On fuzzy valued lacunary ideal convergent sequence spaces defined by a compact operator

1 Introduction

The fuzzy theory was introduced by Zadeh [15] and since then it is became an area of active research for the last fifty years. Whereas the concept of ordinary convergence of a sequence of fuzzy real numbers was firstly introduced by Matloka [9] and some basic properties of these sequences were studied by Nanda [10]. He also showed that the set of all convergent sequences of fuzzy real numbers form a complete metric space. Later, the notion of convergence of the sequence of fuzzy real numbers has been extended to the notion of statistical convergence by Nuray and Savas [11]. The theory of I -convergence of the real valued sequence was introduced by Kostyrko et al. [6] as a generalization of the notion of statistical convergence. Some notions and results from the statistical convergence were extended to the I - convergence by Šalát et al. [12]. Kumar and Kumar [8] extended the idea of ideal convergence to apply on sequences of fuzzy numbers. Afterward, Hazarika [2] has introduced and studied the concept of lacunary I - convergent sequence of fuzzy real numbers. Where a lacunary sequence is an increasing integer sequence θ = (k _r ) such that k₀ = 0 and h _r = k _r - k_r-1→ ∞ as r→ ∞. The intervals are determined by θ and it will be defined by J _r = (k_r-1, k _r ]. (see [1]).

Recall in [7] let X and Y be two normed linear spaces. An operator T: X → Y is said to be a compact linear operator (or completely continuous linear operator) if T is linear and T maps every bounded sequence (x _k ) in X onto a sequence T (x _k ) in Y which has a convergent subsequence. The set of all compact linear operators C ( X , Y ) is closed subspace of B ( X , Y ) (the class of bounded linear operator) and it is Banach space if Y is Banach space. Recently, a compact operator was used to define some new real sequence spaces with the help of the notion of I - convergence by Khan et al. [3–5]. In this paper, we keep the same direction up, in order to prove that a compact operator can be also used to define spaces of lacunary I –convergent sequence of fuzzy real numbers. Further, we study some topological and algebraic properties of these spaces.

2 Definitions and Preliminaries

A fuzzy real number X is a fuzzy set on ℝ i.e, a mapping X : ℝ ⟶ γ ( = [ 0 , 1 ] ) associating each real number t with its grade of membership X (t). A fuzzy number is bounded, convex and normal. Let D denotes the set of all closed and bounded intervals X = [x₁, x₂] on the real line ℝ . For X, Y ∈ D, we define X ≤ Y if and only if x₁ ≤ y₁ and x₂ ≤ y₂. Define a metric d on D by d ( X , Y ) = max { | x 1 - y 1 | , | x 2 - y 2 | } , where Y = [ y 1 , y 2 ] .

It is known that (D, d) is a complete metric space and the relation ‘≤’ is a partial order on D. Let ℝ (γ) denotes the set of all fuzzy numbers. The absolute value |X| of X ∈ ℝ ( γ ) is defined as | X | ( t ) = { max { X ( t ) , X ( - t ) } , if t > 0 0 , if t < 0 . Let d ¯ : ℝ ( γ ) × ℝ ( γ ) ⟶ ℝ be defined by d ¯ ( X , Y ) = sup 0 ≤ α ≤ 1 d ( X , Y ) , for any α ∈ γ . Then d ¯ defines a metric on ℝ(γ). The additive identity and multiplicative identity in ℝ(γ) are denoted by 0 ¯ and 1 ¯ , respectively.

Definition 2.1. [2] A sequence (X _k ) of fuzzy real numbers is said to be I-Cauchy if there exists a subsequence (X _{N ϵ} ) of a sequence (X _k ) such that for each ϵ > 0, the set { k ∈ ℕ : d ¯ ( X k , X N ϵ ) ≥ ϵ } ∈ I .

Definition 2.2. [8] A sequence (X _k ) of fuzzy real numbers is said to be I - convergent if there exists a fuzzy real numbers X₀ such that for each ϵ > 0, the set { k ∈ ℕ : d ¯ ( X k , X 0 ) ≥ ϵ } ∈ I .

Definition 2.3. [8] A sequence (X _k ) of fuzzy real numbers is said to be I - bounded if there exists a fuzzy real number M > 0 such that, the set { k ∈ ℕ : d ¯ ( X k , 0 ¯ ) > M } ∈ I .

