Vol. 10, No. 3, 2017, 574-585
ISSN 1307-5543 – www.ejpam.com Published by New York Business Global
Intuitionistic Fuzzy Zweier I-convergent Double
Sequence Spaces Defined by Orlicz Function
Vakeel A. Khan1, Yasmeen1, Hira Fatima1, Ayaz Ahmad2,∗
1 Department of Mathematics, Aligarh Muslim University,Aligarh, India
2 Department of Mathematics,National Institute of Technology, Patna, India
Abstract. The purpose of this paper is to introduce the intuitionistic fuzzy ZweierI-convergent double sequence spaces2Z(Iµ,ν)(M) and2Z0(Iµ,ν)(M) defined by Orlicz function and study the fuzzy
topology on the said spaces.
Key Words and Phrases: Ideal, filter, doubleI-convergence, intuitionistic fuzzy normed spaces.
1. Introduction and Preliminaries
After the pioneering work of Zadeh [29], a huge number of research papers have been appeared on fuzzy theory and its applications as well as fuzzy analogues of the classical the-ories. Fuzzy set theory is a powerful hand set for modelling uncertainty and vagueness in various problems arising in field of science and engineering. It has a wide range of applica-tions in various fields: population dynamics [3], chaos control [5], computer programming [6], nonlinear dynamical system [7], etc. Fuzzy topology is one of the most important and useful tools and it proves to be very useful for dealing with such situations where the use of classical theories breaks down. The concept of intuitionistic fuzzy normed space [25] and of intuitionistic fuzzy 2-normed space [21] are the latest developments in fuzzy topology. Recently V. A. Khan and Yasmeen([12], [13]) studied the intuitionistic fuzzy ZweierI-convergent sequence spaces defined by modulus function and Orlicz function.
The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical problems/matrices(double sequences) through the cept of density. The notion of I-convergence, which is a generalization of statistical con-vergence [4], was introduced by Kostyrko, Salat and Wilczynski [14] by using the idea ofI
of subsets of the set of natural numbersNand further studied in [22]. Recently, the notion
of statistical convergence of double sequences x= (xij) has been defined and studied by Mursaleen and Edely [20]; and for fuzzy numbers by Sava¸s and Mursaleen [26]. Quite ∗
Corresponding author.
Email addresses: [email protected](V. A. Khan),[email protected](Yasmeen),
[email protected](H. Fatima), [email protected] (A. Ahmad)
recently, Das et al. [15] studied the notion of I and I∗- convergence of double sequences inR.
We recall some notations and basic definitions used in this paper.
Definition 1. Let I ⊂ 2N be a non-trivial ideal in N. Then a sequence x = (x
k) is said
to be I-convergent to a number L if, for every >0, the set {k∈N:|xk−L|≥} ∈I. Definition 2. Let I ⊂ 2N be a non-trivial ideal in N. Then a sequence x = (x
k) is
said to be I-Cauchy if, for each >0,there exists a number N =N() such that the set {k∈N:|xk−xN |≥} ∈I.
Definition 3. The five-tuple (X, µ, ν,∗,) is said to be an intuitionistic fuzzy normed space(for short, IFNS) if X is a vector space, ∗ is a continuous t-norm, is a continuous t-conorm and µ, ν are fuzzy sets on X×(0,∞) satisfying the following conditions for every x, y∈X and s, t >0 :
(a) µ(x, t) +ν(x, t)≤1, (b) µ(x, t)>0,
(c) µ(x, t) = 1 if and only if x= 0, (d) µ(αx, t) =µ(x,|α|t ) for each α6= 0, (e) µ(x, t)∗µ(y, s)≤µ(x+y, t+s), (f ) µ(x, .) : (0,∞)→[0,1]is continuous, (g) lim
t→∞µ(x, t) = 1 and limt→0µ(x, t) = 0,
(h) ν(x, t)<1,
(i) ν(x, t) = 0 if and only if x= 0, (j) ν(αx, t) =ν(x,|α|t ) for eachα 6= 0, (k) ν(x, t)ν(y, s)≥ν(x+y, t+s), (l) ν(x, .) : (0,∞)→[0,1] is continuous, (m) lim
t→∞ν(x, t) = 0 and limt→0ν(x, t) = 1.
In this case (µ, ν) is called an intuitionistic fuzzy norm.
