Generalized statistically convergent sequences of fuzzy numbers

1 Introduction

The concepts of fuzzy sets and fuzzy set operations were first introduced by Zadeh [30]. Subsequently, several authors discussed various aspects of the theory and applications of fuzzy sets such as fuzzy topological spaces, similarity relations and fuzzy orderings, fuzzy measures of fuzzy events, fuzzy mathematical programming and ranking method for generalized fuzzy numbers (cf. [11, 15]). Matloka [19] introduced bounded and convergent sequences of fuzzy numbers and studied some of their properties. The concept of statistical convergence is closely related to the concept of convergence in probability, was introduced by Fast [13] and also independently by Buck [4] and Schoenberg [27] for real and complex sequences which was studied further by Fridy [14], Connor [6] and many other authors. In recent years, generalizations of statistical convergence have appeared in the study of strong integral summability and the structure of ideals of bounded continuous functions on locally compact spaces. The existing literature on statistical convergence appears to have been restricted to real or complex sequences, but in the works by Nanda [24], Savaş [26], Mursaleen and Basarir [20], Kumar and Kumar [17], Kumar et al. [16], and Dündar and Talo [9], the idea of statistical convergence was extended to include sequences of fuzzy numbers. Most recently, statistical convergence has applications in approximation theory which is known as statistical approximation (see the recent works [10, 21] and [29]).

In this paper, we define and study the λ-bounded and λ-convergent, λ-Cauchy, λ-statistically convergent, λ-statistically Cauchy sequences of fuzzy numbers and find their inclusion relations. All these classes provide a helpful tool in studying various types of convergence problems for sequences of fuzzy numbers. Like sequences of real numbers, these summabilty methods of fuzzy sequences can also be applied in fuzzy approximation.

First we recall some basic definitions and notations of fuzzy numbers and sequences of fuzzy numbers.

A fuzzy number is a fuzzy set on the real axis, that is, a mapping X : ℝ n → [ 0 , 1 ] which satisfies the following four conditions:

X is normal, that is, there exists an x 0 ∈ ℝ n such that X (x₀) =1;

Let C ( ℝ n ) = { A : A ⊂ ℝ n and A is compact and convex } .

The space C ( ℝ n ) has a linear structure induced by the following operations: A + B = { a + b : a ∈ A , b ∈ B } and μ A = { μ a : a ∈ A } for A , B ∈ C ( ℝ n ) and μ ∈ ℝ . The Hausdorff distance between A and B of C ( ℝ n ) is defined as follows: δ ∞ ( A , B ) = max ( sup a ∈ A inf b ∈ B ‖ a - b ‖ , sup b ∈ B inf a ∈ A ‖ a - b ‖ ) , where || . || denotes the usual Euclidean norm in ℝ n . It is well known that ( C ( ℝ n ) , δ ∞ ) is a complete (non-separable) metric space.

For 0 < α ≦ 1, the α-level set X α = { x : x ∈ ℝ n and X ( x ) ≧ α } is a nonempty compact convex subset of ℝ n , as is the support X⁰ . Let L ( R n ) denote the set of all fuzzy numbers. The linear structure of L ( R n ) induces addition X + Y and scalar multiplication μX, μ ∈ ℝ , in terms of α-level sets, by [ X + Y ] α = [ X ] α + [ Y ] α and [ μ X ] α = μ [ X ] α for each 0 ≦ α ≦ 1 . We define, for each 1≦ q < ∞, d q ( X , Y ) = ( ∫ 0 1 δ ∞ ( X α , Y α ) q d α ) 1 q and d ∞ ( X , Y ) = sup 0 ≦ α ≦ 1 δ ∞ ( X α , Y α ) . Clearly, we have d ∞ ( X , Y ) = lim q → ∞ d q ( X , Y ) with d _q  ≦ d _r if q ≦ r . Moreover, d _q is a complete, separable and locally compact metric space. In this paper, d will denote d _q with 1 ≦ q ≦ ∞ .

Definition 1. A sequence X = ( X k ) k ∈ ℕ of fuzzy numbers is said to be bounded if the set sup k d ( X k , 0 ) < ∞ .

A sequence X = ( X k ) k ∈ ℕ of fuzzy numbers is said to be convergent to the fuzzy number X₀, written as follows: lim k → ∞ X k = X 0 , if for every ɛ > 0 there exists a positive integer k₀ such that d ( X k , X 0 ) < ɛ ( k > k 0 ) . By l ∞ F and c ^F we denote the set of all bounded and convergent sequences of fuzzy numbers, respectively.

