On the Borel summability method of sequences of fuzzy numbers

Abstract

In the current paper, we define Borel summability of sequences of fuzzy numbers and investigate Tauberian conditions under which Borel summable sequences of fuzzy numbers are convergent. Also, we give analogous Tauberian results for Borel summability method of series of fuzzy numbers.

Keywords

Sequences of fuzzy numbers Borel summability method Tauberian theorems

1 Introduction

A sequence (s_k) of complex numbers is said to be Borel summable to s if the series $\sum_{k = 0}^{\infty} \frac{x^{k}}{k!} s_{k}$ converges for all $x \in ℝ$ and $\begin{matrix} e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} s_{k} \to s, x \to \infty . \end{matrix}$

Many studies have been done on power series methods of summability of sequences of complex numbers [6–8 , 53], and particularly on the Borel summability method [9 , 48]. On the other hand, besides the developments in various branches of fuzzy mathematics [10 , 59], the theory of sequences of fuzzy numbers has become a recent research area. The concept of sequence of fuzzy numbers was defined in [35] and different classes of sequences of fuzzy numbers have been introduced followingly [3 , 50–52]. Furthermore, various weighted mean methods of summability for sequences of fuzzy numbers have been defined and studied recently [2 , 49]. Also, as a power series method, Abel summability method of sequences of fuzzy numbers has been introduced in [54]. In the light of the studies above, in this study we define Borel summability method for sequences and series of fuzzy numbers and give Tauberian conditions for the Borel summability method.

Before continuing with main results, we give some definitions and theorems which are necessary.

A fuzzy number is a fuzzy set on the real axis, i.e. a mapping $u : ℝ \to [0, 1]$ which satisfies the following four conditions:

u is normal, i.e. there exists an $t_{0} \in ℝ$ such that u (t₀) =1.

u is fuzzy convex, i.e. u [λx + (1 - λ) y] ≥ min {u (x) , u (y)} for all $x, y \in ℝ$ and for all λ ∈ [0, 1].

u is upper semi-continuous.

The set $[u]_{0} : = \bar{{t \in ℝ : u (t) > 0}}$ is compact [55], where $\bar{{t \in ℝ : u (t) > 0}}$ denotes the closure of the set ${t \in ℝ : u (t) > 0}$ 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. λ-level set [u] _λ of u ∈ E¹ is defined by $\begin{matrix} [u]_{λ} : = {\begin{matrix} {t \in ℝ : u (t) \geq λ} & , & if 0 < λ \leq 1, \\ \bar{{t \in ℝ : u (t) > λ}} & , & if λ = 0 . \end{matrix} \end{matrix}$

The set [u] _λ is closed, bounded and non-empty interval for each λ ∈ [0, 1] which is defined by [u] _λ : = [u^- (λ) , u⁺ (λ)]. $ℝ$ , the set of real numbers, can be embedded in E¹, since each $r \in ℝ$ can be regarded as a fuzzy number $\bar{r}$ defined by $\begin{matrix} \bar{r} (x) : = {\begin{matrix} 1 & , & if x = r, \\ 0 & , & if x \neq r . \end{matrix} \end{matrix}$

Following theorem is very useful to represent a fuzzy number by means of an interval of real numbers.

Theorem 1(Representation Theorem). [21] Let [u] _λ = [u^- (λ) , u⁺ (λ)] foru ∈ E¹and for each λ ∈ [0, 1]. Then the following statements hold:

u^- (λ) is a bounded and non-decreasing left continuous function on (0, 1].

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

The functionsu^- (λ) andu⁺ (λ) are right continuous at the point λ = 0.

u^- (1) ≤ u⁺ (1).

Conversely, if the pair of functions α and β satisfies the conditions (i)-(iv), then there exists a unique u ∈ E¹ such that [u] _λ : = [α (λ) , β (λ)] for each λ ∈ [0, 1]. The fuzzy number u corresponding to the pair of functions α and β is defined by $u : ℝ \to [0, 1]$ , u (t) : = sup {λ : α (λ) ≤ t ≤ β (λ)}.

