Tauberian theorems for statistically (C,1,1) summable double sequences of fuzzy numbers

Definition 1.1

Consider a fuzzy subset of the real line u : ℝ→[0, 1]. Then the mapping u is a fuzzy number if it satisfies the following additional properties:

1. u is normal, i. e., there exists a t₀ ∊ ℝ such that u(t₀)=1.

2. u is fuzzy convex, i. e., for any t₀, t₁ ∊ ℝ and for any α ∊[0, 1], u (α t₀ + (1− α) t₁) ≥ min {u(t₀), u (t₁)}.

3. u is upper semicontinuous on ℝ.

4. The support of u, [u]0:={t∈R:u(t)>0}¯ is compact, where {t∈R:u(t)>0}¯ denotes the closure of the set {t ∊ ℝ : u(t) > 0} in the usual topology of ℝ.

We denote the set of all fuzzy numbers on ℝ by E¹ and call it the space of fuzzy numbers.

We recall the linear structure of E¹ as follows. For u ∊ E¹, the α-level set of u is defined by

Then, it is easily established (see [21]) that u is a fuzzy number if and only if [u]_α is a closed, bounded and nonempty interval for each α ∊ [0, 1] with [u]_β ⊆ [u]_α if 0 ≤ α ≤ β ≤ 1. From this characterization of fuzzy numbers, it follows that a fuzzy number u is completely determined by the end points of the intervals [u]_α=[u⁻(α), u⁺(α)] where u⁻(α) ≤ u⁺(α) and u⁻(α),u⁺(α)∊ ℝ for each α∊[0, 1].

Immediately after, Goetschel and Voxman [3] presented another representation of a fuzzy number as a pair of functions that satisfy some properties.

Theorem 1.2

(Representation Theorem, [3]). Let u ∊ E¹ and let [u]_α=[u⁻(α),u⁺(α)]. Then the functions u⁻,u⁺:[0, 1]→ ℝ, defining the endpoints of the α-level sets, satisfy the following conditions:

1. u⁻(α) ∊ ℝ is a bounded, non-decreasing and left continuous function on (0,1].

2. u⁺(α)∊ ℝ is a bounded, non-increasing and left continuous function on (0,1].

3. The functions u⁻(α) and u⁺(α) are right continuous at α=0.

4. u⁻(1) ≤ u⁺(1).

Conversely, if the pair of functions f and g satisfies the above conditions (i)-(iv), then there exists a unique fuzzy number u such that [u]_α :=[f(α),g(α)] for each α ∊ [0, 1] and u(x):=supα∈[0,1]{α:f(α)≤x≤g(α)}.

Suppose that u, v ∊ E¹ are represented by [u⁻(α),u⁺(α)] and [v⁻(α), v⁺(α)] for each α ∊ [0, 1], respectively. Then, the operations addition and scalar multiplication on the set of fuzzy numbers are defined as follows:

The set of all real numbers can be embedded in E¹. For r∈R,r¯∈E1 is defined by

The following lemma deals with the algebraic properties of fuzzy numbers.

Lemma 1.3

([22]).

1. The addition of fuzzy numbers is associative and commutative, i. e., u + v = v + u and u+(v + w) = (u + v)+w, for any u, v, w∊ E¹.

2. 0¯∈E1 is neutral element with respect to+, i. e., u+0¯=0¯+u=u, for any u∊ E¹.

3. With respect to+, none of u∊ E¹\ℝ has opposite in E¹.

4. For any a, b ∊ ℝ with ab≥ 0 and any u∊ E¹, we have (a + b)u = au + bu. For general a, b ∊ ℝ, this property does not hold.

5. For any a ∊ ℝ and u, v∊ E¹, we have a(u + v) = au+av.

6. For any a, b ∊ ℝ and u∊ E¹, we have (ab)u = a(bu).

As a consequence of Lemma 1.3, we attain that the space of fuzzy numbers is not a linear space.

The concept of metric space may be defined as an arbitrary fuzzy set in which the distance between all elements of the set are described. It is possible to define several different metrics on the space of fuzzy numbers; however, the most well known and preferential metric among these metrics is the Hausdorff distance for fuzzy numbers based on the classical Hausdorff distance between compact convex subsets of ℝⁿ. Let W denote the set of all closed and bounded intervals. For the particular case when A = [a⁻, a⁺], B = [b⁻, b⁺] are two intervals, the Hausdorff distance on W is defined by

It is known that W is a complete separable metric space in consideration of the Hausdorf distance (cf. Nanda [5]). At the moment, we may define the metric D on the space of fuzzy numbers with the help of the Hausdorff metric d.

Definition 1.4

([22]). Let D : E¹ × E¹→ ℝ₊,

Then D is called the Hausdorff distance between fuzzy numbers u and v.