Definition 2. Let X = ( X k ) k ∈ ℕ be a sequence of fuzzy numbers. Then the sequence X = ( X k ) k ∈ ℕ is said to be λ-statistically convergent to the fuzzy number X₀ if, for every ɛ > 0, lim n → ∞ 1 n | { k ≦ n : d ( Λ k ( X ) , X 0 ) ≧ ɛ } | = 0 , where the vertical bars indicate the number of elements in the enclosed set. In this case we write X k → X 0 ( S F ( λ ) ) or S F ( λ ) lim k → ∞ X k = X 0 . The set of all λ-statistically convergent sequences of fuzzy numbers is denoted by S ^F  (λ) see (see, for details, [3] and [20]). We note that S F ( λ ) lim k → ∞ X k = X 0 .

Definition 3. A sequence X = ( X k ) k ∈ ℕ of fuzzy numbers is said to be a λ-Cauchy sequence if, for every ɛ > 0, there is a positive integer N₀ such that d (Λ _k  (X) , Λ_ℓ (X)) < ɛ for all k, ℓ > N₀ (see [19]).

Definition 4. A sequence X = ( X k ) k ∈ ℕ of fuzzy numbers is said to be a λ-statistically Cauchy if, for every ɛ > 0, there exists a positive integer N = N (ɛ) such that lim n → ∞ 1 n | { k ≦ n : d ( Λ k ( X ) , Λ N ( X ) ) ≧ ɛ } | = 0 . For more details about fuzzy sequence spaces, see ([1, 28]) as well as the references cited therein.

The concept of statistical pre-Cauchy sequence was given by Connor et al. [7] for scalar sequences. It is shown that statistically convergent sequences are statistically pre-Cauchy and any bounded statistically pre-Cauchy sequence with a nowhere dense set of limit points is statistically convergent.

Definition 5. A sequence X = ( X k ) k ∈ ℕ of fuzzy numbers is said to be a λ-statistically pre-Cauchy if, for every ɛ > 0, there exists a positive integer N = N (ɛ) such that lim n → ∞ 1 n 2 | { k ≦ n : d ( Λ k ( X ) , Λ N ( X ) ) ≧ ɛ } | = 0 . The idea of a fuzzy real valued statistically pre-Cauchy sequences is studied in [8].

In this paper, we define the concept of λ-bounded and λ-convergent sequences of fuzzy numbers which for real numbers were defined by Mursaleen and Noman (see [22] and [23]).

Definition 6. Let λ = ( λ k ) k = 1 ∞ be a strictly increasing sequence of positive real numbers tending to infinity, that is, 0 < λ 1 < λ 2 < λ 3 < ⋯ and λ k → ∞ as k → ∞ . We then say that a sequence x = (x _k ) ∈ w isλ-convergent to the number L, called the λ-limit of x, if Λ _m  (x) ⟶ L as m→ ∞, where Λ m ( x ) = 1 λ m ∑ k = 1 m ( λ k - λ k - 1 ) x k , λ 0 = 0 .

A sequence of fuzzy numbers X = ( X k ) k ∈ ℕ is said to be

λ-bounded if the set { Λ k ( X ) : k ∈ ℕ } of fuzzy numbers is bounded;

By l ∞ F ( λ ) and c ^F  (λ) we denote the sets of all λ-bounded and all λ-convergent sequences of fuzzy numbers, respectively.

Definition 7. Let p = ( p k ) k ∈ ℕ be a bounded sequence of positive real numbers. Also let M be an Orlicz function. For some fuzzy number X₀, we define a class of fuzzy sequences as follows: w F ( M , p , λ ) = { X = ( X k ) : lim n → ∞ 1 n ∑ k = 1 n [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k = 0 } where an Orlicz function is a function M : [0, ∞) → [0, ∞), which is continuous, non-decreasing andconvex with M ( 0 ) = 0 , M ( x ) > 0 ( x > 0 ) and lim x → ∞ M ( x ) = ∞ . If convexity of the Orlicz function M (x) is replaced by the property that M (x + y) ≦ M (x) + M (y) , then this function is called the modulus function (see Ruckle [25]). An Orlicz function M is said to satisfy the Δ₂-condition for all values of u if there exists K > 0 such that M (2u) ≦ KM (u)   (u ≧ 0).