Let u, v, w ∈ E¹ and $k \in ℝ$ . Then the operations addition and scalar multiplication are defined on E¹ as $\begin{matrix} u + v = w & \Leftrightarrow & [w]_{λ} = [u]_{λ} + [v]_{λ} all λ \in [0, 1] \\ \Leftrightarrow & w^{-} (λ) = u^{-} (λ) + v^{-} (λ) and \\ w^{+} (λ) = u^{+} (λ) + v^{+} (λ); \end{matrix}$ [ku] _λ = k [u] _λ forall λ ∈ [0, 1] .

Lemma 1. [5] The following statements hold:

$\bar{0} \in E^{1}$ is neutral element with respect to +, i.e., $u + \bar{0} = \bar{0} + u = u$ for allu ∈ E¹.

With respect to $\bar{0}$ , none of $u \neq \bar{r}$ , $r \in ℝ$ has opposite inE¹ .

For any $a, b \in ℝ$ witha, b ≥ 0 or a, b ≤ 0 and anyu ∈ E¹, we have (a + b) u = au + bu. For general $a, b \in ℝ$ , the above property does not hold.

For any $a \in ℝ$ and anyu, v ∈ E¹, we havea (u + v) = au + av .

For any $a, b \in ℝ$ and anyu ∈ E¹, we havea (bu) = (ab) u .

Remark 1. It should be noted that for any $a_{k} \in ℝ$ with a_k ≥ 0 and any u ∈ E¹, we have $u \sum_{k = 0}^{n} a_{k} = \sum_{k = 0}^{n} {ua}_{k}$ by (iii) of Lemma 1.

Let W be the set of all closed bounded intervals A of real numbers with endpoints $\underline{A}$ and $\bar{A}$ , i.e. $A : = [\underline{A}, \bar{A}]$ . Define the relation d on W by $\begin{matrix} d (A, B) : = max {| \underline{A} - \underline{B} |, | \bar{A} - \bar{B} |} . \end{matrix}$

Then it can easily be observed that d is a metric on W (cf. Diamond and Kloeden [18]) and (W, d) is a complete metric space ([37]). Now, we may define the metric D on E¹ by means of the Hausdorff metric d as $\begin{matrix} D (u, v) & = & sup_{λ \in [0, 1]} d ([u]_{λ}, [v]_{λ}) \\ = & sup_{λ \in [0, 1]} max {| u^{-} (λ) - v^{-} (λ) |, \\ | u^{+} (λ) - v^{+} (λ) |} . \end{matrix}$

One may also see [27] for another recent metric defined by means of λ-level set concept.

Proposition 1. [5] Letu, v, w, z ∈ E¹and $k \in ℝ$ . Then,

(E¹, D) is a complete metric space.

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

D (u + v, w + v) = D (u, w).

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

$| D (u, \bar{0}) - D (v, \bar{0}) | \leq D (u, v) \leq D (u, \bar{0}) + D (v, \bar{0})$ .

A sequence u = (u_k) of fuzzy numbers is a function u from the set $ℕ$ , the set of natural numbers, into the set E¹.

A sequence (u_k) of fuzzy numbers is said to be bounded if there exists M > 0 such that $D (u_{n}, \bar{0}) < M$ for all $n \in ℕ$ . We denote the set of all bounded sequences of fuzzy numbers by ℓ_∞ (F).

A sequence (u_k) of fuzzy numbers is said to be convergent to μ ∈ E¹ if for every ɛ > 0 there exists an $k_{0} = k_{0} (ɛ) \in ℕ$ such that D (u_k, μ) < ɛ forall k ≥ k₀ .

The fuzzy number L ∈ E¹ is called the limit of fuzzy-valued function $f : A \subseteq ℝ \to E^{1}$ as x approaches a if for every ɛ > 0 there exists δ > 0 such that D (f (x) , L) < ɛ whenever |x - a| < δ.

Now we introduce the series of fuzzy numbers. We note that, throughout this paper, we mean $\sum_{k = 0}^{\infty} u_{k}$ by ∑u_k.