Definition 4. Let (X, µ, ν,∗,) be an IFNS. Then a sequence x = (xk) is said to be
convergent to L∈X with respect to the intuitionistic fuzzy norm(µ, ν) if, for every >0
and t >0, there exists k0 ∈N such that µ(xk−L, t) >1− and ν(xk−L, t)< for all
k≥k0. In this case we write (µ, ν)−limx=L.
Definition 5. Let (X, µ, ν,∗,) be an IFNS. Then a sequence x = (xk) is said to be a Cauchy sequence with respect to the intuitionistic fuzzy norm(µ, ν)if, for every >0 and t >0, there exists k0 ∈N such thatµ(xk−xl, t)< and ν(xk−xl, t)< for all k, l≥k0.
Definition 6. Let K be the subset of natural numbers N. Then the asymptotic density of K, denoted by δ(K), is defined as δ(K) = lim
n
1
n|{k≤n:k∈K}|, where the vertical bars
A number sequence x= (xk) is said to be statistically convergent to a number` if, for
each >0, the set K() ={k≤n:|xk−`|> } has asymptotic density zero, i.e.
lim n
1
n|{k≤n:|xk−`|> }|= 0. In this case we write st−limx=`.
Definition 7. A number sequence x= (xk) is said to be statistically Cauchy sequence if,
for every >0,there exists a number N =N() such that
lim n
1
n|{j≤n:|xj−xN |≥}|= 0.
The concepts of statistical convergence and statistical Cauchy for double sequences in intuitionistic fuzzy normed spaces have been studied by Mursaleen and Mohiuddine[14].
Definition 8. Let I ⊂ 2N be a non trivial ideal and (X, µ, ν,∗,) be an IFNS. A
se-quence x= (xk) of elements of X is said to be I-convergent to L∈X with respect to the
intuitionistic fuzzy norm (µ, ν) if for every >0 and t >0 , the set {k∈N:µ(xk−L, t)≥1−or ν(xk−L, t)≤} ∈I.
In this caseL is called theI-limit of the sequence(xk) with respect to the intuitionistic
fuzzy norm (µ, ν) and we writeI(µ,ν)−limxk=L.
2. I2-Convergence in an IFNS
Definition 9. Let (X, µ, ν,∗,) be an IFNS. Then, a double sequence x= (xij) is said to
be statistically convergent to L∈X with respect to the intuitionistic fuzzy norm (µ, ν) if, for every >0 and t >0,
δ({(i, j)∈N×N:µ(xij−L, t)≤1− or ν(xij−L, t)≥}) = 0.
or equivalently
lim mn
1
mn|{i≤m, j ≤n,:µ(xij −L, t)≤1−or ν(xij −L, t)≥}|= 0. In this case we write st2(µ,ν)−limx=L.
Definition 9.2 Let (X, µ, ν,∗,) be an IFNS. Then, a double sequence x= (xij) is said
> 0 and t > 0, there exist N = N() and M = M() such that for all i, p ≥ N and j, q≥M,
δ({(i, j)∈N×N:µ(xij −xpq, t)≤1− or ν(xij−xpq, t)≥}) = 0.
Definition 10. LetI2 be a non trivial ideal ofN×Nand(X, µ, ν,∗,) be an intuitionistic
fuzzy normed space. A double sequence x = (xij) of elements of X is said to be I2
convergent to L∈X with respect to the intuitionistic fuzzy norm (µ, ν) if, for each >0
and t >0,
{(i, j)∈N×N:µ(xij−L, t)≤1− or ν(xij−L, t)≥} ∈I2.
In this case we write I2(µ,ν)−limx=L.
The approach of constructing new sequence spaces by means of the matrix domain of a particular limitation method have been recently employed by Altay, Ba¸sar, Mursaleen [1], Malkowsky [19] Ng and Lee [23], and Wang [28]. S¸eng¨onul¨ [27] defined the sequence
y= (yi) which is frequently used as theZp transformation of the sequencex= (xi) i.e,
yi =pxi+ (1−p)xi−1
wherex−1= 0, p6= 1,1< p <∞ and Zp denotes the matrix Zp = (zik) defined by
zik =
( p, if (i=k),
1−p, (i−1 =k); (i, k∈N) 0, otherwise .
Analogous to Ba¸sar and Altay [2], S¸eng¨onu¨l [27] introduced the Zweier sequence spaces
Z andZ0 as follows
Z ={x= (xk)∈ω:Zpx∈c};
Z0 ={x= (xk)∈ω :Zpx∈c0}.
