - C, z; u) > a}) = 0;
(3) ¿2({(m, n) € N x N : 8(1mra - C, z; u) > 1 - a, tf (lmra - C, z; u) < a, ^(1mra - C, z; u) < a}) = 1;
(4) ¿2({(m,n) € N x N : 8(1mra-C,z; u)>1-a}) = ¿2({(m,n) € N x N : tf(1m„-C,z; u) < a}) = ^({(m, n) € N x N : ^(1mra - C, z; u) < a}) = 1;
(5) st2(N2) - ©(Imn - C, z; u) = 1, st2(N2) - - C, z; u) = 0, and «¿2(^2) - limm)„^ ^(Imn - C, z; u) = 0.
Theorem 1. Let {1mn} be a double sequence in an N2-NS Z. If
N2 - lim Imn = C,
then
Proof. Let
St2(N2) - lim 1mra = C.
N2 - lim Imn = C.
Then, for all a € (0,1) and u > 0, there exists no € N such that
©(Imn - C,z; u) > 1 - a, - C, z; u) < a, and - C,z; u) <a
for all m, n > n0 and nonzero z € Z. So, the set
{(m, n) € N x N : ©(¿mn - C, z; u) < 1 - a or tf(1mn - C, z; u) > a and ^(¿mn - C, z; u) > a} has at most finitely many terms. Since double natural density of a finite set is zero, ^2({(m,n) € N x N : ©(¿mn-C,z; u) < 1-a or tf(1mn-C,z; u) > a and ^(¿mn-C,z; u) > a}) = 0. Therefore,
st2(N2) - lim 1mra = C. This completes the proof. □
But in the general case, the converse to Theorem 1 does not have to be true, as shown in the following example.
Example 5. Let Y = R2 with ||x,y|| = |xiy2 - x2yi|, where x = (xi, x2),y = (yi,y2) € R2. Define a continuous ¿-norm □ and a continuous t-conorm * as a □ b = ab and a * b = min{a + b, 1} for a, b € [0,1], respectively. Take a € (0,1), x,y € Y, and u > 0 such that u > ||x,y||. Consider
r^t \ u n. . ||x,y| ,, . ||x,y|
@{x, y, u) =--¡i-¡7, 't){X, y] u) = —-¡i-¡7, ip{x, y, u) =-.
u + ||x,y|| u + ||x,y|| u
Then, N2 = is a neutrosophic 2-norm on Y and the quadruple Z = (Y, N2, □, *) becomes
a neutrosophic 2-normed space. Define a double sequence {1mn} € Z by
J(mn, 0), m = s2, n = t2, s,t € N;
'mn — S , ,
1(0,0), otherwise.
Then, for nonzero z € Z, we have
Ks,t(a, u) = {m < s, n < t : 0(lmn, z; u) < 1 — a or $(lmn, z; u) > a and ^(lmn, z; u) > a}
f / ^ . u ^ i l^mn z|| ^ j ||lmn, z|| ^
= •! m < s,n <t :-—-rr < 1 — a or -—-- > a and - > a
u + H^mn z|| u + H^mn z|| u
= \ m < s, n <t : > V(T or > ua
1 — a
= {m < s, n < t : lmn = (mn, 0)} = {m < s,n < t : m = s2, n = t2, s,t € N}
and
11 . \/s\/t
— |3CS t(a, «)| < — lim < s,n<t : m = s2, n = t2, s, t € N) I < --: ->• 0 as s, t ->• oo;
st st st
i.e.,
st2(N2) — lim lmn = 0.
But {lmn} is not convergent with respect to N2.
Theorem 2. Let {lmn} be a double sequence in an N2-NS Z. If {lmn} is statistically convergent with respect to N2, then an st2(N2)-limit of {lmn} is unique.
Proof. Suppose that
st2 (N2) — lim lmn = 6, st2 (N2) — lim lmn = £2,
where = £2. Given a € (0,1), choose A € (0,1) such that
(1 — A) □ (1 — A) > 1 — a, A * A < a. Now, for all u > 0 and z € Z, we define the sets
A©i(A, u) = {(m,n) € N x N : 6(lmn — Ci,z u/2) < 1 — A},
A©2(A,u) = {(m,n) € N x N : 6(lmn — C2,z u/2) < 1 — A},
Atfi(A,u) = {(m,n) € N x N : #(lmn — Ci, z; u/2) > A},
A^(A,u) = {(m,n) € N x N : #(lmn — C2, z; u/2) > A},
A^i(A,u) = {(m,n) € N x N : ^(lmn — Ci, z; u/2) > A}, A}.
A^(A,u) = {(m,n) € N x N : ^(lmn — C2, z; u/2) >
Since
St2(N2) — lim lmn = 6, St2(N2) — lim lmn = 6,
using Lemma 2, we get
¿2(A©i(A,u)) = ¿2(Am(A,U)) = ¿2(A^i(A,u)) = 0
and
¿2(A©2(A,u)) = ¿2(Atf2(A,u)) = ¿2(A^2 (A,u)) = 0.
Now, let
A©,^(A,u) = [A©i(A,u) U A©2(A,u)] n [AM(A,u) U A^(A,u)] n [A^i(A,u) U A^(A,u)].
