Some fuzzy anti λ -ideal convergent double sequence spaces

1 Introduction and preliminaries

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 ⊂ 2 ^X 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 F ⊂ 2 X is said to be a filter on X if for all A , B ∈ F implies A ∩ B ∈ F and for all A ∈ F with A ⊆ B implies B ∈ F . An ideal I ⊂ 2 ^X is said to be non trivial if I ≠ 2 ^X ; 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 F ( I ) called the filter associate with ideal I, that is F ( I ) = { K ⊆ X : K c ∈ I } , where K c = X ∖ K . Throughout the article, I is an admissible ideal on ℕ × ℕ , and ₂ω denotes the class of all double real sequences. The spaces ₂l_∞, ₂c and₂c₀ are the Banach spaces of bounded, convergent, and null double sequences of reals respectively with the norm ∥ x ∥ = sup i , j ∈ ℕ | x ij | .(1)

Definition 1.1. [16, 17] A double sequence x = (x_ij)∈2ω is said to be I-convergent to a number L if for every ∈ > 0 there are m , n ∈ ℕ , such that { ( i , j ) : | x ij - L | ≥ ∈ , i ≤ m , j ≤ n } ∈ I .(2) In this case, we write I - lim x _ij  = L.

Definition 1.2. [16, 17] A double sequence (x_{ij )∈2}ω is said to be I-Cauchy if for every ∈ > 0 there exist numbers m = m ( ∈ ) , n = n ( ∈ ) ∈ ℕ such that for all i, p ≥ m and j, q ≥ n { ( i , j ) : | x ij - x pq | ≥ ∈ , i ≤ m , j ≤ n } ∈ I .(3)

Definition 1.3. [16, 17] A double sequence (x_{ij )∈2}ω is said to be I-bounded if there exists M > 0 such that { ( i , j ) : | x ij | > M } ∈ I .(4)

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,

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.

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 ν : X × ℝ → ℝ of X × ℝ 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 ;

Example 1.1. Let (X, ∥. ∥) be a normed linear space and let the t-conorm ⋄ be given by a ⋄ b = a + b - ab. Define ν : X × ℝ → [ 0 , 1 ] by ν ( x , t ) = { 0 , if t > ∥ x ∥ 1 , if t ≤ ∥ x ∥ . Then ν is a fuzzy antinormon X with respect to the t-conorm ⋄. This antinorm ν satisfies also the following: ( FaN 6 ) ν ( x , t ) < 1 for each t > 0 implies x = θ .

Example 1.2. Let (X, ∥. ∥) be a normed linear space and consider the t-conorm ⋄ defined by a ⋄ b = min {a + b, 1}. Define ν : X × ℝ → [ 0 , 1 ] by ν ( x , t ) = { ∥ x ∥ 2 t - ∥ x ∥ , if t > ∥ x ∥ 1 , if t ≤ ∥ x ∥ . 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 ℝ and strictly decreasing on the subset {t : 0 < ν (x, t) <1} of ℝ .

Definition 1.6. [12] A sequence ( x n ) n ∈ ℕ 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 n 0 ∈ ℕ such that ν ( x n - x , t ) < ∈ for each n ≥ n 0(5)

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 ∥ x ∥ λ = { t > 0 : ν ( x , t ) ≤ 1 - λ }(6) 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.

Now, in this section we define fuzzy I _λ -convergence, fuzzy I _λ - anti-convergence, fuzzy I _λ - anti-Cauchy and fuzzy I _λ - compactness for double sequences with respect to an ideal I on ℕ × ℕ .

Definition 2.1. A double sequence (x _ij ) in a fuzzy antinormed linear space X is said to be fuzzy I _ν -convergent to a point x ∈ X if for each ∈ > 0 and each t > 0 the set { ( i , j ) : ν ( x ij - x , t ) < ∈ } ∈ I .(7) In this case, we write fuzzy I _ν  - lim x _ij  = x and x is called a fuzzy I _ν -limit of (x _ij ).

Definition 2.2. Let X be a fuzzy antinormed double sequence space and λ ∈ (0, 1). A sequence (x _ij ) ∈ X is said to be fuzzy I _λ -convergent to x ∈ X if for all t > 0, the set { ( i , j ) : ν ( x ij - x , t ) < 1 - λ } ∈ I .(8) In this case, we write fuzzy I _λ  - lim ν (x _ij  - x, t) =0 and x is called a fuzzy I _λ -limit of (x _ij ).

Definition 2.3. Let X be a fuzzy antinormed double sequence space and λ ∈ (0, 1). A sequence (x _ij ) ∈ X is said to be fuzzy I _λ -anti-convergent in X if there exist x ∈ X and M ∈ F ( I ) such that for all t > 0, M = { ( i , j ) : ν ( x ij - x , t ) < 1 - λ } .(9) In this case, we write (x _ij ) → x and x is called a fuzzy I _λ -anti-limit of (x _ij ).