Khan, Ebadullah and Yasmeen [8] introduced the following classes of sequences:
ZI ={(x
k)∈ω :∃L∈Csuch that for a given >0,{k∈N:|x/k−L|≥} ∈I};
ZI
0 ={(xk)∈ω : for a given >0;{k∈N:|x/k|≥} ∈I}
where (x/k) = (Zpx).
Recently V.A. Khan and Yasmeen [13] introduced the following sequence spaces:
ZI
(µ,ν)(M) = n
(xk)∈ω: n
k∈N:Mµ(x
/ k−L,t)
ρ
≤1−orMν(x
/ k−L,t)
ρ
≥o∈Io,
ZI
0(µ,ν)(M) = n
(xk)∈ω : n
k∈N:M µ(x/
k,t) ρ
≤1−orM
ν(x/
k,t) ρ ≥ o ∈I o .
In this article we introduce the intuitionistic Zweier I-convergent dou-ble sequence spaces defined by Orlicz function as follows:
2Z(Iµ,ν)(M) = n
(xij)∈2ω : n
(i, j)∈N×N:Mµ(x
// ij−L,t)
ρ
≤1−orMν(x
// ij−L,t)
ρ
≥o∈I2 o
;
2Z0(Iµ,ν)(M) = n
(xij)∈2ω: n
(i, j)∈N×N:M
µ(x//
ij,t) ρ
≤1− orM
ν(x//
ij,t) ρ
≥
o
∈I2 o
.
3. Main results
Theorem 1. The spaces 2Z(Iµ,ν)(M) and 2Z0(Iµ,ν)(M) are linear spaces.
Proof. We prove the result for 2Z(Iµ,ν)(M). Similarly the result can be proved for 2Z0(Iµ,ν)(M). Let (x//ij), (y
//
ij) ∈ 2ZI(µ,ν)(M) and let α, β be scalars. Then for a given >0, we have
A1= (
(i, j)∈N×N:Mµ x
// ij−L1,2|α|t
ρ1
≤1−orν x // ij−L1,2|α|t
ρ1
≥o∈I2;
A2= (
(i, j)∈N×N:M µ y//
ij−L2, t
2|β|
ρ2
≤1−or M
ν y//
ij−L2, t 2|β| ρ2 ≥ o
∈I2.Thus
Ac1= (
(i, j)∈N×N:M µ x//
ij−L1, t
2|α|
ρ1
>1−orM
ν x//
ij−L1, t 2|α| ρ1 < o
∈ F(I2);
Ac2 = (
(i, j)∈N×N:Mµ y
// ij−L2,2|β|t
ρ2
>1−orMν y
// ij−L2,2|β|t
ρ2
< o ∈ F(I2).
Define the set A3 = A1 ∪A2, so that A3 ∈ I2. It follows that Ac3 is a non-empty set in
F(I2).
We shall show that forρ3 =max{2|α|ρ1,2|β |ρ2}and for each (x
// ij),(y
//
ij) ∈ 2ZI(µ,ν)(M), Ac3 ⊂
(i, j)∈N×N:M
µ (αx//
ij+βy //
ij)−(αL1+βL2),t
ρ3
>1− or
M
ν (αx//ij +βyij//)−(αL1+βL2), t
ρ3
< .
Mµ x
//
mn−L1,2|α|t
ρ3
>1−orMν x
//
mn−L1,2|α|t ρ3 < and M µ y //
mn−L2,2|β|t
ρ3
>1−orM
ν y //
mn−L2,2|β|t ρ3 < . we have M
µ (αx//
mn+βymn// )−(αL1+βL2),t
ρ3
≥M
µ αx//
mn−αL1,2t
ρ3
∗M
µ βy//
mn−βL2,2t
ρ3
=M
µ x//mn−L1,2|α|t
ρ3
∗M
µ ymn// −L2,2|β|t
ρ3
>(1−)∗(1−) = (1−) and
M
ν (αx//
mn+βy//mn)−(αL1+βL2),t
ρ3
≤M
ν αx//
mn−αL1,2t
ρ3
M
ν βy//
mn−βL2,2t
ρ3
=M
ν x//mn−L1,2|α|t
ρ3
M
ν y//mn−L2,2|β|t
ρ3
> =.
This implies that
Ac3 ⊂
(i, j)∈N×N:M
µ (αx//
ij+βy //
ij)−(αL1+βL2),t
ρ3
>1− or
M
ν (αx//ij +βy//ij)−(αL1+βL2), t ρ3 < o .
Hence 2Z(Iµ,ν)(M) is a linear space.