2 Main results

We begin by stating the following result.

Theorem 1. Let X = ( X k ) k ∈ ℕ be a sequence of fuzzy numbers and M be a bounded Orlicz function. Then X is a fuzzy λ-statistically pre-Cauchy sequence if and only if lim n → ∞ 1 n 2 ∑ j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ] = 0 .(1)

Proof. First of all, we suppose that the condition (1) is satisfied. Then, for each given ɛ > 0 and n ∈ ℕ, we have 1 n 2 ∑ j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] = 1 n 2 ∑ [ d ( Λ k ( X ) , Λ j ( X ) ) < ɛ ] j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] + 1 n 2 ∑ [ d ( Λ k ( X ) , Λ j ( X ) ) ≧ ɛ j , k ≦ n ] [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] ≧ 1 n 2 ∑ [ d ( Λ k ( X ) , Λ j ( X ) ) ≧ ɛ ] j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] ≧ M ( ɛ ) 1 n 2 | { ( j , k ) : j , k ≦ n , d ( Λ k ( X ) , Λ j ( X ) ) ≧ ɛ } | . Thus, clearly, X is a fuzzy λ-statistically pre-Cauchy sequence.

Conversely, let us suppose that X is a λ-statistically pre-Cauchy sequence and ɛ > 0 be given. We then choose δ > 0 such that M ( δ ) < ɛ 2 . Since the Orlicz function M is bounded, there exists an integer T such that M ( d ( Λ k ( X ) , 0 ¯ ) ) < T for all d ( Λ k ( X ) , 0 ¯ ) , where 0 ¯ is the zero sequence. For each n ∈ ℕ, we now write

1 n 2 ∑ j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] = 1 n 2 ∑ [ d ( Λ k ( X ) , Λ j ( X ) ) < δ ] j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] + 1 n 2 ∑ [ d ( Λ k ( X ) , Λ j ( X ) ) ≧ δ ] j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] ≦ M ( δ ) + 1 n 2 ∑ [ d ( Λ k ( X ) , Λ j ( X ) ) ≧ δ ] j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] ≦ ɛ 2 + T ( 1 n 2 | { ( j , k ) : j , k ≦ n , d ( Λ k ( X ) , Λ j ( X ) ) ≧ δ } | ) .(2)

Since X is λ-statistically pre-Cauchy sequence, there is an N such that the last member of (2) is less than ɛ for all n > N . Hence we have the condition(1).

This completes the proof of Theorem 1.

Theorem 2. Let M be an Orlicz function and p = ( p k ) k ∈ ℕ be any bounded sequence of positive real numbers. Suppose also that 0 < h = inf k ∈ ℕ p k ≦ p k ≦ sup k ∈ ℕ p k = H < ∞ .

Then for any sequence X of fuzzy numbers, we have w F ( M , p , λ ) ⊂ S F ( λ ) .

Proof. Let X ∈ w ^F  (M, p, λ) . For given ɛ > 0, let ∑₁ and ∑₂ denote the sum over k ≦ n with d (Λ _k  (X) , X₀) ≧ ɛ and the sum over k ≦ n with d (Λ _k  (X) , X₀) < ɛ, respectively. Then 1 n ∑ j , k ≦ n [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k = 1 n ( ∑ 1 [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k + ∑ 2 [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k ) ≧ 1 n ∑ 1 [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k ≧ 1 n ∑ 1 [ M ( ɛ ) ] p k ≧ 1 n ∑ 1 min ( [ M ( ɛ ) ] h , [ M ( ɛ ) ] H ) = 1 n | { k ≦ n : d ( Λ k ( X ) , X 0 ) ≧ ɛ } | × min ( [ M ( ɛ ) ] h , [ M ( ɛ ) ] H ) , which shows that X ∈ S ^F  (λ) . This completes the proof of Theorem 2.

