The concept of fuzzy sets was introduced by Zadeh as a means of representing data that was not precise but rather fuzzy. Recently, Koinac [12] studied some topological properties of fuzzy antinormed linear spaces. This has motivated us to introduce and study the fuzzy antinormed double sequence spaces with respect to ideal by using a compact linear operator and prove some theorems, in particular convergence and completeness theorems on these new double sequence spaces.
Fuzzy set theory was formalised by Professor Lofti Zadeh [21] at the University of California in 1965. Thereafter, fuzzy set theory found many applications in different areas of mathematics and in other fields. The concept of fuzzy norm was introduced by Katsaras [9] in 1984. In 1992, by using fuzzy numbers, Felbin [7] introduced the fuzzy norm on a linear space. Cheng and Mordeson [3] introduced another idea of fuzzy norm on a linear space, and in 2003 Bag and Samanta [1] modified the defnition of fuzzy norm of Cheng-Mordeson [3]. In [2] a comparative study of the fuzzy norms defined by Katsaras [9], Felbin [7] and Bag and Samanta [1] was given.
Later on, Jebril and Samanta [8] introduced the concept of fuzzy anti-norm on a linear space depending on the idea of fuzzy anti norm, introduced by Bag and Samanta [2]. The motivation of introducing fuzzy anti-norm is to study fuzzy set theory with respect to the non-membership function. It is useful in the process of decision making problems.
The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [6] and Schoenberg [20]. There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces (see, for example, [5] and references therein). One such very important generalization of this notion was introduced by Kostyrko et al. [14] by using an ideal I of subsets of the set of natural numbers, which they called I-convergence. After that the idea of I-convergence for double sequence was introduced by Das et al. [4] in 2008 (see also [13, 18] for ideal convergence in fuzzy context).
Now, we recall some terms and definitions which will be used throughout the article.
Let X be a non empty set then a family I ⊂ 2X is said to be an ideal in X if ∅ ∈ I, I is additive i.e for all A, B ∈ I ⇒ A ∪ B ∈ I and I is hereditary i.e for all A ∈ I, B ⊆ A ⇒ B ∈ I [10, 11]. A non empty family of sets is said to be a filter on X if for all implies and for all with A ⊆ B implies . An ideal I ⊂ 2X is said to be non trivial if I ≠ 2X; a non trivial ideal is said to be admissible if I ⊇ {{x} : x ∈ X} and is said to be maximal if there cannot exist any non trivial ideal J ≠ I containing I as a subset. For each ideal I there is a filter called the filter associate with ideal I, that is
Throughout the article, I is an admissible ideal on , and 2ω denotes the class of all double real sequences. The spaces 2l∞, 2c and 2c0 are the Banach spaces of bounded, convergent, and null double sequences of reals respectively with the norm
Definition 1.1. [16, 17] A double sequence x = (xij)∈2ω is said to be I-convergent to a number L if for every ∈ > 0 there are , such that
In this case, we write I - lim xij = L.
Definition 1.2. [16, 17] A double sequence (xij
)∈2ω is said to be I-Cauchy if for every ∈ > 0 there exist numbers such that for all i, p ≥ m and j, q ≥ n
Definition 1.3. [16, 17] A double sequence (xij
)∈2ω is said to be I-bounded if there exists M > 0 such that
Definition 1.4. [15, 19] A binary operation ⋄ : [0, 1] × [0, 1] ⟶ [0, 1] is said to be a continuous t-conorm if it satisfies the following conditions:
⋄ is associative and commutative,
⋄ is continuous,
a ⋄ 0 = a for all a ∈ [0, 1],
a ⋄ b ≤ c ⋄ d whenever a ≤ c and b ≤ d for each a, b, c, d ∈ [0, 1].
Some example of continuous t-conorm are a ⋄ b = {a + b - ab}, a ⋄ b = max{a, b}, a ⋄ b = min a + b, 1.
Remark 1.1.
For any r1, r2 ∈ (0, 1) with r1 > r2, there exist r3 ∈ (0, 1) such that r1 > r4 ⋄ r2.
For any r4 ∈ (0, 1), there exist r5 ∈ (0, 1) such that and r5 ⋄ r5 ≤ r4.
Recall now the notion of fuzzy antinorm in a linear space with respect to a continuous t-conorm following [9].