Definition 2.4. [6] A family of sets I ⊆ 2 ℕ is called an ideal if and only if

∅ ∈ I ,

Definition 2.5. [6] A family of sets F ⊆ 2 ℕ is called a filter in ℕ if and only if

∅ ∉ F ,

Definition 2.6. [8] A sequence (X _k ) of fuzzy real numbers is said to be I - null if there exists a fuzzy real numbers X₀ such that for each ϵ > 0, the set { k ∈ ℕ : d ¯ ( X k , 0 ¯ ) ≥ ϵ } ∈ I .

Definition 2.7. [12] Let K = { k i ∈ ℕ : k 1 < k 2 < … } ⊆ ℕ and E be a sequence space. A K–step space of E is a sequence space λ K E = { ( x k i ) ∈ ω : ( x k ) ∈ E } . A canonical pre–image of a sequence ( x k i ) ∈ λ K E is a sequence (y _k ) ∈ ω defined as follows: y k = { x k , if k ∈ K 0 , otherwise . A canonical pre-image of a step space λ K E is a set of canonical pre-images of all elements in λ K E . i.e., y is in canonical pre-image of λ K E iff y is canonical pre-image of some element x ∈ λ K E .

Definition 2.8. [12] A sequence space E is said to be monotone, if it is contains the canonical pre-images of it is step space. (i.e., if for all infinite K ⊆ ℕ and (x _k ) ∈ E the sequence (α _k x _k ), where α _k = 1 for k ∈ K and α _k = 0 otherwise, belongs to E).

Definition 2.9. [12] A sequence space E is said to be convergence free, if (x _k ) ∈ E whenever (y _k ) ∈ E and (y _k ) =0 implies that (x _k ) =0 for all k ∈ ℕ .

Lemma 2.1. [12] Every solid space is monotone.

Lemma 2.2. [12] If I ⊂ 2 ℕ is a maximal ideal, then for each A ⊂ ℕ we have either A ∈ I or ℕ ∖ A ∈ I .

3 Main Results

In this section, before we state our main results, by ω ^F we denote the class of all sequences of fuzzy real numbers and I be an admissible ideal of subset of the natural numbers ℕ . Also, we will consider T as a compact operator on the real space ℝ . Now, for ϵ > 0, we introduce the sequence spaces F C I θ ( T ) = { X = ( X k ) ∈ ω F : { r ∈ ℕ : h r − 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 0 ) ≥ ϵ } ∈ ℐ for some X 0 ∈ R ( γ ) } , F C 0 I θ ( T ) = { X = ( X k ) ∈ ω F : { r ∈ ℕ : h r − 1 ∑ k ∈ J r d ¯ ( T ( X k ) , 0 ¯ ) ≥ ϵ } ∈ ℐ } , F ℓ ∞ I θ ( T ) = { X = ( X k ) ∈ ω F : ∃ M > 0 such that { r ∈ ℕ : h r − 1 ∑ k ∈ J r d ¯ ( T ( X k ) , 0 ¯ ) ≥ M } ∈ ℐ }, F ℓ ∞ I θ ( T ) = { X = ( X k ) ∈ ω F : sup r h r − 1 ∑ k ∈ J r d ¯ ( T ( X k ) , 0 ¯ ) < ∞ } .

We write F m I θ ( T ) = F C I θ ( T ) ∩ F ℓ ∞ θ ( T ) F m 0 I θ ( T ) = F C 0 I θ ( T ) ∩ F ℓ ∞ θ ( T ) .

Example 3.1. Let I = I d = { A ⊆ N : d ( A ) = 0 } , where d (A) denotes the asymptotic density of A. In this case c F I θ ( T ) = S F θ ( T ), where F S θ ( T ) is the space of all lacunary statistically convergent sequence of fuzzy real numbers we define by a compact operator as follows: F S θ ( T ) = { X k ∈ ω F : d ( { r ∈ ℕ : h r − 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 0 ) ≥ ϵ } ) = 0 for some X 0 ∈ R ( γ ) } .

Example 3.2. Let I = I f = { A ⊆ N : A is finite } . I f is an admissible ideal in ℕ and c F I θ ( T ) = c F θ ( T ) (The space of all lacunary convergent sequence of fuzzy real numbers which is defined by a compact operator).

Proposition 3.1. The sequence spaces c F I θ ( T ) and F c 0 I θ ( T ) are not convergent free.

Proof. For this result consider the following example.

Example 3.3. Let I = I δ and T (X _k ) = X _k .

