A Tauberian theorem for the weighted mean method of improper integrals of fuzzy-number-valued functions

Abstract

Let 0 ≠ p (x) be a nondecreasing real valued function on [0, ∞) such that p (0) =0 and $\underset{x \to \infty}{lim inf} \frac{p (λ x)}{p (x)} > 1 for every λ > 1 .$

Given a fuzzy-number-valued continuous function f (x) on [0, ∞), we define $s (x) : = \int_{0}^{x} f (t) dt and σ (x) : = \frac{1}{p (x)} \int_{0}^{x} s (t) dp (t), x > 0 .$

It is known that the limit $lim_{x \to \infty} s (x) = μ$ exists, then the limit $lim_{x \to \infty} σ (x) = μ$ also exists. But the converse of this implication need not be satisfied in general.

In this paper, our goal is to find a condition under which the existence of $lim_{x \to \infty} σ (x) = μ$ follows from that of $lim_{x \to \infty} s (x) = μ$ .

As special cases, we obtain some Tauberian conditions of slowly decreasing type and Landau type for the Cesàro summability method of improper integrals of fuzzy-number-valued functions.

Keywords

Fuzzy-number-valued functions fuzzy Riemann-Stieltjes integral Tauberian theorems weighted mean method of integrals slowly decreasing function

1 Introduction

Since Zadeh [28] introduced the concept of fuzzy sets, the theory of fuzzy sets has attracted the attention of many researchers from various branches of mathematics. Recently, there has been an increasing interest on the summability methods of sequences of fuzzy numbers. Dubois and Prade [9] introduced the fuzzy numbers and defined the basic operations of addition, subtraction, multiplication and division. Matloka [17] introduced bounded and convergent sequences of fuzzy numbers and proved that every convergent sequence of fuzzy numbers is bounded. Subrahmanyam [23] defined the Cesàro summability method of fuzzy numbers and obtained fuzzy analogues of some Tauberian theorems for Cesàro summability method. Talo and Çakan [25], Çanak [5 –7] and Talo and Başar [26] proved some Tauberian theorems for Cesàro summability of sequences of fuzzy numbers. Tripathy and Baruah [24] introduced the weighted mean method of sequences of fuzzy numbers and obtained fuzzy analogues of classical Tauberian theorems for the weighted mean method. Later, Çanak [8] obtained a fuzzy analogue of a Tauberian theorem due to Móricz and Rhoades [19].

Yavuz et al. [27] introduced Cesàro summability of integrals of fuzzy-number-valued functions and recovered convergence of an improper fuzzy Riemann integral from its Cesàro summability under one-sided Tauberian conditions. They also presented fuzzy analogues of Landau’s one-sided type Tauberian conditions and slow decreasing.

Dubios and Prade [11 –13] defined the integral for fuzzy-number-valued functions. Nanda [21] and Hseing- Chung Wu [16] defined the Riemann-Stieltjes integral for fuzzy-number-valued functions. Since their definitions were too complex, Ren and Wu [22] redefined the Riemann-Stieltjes integral for fuzzy-number-valued functions using the integral sums as in the classical Riemann-Stieltjes integral.

In [27], Yavuz et al. introduced Cesàro summability of integrals of fuzzy-number-valued functions and recovered convergence of an improper fuzzy Riemann integral from its Cesàro summability under one-sided Tauberian conditions. They also presented fuzzy analogues of Landau’s one-sided type Tauberian conditions and slow decreasing.

In this paper, we recall some basic results of fuzzy numbers and fuzzy-number-valued functions and introduce the concept of the weighted mean method of improper integrals of fuzzy-number-valued functions in Section 2. In Section 3, we obtain a necessary and sufficient condition for the method $(\bar{N}, p)$ to be regular, and give a Tauberian theorem for the weighted mean method of improper integrals of fuzzy-number-valued functions. We also list fuzzy analogues of some Tauberian conditions of slowly decreasing type and Landau type for the Cesàro summability method of improper integrals of fuzzy-number-valued functions.

2 Preliminaries

We now give the following definitions and notations that are needed in the sequel. For the sake of completeness of the paper we give our study in Section 3.

In [14], Goetschel and Voxman introduced the concept of fuzzy numbers as follows:

A fuzzy number is a fuzzy set on $ℝ$ , i.e. a mapping $u : ℝ \to [0, 1]$ with the following properties:

u is normal; i.e. there is a unique element $x_{0} \in ℝ$ such that u (x₀) =1 .

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

u is upper semicontinuous.

The support of u, $[u]_{0} : = \bar{{x \in ℝ : u (x) > 0}}$ is compact, where $\bar{{x \in ℝ : u (x) > 0}}$ denotes the closure of the set ${x \in ℝ : u (x) > 0}$ 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 $[u]_{α} : = {\begin{matrix} {x \in ℝ : u (x) \geq α}, & 0 < α \leq 1 \\ \bar{{x \in ℝ : u (x) > α}}, & α = 0 . \end{matrix}$

It is well known (see [10]) that [u] _α is a closed, bounded, and nonempty interval for each α ∈ [0, 1] with [u] _β ⊂ [u] _α if α < β . The set [u] _α is also defined by [u] _α : = [u^- (α) , u⁺ (α)] .