Definition 1. [29] Let (u_k) be a sequence of fuzzy numbers. Then the expression ∑u_k is called a series of fuzzy numbers. Denote $s_{n} = \sum_{k = 0}^{n} u_{k}$ for all $n \in ℕ$ . If the sequence (s_n) converges to a fuzzy number u, then we say that the series ∑u_k of fuzzy numbers converges to u and write ∑u_k = u which implies as n→ ∞ that $\begin{matrix} \sum_{k = 0}^{n} u_{k}^{-} (λ) \to u^{-} (λ) and \sum_{k = 0}^{n} u_{k}^{+} (λ) \to u^{+} (λ) \end{matrix}$ uniformly in λ ∈ [0, 1]. Conversely, if the fuzzy numbers $u_{k} = {(u_{k}^{-} (λ), u_{k}^{+} (λ)) : λ \in [0, 1]}$ , $\sum_{k} u_{k}^{-} (λ) = u^{-} (λ)$ and $\sum_{k} u_{k}^{+} (λ) = u^{+} (λ)$ converge uniformly in λ, then u = {(u^- (λ) , u⁺ (λ)) : λ ∈ [0, 1]} defines a fuzzy number such that u = ∑u_k.

We say otherwise the series of fuzzy numbers diverges. Additionally, if the sequence (s_n) is bounded then we say that the series ∑u_k of fuzzy numbers is bounded. We denote the set of all bounded series of fuzzy numbers by bs (F).

Theorem 2. [45] If ∑u_kand ∑v_kconverge, thenD (∑u_k, ∑v_k) ≤ ∑D (u_k, v_k) .

Theorem 3. [45] If $\sum D (u_{k}, \bar{0}) < \infty$ , then the series ∑u_kis convergent.

3 Main results

Definition 2. A sequence (u_k) of fuzzy numbers is said to be Borel summable to μ ∈ E¹ if the series $\begin{matrix} f (x) = \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{k} \end{matrix}$ converges for x ∈ (0, ∞) and $lim_{x \to \infty} e^{- x} f (x) = μ$ .

Theorem 4.If sequence (u_k) of fuzzy numbers converges toμ ∈ E¹, then (u_k) is Borel summable toμ.

Proof. Let u_k → μ. Since (u_k) is bounded, there exists K > 0 such that $D (u_{k}, \bar{0}) \leq K$ for all $k \in ℕ$ . Then we can get $\begin{matrix} \sum_{k = 0}^{\infty} D (\frac{x^{k}}{k!} u_{k}, \bar{0}) = \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} D (u_{k}, \bar{0}) \leq K \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} = {Ke}^{x}, \end{matrix}$ and so $\sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{k}$ is convergent by Theorem 3. Also, for given ɛ > 0 there exists N = N (ɛ) such that $D (u_{n}, μ) < \frac{ɛ}{2}$ whenever n > N, and for n ≤ N there exists M > 0 such that D (u_n, μ) ≤ M. So, by Remark 1 and Theorem 2, we get that $\begin{matrix} D (e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{k}, μ) & = & D (e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{k}, e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} μ) \\ \leq & e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} D (u_{k}, μ) \\ = & e^{- x} \sum_{k = 0}^{N} \frac{x^{k}}{k!} D (u_{k}, μ) \\ + e^{- x} \sum_{k = N + 1}^{\infty} \frac{x^{k}}{k!} D (u_{k}, μ) \\ < & {Me}^{- x} \sum_{k = 0}^{N} \frac{x^{k}}{k!} + \frac{ɛ}{2} \cdot \end{matrix}$ Since $e^{- x} \sum_{k = 0}^{N} \frac{x^{k}}{k!} \to 0$ as x→ ∞, there exists an x₀ > 0 such that $e^{- x} \sum_{k = 0}^{N} \frac{x^{k}}{k!} < \frac{ɛ}{2 M}$ whenever x > x₀. So $\begin{matrix} lim_{x \to \infty} D (e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{k}, μ) = 0, \end{matrix}$ which completes the proof.□

The following example shows that a Borel summable sequence does not have to converge.