Definition 2.4. Let λ ∈ (0, 1). A sequence (x _ij ) in a fuzzy antinormed double sequence space X is said to be fuzzy I _λ -anti-Cauchy if there exist numbers m , n ∈ ℕ and M ∈ F ( I ) such that for all i, p ≥ m, j, q ≥ n and all t > 0, M = { ( i , j ) : ν ( x ij - x pq , t ) < 1 - λ } .(10)

Definition 2.5. A fuzzy antinormed double sequence space X is said to be fuzzy I _λ -anti-complete, λ ∈ (0, 1), if for every fuzzy I _λ -anti-Cauchy sequence in X is fuzzy I _λ - anti-convergent in X.

Now, we define two fuzzy antinormed double sequence spaces with the help of a compact operator T:

2 X ν I ( T ) = { ( x ij ) ∈ 2 ℓ ∞ : ( i , j ) : ν ( T ( x ij - x ) , t ) < ∈ } ;(11) 2 X 0 ν I ( T ) = { ( x ij ) ∈ 2 ℓ ∞ : ( i , j ) : ν ( T ( x ij ) , t ) < ∈ } .(12) 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:

2 B x ( r , t ) ( T ) = { ( y i j ) ∈ 2 ℓ ∞ : v ( T ( x i j ) − T ( y i j ) , t ) < r } .(13)

Theorem 2.1. In the fuzzy antinormed linear double sequence space 2 X ν I ( T ) with respect to an idempotent t-conorm ⋄ satisfying (FaN6) and (FaN7) a sequence is I _ν -convergent if and only if it is I _λ -convergent for each λ ∈ (0, 1).

Proof. Let (x _ij ) be a sequence in 2 X ν I ( T ) such that (x _ij ) is I _ν - convergent to x, i.e., for each t > 0 I ν - lim i , j → ∞ ν ( T ( x ij - x ) , t ) = 0 .(14) Fix λ ∈ (0, 1). So, I ν - lim i , j → ∞ ν ( T ( x ij - x ) , t ) = 0 < 1 - λ . There exists a set P ∈ I such that for each (m, n) ∈ P, ν ( T ( x mn - x ) , t ) < 1 - λ(15) Since ∥T (x _mn ) - T (x) ∥  _λ  = v {t > 0 : ν (T (x _mn  - T (x) , t) <1 - λ}, we have ∥T (x _mn ) - 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 ∥x _mn  - x ∥  _λ I- converges to 0. Conversely, suppose now that for each λ ∈ (0, 1), ∥x _ij  - x ∥  _λ I-converges to 0. This means that for each λ ∈ (0, 1) and each ∈ > 0 there is a set P _λ  ∈ I such that, for each (i, j) ∈ P ∥ T ( x ij - x ) ∥ λ ≤ ∈ .(16) Therefore, ν ( T ( x ij - x ) , ∈ ) = v { 1 - λ : ∥ T ( x ij - x ) ∥ λ ≤ ∈ }(17) implies ν (T (x _ij  - x) , ∈) ≤1 - λ for each λ ∈ (0, 1) and each (i, j) ∈ P, which means I ν - lim ν ( T ( x ij - x ) , ∈ ) = 0(18) that is, (x _ij ) is I _ν -convergent to x as i, j → ∞.■

Theorem 2.2. Let 2 X ν I ( T ) be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄ satisfying (FaN6). Then fuzzy I _λ -anti-limit of a fuzzy I _λ -anti-convergent sequence is unique.

Proof. Let ( x ij ) ∈ 2 X ν I ( T ) be fuzzy I _λ - convergent double sequence converging two distinct points x and y in 2 X ν I ( T ) . This means that for each t > 0, there exist x, y ∈ X and A 1 , A 2 ∈ F ( I ) such that

A 1 = { ( i , j ) : ν ( T ( x ij - x ) , t ) < 1 - λ } ∈ F ( I ) ;(19) A 2 = { ( i , j ) : ν ( T ( x ij - y ) , t ) < 1 - λ } ∈ F ( I ) .(20) The set A = A 1 ∩ A 2 ∈ F ( I ) and by the assumption on ⋄ for each (i, j) ∈ A, we have ν ( x - y , t ) ≤ ν ( T ( x ij - x ) , t ) ⋄ ν ( T ( x ij - y ) , t ) < ( 1 - λ ) ⋄ ( 1 - λ ) = ( 1 - λ ) . So we have

{ ( i , j ) : ν ( T ( x - y ) , t ) < 1 - λ } ⊇ { ( i , j ) : ν ( T ( x ij - x ) , t ) < 1 - λ } ∩ { ( i , j ) : ν ( T ( x ij - y ) , t ) < 1 - λ }(21) Thus, the sets on right hand side of the above equation (21) belong to F ( I ) . 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 2 X ν I ( T ) and 2 X 0 ν I ( T ) be a fuzzy antinormed double sequence spaces with respect to an idempotent t-conorm ⋄ satisfying (FaN6). Then

if I _λ  - anti - lim x _ij  = x and I _λ  - anti - lim y _ij  = y, then I _λ  - anti - lim(x _ij  + y _ij ) = x + y

Proof. Since I _λ  - anti - lim x _ij  = x and I _λ  - anti - lim y _ij  = y, there exist M 1 , M 2 ∈ F ( I ) such that for all t > 0 we have