The operations addition, subtraction and scalar multiplication on the set of fuzzy numbers are defined as follows:

Let u, v ∈ E¹, $k \in ℝ$ and α ∈ [0, 1]. Then $\begin{matrix} [u + v]_{α} & : = & [u^{-} (α) + v^{-} (α), u^{+} (α) + v^{+} (α)], \\ [u - v]_{α} & : = & [u^{-} (α) - v^{+} (α), u^{+} (α) - v^{-} (α)], \\ [ku]_{α} & : = & k [u]_{α} . \end{matrix}$

The set of all real numbers can be embedded in E¹. For $r \in ℝ$ , $\bar{r} \in E^{1}$ is defined by $\bar{r} (x) : = {\begin{matrix} 1, & x = r \\ 0, & x \neq r . \end{matrix}$

In [14], Goetschel and Voxman presented another representation of a fuzzy number as a pair of functions that satisfy some properties.

Theorem 2.1. [14] Let u ∈ E¹ and let [u] _α = [u^- (α) , u⁺ (α)]. Then the functions $u^{-}, u^{+} : [0, 1] \to ℝ$ , defining the endpoints of the α-level sets, satisfy following conditions:

$u^{-} (α) \in ℝ$ is a bounded, non-decreasing and left continuous function on (0, 1].

$u^{+} (α) \in ℝ$ is a bounded, non-increasing and left continuous function on (0, 1].

The functionsu^- (α) andu⁺ (α) are right continuous atα = 0.

u^- (1) ≤ u⁺ (1).

Conversely, if the pair of functions f and g satisfy 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 $u (x) : = sup_{α \in [0, 1]} {α : f (α) \leq x \leq g (α)} .$

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

Lemma 2.2. [4]

The addition of fuzzy numbers is associative and commutative, i.e., u + v = v + uandu + (v + w) = (u + v) + w, for allu, v, w ∈ E¹ .

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

With respect to +, none of $u \in E^{1} \ ℝ$ has opposite inE¹.

For any $a, b \in ℝ$ witha . b ≥ 0 and any u ∈ E¹, we have (a + b) u = a . u + b . u. For general $a, b \in ℝ$ this property does not hold.

For any $a \in ℝ$ andu, v ∈ E¹, we havea . (u + v) = a . u + a . v.

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

As a conclusion we obtain by Lemma 2.2 that the space of fuzzy numbers is not a linear space.

The most well known, and also the most employed metric in the space of fuzzy numbers is the Hausdorff distance. The Hausdorff fuzzy distance for fuzzy numbers is based on the classical Hausdorff distance between compact convex subsets of $ℝ^{n}$ . Let us recall the definition of the Hausdorff distance in the case when A = [A^-, A⁺], B = [B^-, B⁺] are two intervals (we denote A, B ∈ W, where W is the set of all closed and bounded intervals) the Hausdorf distance on W is $d (A, B) : = max {| A^{-} - B^{-} |, | A^{+} - B^{+} |} .$

It is known that with respect to the Haussdorf distance, W is a complete separable metric space (cf. Nanda [20]). Now, we may define the metric D on the space of fuzzy numbers by means of the Hausdorff metric d.

Definition 2.3. [4] Let $D : E^{1} \times E^{1} \to ℝ_{+}$ , $\begin{matrix} D (u, v) = sup_{α \in [0, 1]} d ([u]_{α}, [v]_{α}) . \end{matrix}$

Then D is called Hausdorff distance between fuzzy numbers.

Proposition 2.4.[4] Let u, v, w, z ∈ E¹ and $k \in ℝ$ . Then the following statements hold true.

(E¹, D) is a complete metric space.

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

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

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

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

For u, v ∈ E¹, partial ordering relation on E¹ is defined in [3] as follows:

u ⪯ v if and only if [u] _α ⪯ [v] _α, i.e. u^- (α) ≤ v^- (α) and u⁺ (α) ≤ v⁺ (α) for any α ∈ [0, 1] .

We say that u ≺ v if u ⪯ v and there exists α₀ ∈ [0, 1] such that u^- (α₀) < v^- (α₀) or u⁺ (α₀) < v⁺ (α₀) .

We say that u, v ∈ E¹ are incomparable if neither u ⪯ v nor v ⪯ u.

Lemma 2.5. [3] Given two fuzzy numbers u and v the following statements are equivalent:

D (u, v) ≤ ɛ,

$u - \bar{ɛ} ⪯ v ⪯ u + \bar{ɛ}$ , whereɛ > 0 .