Example 1. Let u = (u_k) be a sequence of fuzzy numbers such that $\begin{matrix} u_{k} (t) = {\begin{matrix} 3^{k} (t + (- 1)^{k + 1} k), & if (- 1)^{k} k \leq t \leq (- 1)^{k} k + \frac{1}{3^{k}}, \\ 2 - 3^{k} (t + (- 1)^{k + 1} k), & if (- 1)^{k} k + \frac{1}{3^{k}} \leq t \leq (- 1)^{k} k + \frac{2}{3^{k}}, \\ 0, & otherwise . \end{matrix} \end{matrix}$

Then λ-level set of u_k is $\begin{matrix} [u_{k}]_{λ} = [(- 1)^{k} k + \frac{λ}{3^{k}}, (- 1)^{k} k + \frac{2 - λ}{3^{k}}] . \end{matrix}$

Now since $\begin{matrix} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{k}^{-} (λ) & = & - x \sum_{k = 1}^{\infty} \frac{(- x)^{k - 1}}{(k - 1)!} + λ \sum_{k = 0}^{\infty} (\frac{(x / 3)^{k}}{k!}) \\ = & - {xe}^{- x} + λ e^{x / 3} \end{matrix}$ and $\begin{matrix} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{k}^{+} (λ) & = & - x \sum_{k = 1}^{\infty} \frac{(- x)^{k - 1}}{(k - 1)!} + (2 - λ) \sum_{k = 0}^{\infty} (\frac{(x / 3)^{k}}{k!}) \\ = & - {xe}^{- x} + (2 - λ) e^{x / 3} \end{matrix}$

converge uniformly in λ, $f (x) = \sum \frac{x^{k}}{k!} u_{k}$ converges by Definition 1 and

$[f (x)]_{λ} = [- {xe}^{- x} + λ e^{x / 3}, - {xe}^{- x} + (2 - λ) e^{x / 3}] .$

Since $\begin{matrix} D (e^{- x} f (x), \bar{0}) & = & sup_{λ \in [0, 1]} d ([e^{- x} f (x)]_{λ}, [\bar{0}]_{λ}) \\ = & sup_{λ \in [0, 1]} max {| - {xe}^{- 2 x} + λ e^{- 2 x / 3} |, \\ | - {xe}^{- 2 x} + (2 - λ) e^{- 2 x / 3} |} \\ = & sup_{λ \in [0, 1]} (- {xe}^{- 2 x} + (2 - λ) e^{- 2 x / 3}) \\ = & - {xe}^{- 2 x} + 2 e^{- 2 x / 3}, \end{matrix}$ we have that $lim_{x \to \infty} e^{- x} f (x) = \bar{0}$ .

This shows that sequence (u_k) is Borel summable to fuzzy number $\bar{0}$ , but not convergent. □

Now, we give the conditions under which Borel summable sequences of fuzzy numbers are also convergent and bounded.

Theorem 5.If a sequence (u_k) of fuzzy numbers is Borel summable to a fuzzy numberμand $\sqrt{k} D (u_{k - 1}, u_{k})$ = o (1) , then (u_k) converges toμ.