The following proposition presents some properties of the Hausdorff distance between fuzzy numbers.

Proposition 1.5

([22]). Let u, v, w, z ∊ E¹ and k∊ ℝ. Then the following statements hold true.

1. (E¹, D) is a complete metric space.

2. D(u + w, v +w) = D(u, v), i.e., D is translation invariant.

3. D(ku, kv) = |k| D(u, v).

4. D(u + v, w +z) ≤ D(u, w) + D(v, z).

5. |D(u,0¯)−D(v,0¯)|≤D(u,v)≤D(u,0¯)+D(v,0¯).

Definition 2.1

A double sequence u = (u_mn) of fuzzy numbers is a function u from ℕ × ℕ (ℕ is the set of all natural numbers) into the set E¹. The fuzzy number u_mn denotes the value of the function at a point (m, n)∊ ℕ× ℕ and is called the (m, n)-term of the double sequence.

We denote the set of all double sequences of fuzzy numbers by w²(F).

Definition 2.2

A double sequence u = (u_mn) of fuzzy numbers is said to be convergent in Pringsheim’s sense (or P-convergent) to the fuzzy number μ₀, written as limm,n→∞umn=μ0, integer n₀(ϵ) such that D(u_mn, μ₀)ϵ whenever m, n ≥ n₀, and we denote by P-lim u = μ₀. The number μ₀ is called the Pringsheim limit of u.

More exactly, we say that a double sequence (u_mn) converges to a fuzzy number μ₀ if (u_mn) tends to μ₀ as both m and n tend to infinity independently of one another.

We denote the space of all P-convergent double sequences of fuzzy numbers by c²(F). Note that throughout this paper, we always mean convergence in Pringsheim’s sense.

Definition 2.3

A double sequence u=(u_mn) of fuzzy numbers is bounded if there exists a positive number M such that D(umn,0¯)<M for all m and n, i.e, if

We denote the set of all bounded double sequences of fuzzy numbers by ℓ∞2(F).

Note that unlike single sequences of fuzzy numbers, every P-convergent double sequences of fuzzy numbers need not be bounded.

Example 2.4

Consider the double sequence u=(u_mn) of fuzzy numbers defined by

where

It is clear that (u_mn) is P-convergent to 0¯. One can check that the endpoints of the α-level set of the sequence (u_mn) are

for all m, n ∊ ℕ. From this point of view, we have

Since the sequence (ωm+(0))=(∑j=0ml(j+2)log⁡(j+2)) is divergent, the double sequence u=(u_mn) is not bounded.

For a double sequence (u_mn) of fuzzy numbers, its (C, 1, 1) means are defined by

for all nonnegative integers m and n (see [23]).

Definition 2.5

A double sequence u=(u_mn) of fuzzy numbers is said to be (C, 1, 1) summable to a fuzzy number μ₀ if

As in the single sequence of fuzzy numbers, we give the definition of natural density of K⊂ ℕ × ℕ and we present the statistically convergent double sequence of fuzzy numbers by using this concept.

Let K ⊂ ℕ× ℕ be a two dimensional set of positive integers and let

We say that K has a double natural density if the sequence (|Kmn|(m+1)(n+1)) has alimit in Pringsheim’s sense. In this case, we write

where the vertical bars denote the cardinality of the enclosed set.

In [23], Savaş and Mursaleen introduced the concept of statistical convergence for double sequences of fuzzy numbers as follows:

Definition 2.6

A double sequence u = (u_mn) of fuzzy numbers is said to be statistically convergent in Pringsheim’s sense to the fuzzy number μ 0, written as st₂− limm,n→∞ u_mn = μ 0, if for every ϵ>0,

We denote the set of all statistically convergent double sequences of fuzzy numbers by st²(F).

We note that if a double sequence of fuzzy numbers is P-convergent to the fuzzy number μ₀, then it is also statistically convergent to same number (see [23]). However, the converse is not necessarily true. In other words, a double sequence of fuzzy numbers which is statistically convergent need not be P-convergent. Now, we construct an example of a double sequence of fuzzy numbers which is statistically convergent, but not P-convergent as follows.

Example 2.7

Consider the double sequence u=(u_mn) of fuzzy numbers defined by

It is clear that (u_mn) is divergent. On the other hand, it is statistically convergent to 0¯, since

for every ϵ > 0.

We say that the double sequence (u_mn) of fuzzy numbers is called statistically (C, 1, 1) summable to μ₀ if st₂− limm,n→∞ σ_mn = μ₀.

One of the main theorems of this paper shows that if a double sequence of fuzzy numbers is statistically convergent, then it is also statistically (C, 1, 1) summable provided that it is bounded.