Lemma 2.6. [26] Let u, v, w ∈ E¹. If u + w ⪯ v + w, then u ⪯ v .

We now quote the following definitions which will be needed in the sequel. It should be noted that throughout this paper the concepts of limit and continuity of a fuzzy-number-valued function will be considered in the metric space (E¹, D).

Definition 2.7. [2] Let f : [a, b] → E¹ be a fuzzy-number-valued function. Then f is said to be continuous at x₀ ∈ [a, b] if for every ɛ > 0 there exists a δ (ɛ) >0 such that D (f (x) , f (x₀)) < ɛ whenever x ∈ [a, b] with |x - x₀| < δ. Also, we say that f is continuous on [a, b] if it is continuous at every point x ∈ [a, b]. The space of all continuous functions on [a, b] is denoted by $C ([a, b], E^{1})$ .

It should be noted that if f : [a, b] → E¹ is continuous, then f is bounded, i.e. there is a C > 0 such that D (f (x) , μ) ≤ C for all x ∈ [a, b] and μ ∈ E¹ (cf. Anastassiou [1]).

In [14], Goetschel and Voxman defined the integration of fuzzy-number-valued functions in ways that parallel closely the corresponding definitions for real integration. In the sequel, they extended the elementary results which were given for real valued ordinary functions in the analysis to fuzzy-number-valued functions. We now give this definition and elementary results for fuzzy-number-valued functions.

Definition 2.8. [14] Let f : [a, b] → E¹ be bounded. Then f is said to be the fuzzy Riemann integrable to I ∈ E¹ if for every ɛ > 0 there exists a δ (ɛ) >0 such that for each partition P = {a = x₀ < x₁ < . . . < x_n = b} of [a, b] with the norms |P| < δ (ɛ) and for any points ξ_i ∈ [x_i, x_i+1], we have $D (\sum_{i = 0}^{n - 1} f (ξ_{i}) (x_{i + 1} - x_{i}), I) < ɛ .$

In this case, we denote $I = \int_{a}^{b} f (x) dx .$

Theorem 2.9. [14] If f : [a, b] → E¹, which has the parametric representation $[f (x)]_{α} = [f_{α}^{-} (x), f_{α}^{+} (x)]$ for each x ∈ [a, b], is continuous, then $\int_{a}^{b} f (x) dx$ exists and belongs to E¹, furthermore it holds ${[\int_{a}^{b} f (x) dx]}_{α} = [\int_{a}^{b} f_{α}^{-} (x) dx, \int_{a}^{b} f_{α}^{+} (x) dx] .$

Lemma 2.10. [1] If f : [a, b] → E¹ is continuous, then $s (x) = \int_{a}^{x} f (t) dt$ is continuous in x ∈ [a, b].

Definition 2.11. [2] Let f (x) be a fuzzy-number-valued function defined on the unbounded interval [a, ∞). Then, we define $\int_{a}^{\infty} f (x) dx = lim_{t \to \infty} \int_{a}^{t} f (x) dx$ provided the limit on the right-hand side exists in E¹, in which case we say the improper fuzzy integral converges and is equal to the value of limit. Otherwise, we say the integral diverges.

Instead of investigating the properties which are given for the fuzzy Riemann integrable functions separately, we shall immediately consider a more general situation.

Definition 2.12. [22] Let f : [a, b] → E¹ be a bounded fuzzy-number-valued function and g (x) be a nondecreasing real valued function on [a, b]. Then f is said to be the fuzzy Riemann-Stieltjes integrable with respect to g (x) to I ∈ E¹ if for every ɛ > 0 there exists a δ (ɛ) >0 such that for each partition P = {a = x₀ < x₁ < . . . < x_n = b} of [a, b] with the norms |P| < δ (ɛ) and for any points ξ_i ∈ [x_i, x_i+1], we have $D (\sum_{i = 0}^{n - 1} f (ξ_{i}) [g (x_{i + 1}) - g (x_{i})], I) < ɛ .$

In this case, we denote $I = \int_{a}^{b} f (x) dg (x)$ or briefly, $I = \int_{a}^{b} fdg .$

By taking g (x) = x, we see that the fuzzy Riemann integral is a special case of the fuzzy Riemann-Stieltjes integral.

Lemma 2.13. [15] If f₁, f₂ : [a, b] → E¹ are continuous and g is a nondecreasing real-valued function on [a, b], then the function $F : [a, b] \to ℝ_{+}$ defined by F (x) = D (f₁ (x) , f₂ (x)) is continuous on [a, b] and $D (\int_{a}^{b} f_{1} dg, \int_{a}^{b} f_{2} dg) \leq \int_{a}^{b} Fdg .$

We now define the concept of weighted mean method of improper integrals of fuzzy-number-valued functions and slow decreasing of fuzzy-number-valued functions.