M 1 = { ( i , j ) : ν ( T ( x ij - x ) , t 2 ) < 1 - λ } ∈ F ( I ) ;(22) M 2 = { ( i , j ) : ν ( T ( y ij - y ) , t 2 ) < 1 - λ } ∈ F ( I ) .(23) The set M = M 1 ∩ M 2 ∈ F ( I ) and by the assumption on ⋄ for each (i, j) ∈ M, we have ν ( T ( x ij + y ij ) - T ( x + y ) , t ) ≤ ν ( T ( x ij - x ) , t 2 ) ⋄ ν ( T ( y ij - y ) , t 2 ) < ( 1 - λ ) ⋄ ( 1 - λ ) = ( 1 - λ ) . So we have

{ ( i , j ) : ν ( T ( x ij + y ij ) - ( x + y ) , t ) < 1 - λ } ⊇ { ( i , j ) : ν ( T ( x ij - x ) , t 2 ) < 1 - λ } ∩ { ( i , j ) : ν ( T ( x ij - y ) , t 2 ) < 1 - λ }(24) Thus, the sets on right hand side of the above equation (24) belong F ( I ) . So we have M = {(i, j) : ν (T (x _ij  + y _ij ) - (x + y) , t) <1 - λ} ∉ I which means that I _λ  - anti - lim(x _ij  + y _ij ) = x + y.

(2) The fact I _λ  - anti - lim x _ij  = x implies that there exists M ∈ F ( I ) such that for all t > 0 we have

M = { ( i , j ) : ν ( T ( x ij - x ) , t ) < 1 - λ } ∈ F ( I ) .(25)

Therefore, for each (i, j) ∈ M, we have ν ( rT ( x ij ) - rT ( x ) , t ) = ν ( T ( x ij - x ) , t | r | ) < 1 - λ . We have

{ ( i , j ) : ν ( rT ( x ij ) - rT ( x ) , t ) < 1 - λ } ⊇ { ( i , j ) : ν ( T ( x ij ) - T ( x ) , t ) < 1 - λ }(26) From this we get, {(i, j) : ν (rT (x _ij ) - rT (x) , t) ≥1 - λ} ∉ I which shows that I _λ  - anti - lim  rx _ij = rx.■

Theorem 2.4. Let 2 X ν I ( T ) be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄. If ( x ij ) ∈ 2 X ν I ( T ) is I _λ -anticonvergent to x ∈ 2 X ν I ( T ) , then ∥x _ij  - x ∥  _λ is I-convergent to 0.

Theorem 2.5. Let 2 X ν I ( T ) be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄ satisfying (FaN6) and λ ∈ (0, 1). Then every I _λ -anti-convergent double sequence ( x ij ) ∈ 2 X ν I ( T ) is fuzzy I _λ -anti-Cauchy.

Proof. Let ( x ij ) ∈ 2 X ν I ( T ) be fuzzy I _λ - anti-convergent double sequence. This shows that there exists M ∈ F ( I ) such that for all t > 0 we have { ( i , j ) : ν ( T ( x ij - x ) , t 2 ) < 1 - λ } ∈ F ( I ) .(27) Therefore for each (i, j) , (m, n) ∈ M, we have ν ( T ( x ij ) - T ( x mn ) , t ) ≤ ν ( T ( x ij - x ) , t 2 ) ⋄ ν ( T ( x ij - x ) , t 2 ) . < ( 1 - λ ) ⋄ ( 1 - λ ) = ( 1 - λ ) which means that (x _ij ) is fuzzy I _λ -anti-quasi- Cauchy in 2 X ν I ( T ) .■

Theorem 2.6. Let ( 2 X ν I ( T ) , v ) be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄. Then I-Cauchy sequence (x _ij ) in ( 2 X ν I ( T ) , ∥ . ∥ λ ) , λ ∈ ( 0 , 1 ) , is fuzzy I _λ -anti-quasi-Cauchy in ( 2 X ν I ( T ) , v ) .

Theorem 2.7. Let 2 X ν I ( T ) be a fuzzy antinormed double sequence space with respect to an idempotent t-conorm ⋄. If 2 X ν I ( T ) is fuzzy I _λ -anti-complete, then 2 X ν I ( T ) is I- complete with respect to ∥. ∥  _λ , λ ∈ (0, 1).

Proof. Let (x _ij ) be fuzzy I _λ - anti-Cauchy sequence in 2 X ν I ( T ) . As 2 X ν I ( T ) is fuzzy I _λ -anti-complete then fuzzy I _λ - anti-Cauchy sequence (x _ij ) is fuzzy I _λ - anti-convergent to x (say). By Theorem (2.4), this means that ∥T (x _ij  - x) ∥  _λ is convergent to 0; i.e., as T is a compact operator, (x _ij ) is I _λ - is convergent to 0. Hence 2 X ν I ( T ) is I _λ -complete with respect to ∥. ∥  _λ ,  λ ∈ (0, 1). Therefore ( 2 X ν I ( T ) , ∥ . ∥ λ ) is I- complete.■

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.