Proof. Let the sequence (u_k) be Borel summable to a fuzzy number μ. Since (u_k) is Borel summable, $\sum \frac{x^{k}}{k!} u_{k} = f (x)$ exists in E¹ and $lim_{x \to \infty} e^{- x}$ f (x) = μ. By Remark 1 and Theorem 2 we get $\begin{matrix} D (u_{n}, μ) & \leq & D (u_{n}, e^{- x} f (x)) + D (e^{- x} f (x), μ) \\ = & D (e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{n}, e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} u_{k}) \\ + D (e^{- x} f (x), μ) \\ \leq & e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} D (u_{n}, u_{k}) + D (e^{- x} f (x), μ) \end{matrix}$ From the hypothesis of theorem for given ɛ > 0 there exist n₁ such that $D (u_{k - 1}, u_{k}) < \frac{ɛ}{6 \sqrt{k}}$ whenever k > n₁, and there exist M > 0 such that $D (u_{k - 1}, u_{k}) < \frac{M}{\sqrt{k}}$ for $k \in ℕ$ . So for k > n₁ we have $D (u_{k}, u_{n}) < \frac{ɛ}{3} \frac{| n - k |}{\sqrt{n}}$ and for $k \in ℕ$ $D (u_{k}, u_{n}) < 2 M \frac{| n - k |}{\sqrt{n}} \cdot$ Then $\begin{matrix} e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} D (u_{n}, u_{k}) & = & e^{- x} \sum_{k = 0}^{n_{1}} \frac{x^{k}}{k!} D (u_{n}, u_{k}) \\ + e^{- x} \sum_{k = n_{1} + 1}^{\infty} \frac{x^{k}}{k!} D (u_{n}, u_{k}) \\ < & 2 {Me}^{- x} \sum_{k = 0}^{n_{1}} \frac{x^{k}}{k!} \frac{| n - k |}{\sqrt{n}} \\ + \frac{ɛ}{3} e^{- x} \sum_{k = n_{1} + 1}^{\infty} \frac{x^{k}}{k!} \frac{| n - k |}{\sqrt{n}} \end{matrix}$ is obtained. Now for x = n we get that $\begin{matrix} D (u_{n}, μ) & < & 2 {Me}^{- n} \sum_{k = 0}^{n_{1}} \frac{n^{k}}{k!} \frac{| n - k |}{\sqrt{n}} \\ + \frac{ɛ}{3} e^{- n} \sum_{k = n_{1} + 1}^{\infty} \frac{n^{k}}{k!} \frac{| n - k |}{\sqrt{n}} + D (e^{- n} f (n), μ) \\ < & 2 {Me}^{- n} \sum_{k = 0}^{n_{1}} \frac{n^{k + 1 / 2}}{k!} \\ + \frac{ɛ}{3} \frac{e^{- n}}{\sqrt{n}} (\sum_{k = n_{1} + 1}^{n} \frac{n^{k}}{k!} (n - k) \\ + \sum_{k = n}^{\infty} \frac{n^{k}}{k!} (k - n)) + D (e^{- n} f (n), μ) \\ < & 2 M (n_{1} + 1) e^{- n} n^{n_{1} + 1 / 2} \\ + \frac{ɛ}{3} \frac{e^{- n}}{\sqrt{n}} {\frac{2 n^{n}}{(n - 1)!} - \frac{n^{n_{1} + 1}}{n_{1}!}} \\ + D (e^{- n} f (n), μ) \\ < & 2 M (n_{1} + 1) e^{- n} n^{n_{1} + 1 / 2} + \frac{ɛ}{3} \frac{2 e^{- n} n^{n}}{\sqrt{n} (n - 1)!} \\ + D (e^{- n} f (n), μ) . \end{matrix}$

Since 2M (n₁ + 1) e^-nn^n₁+1/2 → 0, $\frac{2 e^{- n} n^{n}}{\sqrt{n} (n - 1)!} \to \sqrt{\frac{2}{π}}$ and (u_k) is Borel summable to μ, the followings hold:

sep 5pt

There exists $n_{2} \in ℕ$ such that

$2 M (n_{1} + 1) e^{- n} n^{n_{1} + 1 / 2} < \frac{ɛ}{3}$ whenever n > n₂.

There exists $n_{3} \in ℕ$ such that $\frac{2 e^{- n} n^{n}}{\sqrt{n} (n - 1)!} < 1$ whenever n > n₃.

There exists $n_{4} \in ℕ$ such that $D (e^{- n} f (n), μ) < \frac{ɛ}{3}$ whenever n > n₄.

This means that there exists n₀ = max {n₁, n₂, n₃, n₄} such that for n > n₀ we have D (u_n, μ) < ɛ, which completes the proof.□

Theorem 6.If sequence (u_k) of fuzzy numbers is Borel summable to fuzzy numberμand $\sqrt{k} D (u_{k - 1}, u_{k})$ = O (1) , then (u_k) ∈ ℓ _∞ (F).