Definition 2.14. Let 0 ≠ p (x) be a nondecreasing real valued function on [0, ∞) such that p (0) =0 and $\underset{x \to \infty}{lim inf} \frac{p (λ x)}{p (x)} > 1 for every λ > 1 .$

Given a fuzzy-number-valued continuous function f (x) on [0, ∞), we define $s (x) : = \int_{0}^{x} f (t) dt$ . The weighted mean of the function s (x) is defined by $σ (x) = \frac{1}{p (x)} \int_{0}^{x} s (t) dp (t) x > 0 .$

The improper integral $\int_{0}^{\infty} f (t) dt$ is said to be integrable by weighted mean method determined by the function p (x) or briefly, $(\bar{N}, p)$ integrable to μ ∈ E¹, if for every ɛ > 0 there exists a x₀ = x₀ (ɛ) ≥0 such that $D (σ (x), μ) \leq ɛ {for x > x_{0} .$

Weighted mean methods also called Riesz methods or $(\bar{N}, p)$ methods in the literature. If p (x) = x for all x, then $(\bar{N}, p)$ integrability method reduces to Cesàro integrability method.

Since the existence of the improper integral $\int_{0}^{\infty} f (t) dt = μ$ implies that of $lim_{x \to \infty} σ (x) = μ$ , we say that the $(\bar{N}, p)$ method is regular.

Definition 2.15. [27] A fuzzy-number-valued function s (x) is said to be slowly decreasing if for every ɛ > 0 there exist x₀ ≥ 0 and λ > 1, such that $s (t) ⪰ s (x) - \bar{ɛ}$ (1) whenever x₀ < x ≤ t ≤ λx or equivalently, for every ɛ > 0 there exist x₀ ≥ 0 and 0 < λ < 1 such that for every x > x₀ $s (x) ⪰ s (t) - \bar{ɛ}$ (2) whenever λx ≤ t ≤ x .

The next lemma which restricts p (x) is given by Móricz [18] and we use it for the proof of our results.

Lemma 2.16.[18] If p (x) is a nondecreasing real valued function on [0, ∞), then the conditions $\underset{x \to \infty}{lim inf} \frac{p (λ x)}{p (x)} > 1 for every λ > 1$ (3) and $\underset{x \to \infty}{lim inf} \frac{p (x)}{p (λ x)} > 1 for every 0 < λ < 1$ (4)are equivalent.

It should be note that (3) implies that $lim_{x \to \infty} p (x) = \infty$ .

3 Main results

In this section, we first prove that the method $(\bar{N}, p)$ is regular under the condition p (x)→ ∞ as x→ ∞. Then we give a Tauberian theorem for the weighted mean method of improper integrals of fuzzy-number-valued functions.

Theorem 3.1. The method $(\bar{N}, p)$ is regular if p (x)→ ∞ as x→ ∞.

Proof. Assume that $\int_{0}^{\infty} f (t) dt$ converges to μ ∈ E¹. In this case, we can write $lim_{x \to \infty} s (x) = \int_{0}^{\infty} f (t) dt = μ$ . Since we have $lim_{x \to \infty} D (s (x), μ) = 0$ , for any given ɛ > 0 there exists a x₀ = x₀ (ɛ) ≥0 such that $D (s (x), μ) \leq \frac{ɛ}{2}$ for x > x₀. In addition to this, since every fuzzy-number-valued continuous function s (x) is fuzzy bounded in x ∈ [0, ∞), there exists a C > 0 such that D (s (x) , μ) ≤ C for all x ∈ [0, ∞). Therefore, by Lemma 2.13 we obtain $\begin{matrix} D (σ (x), μ) \\ = D (\frac{1}{p (x)} \int_{0}^{x} s (t) dp (t), μ) \\ = D (\frac{1}{p (x)} \int_{0}^{x} s (t) dp (t), \frac{1}{p (x)} \int_{0}^{x} μ dp (t)) \\ = \frac{1}{p (x)} D (\int_{0}^{x} s (t) dp (t), \int_{0}^{x} μ dp (t)) \\ \leq \frac{1}{p (x)} \int_{0}^{x} D (s (t), μ) dp (t) \\ = \frac{1}{p (x)} \int_{0}^{x_{0}} D (s (t), μ) dp (t) \\ + \frac{1}{p (x)} \int_{x_{0}}^{x} D (s (t), μ) dp (t) \\ \leq \frac{1}{p (x)} \int_{0}^{x_{0}} Cdp (t) + \frac{1}{p (x)} \int_{x_{0}}^{x} \frac{ɛ}{2} dp (t) \\ = C \frac{p (x_{0})}{p (x)} + \frac{ɛ}{2} (1 - \frac{p (x_{0})}{p (x)}) \\ \leq C \frac{p (x_{0})}{p (x)} + \frac{ɛ}{2} \end{matrix}$

Since $lim_{x \to \infty} \frac{p (x_{0})}{p (x)} = 0$ , there exists a x₁ = x₁ (ɛ) ≥0 such that $| \frac{p (x_{0})}{p (x)} | < \frac{ɛ}{2 C}$ for x > x₁. Hence, for any given ɛ > 0 there exists a x₂ = max {x₀, x₁} such that $D (σ (x), μ) \leq \frac{ɛ}{2}$ for x > x₂. This implies that $lim_{x \to \infty} σ (x) = μ .$ □

In the next example, we construct a fuzzy-number-valued function f on [0, ∞) and show that if the improper integral $\int_{0}^{\infty} f (u) du$ converges, then it is integrable by the weighted mean method determined by the function p (x) = x.