Then, clearly, ^(A©,^(A,u)) = 0; i.e., ^(A©,^(A,u)) = 1.
Let (p, q) € A© ^(A, u). Then, the following three cases are possible.
Case i. If (p, q) € A©1(A, u) n A©2(A,u), then
8(Ci - z; u) > B(1Pq - Ci, z; u/2) □ 8(1Pq - {2, z; u/2) > (1 - A) □ (1 - A) > 1 - a.
Since a € (0,1) is arbitrary, we have 6({i - C2,z; u) = 1, which yields = C2. Case ii. If (p, q) € A^i(A,u) n A^2(A, u), then
tf (Ci - C2, z; u) < tf - Ci, z; u/2) * tf (^ - {2, z; u/2) < A * A < a.
Since a € (0,1) is arbitrary, we have tf (^ - {2, z; u) = 0, which yields Ci = {2.
Case iii. If (p, q) € A^i(A,u) n A^2(A, u), then, similarly to Case ii, we get Ci = {2. Hence, an st2(N2)-limit of {1mn} is unique. This completes the proof. □
Theorem 3. Let Y be a real vector space, and let {1mn} and {wmn} be two double sequences in an N2-NS Z. Then, the following statements hold:
(1) if st2(N2) - Imn = Ci and si2(^2) - wmra = C2, then
(N2) - lim 1mra + Wmra = Ci + {2;
(2) if st2(N2) - Imn = Ci and c = 0, then st2(N2) -
c1mn — cCi •
Proof. It is easy. So, we omit the details. □
Theorem 4. Let {1mn} be a double sequence in an N2-NS Z. Then,
^(N2) - lim 1mra = C
if and only if there exists a subset
K = {mi <m2 < ••• <mp < ••• ; ni < n < ••• < n < •••}c N x N
such that = 1 and N2 - 1TOpnq = C.
Proof. First, suppose that st2(N2) - 1mn = C. Now, for all u > 0, k € N, and
nonzero z € Z, define
A^2{k,v) = |(m,n)eNxN : B(/mra-{,z;u)>l-i, tf(Zmra-£,z;^(Zmra-£,,z;u)<i}, (3.1) and
= |(m,n)eNxN : @(lmn-£,z;u) < 1—^ or &(lmn-£,z;u)>^ and ^(Un -C,z;u)>i|.
Then, clearly, AN2(k + 1,u) C AN2(k,u) and, by our assumption, we have ¿2(®N2(k,u)) = 0. Also, from (3.1), we get ¿2(AN2 (k,u)) = 1. Now, let us show that, for (m,n) € AN2 (k,u),
N2 - lim Imn = C
Suppose that {1mn|(ra,n)eAN2(fc,u) is not convergent with respect to N2. Then, for some a € (0,1), we have
©(¿mn - C,z; u) < 1 - a, tf(1mn - C,z; u) > a, - C,z; u) > a
except for at most finite number of terms (m,n) € AN2 (k,u) and nonzero z € Z. Define
CN2(a,u) = {(m,n)€NxN : ©(¿mn-C,z;u) > 1-a and tf(1mn-C,z;u) < a, ^(1mn-C,z;u) < a},
where a > 1/k. Clearly, ¿2(CN2 (a, u)) = 0. Since a > 1/k, we have AN2 (k,u) C CN2 (a, u) and, hence, ¿2(AN2(k,u)) = 0, which contradicts ¿2(AN2(k,u)) = 1. Therefore, for (m,n) € AN2(k,u), we have
N2 - lim Imn = C Conversely, suppose that there exists a subset
K = {mi <m2 < ••• <mp < ••• ; « <«,2 < ••• < « < •••}C N x N
such that
¿2(K) = 1, N2 - lim 1mpraq = C-
Then, for all a € (0,1) and u > 0, there exists p0 € N such that
©(¿mpnq - C, z; u) > 1 - a, tf (1mpnq - C, z; u) < a, nq - C, z; u) < a
for all p, q > p0 and nonzero z € Z. Therefore,
{(m, n) € N x N : 0(1mn - C, z; u) < 1 - a or tf(1mn - C, z; u) > a and ^(1mn - C, z; u) > a} C N x N \ {mP0+i < mP0+2,...; nPo+i < nPo+2,...}.
Hence,
£2({(m,n)€NxN : 6(1mn-C, z; u) < 1-a or tf(1mn-C,z; u) > a and ^(1mn-C, z; u) > a}) = 0; i.e., st2(N2) - limm,n^ Imn = C. n
Definition 15. Let {1mn} be a double sequence in an N2-NS Z, a € (0,1), and let u > 0. Then, {¿mn} is called statistically Cauchy with respect to N2 if there exist m0 = m0(a) and n0 = n0(a) € N such that
(m,n) € N x N : ©(Zmn - ¿mono,z; u) < 1 - a or tf(1mn - ¿mono,z; u) > a and ^(Imn - ¿mono, z; u) > a}) =0
for nonzero z € Z.