Definition 1.5. [12] Let X be a real linear space and ⋄ a t-conorm. A fuzzy subset of is called a fuzzy antinorm on X with respect to the t-conorm if, for all x, y ∈ X
for each t∈ (- ∞ , 0] , ν (x, t) =1 ;
for each t ∈ (0, ∞) , ν (x, t) =0 if and only if x = θ ;
for each t ∈ (0, ∞) , ν (αx) = ν (x, |α|) if α ≠ 0;
for all
.
Note that if ν is the antinorm in the definition above, then ν (x, t) is nonincreasing with respect to t for each x ∈ X. The following are example of fuzzy antinorms with respect to a corresponding t-conorm and show how a fuzzy antinorm can be obtained from a norm.
Example 1.1. Let (X, ∥. ∥) be a normed linear space and let the t-conorm ⋄ be given by a ⋄ b = a + b - ab. Define by
Then ν is a fuzzy antinormon X with respect to the t-conorm ⋄. This antinorm ν satisfies also the following:
Example 1.2. Let (X, ∥. ∥) be a normed linear space and consider the t-conorm ⋄ defined by a ⋄ b = min {a + b, 1}. Define by
Then ν is a fuzzy antinorm on X with respect to the t-norm ⋄. Note that this ν satisfies the condition (FaN6) and also the following:
(FaN7)ν (x, .) is a continuous function on and strictly decreasing on the subset {t : 0 < ν (x, t) <1} of .
Definition 1.6. [12] A sequence in a fuzzy antinormed linear space (X, ν, ⋄) is said to beν-convergent to a point x ∈ X if for each ∈ > 0 and each t > 0 there is such that
Let (X, ν, ⋄) be a fuzzy antinormed linear space with respect to an idempotent t-conorm ⋄, and let ν satisfy (FaN6). Then for each λ ∈ (0, 1) the function ∥x ∥ λ : X → [0, ∞) defined by
is a norm on X and v = {∥ x ∥ λ : λ ∈ (0, 1)} is an asscending family of norms on X. In this paper, we generalize the definition of fuzzy anti-norm on a linear space. Later on, we study some relations and results on them.
Fuzzy(anti) Iλ- convergence
Now, in this section we define fuzzy Iλ-convergence, fuzzy Iλ- anti-convergence, fuzzy Iλ- anti-Cauchy and fuzzy Iλ- compactness for double sequences with respect to an ideal I on
Definition 2.1. A double sequence (xij) in a fuzzy antinormed linear space X is said to be fuzzy Iν-convergent to a point x ∈ X if for each ∈ > 0 and each t > 0 the set
In this case, we write fuzzy Iν - lim xij = x and x is called a fuzzy Iν-limit of (xij).
Definition 2.2. Let X be a fuzzy antinormed double sequence space and λ ∈ (0, 1). A sequence (xij) ∈ X is said to be fuzzy Iλ-convergent to x ∈ X if for all t > 0, the set
In this case, we write fuzzy Iλ - lim ν (xij - x, t) =0 and x is called a fuzzy Iλ-limit of (xij).
Definition 2.3. Let X be a fuzzy antinormed double sequence space and λ ∈ (0, 1). A sequence (xij) ∈ X is said to be fuzzy Iλ-anti-convergent in X if there exist x ∈ X and such that for all t > 0,
In this case, we write (xij) → x and x is called a fuzzy Iλ-anti-limit of (xij).
Definition 2.4. Let λ ∈ (0, 1). A sequence (xij) in a fuzzy antinormed double sequence space X is said to be fuzzy Iλ-anti-Cauchy if there exist numbers and such that for all i, p ≥ m, j, q ≥ n and all t > 0,
Definition 2.5. A fuzzy antinormed double sequence space X is said to be fuzzy Iλ-anti-complete, λ ∈ (0, 1), if for every fuzzy Iλ-anti-Cauchy sequence in X is fuzzy Iλ- anti-convergent in X.
Now, we define two fuzzy antinormed double sequence spaces with the help of a compact operator T:
It is easy to check that these are really fuzzy antinormed double sequence spaces. In what follows T is a compact linear operator. We also define an open ball with centre x and radius r with respect to t as follows:
Theorem 2.1.In the fuzzy antinormed linear double sequence space with respect to an idempotent t-conorm ⋄ satisfying (FaN6) and (FaN7) a sequence is Iν-convergent if and only if it is Iλ-convergent for each λ ∈ (0, 1).