Proof. Let (u_k) be Borel summable to μ ∈ E¹ and $\sqrt{k} D (u_{k - 1}, u_{k}) = O (1)$ be satisfied. So as that in the proof of Theorem 5, taking x=n, we have $\begin{matrix} D (u_{n}, μ) & \leq & e^{- x} \sum_{k = 0}^{\infty} \frac{x^{k}}{k!} D (u_{n}, u_{k}) + D (e^{- x} f (x), μ) \\ < & 2 M \frac{e^{- n}}{\sqrt{n}} \sum_{k = 0}^{\infty} \frac{n^{k}}{k!} | n - k | + D (e^{- n} f (n), μ) \\ = & 2 M \frac{e^{- n}}{\sqrt{n}} (2 \frac{n^{n + 1}}{n!}) + D (e^{- n} f (n), μ) . \end{matrix}$ Since limit of expression on the right hand side exists as n→ ∞, we have that D (u_n, μ) is bounded and so is $D (u_{n}, \bar{0})$ . This completes the proof.□

In view of the results obtained above for the sequences of fuzzy numbers, as corollaries, we may define Borel summability method of series of fuzzy numbers and give related Tauberian results.

Definition 3. A series ∑u_k of fuzzy numbers is said to be Borel summable to a fuzzy number μ if the sequence of partial sums of the series ∑u_k is Borel summable to μ.

Corollary 1.If series ∑u_kof fuzzy numbers converges to fuzzy numberμ, then it is Borel summable toμ.

Corollary 2.If series ∑u_kof fuzzy numbers is Borel summable to fuzzy numberμand $\sqrt{k} D (u_{k}, \bar{0}) = o (1),$ then ∑u_k = μ.

Corollary 3.If series ∑u_kof fuzzy numbers is Borel summable to fuzzy numberμand $\sqrt{k} D (u_{k}, \bar{0}) = O (1),$ then (u_k) ∈ bs (F).

3 Conclusion

In this paper, we have introduced the Borel summability method for sequences of fuzzy numbers and given some conditions for sequences of fuzzy numbers under which convergence follows from Borel summability. Also, as corollaries, we have obtained similar results for Borel summability of series of fuzzy numbers. Advantage of the Borel summability method is that one can handle fast-growing divergent series of fuzzy numbers even with factorially growing terms. Such divergent series are remarkably ubiquitous in the classical analysis and researchers used Borel type summability methods as a tool to sum these series occuring as solutions of problems on the theory of differential equations [4 , 36]. On the other hand, the theory of fuzzy differential equations has been introduced recently and some concepts known from the classical case have been extended to the theory [1 , 58]. At this point, current paper, pioneering the concept of Borel type summability methods on fuzzy number space, is likely to help researchers who deal with fuzzy analogues of above mentioned problems on the theory of fuzzy differential equations.

Footnotes

Acknowledgments

The authors would like to express their gratitude to associate editor and referees for many helpful and detailed suggestions on the main results and presentation of the paper.

References

Allahviranloo

and Hooshangian

, Fuzzy generalized H-differential and applications to fuzzy differential equations of second-order, Journal of Intelligent and Fuzzy Systems26(4) (2014), 1951–1967.

Altin

, Mursaleen

and Altinok

, Statistical summability (C; 1)-for sequences of fuzzy real numbers and a Tauberian theorem, Journal of Intelligent and Fuzzy Systems21 (2010), 379–384.

Altinok

and Çolak

and Altin

, On the class of λ-statistically convergent difference sequences of fuzzy numbers, Soft Computing16(6) (2012), 1029–1034.

Balser

, Summability of formal power series solutions of ordinary and partial differential equations, Functional Differential Equations8(1-2) (2001), 11–24.

Bede

and Gal

S.G.

, Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems147 (2004), 385–403.

Boos

, Oxford Mathematical Monographs, Classical and modern methods in summability2000.

Borwein

, On method of summability based on power series, Proc Roy Soc Edinburgh64 (1957), 342–349.

Borwein

, On Relations between weighted Mean and power series methods of summability, Journal of Mathematica Analysis and Applications139 (1989), 178–186.

Borwein

and Kratz

, A one-sided tauberian theorem for the Borel summability method, Journal of Mathematical Analysis and Applications293 (2004), 285–292.

10.

Buckley

J.J.

and Feuring

, Introduction to fuzzy partial differential equations, Fuzzy Sets and Systems105 (1999), 241–248.