We now define the concepts of slow oscillation for the double sequences (u_mn) of fuzzy numbers in certain senses as follows.

We say that a double sequence (u_mn) of fuzzy numbers is slowly oscillating in sense (1, 1) if

or equivalently,

if for every ϵ>0 there exist n₀ = n₀(ϵ) and λ = λ(ϵ)>1 such that

whenever n₀ < m < j ≤ λ_m and n₀ < n < k ≤ λ_n.

Here, by λ_n we denote the integral part of the product λ n.

It easily follows from (3) that every P-convergent double sequence of fuzzy numbers is slowly oscillating in sense (1, 1), but the converse is not true in general. An example indicating that the converse is not true was constructed by Çanak et al. [20].

We say that a double sequence (u_mn) of fuzzy numbers is said to be slowly oscillating in sense (1, 0) if

We say that a double sequence (u_mn) of fuzzy numbers is said to be slowly oscillating in the strong sense (1, 0) if (4) is satisfied with

We say that a double sequence (u_mn) of fuzzy numbers satisfies the two-sided Tauberian condition of Hardy type in sense (1, 0) if there exist positive constants n₀ and H such that

It is clear that if (6) holds, then (u_mn) is slowly oscillating in both sense (1, 0) and the strong sense (1, 0).

Similarly, the slow oscillation of the double sequence (u_mn) of fuzzy numbers in sense (0,1) and the strong sense (0,1) can be analogously defined. In addition to these, a double sequence (u_mn) of fuzzy numbers which satisfies the two-sided Tauberian condition of Hardy type in sense (0,1) can be defined and it is slowly oscillating in both sense (0,1) and strong sense (0,1).

We note that if the double sequence (u_mn) of fuzzy number is slowly oscillating in senses (1, 0), (0, 1) and slowly oscillating in the strong sense (1, 0) or (0, 1), then (u_mn) is slowly oscillating in sense (1, 1).

As a matter of fact, without loss of generality we suppose that the double sequence (u_mn) is slowly oscillating in senses (1, 0), (0, 1) and in the strong sense (1, 0). For every large enough m and n, that is, m, n≥n₀ and λ > 1, we have

Taking the lim sup and the limit of both sides of this inequality as m, n → ∞ and λ→ 1⁺ respectively, we obtain that the terms on the right-hand side of this inequality tends to 0. Therefore, we obtain that (u_mn) is slowly oscillating in sense (1, 1).

On the other hand, a double sequence (u_mn) of fuzzy numbers is said to be statistically slowly oscillating in sense (1, 0) if for every ϵ > 0,

A double sequence (u_mn) of fuzzy numbers is statistically slowly oscillating in the strong sense (1, 0) if, for every ϵ > 0, (7) is satisfied with (5).

Similarly, the statistical slow oscillation of the double sequence (u_mn) of fuzzy numbers in sense (0, 1) and the strong sense (0, 1) can be analogously defined.

We note that if the double sequence (u_mn) of fuzzy numbers is slowly oscillating in sense (1, 0), then (u_mn) is statistically slowly oscillating in sense (1, 0).

As a matter of fact, suppose that the double sequence (u_mn) is slowly oscillating in sense (1, 0) . Given any ϵ > 0. For every large enough m and n, that is, m, n ≥ n₀(ϵ), we have

Taking the lim sup of both sides of the inequality (8) as M, N → ∞, we obtain that the term on the right-hand side of the inequality (8) tends to 0. Then taking the limit of both sides of the last inequality as λ→ 1⁺, we conclude that (u_mn) is statistically slowly oscillating in sense (1, 0).

Note that there is a similar relation between the concepts of the slow oscillation and the statistical slow oscillation in sense (0,1). In a similar way, we can also indicate that the double sequence (u_mn) of fuzzy numbers which is slowly oscillating in the strong sense (1, 0) (or (0, 1)) is statistically slowly oscillating in the strong sense (1, 0) (or (0,1)).

Lemma 3.1

([19, Theorem 1 The following statements are equivalent:

1. The double sequence (u_mn) of fuzzy numbers is statistically convergent to a fuzzy number μ₀.

2. There exists a P-convergent double sequence (w_mn) of fuzzy numbers such that

   δ2({(m,n)∈N×N:umn≠wmn})=0.

3. There exists a subset K = {(m_j, n_k)∊ ℕ× ℕ : j, k ∊ ℕ}⊆ ℕ× ℕ such that the natural density of K is 1 and lim u_mjnk = μ₀.

In the following lemma, we present two representations for the distance between the general terms of the double sequences (u_mn) and (σ_mn).