Example 3.2. Consider the fuzzy-number-valued function f on the unbounded interval [0, ∞) defined by

$\begin{matrix} (f (u)) (t) \\ = {\begin{matrix} 1 + \frac{{((u + 3) (u + 1))}^{2}}{u^{2} + 4 u + 5} t & if t \in [\frac{- (u^{2} + 4 u + 5)}{{((u + 3) (u + 1))}^{2}}, 0] \\ 1 - \frac{{((u + 3) (u + 1))}^{2}}{u^{2} + 4 u + 5} t & if t \in [0, \frac{u^{2} + 4 u + 5}{{((u + 3) (u + 1))}^{2}}] \\ 0 & otherwise \end{matrix} \end{matrix}$ for all u ≥ 0. Since the fuzzy-number-valued function f with the parametric representation $[f (u)]_{α} = [f_{α}^{-} (u), f_{α}^{+} (u)]$ for each u ∈ [0, ∞) is continuous, it is integrable on the same interval. Using the components $f_{α}^{-} (u)$ and $f_{α}^{+} (u)$ of the parametric representation of f, where $f_{α}^{-} (u) = (α - 1) (u^{2} + 4 u + 5) {((u + 3) (u + 1))}^{- 2}$ and $f_{α}^{+} (u) = (1 - α) (u^{2} + 4 u + 5) {((u + 3) (u + 1))}^{- 2},$ we find that $\begin{matrix} s_{α}^{-} (t) & = & \int_{0}^{t} f_{α}^{-} (u) du \\ = & \frac{(1 - α)}{2} (\frac{1}{t + 3} + \frac{1}{t + 1} - \frac{3}{4}) \end{matrix}$ and $\begin{matrix} s_{α}^{+} (t) & = & \int_{0}^{t} f_{α}^{+} (u) du \\ = & \frac{(α - 1)}{2} (\frac{1}{t + 3} + \frac{1}{t + 1} - \frac{3}{4}) . \end{matrix}$

It easily follows that the improper integrals $\int_{0}^{\infty} f_{α}^{-} (u) du$ and $\int_{0}^{\infty} f_{α}^{+} (u) du$ converge to 3 (α - 1)/8 and 3 (1 - α)/8, respectively. If we take [μ] _α = [3 (α - 1)/8, 3 (1 - α)/8], then the improper integral $\int_{0}^{\infty} f (u) du$ converges to the fuzzy number μ, which corresponds to the α-level set [μ] _α, defined by $μ (t) = {\begin{matrix} \frac{8 t}{3} + 1 & if t \in [- \frac{3}{8}, 0] \\ 1 - \frac{8 t}{3} & if t \in [0, \frac{3}{8}] \\ 0 & otherwise . \end{matrix}$

On the other hand, if we choose p (x) = x, then we obtain the components of the parametric representation of weighted mean of the function s (t) as follows: $\begin{matrix} σ_{α}^{-} (x) & = & \frac{1}{x} \int_{0}^{x} s_{α}^{-} (t) dt \\ = & \frac{(1 - α)}{2 x} ln (\frac{(x + 1) (x + 3)}{3}) + \frac{3 (α - 1)}{8} \end{matrix}$ and $\begin{matrix} σ_{α}^{+} (x) & = & \frac{1}{x} \int_{0}^{x} s_{α}^{-} (t) dt \\ = & \frac{(α - 1)}{2 x} ln (\frac{(x + 1) (x + 3)}{3}) + \frac{3 (1 - α)}{8} . \end{matrix}$

It then easily follows that $σ_{α}^{-} (x)$ and $σ_{α}^{+} (x)$ converge to 3 (α - 1)/8 and 3 (1 - α)/8 as x→ ∞, respectively. Hence, we obtain $lim_{x \to \infty} D (σ (x),$ μ) =0, that is, σ (x) converge to μ. In other words, the improper integral $\int_{0}^{\infty} f (u) du$ is integrable by the weighted mean method determined by the function p (x) = x to the fuzzy number μ.

It is an immediate consequence of Example 3.2 that if the improper integral $\int_{0}^{\infty} f (u) du$ converges to a fuzzy number, then this integral is integrable by the weighted mean method determined by the function p (x) to the same number. However, that the converse is not always true is seen by a careful examination of the following example.