Theorem 5. Let {lmn} be a double sequence in an N2-NS Z. If
si2(N2) - lim lmra =
then {lmn} is statistically Cauchy with respect to N2. Proof. Let
(N2) - lim lmn = £
and a € (0,1) be given. Choose A € (0,1) such that
(1 - A) □ (1 - A) > 1 - a, A * A < a.
Then, for A € (0,1), u > 0, and nonzero z € Z, we have 52(AN2(A, u)) = 0, where
AN2 (A, u) = {(m,n) € N x N : ©(lmn - £,z; u/2) < 1 - A or - £,z; u/2) > A
and - £,z; u/2) > A}.
Then, ¿2(N x N \ AN2 (A,u)) = 1. Let (mo, no) € AN2 (a,u). So,
©(lmono - C z; u/2) > 1 - A, ^(lm0nc - C
z; u/2) < A and no - £,z; u/2) < A.
Now, we define
Bn2 (a, u) = {(m,n) € N x N : ©(lmn - ¿mono ,z; u) < 1 - a or tf(lmn - ¿mono ,z; u) > a
and - Imono, z; u) > a}
for every nonzero z € Z. Let us show that ®N2(a, u) C AN2 (A,u). Let (p, q) € ®N2 (a, u). Then, we get
©(lpq - lmono, z; u) < 1 - a, $(lpq - lmono, z; u) > a and - lmono , z; u) > a.
Case i. Consider ©(lpq - lmono, z; u) < 1 - a. Let us show that
©(lpq - z; u/2) < 1 - A.
Suppose that
©(lpq - C,z; u/2) > 1 - A.
Then, we have
1 - a > 8(lpq - lmono, z; u) > B(lpq-£, z; u/2) □ ©(¿mono-C, z; u/2) > (1 - A) □ (1 - A) > 1 - a, which is impossible. Therefore,
8(lpq - z; u/2) < 1 - A. Case ii. Consider $(lpq - lmono, z; u) > a. Let us show that
- C,z; u/2) > A.
Suppose that
- e,z; u/2) < A.
Then, we have
a < - lmono, z; u) < ^(lpq - z; u/2) □ ^(lmono - C, z; u/2) < A * A < a,
which is impossible. Therefore, we have
tf(lpq - £,z; u/2) > A.
Case iii. If we consider ^(lpq - lmono, z; u) > a, then, similarly to Case ii, we can show that
^(lpq - £,z; u/2) > A.
Therefore, (p, q) € AN2 (A, u). Hence, ®N2 (a, u) C AN2 (A,u). Since ¿2(AN2 (A,u)) = 0, we have ¿2(®N2(a, u)) = 0. So, {lmn} is statistically Cauchy with respect to N2. □
Theorem 6. Let {lmn} be a double sequence in an N2-NS Z. If {lmn} is statistically Cauchy with respect to N2, then it is statistically convergent with respect to N2.
Proof. Suppose that {lmn} is statistically Cauchy with respect to N2 but not statistically convergent to any £ € Z with respect to N2. Then, for a € (0,1), u > 0, and nonzero z € Z, there exist m0 = m0(a) and n0 = n0(a) € N such that ¿2(X) = 0, where
K = {(m,n) € N x N : ©(lmn - ¿mono ,z; u) < 1 - a or tf(lmn - ¿mono, z; u) > a
and ^(lmn - lmono, z; u) > a},
and ¿2(M) = 0, where
M = {(m,n) € N x N : ©(lmn - £,z; u/2) > 1 - a or #(lmn - £,z; u/2) < a
and ^(lmn - £,z; u/2) < a}.
Since and
if
and
©(lmn — lmono, z; u) > 20(lmn — C, z; u/2) > 1 — a
^(lmn lmono, z; u) < 2^(lmn C, z; u/2) < a
^(lmn — lmono, z; u) < 2^(lmn — C, z; u/2) < a,
r^n u, 1 — a
@(lmn - i, Z] -) > -
ti(lmn - Z; < ^(¿mn - z! «) <
we have
^d(m, n) € N x N : ©(lmn — lmono,z; u) > 1 — a and tf(lmn — lmono, z; u) < a, ^(lmn — lmono, z; u) < a}) = 0.
This gives ¿2(Kc) = 0 and so ¿2(K) = 1, a contradiction. Therefore, {lmn} is statistically convergent to some C. D
Definition 16. An N2-NS Z is called! statistically complete with respect to N2 if every statistically Cauchy sequence is statistically convergent with respect to N2.
Remark 1. In the light of Theorems 5 and 6, we see that every N2-NS is statistically complete for double sequences.
Conclusion and future developments
In this paper, we have dealt with statistical convergent double sequences in an N2-NS and have shown that every N2-NS is statistically complete. Later on, these results may be the opening of new tools to generalize this notion in various directions such as I2-statistical and I2-lacunary statistical convergence with respect to N2. Also, this idea can be used in convergence-related problems in many branches of science and engineering.
Acknowledgements
We express great gratitude and deep respect to the reviewers and the managing editor for their valuable comments which improved the quality of the paper. The second author is grateful to The Council of Scientific and Industrial Research (CSIR), HRDG, India, for the award of Senior Research Fellow during the preparation of this paper.
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