Theorem 2. Every open ball2Bx//(r, t)(M) is an open set in 2Z(Iµ,ν)(M).
Proof. Let 2Bx//(r, t)(M) be an open ball with centre x// and radius r with respect tot.
That is
2Bx//(r, t)(M) ={(i, j)∈N×N:M
µ(x //
ij −L, t)
ρ
≤1−r orMν(x
//
ij −L, t)
ρ
≥r} ∈I2.
Lety//∈2Bxc//(r, t)(M). Then
Mµ(x
//−y//, t)
ρ
>1−randMν(x
//−y//, t)
ρ
.
Since Mµ(x//−yρ //,t)>1−r, there existst0 ∈(0,1) such that
Mµ(x
//−y//, t
0) ρ
>1−r and Mν(x
//−y//, t
0) ρ
< r.
Putting r0 =M
µ(x//−y//,t
0)
ρ
. We have r0 >1−r, there exists s ∈(0,1) such that r0 >1−s >1−r.
Forr0 >1−s, we haver1, r2∈(0,1) such thatr0∗r1 >1−sand (1−r0)(1−r2)≤s.
Putting r3 =max{r1, r2}, consider the ball 2Byc//(1−r3, t−t0)(M). We prove that
2Byc//(1−r3, t−t0)(M)⊂2B c
x//(r, t)(M). Let z//∈2Bcy//(1−r3, t−t0)(M).
M
µ(y//−z//,t−t0) ρ
> r3 and M
ν(y//−z//,t−t0) ρ
< r3.
Therefore,
M
µ(x//−z//, t)
ρ
≥M
µ(x//−y//, t0)
ρ
∗M
µ(y//−z//, t−t0)
ρ
≥(r0∗r3)≥(r0∗r1)≥(1−s)>(1−r).
and
M
ν(x//−z//, t)
ρ
≤M
ν(x//−y//, t0)
ρ
M
ν(y//−z//, t−t0)
ρ
≤(1−r0)(1−r3)≤(1−r0)>(1−r2)< s < r.
Thus z//∈2Bcx//(r, t)(M) and hence2Bcy//(1−r3, t−t0)(M)⊂2B c
x//(r, t)(M). Remark 1. 2Z(Iµ,ν)(M) is IFNS.
Define
2τ(µ,ν)(M) ={A⊂2ZI(µ,ν)(M) : for each x∈A there exists t >0 andr ∈(0,1) s. t. 2Bcx//(r, t)(M)⊂A}.
Then 2τ(µ,ν)(M) is a topology on 2Z(Iµ,ν)(M).
Theorem 3. The topology 2τ(µ,ν)(M) on2Z(Iµ,ν)(M) is first countable.
Proof.
2Bx// n1,n1
(M) :n= 1,2,3, ... is a local base atx//the topology
2τ(µ,ν)(M) on 2Z(Iµ,ν)(M) is first countable.
Proof. We prove the result for 2Z(Iµ,ν)(M). Similarly the result can be proved for 2Z0(Iµ,ν)(M).
Letx//, y//∈2ZI(µ,ν)(M) such thatx//6=y//.
Then 0< Mµ(x//−yρ //,t)<1 and 0< Mν(x//−yρ //,t)<1.
Puttingr1=M
µ(x//−y//,t) ρ
andr2=M
ν(x//−y//,t) ρ
and r=max{r1,1−r2}.
For eachr0∈(r,1),there existsr3 and r4 such that
r3∗r4 ≥r0 and (1−r3)(1−r4)≤(1−r0).
Puttingr5=max{r3,1−r4}and consider the open balls2Bx//(1−r5,2t)(M) and2By//(1−
r5,t2)(M).
Then clearly2Bx//(1−r5,2t) ∩2By//(1−r5,2t)(M) =φ.
For if there existsz//∈2Bx//(1−r5,2t) ∩ 2By//(1−r5,2t)(M), then
r1 =M
µ(x//−y//, t)
ρ
≥Mµ(x
//−z//,t
2) ρ
∗Mµ(z
//−y//,t
2) ρ
≥r5∗r5≥r3∗r3 ≥r0 > r1.
and
r2=M
ν(x//−y//, t)
ρ
≤M
µ(x//−z//,t 2) ρ
M
ν(z//−y//,t 2) ρ
≤(1−r5)(1−r5)≤(1−r4)(1−r4)≤(1−r0)< r,
which is a contradiction. Hence 2Z(Iµ,ν)(M) is Housdorff.