11.

Çanak

İ.

and Totur

Ü.

, Some Tauberian theorems for Borel summability methods, Applied Mathematics Letters23 (2010), 302–305.

12.

Çanak

İ.

, On the Riesz mean of sequences of fuzzy real numbers, Journal of Intelligent and Fuzzy Systems26(6) (2014), 2685–2688.

13.

ÇCanak

İ.

, Tauberian theorems for Cesáro summability of sequences of fuzzy number, Journal of Intelligent and Fuzzy Systems27(2) (2014), 937–942.

14.

Çanak

İ.

, Some conditions under which slow oscillation of a sequence of fuzzy numbers follows from Cesaro summability of its generator sequence, Iranian Journal of Fuzzy Systems11(4) (2014), 15–22.

15.

Çanak

İ.

, Hölder summability method of fuzzy numbers and a Tauberian theorem, Iranian Journal of Fuzzy Systems11(4) (2014), 87–93.

16.

Çanak

İ.

, Tauberian theorems for the product of Borel and Holder summability methods, Analele Ştiintifice ale Universitatii “Al I Cuza” din Iaşi (2015). DOI: 10.2478/aicu-2014-0039

17.

Çolak

, Altin

and Mursaleen

, On some sets of difference sequences of fuzzy numbers, Soft Computing15(4) (2011), 787–793.

18.

Diamond

and Kloeden

, Metric spaces of fuzzy sets, Fuzzy Sets and Systems35 (1990), 241–249.

19.

Dubois

and Prade

, (Eds.), The Handbooks of Fuzzy Sets Series, Springer, 1998–2000.

20.

Dubois

and Prade

, The legacy of 50 years of fuzzy sets: A discussion, Fuzzy Sets and Systems (2015). DOI: 10.1016/j.fss.2015.09.004

21.

Goetschel

and Voxman

, Elementary fuzzy calculus, Fuzzy Sets and Systems18 (1986), 31–43.

22.

Hardy

G.H.

, Divergence Series, Oxford, 1949.

23.

Hashemi

M.S.

and Malekinagad

, Series solution of fuzzy wavelike equations with variable coefficients, Journal of Intelligent and Fuzzy Systems25(2) (2013), 415–428.

24.

Hazarika

, Lacunary difference ideal convergent sequence spaces of fuzzy numbers, Journal of Intelligent and Fuzzy Systems25(1) (2013), 157–166.

25.

Hazarika

, Lacunary ideal convergent double sequences of fuzzy real numbers, Journal of Intelligent and Fuzzy Systems27(1) (2014), 495–504.

26.

Hibino

, Borel summability of divergent solutions for singular first-order partial differential equations with variable coefficients, Journal of Differential Equations227 (2006), 499–533.

27.

Janizade-Haji

, Zare

H.K.

, Eslamipoor

and Sepehriar

, A developed distance method for ranking generalized fuzzy numbers, Neural Computing and Applications25(3-4) (2014), 727–731.

28.

Karsli

and Tiryaki

I.U.

, Convergence of Nonlinear Singular Integral Operators to the Borel Differentiable Functions, International Congress in Honour of Professor Ravi P. Agarwal, 2014.

29.

Kim

Y.K.

and Ghil

B.M.

, Integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems86 (1997), 213–222.

30.

Kratz

, A tauberian theorem for the Borel method, Indian Journal of Mathematics42 (2000), 297–307.

31.

, Zhao

and Yan

, The Cauchy problem of fuzzy differential equations under generalized differentiability, Fuzzy Sets and Systems200 (2012), 1–24.

32.

Lutz

D.A.

, Miyake

and Schäfke

, On the Borel summability of divergent solutions of the heat equation, Nagoya Mathematical Journal154 (1999), 1–29.

33.

Łysik

, The Borel summable solutions of heat operators on a real analytic manifold, Journal of Mathematical Analysis and Applications410 (2014), 117–123.

34.

Malek

, On the summability of formal solutions of linear partial differential equations, Journal of Dynamical and Control Systems11(3) (2005), 389–403.

35.

Matloka

, Sequences of fuzzy numbers, Busefal28 (1986), 28–37.