Example 3.3. Consider the fuzzy-number-valued function f on the unbounded interval [0, ∞) defined by $\begin{matrix} (f (u)) (t) \\ = {\begin{matrix} 1 + \frac{e^{u}}{2 u + 1} (t - ω (u)) & if t \in [ω (u) - \frac{(2 u + 1)}{e^{u}}, ω (u)] \\ 1 - \frac{e^{u}}{2 u + 1} (t - ω (u)) & if t \in [ω (u), ω (u) + \frac{(2 u + 1)}{e^{u}}] \\ 0 & otherwise \end{matrix} \end{matrix}$

where $ω (u) = \sqrt{1 + cos 3 u}$ for all u ≥ 0. Since the fuzzy-number-valued function f with the parametric representation $[f (u)]_{α} = [f_{α}^{-} (u), f_{α}^{+} (u)]$ for each u ∈ [0, ∞) is continuous, it is integrable on the same interval. Using the components $f_{α}^{-} (u)$ and $f_{α}^{+} (u)$ of the parametric representation of f, where $f_{α}^{-} (u) = \sqrt{1 + cos 3 u} + (α - 1) (2 u + 1) e^{- u}$ and $f_{α}^{+} (u) = \sqrt{1 + cos 3 u} + (1 - α) (2 u + 1) e^{- u},$ we find that $\begin{matrix} s_{α}^{-} (t) & = & \int_{0}^{t} f_{α}^{-} (u) du \\ = & \frac{2 \sqrt{2}}{3} sin \frac{3 t}{2} \\ + (1 - α) ((2 t + 1) e^{- t} + 2 e^{- t} - 3) \end{matrix}$ and $\begin{matrix} s_{α}^{+} (t) & = & \int_{0}^{t} f_{α}^{+} (u) du \\ = & \frac{2 \sqrt{2}}{3} sin \frac{3 t}{2} \\ + (α - 1) ((2 t + 1) e^{- t} + 2 e^{- t} - 3) . \end{matrix}$

Therefore, it can be seen that $\int_{0}^{\infty} f (u) du = lim_{t \to \infty} s (t)$ is divergent. On the other hand, if we take p (x) = x, then we obtain the components of the parametric representation of weighted mean of the function s (t) as follows: $\begin{matrix} σ_{α}^{-} (x) & = & \frac{1}{x} \int_{0}^{x} s_{α}^{-} (t) dt = \frac{4 \sqrt{2}}{9 x} (1 - cos \frac{3 x}{2}) \\ + \frac{(1 - α)}{x} (5 - 3 x - (2 x + 5) e^{- x}) \end{matrix}$ and $\begin{matrix} σ_{α}^{+} (x) & = & \frac{1}{x} \int_{0}^{x} s_{α}^{+} (t) dt = \frac{4 \sqrt{2}}{9 x} (1 - cos \frac{3 x}{2}) \\ + \frac{(1 - α)}{x} ((2 x + 5) e^{- x} + 3 x - 5) . \end{matrix}$

It then easily follows that $σ_{α}^{-} (x)$ and $σ_{α}^{+} (x)$ converge to 3 (α - 1) and 3 (1 - α) as x→ ∞, respectively. If we take [μ] _α = [3 (α - 1) , 3 (1 - α)], we get $lim_{x \to \infty} D (σ (x), μ) = 0$ , that is, σ (x) converges to μ. In other words, the improper integral $\int_{0}^{\infty} f (u) du$ is integrable by the weighted mean method determined by the function p (x) = x to the fuzzy number μ defined by $μ (t) = {\begin{matrix} \frac{t}{3} + 1 & if t \in [- 3, 0] \\ 1 - \frac{t}{3} & if t \in [0, 3] \\ 0 & otherwise . \end{matrix}$

By the next theorem, we prove that the existence of $lim_{x \to \infty} σ (x) = μ$ may imply that of $lim_{x \to \infty} s (x) = μ$ under some suitable conditions on s (x). Such a condition is called a Tauberian condition and the resulting theorem is called a Tauberian theorem.

Theorem 3.4. Let f : [0, ∞) → E¹ be continuous and the condition (3) is satisfied. If $\int_{0}^{\infty} f (t) dt$ is $(\bar{N}, p)$ integrable to μ ∈ E¹ and s (x) is slowly decreasing, then $\int_{0}^{\infty} f (t) dt$ converges to μ.