Example 1. [5] Take M (X) = X, λ n = n (n = 1, 2, 3, . . .) and p _k  = 1 . Define the sequence (X _k ) as follows: X k ( x ) = { x k 3 + 1 , if - k 3 ≤ x ≤ 0 - x k 3 + 1 , if 0 < x ≤ k 3 0 , otherwise } , for k = n 3 x - 4 , if 4 ≤ x ≤ 5 - x + 6 , if 5 < x ≤ 6 0 , otherwise } , k ≠ n 3 Then we obtain [ X k ] α = { [ - k 3 ( 1 - α ) , k 3 ( 1 - α ) ] , if k = n 3 [ 4 + α , 6 - α ] , if k ≠ n 3 and [ Δ X k ] α = { [ - k 3 - 6 + α ( k 3 + 1 ) , k 3 - 4 - α ( k 3 + 1 ) ] , if k = n 3 ; [ - ( k + 1 ) 3 ( 1 - α ) + 4 + α , ( k + 1 ) 3 ( 1 - α ) + 6 - α ] , if k + 1 = n 3 ; [ - 2 + 2 α , 2 - 2 α ] , otherwise .

Then X _k  ∈ S ^F  (λ), but X _k  ∉ w ^F  (M, p, λ). Hence the inclusion in Theorem 2 is strict.

Theorem 3. Let M be a bounded Orlicz function and let p = ( p k ) k ∈ ℕ be any bounded sequence of positive real numbers. If 0 < h = inf k ∈ ℕ p k ≦ p k ≦ sup k ∈ ℕ p k = H < ∞ , then for a bounded sequence X of fuzzy numbers, we have S F ( λ ) ⊂ w F ( M , p , λ ) .

Proof. Let X ∈ S ^F  (λ) . Suppose that the Orlicz function M is bounded. Let ɛ > 0 be given and let ∑₁ and ∑₂ denote the sum over k ≦ n with d (Λ _k  (X) , X₀) ≧ ɛ and the sum over k ≦ n with d (Λ _k  (X) , X₀) < ɛ, respectively. Since M is bounded, there exists an integer T such that [M (d (Λ _k  (X) , X₀))] < T . Thus we have 1 n ∑ k ≦ n [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k = 1 n ( ∑ 1 [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k + ∑ 2 [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k ) ≦ 1 n ∑ 1 [ T ] p k + 1 n ∑ 2 [ M ( ɛ ) ] p k ≦ max ( 1 , [ T ] H ) 1 n | { k ≦ n : d ( Λ k ( X ) , X 0 ) ≧ ɛ } | + max ( [ M ( ɛ ) ] h , [ M ( ɛ ) ] H )

Hence X ∈ w ^F  (M, p, λ) . This completes the proof of Theorem 3.

Example 2. [5] Take M (X) = X, λ _n  = n and p _k  = 1 . Consider the fuzzy sequence X = (X _k ) as follows

X k ( x ) = { 1 ¯ , k even - 1 ¯ , k odd where Δ X k ( x ) = { 2 ¯ , k even - 2 ¯ , k odd

Now d ( Δ X 2 k , Δ X 2 k + 1 ) = sup δ ∞ ( [ 2 ¯ ] α , [ - 2 ¯ ] α ) = 4 ≠ 0 so ΔX_2k (x) does not convergence in L ( ℝ n ) .

However d ( S 2 k , 0 ¯ ) = d ( 1 2 k ∑ n = 1 2 k Δ X n , 0 ¯ ) = 0 , d ( - 2 + 2 - 2 + 2 - . . . + 2 2 k , 0 ¯ ) = d ( 0 ¯ , 0 ¯ ) = 0 and d ( S 2 k + 1 , 0 ¯ ) = d ( 1 2 k + 1 ∑ n = 1 2 k + 1 Δ X n , 0 ¯ ) = d ( - 2 + 2 - 2 + 2 - . . . + 2 2 k + 1 , 0 ¯ ) = d ( - 2 + 2 - 2 + 2 - . . . + 2 2 k + 1 , 0 ¯ ) = 2 2 k + 1 where S k = 1 k ∑ n = 1 k Δ X n so d ( S 2 k , 0 ¯ ) → 0 as k → ∞ . Thus (ΔX _k ) is w ^F  (M, p, λ) summable to 0 ¯ , but (ΔX _k ) is not in S ^F  (λ) .

Theorem 4. Let X = ( X k ) k ∈ ℕ be a sequence of fuzzy numbers and p = (p _k ) be any bounded sequence of positive real numbers. If 0 < h = inf k ∈ ℕ p k ≦ p k ≦ sup k ∈ ℕ p k = H < ∞ , then X is fuzzy λ-statistically convergent to X₀ if and only if lim n → ∞ 1 n 2 ∑ j , k ≦ n [ M ( d ( Λ k ( X ) , X 0 ) ) ] p k = 0(3) for any Orlicz function M.