36.

Miyake

, Borel summability of divergent solutions of the Cauchy problem to non-Kovaleskian equations, In: Partial Differential Equations and Their Applications, 1999, pp. 225–239.

37.

Nanda

, On sequence of fuzzy numbers, Fuzzy Sets and Systems33 (1989), 123–126.

38.

Önder

, Sezer

S.A.

and Çanak

, A Tauberian theorem for the weighted mean method of summability of sequences of fuzzy numbers, Journal of Intelligent and Fuzzy Systems28 (2015), 1403–1409.

39.

Patterson

R.F.

, Sen

and Rhoades

B.E.

, A Tauberian theorem for a generalized power series method, Applied Mathematics Letters18 (2005), 1129–1133.

40.

Salahshour

and Haghi

, Solving fuzzy heat equation by fuzzy laplace transforms, Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications Communications in Computer and Information, Volume 81, 2010, pp. 512–521.

41.

Song

and Wu

, Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations, Fuzzy Sets and Systems110 (2000), 55–67.

42.

Stadtmüller

and Tali

, On certain families of generalized nörlund methods and power series methods, Journal of Mathematical Analysis and Applications238 (1999), 44–66.

43.

Stadtmüller

and Tali

, A family of generalized Nörlund methods and related power series methods applied to double sequences, Mathematische Nachrichten282(2) (2009), 288–306.

44.

Subrahmanyam

P.V.

, Cesáro summability of fuzzy real numbers, Journal of Analysis7 (1999), 159–168.

45.

Talo

Ö.

and Başar

, Determination of the duals of classical sets of sequences of fuzzy numbers and related matrix transformations, Computers and Mathematics with Applications58(4) (2009), 717–733.

46.

Talo

Ö.

and Çakan

, On the Cesáro convergence of sequences of fuzzy numbers, Applied Mathematics Letters25 (2012), 676–681.

47.

Talo

Ö.

and Başar

, On the slowly decreasing sequences of fuzzy numbers, Abstract and Applied Analysis (2013), 1–7. Article ID 868919, doi: 10.1155/2013/891986

48.

Tam

, Cesáro and general euler-borel summability, Proceedings of the American Mathematical Society115(3) (1992), 747–755.

49.

Tripathy

B.C.

and Baruah

, Nörlund and Riesz mean of sequences of fuzzy real numbers, Applied MathematicsLetters23 (2010), 651–655.

50.

Tripathy

B.C.

and Dutta

A.J.

, Lacunary bounded variation sequence of fuzzy real numbers, Journal of Intelligent and Fuzzy Systems24(1) (2013), 185–189.

51.

Tripathy

B.C.

and Sen

, On fuzzy I-convergent difference sequence space, Journal of Intelligent and Fuzzy Systems25(3) (2013), 643–647.

52.

Tripathy

B.C.

, Braha

N.L.

and Dutta

A.J.

, A new class of fuzzy sequences related to the ℓ_p space defined by Orlicz function, Journal of Intelligent and Fuzzy Systems26(3) (2014), 1273–1278.

53.

Watson

, A Tauberian theorem for discrete power series methods, Analysis22 (2002), 361–365.

54.

Yavuz

and Talo

Ö.

, Abel summability of sequences of fuzzy numbers, Soft Computing (2014). DOI: 10.1007/s00500-014-1563-7

55.

Zadeh

L.A.

, Fuzzy sets, Information and Control8 (1965), 29–44.

56.

Zhan

, Liu

and Davvaz

, A new rough set theory: Rough soft hemirings, Journal of Intelligent and Fuzzy Systems28(4) (2015), 1687–1697.

57.

Zhan

, The Uncertainties of Ideal Theory on Hemirings, Science Press, 2015.

58.

Zhang

and Sun

, Practical stability of fuzzy differential equations with the second type of Hukuhara derivative, Journal of Intelligent and Fuzzy Systems (2015), 1–7. DOI: 10.3233/IFS-151596

59.

Zimmermann

H.-J.

, Fuzzy Set Theory - and its Applications, 4th ed, Kluwer Academic Publishers, 2001.