Proof. Assume that $\int_{0}^{\infty} f (t) dt$ is $(\bar{N}, p)$ integrable to μ ∈ E¹ and s (x) is slowly decreasing. Let λ > 1. For each x such that p (λx) > p (x),

$\begin{matrix} \frac{p (λ x)}{p (λ x) - p (x)} σ (λ x) + σ (x) \\ = \frac{p (λ x)}{p (λ x) - p (x)} (\frac{1}{p (λ x)} \int_{0}^{λ x} s (t) dp (t)) \\ + \frac{1}{p (x)} \int_{0}^{x} s (t) dp (t) \\ = \frac{1}{p (λ x) - p (x)} (\int_{0}^{x} s (t) dp (t) + \int_{x}^{λ x} s (t) dp (t)) \\ + \frac{1}{p (x)} \int_{0}^{x} s (t) dp (t) \\ = (\frac{1}{p (λ x) - p (x)} + \frac{1}{p (x)}) \int_{0}^{x} s (t) dp (t) \\ + \frac{1}{p (λ x) - p (x)} \int_{x}^{λ x} s (t) dp (t) \\ = \frac{p (λ x)}{p (λ x) - p (x)} (\frac{1}{p (x)} \int_{0}^{x} s (t) dp (t)) \\ + \frac{1}{p (λ x) - p (x)} \int_{x}^{λ x} s (t) dp (t) \\ = \frac{p (λ x)}{p (λ x) - p (x)} σ (x) \\ + \frac{1}{p (λ x) - p (x)} \int_{x}^{λ x} s (t) dp (t) . \end{matrix}$ (5)

Using (3), we have

$\begin{matrix} \underset{x \to \infty}{lim sup} \frac{p (λ x)}{p (λ x) - p (x)} \\ = \underset{x \to \infty}{lim sup} \frac{1}{1 - \frac{p (x)}{p (λ x)}} \\ = {\underset{x \to \infty}{lim inf} (1 - \frac{p (x)}{p (λ x)})}^{- 1} \\ = {1 - \underset{x \to \infty}{lim sup} \frac{p (x)}{p (λ x)}}^{- 1} \\ = {1 - \frac{1}{{lim inf}_{x \to \infty} \frac{p (λ x)}{p (x)}}}^{- 1} < \infty . \end{matrix}$ (6)

Then by taking into account (6) and the assumed integrability $(\bar{N}, p)$ of $\int_{0}^{\infty} f (t) dt$ to μ, we obtain $\begin{matrix} \underset{x \to \infty}{lim sup} D (\frac{p (λ x)}{p (λ x) - p (x)} σ (λ x), \frac{p (λ x)}{p (λ x) - p (x)} σ (x)) \\ = \underset{x \to \infty}{lim sup} \frac{p (λ x)}{p (λ x) - p (x)} D (σ (λ x), σ (x)) \\ = \underset{x \to \infty}{lim sup} \frac{p (λ x)}{p (λ x) - p (x)} D (σ (λ x) + μ, σ (x) + μ) \\ \leq \underset{x \to \infty}{lim sup} \frac{p (λ x)}{p (λ x) - p (x)} [D (σ (λ x), μ) + D (σ (x), μ)] \\ \leq \underset{x \to \infty}{lim sup} \frac{p (λ x)}{p (λ x) - p (x)} \underset{x \to \infty}{lim sup} [D (σ (λ x), μ) \\ + D (σ (x), μ)] \\ \leq \underset{x \to \infty}{lim sup} \frac{p (λ x)}{p (λ x) - p (x)} [\underset{x \to \infty}{lim sup} D (σ (λ x), μ) \\ + \underset{x \to \infty}{lim sup} D (σ (x), μ)] \\ \leq \underset{x \to \infty}{lim sup} \frac{p (λ x)}{p (λ x) - p (x)} [lim_{x \to \infty} D (σ (λ x), μ) \\ + \lim_{x \to \infty} D (σ (x), μ)] \\ = 0 . \end{matrix}$

From this point of view, for any given ɛ > 0 there exists a x₀ = x₀ (ɛ) ≥0 such that for x > x₀ $D (\frac{p (λ x)}{p (λ x) - p (x)} σ (λ x), \frac{p (λ x)}{p (λ x) - p (x)} σ (x)) \leq \frac{ɛ}{3} .$

Thus, by Lemma 2.5, we obtain that

$\begin{matrix} \frac{p (λ x)}{p (λ x) - p (x)} σ (x) - \frac{\bar{ɛ}}{3} \\ ⪯ \frac{p (λ x)}{p (λ x) - p (x)} σ (λ x) \\ ⪯ \frac{p (λ x)}{p (λ x) - p (x)} σ (x) + \frac{\bar{ɛ}}{3} . \end{matrix}$ (7)

Since $\int_{0}^{\infty} f (t) dt$ is $(\bar{N}, p)$ integrable to μ, we have $lim_{x \to \infty} D (σ (x), μ) = 0$ . Then, there exists a x₁ = x₁ (ɛ) ≥0 such that $D (σ (x), μ) \leq \frac{ɛ}{3}$ for x > x₁. Hence, by Lemma 2.5, we obtain $μ - \frac{\bar{ɛ}}{3} ⪯ σ (x) ⪯ μ + \frac{\bar{ɛ}}{3} .$ (8)

In addition, since s (x) is slowly decreasing, there exist x₂ = x₂ (ɛ) ≥0 and λ = λ (ɛ) >1 such that for x > x₂