Proof. Let (xij) be a sequence in such that (xij) is Iν- convergent to x, i.e., for each t > 0
Fix λ ∈ (0, 1). So, . There exists a set P ∈ I such that for each (m, n) ∈ P,
Since ∥T (xmn) - T (x) ∥ λ = v {t > 0 : ν (T (xmn - T (x) , t) <1 - λ}, we have ∥T (xmn) - T (x) ∥ λ ≤ t for all (m, n) ∈ P. As t > 0 was arbitrary and T is a compact linear operator, for each λ ∈ (0, 1), by (FaN6), we have ∥xmn - x ∥ λI- converges to 0. Conversely, suppose now that for each λ ∈ (0, 1), ∥xij - x ∥ λI-converges to 0. This means that for each λ ∈ (0, 1) and each ∈ > 0 there is a set Pλ ∈ I such that, for each (i, j) ∈ P
Therefore,
implies ν (T (xij - x) , ∈) ≤1 - λ for each λ ∈ (0, 1) and each (i, j) ∈ P, which means
that is, (xij) is Iν-convergent to x as i, j → ∞.■
Theorem 2.2.Let be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄ satisfying (FaN6). Then fuzzy Iλ-anti-limit of a fuzzy Iλ-anti-convergent sequence is unique.
Proof. Let be fuzzy Iλ- convergent double sequence converging two distinct points x and y in . This means that for each t > 0, there exist x, y ∈ X and such that
The set and by the assumption on ⋄ for each (i, j) ∈ A, we have
So we have
Thus, the sets on right hand side of the above equation (21) belong to . Therefore, ν (T (x - y) , t) <1 - λ for each t > 0. Since T is a compact operator, by (FaN6) one obtains x - y = θ i.e., x = y.■
Theorem 2.3.Let and be a fuzzy antinormed double sequence spaces with respect to an idempotent t-conorm ⋄ satisfying (FaN6). Then
if Iλ - anti - lim xij = x and Iλ - anti - lim yij = y, then Iλ - anti - lim(xij + yij) = x + y
if Iλ - anti - lim xij = x and , then Iλ - anti - lim rxij = rx.
Proof. Since Iλ - anti - lim xij = x and Iλ - anti - lim yij = y, there exist such that for all t > 0 we have
The set and by the assumption on ⋄ for each (i, j) ∈ M, we have
So we have
Thus, the sets on right hand side of the above equation (24) belong . So we have M = {(i, j) : ν (T (xij + yij) - (x + y) , t) <1 - λ} ∉ I which means that Iλ - anti - lim(xij + yij) = x + y.
(2) The fact Iλ - anti - lim xij = x implies that there exists such that for all t > 0 we have
Therefore, for each (i, j) ∈ M, we have
We have
From this we get, {(i, j) : ν (rT (xij) - rT (x) , t) ≥1 - λ} ∉ I which shows that Iλ - anti - lim rxij= rx.■
Theorem 2.4.Let be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄. If is Iλ-anticonvergent to , then ∥xij - x ∥ λ is I-convergent to 0.
Theorem 2.5.Let be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄ satisfying (FaN6) and λ ∈ (0, 1). Then every Iλ-anti-convergent double sequence is fuzzy Iλ-anti-Cauchy.
Proof. Let be fuzzy Iλ- anti-convergent double sequence. This shows that there exists such that for all t > 0 we have
Therefore for each (i, j) , (m, n) ∈ M, we have
which means that (xij) is fuzzy Iλ-anti-quasi- Cauchy in .■
Theorem 2.6.Let be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄. Then I-Cauchy sequence (xij) in , is fuzzy Iλ-anti-quasi-Cauchy in .
Theorem 2.7.Let be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄. If is fuzzy Iλ-anti-complete, then is I- complete with respect to ∥. ∥ λ, λ ∈ (0, 1).
Proof. Let (xij) be fuzzy Iλ- anti-Cauchy sequence in . As is fuzzy Iλ-anti-complete then fuzzy Iλ- anti-Cauchy sequence (xij) is fuzzy Iλ- anti-convergent to x (say). By Theorem (2.4), this means that ∥T (xij - x) ∥ λ is convergent to 0; i.e., as T is a compact operator, (xij) is Iλ- is convergent to 0. Hence is Iλ-complete with respect to ∥. ∥ λ, λ ∈ (0, 1). Therefore is I- complete.■
Conclusion
The concept of fuzzy sets is well established as an important and practical construct for modeling. In the present paper, we have studied the concept of fuzzy antinormed double sequence spaces with the help of an ideal and defined by a compact linear operator and studied the fuzzy topology on the said spaces.