Consider the sequence X k ∈ F c 0 I θ ( T ) ⊂ F c I θ ( T ) as follows.

For k ≠ i 2 , i ∈ ℕ X k ( t ) = { 1 , if 0 ≤ t ≤ k - 1 0 , otherwise . and for k = i 2 , i ∈ ℕ , X k ( t ) = 0 ¯ . Then we have X k = { [ 0 , 0 ] , if k 2 = i [ 0 , k - 1 ] , k ≠ i 2 . Hence, X k → 0 ¯ as k → ∞ . Thus X k ∈ F c 0 I θ ( T ) ⊂ F c I θ ( T ) . Let Y _k be another sequence such that, for k = i 2 , i ∈ ℕ , X k ( t ) = 0 ¯ . Then we have Y k ( t ) = { 1 , if 0 ≤ t ≤ k 0 , otherwise . and for k = i 2 , i ∈ ℕ , X k ( t ) = 0 ¯ . Then we have Y k = { [ 0 , 0 ] , if k 2 = i [ 0 , k ] , k ≠ i 2 . This implies that, Y k ∉ F c 0 I θ ( T ) ⊂ F c I θ ( T ) .

Hence c F I θ ( T ) , c F 0 I θ ( T ) are not convergence free.

Remark 3.1. If I = I f then the sequence spaces c F I θ ( T ) , c F 0 I θ ( T ) , ℓ F 0 I θ ( T ) coincide with the sequence spaces _F c, _F c₀ and _F ℓ _∞, respectively which were introduced in [13, 14].

Theorem 3.1. The space c F I θ ( T ) , c F 0 I θ ( T ) and ℓ F ∞ I θ ( T ) are linear spaces.

Proof. Let X = (X _k ), Y = (Y _k )∈ _F c ^{I _θ} (T) and let α, β be scalars. Since X _k , Y _{k ∈ F} c ^{I _θ} (T) for a given ϵ > 0,there exists X 1 , X 2 ∈ ℂ , such that { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 1 ) ≥ ϵ 2 } ∈ I . and { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 2 ) ≥ ϵ 2 } ∈ I . Now, let A 1 = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 1 ) < t 2 | α | } ∈ F ( I ) . A 2 = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 2 ) < t 2 | β | } ∈ F ( I ) . be such that A 1 c , A 2 c ∈ I . Therefore, the set A 3 = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( α ( X k ) + β ( Y k ) ) , ( α L 1 + β L 2 ) ) < ϵ } ⊇ { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , L 1 ) < ϵ 2 | α | } ∩ { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , L 2 ) < ϵ 2 | β | } .(1) Thus, the sets on right hand side of above equation(1) belong to F ( I ) . By definition of filter associated with ideal, the complement of the set on left hand side of above equation(1) belongs to I . This implies that α (X _k ) + β (Y _k )∈ _F c ^{I _θ} (T). Hence c F I θ ( T ) is linear space. The rest of the results can be proved similarly.

Theorem 3.2. A sequence X = ( X k ) ∈ ℓ F ∞ θ ( T ) is I - converges if and only if for every ϵ > 0, there exists N ϵ ∈ ℕ such that { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , T ( X N ϵ ) ) < ϵ } ∈ F ( I ) .(2)

Proof. Let X = ( X k ) ∈ F ℓ ∞ θ ( T ) . Suppose that X 0 = I - lim X . Then for all ϵ > 0, the set B ϵ = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 0 ) < ϵ 2 } ∈ F ( I ) .

Fix an N _ϵ ∈ B _ϵ . Then, we have d ¯ ( T ( X k ) , T ( X N ϵ ) ) ≤ d ¯ ( T ( X k ) , X 0 )

+ d ¯ ( T ( X N ϵ ) , X 0 ) < ϵ 2 + ϵ 2 = ϵ .

Which holds for all N _ϵ ∈ B _ϵ . Hence { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , T ( X N ϵ ) ) < ϵ } ∈ F ( I ) .

Conversely, suppose that { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , T ( X N ϵ ) ) < ϵ } ∈ F ( I ) .

Then, the set C ϵ = { k ∈ ℕ : T ( X k ) ∈ [ T ( X N ϵ ) - ϵ , T ( X N ϵ ) + ϵ ] } ∈ F ( I ) forall ϵ > 0.