Proof. It is easy to prove Theorem 4 by means of Theorems 2 and 3. The details involved are being omitted here.

In this section, we prove several lemmas and apply them to deduce further inclusion relatioships.

Lemma 1. Let M be an Orlicz function. Then there exist K, K′ > 0 such that K | x k | ≦ M ( | x k | ) ≦ K ′ | x k | ( k ∈ ℕ ) .(4)

Proof. Suppose that ( x k ) k ∈ ℕ is a zero sequence. Then, clearly, the inequalities in (4) are satisfied. Let K₁ ≦ |x _k | ≦ L₁ hold true when K₁ > 0 . Since M is non-decreasing, we get M ( K 1 ) ≦ M ( | x k | ) ≦ M ( K 1 ′ ) . Furthermore, since | x k | K 1 ≧ 1 and | x k | K 1 ′ ≦ 1 ( k ∈ ℕ ) , we may write | x k | K 1 ′ M ( K 1 ) ≦ M ( | x k | ) ≦ M ( K 1 ′ ) | x k | K 1 .

Then, taking M ( K 1 ) K 1 ′ = K and M ( K 1 ′ ) K 1 = K ′ , we get (4). This completes the proof of Lemma 1.

Theorem 5. Let M be an Orlicz function and let p = ( p k ) k ∈ ℕ be any bounded sequence of positive real numbers. If 0 < p _k  ≦ q _k and (q _k /p _k ) is bounded, then w F ( M , q , λ ) ⊂ w F ( M , p , λ ) .

Proof. In light of Lemma 1, it is easy to proveTheorem 5, so we omit the details involved.

Theorem 6. Let X = ( X k ) k ∈ ℕ be a sequence of fuzzy numbers. Then X is a fuzzy λ-statistically pre-Cauchy sequence if and only if lim n → ∞ 1 n 2 ∑ j , k ≦ n [ ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] = 0 .(5)

Proof. By Lemma 1, there exist two numbers K and L such that Kd ( Λ k ( X ) , Λ j ( X ) ) ≦ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ≦ Ld ( Λ k ( X ) , Λ j ( X ) ) .

Consequently, we have lim n → ∞ 1 n 2 ∑ j , k ≦ n [ ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] = 0 ⇔ lim n → ∞ 1 n 2 ∑ j , k ≦ n [ M ( d ( Λ k ( X ) , Λ j ( X ) ) ) ] = 0 , which obviously proves Theorem 6.

Remark 1. From Theorems 1 and 6, we conclude that Conditions (1) and (5) are equivalent.

Lemma 2. Let M be an Orlicz function. Then each of the following assertions holds true:

(i) lim t → ∞ M ( t ) t = β exists;

(ii) If 0 < δ < 1, then M (x) ≦2M (1) δ^-1x for eachx ≧ δ.

Proof. Lemma 2 can be proved fairly easily (see, for details, [18]).

Theorem 7. Let M be an Orlicz function. If lim t → ∞ M ( t ) t = β > 0 , then w F ( M , λ ) = w F ( λ ) .

Proof. It is easy to prove that w F ( M , λ ) ⊂ w F ( λ ) . In order to show that w F ( λ ) ⊂ w F ( M , λ ) , let X ∈ w ^F  (λ) . Then S n = 1 n ∑ k = 1 n d ( Λ k ( X ) , X 0 ) → 0 as n → ∞(6) for some X₀ . Now let ɛ > 0 be given and choose δ with 0 < δ < 1 such that M (t) < ɛ for every t with0 ≦ t ≦ 1 . Therefore, we have 1 n ∑ k = 1 n [ M ( d ( Λ k ( X ) , X 0 ) ) ] = 1 n ∑ d ( Λ k ( X ) , X 0 ) < δ [ M ( d ( Λ k ( X ) , X 0 ) ) ] + 1 n ∑ d ( Λ k ( X ) , X 0 ) ≥ δ [ M ( d ( Λ k ( X ) , X 0 ) ) ] ≦ ɛ n + 2 M ( 1 ) δ - 1 S n = ɛ n + 2 M ( 1 ) δ - 1 S n , where we have made use of the assertion (ii) of Lemma 2. It follows that X ∈ w F ( M , λ ) . This completes the proof of Theorem 8.