Authors contributions
All authors of the manuscript have read and agreed to its content and are accountable for all aspects of the accuracy and integrity of the manuscript.
Author details
Vakeel A. Khan received the M.Phil. and Ph.D. degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently he is a Associate Professor at Aligarh Muslim University, Aligarh, India.
Hira Fatima received B.Sc and M.Sc. degrees from Aligarh Muslim University, and is currently a Ph.D. scholar at Aligarh Muslim University, Aligarh,India.
Ayaz Ahmad is working as an Assistant Professor in the Department of Mathematics, National Institute of Technology, Patna, India.
Mohd. Imran is a Ph.D. scholar at Aligarh Muslim University, Aligarh, India.
Competing interests
The authors declare that they have no competing interests.
Footnotes
Acknowledgment
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.
References
1.
BagT., SamantaS.K., Finite dimensional fuzzy normed linear spaces, Journal of Fuzzy Mathematics11(3) (2003), 687–705.
2.
BagT., SamantaS.K., A comparative study of fuzzy norms on a linear space, Fuzzy Sets and Systems159(6) (2008), 670–684.
3.
ChengS.C., MordesonJ.N., Fuzzy linear operators and fuzzy normed linear spaces, Bulletin of the Calcutta Mathematical Society86(5) (1994), 429–436.
4.
DasP., KostyrkoP., WilczynskiW., MalikP., I and I*- convergence of double sequences, Mathematica Slovaca58 (2008), 605–620.
5.
Di MaioG. and KočinacL.J.D.R., Statistical convergence in topology, Topology and its Applications156 (2008), 28–45.
6.
FastH., Sur la convergence statistique, Colloquium Mathematicum2 (1951), 241–244.
7.
FelbinC., The completion of a fuzzy normed linear space, Journal of Mathematical Analysis and Applications174(2) (1993), 428–440.
8.
JebrilI.H., SamantaT.K., Fuzzy anti-normed linear space, Journal of Mathematics and Technology (2010), 66–77.
KhanV.A., Yasmeen, EsiA. and FatimaH., Intuitionistic fuzzy I-convergent double sequence spaces defined by compact operator and modulus function, Journal of Intelligent and Fuzzy Systems33(6) (2017), 3905–3911.
11.
KhanV.A., ShafiqM., Lafuerza-GuillenB., On paranorm}-convergent sequence spaces defined by a compact operator, Afrika Matematika25(4) (2014), 12. DOI 10.1007/s13370-014-0287-2.
12.
KočinacL.J.D.R., Some topological properties of fuzzy antinormed linear spaces, Journal of Mathematics2018, 6. Article ID 9758415. DOI 10.1155/2018-9758415.
13.
KočinacL.J.D.R. and RashidM., On ideal convergence of double sequencea in the topology induced by a fuzzy 2-norm, TWMS Journal of Pure and Applied Mathematics8(1) (2017), 97–111.
14.
KostyrkoP., SalatT., WilczynskiW., I-convergence, Real Analysis Exchange26(2) (2000), 669–686.
15.
MohiuddineS.A., Alotaibi and AlsulamiS.M., Ideal convergence of double sequences in random 2-normd spaces, Advances in Difference Equations2012 (2012), 8. article 149.
16.
MursaleenM., EdelyO.H.H., Statistical convergence of double sequences, Journal of Mathematical Analysis and Applications288 (2003), 223–231.
17.
MursaleenM., MohiuddineS.A., On ideal convergence of double sequences in probabilistic normed spaces, Mathematical Reports12(62) (2010), 359–371.
18.
RashidM., KočinacL.JD.R., Ideal convergence in 2-fuzzy 2-normed spaces, Hacettepe Journal of Mathematics and Statistics46(1) (2017), 149–162.
19.
SasdatiR., VaezpourS.M., Some results of fuzzy Banach spaces, Journal of Applied Mathematics and Computing17 (2005), 475–484.
20.
SchoenbergI.J., The integrability of certain functions and related summability methods, American Mathematical Monthly66 (1959), 361–375.
21.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 338–353.