Let J _ϵ = [T (X _{N ϵ} ) - ϵ, T (X _{N ϵ} ) + ϵ]. If we fix an ϵ > 0, then we have C ϵ ∈ F ( I ) as well as C ϵ 2 ∈ F ( I ) . Hence C ϵ 2 ∩ C ϵ ∈ F ( I ) . This implies that J = J ϵ ∩ J ϵ 2 ≠ ∅ . That is { k ∈ ℕ : T ( X k ) ∈ J } ∈ F ( I ) . That is diam J ≤ 1 2 diam J ϵ . Where the diam of J denotes the length of interval J. continuing in this way, by induction, we get the sequence of closed intervals. J ϵ = I 0 ⊇ I 1 ⊇ … ⊇ I k ⊇ … With the property that diam I k ≤ 1 2 diam I k - 1 for ( k = 2 , 3 , 4 , … ) and { k ∈ ℕ : T ( X k ) ∈ I k } ∈ F ( I ) for ( k = 2 , 3 , 4 , … ) . Then there exists a ξ ∈ ∩ I k where k ∈ ℕ such that ξ = I - lim T ( X k ) . Hence the result holds.

Theorem 3.3. The inclusions c F 0 I θ ( T ) ⊂ c F I θ ( T ) ⊂ ℓ F ∞ I θ ( T ) hold.

Proof. The inclusions c F 0 I θ ( T ) ⊂ c F I θ ( T ) is obvious. Let X = (X _k )∈ _F c ^{I _θ} (T). Then there exists a number X 0 ∈ ℝ such that ℐ θ − lim d ¯ ( T ( X k ) , X 0 ) = 0 That is, the set { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 0 ) ≥ ϵ } ∈ I . Here h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , 0 ¯ ) = h r - 1 ∑ k ∈ I r d ¯ ( T ( X k ) , - X 0 + X 0 ) ≤ h r - 1 ∑ k ∈ I r d ¯ ( T ( X k ) , - X 0 ) + h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 0 ) . From this it easily follows that the sequence (X_k) must belong to ℓ F I θ ( T ). Hence c F 0 I θ ( T ) ⊂ c F I θ ( T ) ⊂ ℓ F ∞ I θ ( T ).

Theorem 3.4. If I is not maximal ideal then the space is neither normal nor monotone.

Example 3.4. Consider a sequence of fuzzy number Y k ( t ) = { 1 + t 2 , if - 1 ≤ t ≤ 1 3 - t 2 , if 1 ≤ t ≤ 3 0 , otherwise . Then Y _k (t)∈ _F c ^{I _θ} (T). Since I is not maximal, by Lemma 2.2 there exists a subset K of ℕ such that K ∉ I and ℕ \ K ∉ I . Let us define a sequence Y = (Y _k ) by Y k = { X k , k ∈ K 0 , otherwise . Then (Y _k ) belong to the canonical pre- image of the K- step spaces of. But (Y _k )∉ _F c ^{I _θ} (T). Hence is not monotone. Therefore by Lemma 2.1 is not normal.

Theorem 3.6. The space c F 0 I θ ( T ) is solid and monotone.

Proof. Let X = ( X k ) ∈ F c 0 I θ ( T ) , for ϵ > 0, the set { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 0 ) ≥ ϵ } ∈ I .(3) Let (α _k ) be a sequence of scalars with |α _k |≤1 for all k ∈ ℕ . Therefore d ¯ ( T ( α k X k ) , X 0 ) = d ¯ ( α k T ( X k ) , X 0 )

≤ | α k | d ¯ ( T ( X k ) , X 0 ) for all k ∈ ℕ . Thus, from the above inequality and equation (3), we have { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( α k X k ) , X 0 ) ≥ ϵ } ⊆ { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k ) , X 0 ) ≥ ϵ } ∈ I . This implies that { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( α k X k ) , X 0 ) ≥ ϵ } ∈ I . Therefore, ( α k X k ) ∈ F c 0 I θ ( T ) . Hence, the space is solid and hence by Lemma 2.1, we get is monotone.

Theorem 3.7. The set c F I θ ( T ) is closed subspace of F ℓ ∞ θ ( T ) .