Lemma 3.2

i) If λ > 1, λ_m>m, and λ_n>n, then

ii) If 0 > λ > 1, λ_m < m, and λ_n < n, then

Proof

1. For λ >1, we obtain by the definition of the (C, 1, 1) means of (u_mn) that

   D(umn,σmn)=D(1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnumn+1(λm−m)(λn−n)(∑j=0m∑k=n+1λn+∑j=m+1λm∑k=0n+∑j=0m∑k=0n+∑j=m+1λm∑k=n+1λn)ujk,σmn+1(λm−m)(λn−n)(∑j=0m∑k=n+1λn+∑j=m+1λm∑k=0n+∑j=0m∑k=0n+∑j=m+1λm∑k=n+1λn)ujk)=D(1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnumn+1(λm−m)(λn−n)∑j=0m∑k=0λnujk+1(λm−m)(λn−n)∑j=m+1λm∑k=0λnujk+1(λm−m)(λn−n)∑j=0m∑k=0nujk,σmn+1(λm−m)(λn−n)∑j=0m∑k=0λnujk+1(λm−m)(λn−n)∑j=0λm∑k=0nujk+1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnujk)=D(1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnumn+1(λm−m)(λn−n)∑j=0λm∑k=0λnujk+1(λm−m)(λn−n)∑j=0m∑k=0nujk,+σmn+1(λm−m)(λn−n)∑j=0m∑k=0λnujk+1(λm−m)(λn−n)∑j=0λm∑k=0nujk+1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnujk)=D(1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnumn+(λm+1)(λn+1)(λm−m)(λn−n)σλm,λn+(m+1)(n+1)(λm−m)(λn−n)σmn+λn+1λn−nσm,λn+λm+1λm−mσλm,n,σmn+(λm+1)(λn+1)(λm−m)(λn−n)(σm,λn+σλm,n)+1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnujk)≤(λm+1)(λn+1)(λm−m)(λn−n)D(σλm,λn,σλm,n)+D((m+1)(n+1)(λm−m)(λn−n)σmn+λn+1λn−nσm,λn+λm+1λm−mσλm,n+n+1λn−nσmn+m+1λm−mσmn+σmn,σmn+(λm+1)(λn+1)(λm−m)(λn−n)σm,λn+n+1λn−nσmn+m+1λm−mσmn+σmn)+1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnD(umn,ujk)≤(λm+1)(λn+1)(λm−m)(λn−n)D(σλm,λn,σλm,n)+D((λm+1)(λn+1)(λm−m)(λn−n)σmn+λn+1λn−nσm,λn+λm+1λm−mσλm,n,(λm+1)(λn+1)(λm−m)(λn−n)σm,λn+λn+1λn−nσmn+λm+1λm−mσmn)+1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnD(umn,ujk)≤(λm+1)(λn+1)(λm−m)(λn−n)D(σλm,λn,σλm,n)+(λm+1)(λn+1)(λm−m)(λn−n)D(σmn,σm,λn)+λm+1λm−mD(σλm,n,σmn)+λn+1λn−nD(σm,λn,σmn)+1(λm−m)(λn−n)∑j=m+1λm∑k=n+1λnD(umn,ujk),

   which completes the proof of Lemma 3.2 (i).

2. For 0 < λ < 1, following a procedure which is similar to the proof of Lemma 3.2 (i), we can reach the conclusion in Lemma 3.2 (ii).

In the following two lemmas, we state the fuzzy analogues of the lemmas given for single sequences of complex numbers by Móricz (see [24], Lemma 9) for double sequences of fuzzy numbers. These lemmas play a crucial role in the proofs of subsequent lemmas which are required for the proofs of our main theorems.

Lemma 3.3

Let (u_mn) be a double sequence of fuzzy numbers. If there exist a positive integer n₀ and λ > 1 such that

then there exists a constant H such that

Proof

Assume that n₀ is large enough to satisfy the condition

without loss of generality. Let n₀ < j. We define the subsequence

where r is determined by the condition j_{r + 1} ≤ n₀ < j_r. It follows from the definition of the subsequence (j_p) that we find

Choose m such that 1≤m≤jλ. We examine chosen m in two cases such that n0≤m≤jλand1≤m<n0. We firstly consider the case n0≤m≤jλ. Then, we define p such that

Taking into account the assumption above, we obtain for n₀ ≤ n

By definition of the subsequence (j_p), we find

From this point of view, we arrive

by using (9) and (10). It follows from the fact that we find

If we combine (11) and (12), then we conclude for n₀ < n that

On the other hand, we consider the case 1 ≤ m < n₀. Then going through similar process as above and taking the assumption into account, for n₀ < n we get

where c:=max1≤m<n0D(un0n,umn). If we follow a similar process to (12), then we find

If we combine (14) and (15), then we conclude for n₀ < n that

It follows from (13) and (16) that we have for n₀ < n

whenever 1≤m≤jλ provided that