Theorem 5. 2Z(Iµ,ν)(m) is an IFNS. 2τ(µ,ν)(M) is a topology on 2Z(Iµ,ν)(M) . Then a
sequence(x//ij)∈2Z(Iµ,ν)(M), x//ij →x// if and if µ(x //
ij −x//, t)(M) and
Mν(x
//
ij −x//, t)
ρ
→0
as i→ ∞, j → ∞.
Proof. Fix t0 >0. Suppose x//ij → x//. Then for r ∈ (0,1), there exists n0 ∈ Nsuch
thatx//ij ∈2Bx//(r, t)(M) for alli≥n0, j≥n0.
2Bx//(r, t)(M) ={(i, j)∈N×N:M
µ(x //
ij −x//, t)
ρ
≤1−r orMν(x
//
ij −x//, t)
ρ
such that2Bxc//(r, t)(M)∈ F(I2). Then
1−Mµ(x
//
ij −x//, t)
ρ
< r
and Mν(x
// ij−x//,t)
ρ
→0 asi→ ∞, j→ ∞
Conversely, if for each t >0, Mµ(x
// ij−x//,t)
ρ
→1 and
M
ν(x//ij −x//, t)
ρ
→0
asi→ ∞, j→ ∞,thenf orr∈(0,1), there existsn0∈Nsuch that 1−M µ(x//
ij−x//,t) ρ
< r
for all i≥n0, j ≥n0.
Thus x//ij ∈2Bxcij(r, t)(M) for alli≥n0, j0≥n0 and hencex //
ij →x//.
Theorem 6. A double sequence x = (x//ij) ∈ 2Z(Iµ,ν)(M). I-converges if and only if for
every >0 and t >0 there exists a number M =M(x, , t), N =N(x, , t) such that {(M, N)∈N×N:M
µ(x//
M N−L, t
2)
ρ
>1−or M
ν(x//
M N−L, t
2)
ρ
< } ∈ F(I2).
Proof. Suppose that 2I(µ,ν)−limx = L and let > 0 and t >0. For a given > 0
choose,s >0 such that (1−)∗(1−)>1−sand < s. Then for eachx∈2Z(Iµ,ν)(M), Ax(, t)(M) ={(i, j)∈N×N:M
µ(x//
ij−L, t
2)
ρ
≤1−orM
ν(x//
ij−L, t
2)
ρ
≥} ∈I2
which implies that
Acx(, t)(M) ={(i, j)∈N×N:M
µ(x//
ij−L, t
2)
ρ
>1−orM
ν(x//
ij−L, t
2)
ρ
< } ∈ F(I2).
Conversely let us choose N ∈Acx(, t)(M).Then
Mµ(x
//
N −L,t2) ρ
>1−orMν(x
//
N −L,t2) ρ
< .
Now we want to show that there exists a number N =N(x, , t) such that
{(i, j)∈N×N:Mµ(x
// ij −x
// N, t)
ρ
≤1−sorMν(x
// ij −x
// N, t)
ρ
≥s} ∈I2.
For this, define for each x∈2Z(Iµ,ν)(M).
2Bx(, t)(M) ={(i, j)∈N×N:M µ(x
// ij −x
// N, t)
ρ
≤1−sorMν(x
// ij −x
// N, t)
ρ
≥s} ∈I2.
Then there exists (m, n)∈2Bx(, t)(M) and (m, n)∈/ 2Ax(, t)(M). Therefore we have
Mµ(x
// ij −x
// N, t)
ρ
<1−s
and Mµ(x
// ij−L,
t
2)
ρ
>1−.
In particularMµ(x
// N−L,
t
2)
ρ
>1−.
Therefore we have
1−s≥Mµ(x
//
mn−x//N, t)
ρ
≥Mµ(x
//
mn−L,2t)
ρ
∗Mµ(x
// N −L,
t
2) ρ
≥(1−)∗(1−)>1−s,
which is not possible.
On the other hand Mν(x
// ij−x
// N,t) ρ
≥sand Mν(x
// ij−L,
t
2)
ρ
< .
In particularMν(x
// N−L,
t
2)
ρ
< .
Therefore we have
s≤M
ν(x//mn−x// N, t)
ρ
≤M
ν(x//mn−L,t 2) ρ
M
ν(x// N −L,
t
2) ρ
≤ < s,
which is not possible.
Hence 2Bx(, t)(M)⊂2Ax(, t)(M),2Ax(, t)(M)∈I2.
This implies that2Bx(, t)(M)∈I2.Hence proved.
Acknowledgements
The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.
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