Proof. Let ( X k p ) be a Cauchy sequence in c F I θ ( T ). Then it have X k p → X in ℓ F ∞ θ ( T ) as p→ ∞, since ( X k p ) ∈ F c I θ ( T ) , then for each ϵ > 0 there exists a _p such that converges to a (say). I - lim X = a . Since ( X k p ) be a Cauchy sequence in. Then for each ϵ > 0 there exists n 0 ∈ ℕ , such that sup k d ¯ ( T ( X k p ) , T ( X k s ) ) < ϵ 3 for all p , s ≥ n 0 . That is for a given ϵ > 0, it has B p , s = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k p ) , T ( X k s ) ) < ϵ 3 } . Now, it has sets B p = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k p ) , a p ) < ϵ 3 } and B s = { r ∈ ℕ : h r - 1 ∑ k ∈ I r d ¯ ( T ( X k s ) , a s ) < ϵ 3 } . Then B p , s c , B p c , B s c ∈ I . Let B c = B p , s c ∪ B p c ∪ B s c . Where B = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( a p , a s ) < ϵ } then B c ∈ I . Consider n₀ ∈ B ^c . Then for each p, s ≥ n₀ it has d ¯ ( a p , a s ) ≤ d ¯ ( T ( X k p ) , a p ) + d ¯ ( T ( X k s ) , a s ) + d ¯ ( T ( X k p ) , T ( X k s ) ) < ϵ 3 + ϵ 3 + ϵ 3 = ϵ . Thus, a _p is a Cauchy sequence of scalars in ℂ , so there exists scalar a ∈ ℂ such that (a _p ) → a as p→ ∞. For this step, let 0 < α < 1 be given. Then it showed that if U = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X ) , a ) < α } then U c ∈ I .(4) Since ( X k p ) → X , there exists p 0 ∈ ℕ such that P = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X p 0 ) , T ( X ) ) < α 3 } . Which implies that P c ∈ I . The number, p _o can be chosen together with equation (4), it have Q = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( a p 0 , a ) < α 3 } . Which implies that Q c ∈ I . Since { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k p 0 ) , a p 0 ) ≥ α 3 } ∈ I . Then it has a subset S such that S c ∈ I , where S = { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k p 0 ) , a p 0 ) < α 3 } . Let U ^c = P ^c ∪ Q ^c ∪ S ^c . Therefore, for each k ∈ U ^c it has { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X ) , a ) < α } ⊇ { r ∈ ℕ : h r - 1 ∑ k ∈ I r d ¯ ( T ( X p 0 ) , T ( X ) ) < α 3 } ∩ { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( a p 0 , a ) < α 3 } ∩ { r ∈ ℕ : h r - 1 ∑ k ∈ J r d ¯ ( T ( X k p 0 ) , a p 0 ) < α 3 } .(5) The sets on the right hand side of equation (5) belong to F ( I ) . Therefore, the set on the left hand side of equation (5) belongs to F ( I ) . Hence its complement belongs to I . Thus, I - lim d ¯ ( a p 0 , a ) = 0 .

Now, we construct the following example to show that the space _F c ^{I _θ} (T) is closed subspace of F ℓ ∞ θ ( T ) .

Example 3.5. Let us consider a sequence of fuzzy number defined by X k ( t ) = { 2 - 1 ( 1 + t ) , for t ∈ [ - 1 , 1 ] 2 - 1 ( 3 - t ) , for t ∈ [ 1 , 3 ] 0 , otherwise . Then d ¯ ( X k , 0 ¯ ) = sup d ( X k , 0 ¯ ) . Hence (X _k )∈ _F c ^{I _θ} (T) and L = 0 ¯ . So, we say that c F I θ ( T ) is closed subspace of F ℓ ∞ θ ( T ) .

The spaces of fuzzy valued lacunary ideal convergence of sequence have been defined here by compact operator and some algebraic and topological properties of these spaces are investigated. These definitions and results provide new tools to deal with the convergence problems of sequences in the fuzzy settings, occurring in many branches of science and engineering.

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 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. A vigorous researcher in the area of Sequence Spaces, he has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylors and Francis), Information Sciences (Elsevier), Applied Mathematics Letters (Elsevier), A Journal of Chinese Universities (Springer- Verlag, China).

Abdullah. A. H. Makharesh received M.Sc., from Dr. Babasaheb Ambedkar Marathwada University, and is currently a Ph.D., scholar at Aligarh Muslim University.

Kamal. M. A. S. Alshlool received M.Sc., from Aligarh Muslim University, and is currently a Ph.D., scholar at Aligarh Muslim University.

Sameera A. A. Abdullah received M.Sc., from Aligarh Muslim University, and is currently a Ph.D., scholar at Aligarh Muslim University.

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.

Competing Interests

The authors declare that they have no competing interests.