$\begin{matrix} \frac{1}{p (λ x) - p (x)} \int_{x}^{λ x} s (t) dp (t) ⪰ \\ \frac{1}{p (λ x) - p (x)} \int_{x}^{λ x} (s (x) - \frac{\bar{ɛ}}{3}) dp (t) \\ = & s (x) - \frac{\bar{ɛ}}{3} . \end{matrix}$ (9)

Using the identity (5) and combining (7), (8) and (9) we obtain that for any given ɛ > 0 there exists a x₃ = max {x₀, x₁, x₂} such that for x > x₃ $\begin{matrix} \frac{p (λ x)}{p (λ x) - p (x)} σ (x) + \frac{2 \bar{ɛ}}{3} + μ \\ ⪰ \frac{p (λ x)}{p (λ x) - p (x)} σ (λ x) + σ (x) \\ = \frac{p (λ x)}{p (λ x) - p (x)} σ (x) \\ + \frac{1}{p (λ x) - p (x)} \int_{x}^{λ x} s (t) dp (t) \\ ⪰ \frac{p (λ x)}{p (λ x) - p (x)} σ (x) + s (x) - \frac{\bar{ɛ}}{3} . \end{matrix}$

Therefore, by Lemma 2.6 we get $\bar{ɛ} + μ ⪰ s (x) .$ (10)

On the other hand, let 0 < λ < 1. For each x such that p (λx) < p (x),

$\begin{matrix} \frac{p (λ x)}{p (x) - p (λ x)} σ (λ x) + \frac{1}{p (x) - p (λ x)} \int_{λ x}^{x} s (t) dp (t) \\ = \frac{p (λ x)}{p (x) - p (λ x)} (\frac{1}{p (λ x)} \int_{0}^{λ x} s (t) dp (t)) \\ + \frac{1}{p (x) - p (λ x)} \int_{λ x}^{x} s (t) dp (t) \\ = \frac{1}{p (x) - p (λ x)} (\int_{0}^{λ x} s (t) dp (t) + \int_{λ x}^{x} s (t) dp (t)) \\ = \frac{1}{p (x) - p (λ x)} \int_{0}^{x} s (t) dp (t) \\ = \frac{p (x)}{p (x) - p (λ x)} σ (x) \\ = (1 + \frac{p (λ x)}{p (x) - p (λ x)}) σ (x) \\ = \frac{p (λ x)}{p (x) - p (λ x)} σ (x) + σ (x) . \end{matrix}$ (11)

Using (3) and Lemma 2 we get

$\begin{matrix} \underset{x \to \infty}{lim sup} \frac{p (λ x)}{p (x) - p (λ x)} \\ = \underset{x \to \infty}{lim sup} \frac{1}{\frac{p (x)}{p (λ x)} - 1} \\ = {\underset{x \to \infty}{lim inf} \frac{p (x)}{p (λ x)} - 1}^{- 1} < \infty \end{matrix}$ (12)

Then going through the similar process above and by taking into account (62), we get

$lim_{x \to \infty} D (\frac{p (λ x)}{p (x) - p (λ x)} σ (λ x), \frac{p (λ x)}{p (x) - p (λ x)} σ (x)) = 0 .$

Hence, for any given ɛ > 0 there exists a $\tilde{x_{0}} = \tilde{x_{0}} (ɛ) \geq 0$ such that for $x > \tilde{x_{0}}$ $D (\frac{p (λ x)}{p (x) - p (λ x)} σ (λ x), \frac{p (λ x)}{p (x) - p (λ x)} σ (x)) \leq \frac{ɛ}{3} .$

So, By Lemma 2.5, we obtain that

$\begin{matrix} \frac{p (λ x)}{p (x) - p (λ x)} σ (λ x) - \frac{\bar{ɛ}}{3} ⪯ \frac{p (λ x)}{p (x) - p (λ x)} σ (x) \\ ⪯ \frac{p (λ x)}{p (x) - p (λ x)} σ (λ x) + \frac{\bar{ɛ}}{3} . \end{matrix}$ (13)

Furthermore, since s (x) is slowly decreasing, there exist $\tilde{x_{1}} = \tilde{x_{1}} (ɛ) \geq 0$ and 0 < λ = λ (ɛ) <1 such that for $x > \tilde{x_{1}}$

$\begin{matrix} \frac{1}{p (x) - p (λ x)} \int_{λ x}^{x} s (t) dp (t) \\ ⪯ \frac{1}{p (x) - p (λ x)} \int_{λ x}^{x} (s (x) + \frac{\bar{ɛ}}{3}) dp (t) \\ = s (x) + \frac{\bar{ɛ}}{3} . \end{matrix}$ (14)

Combining (8), (13) and (14) we obtain by the identity (11) that for any given ɛ > 0 there exists a $\tilde{x_{2}} = max {\tilde{x_{0}}, \tilde{x_{1}}, x_{1}}$ such that for $x > \tilde{x_{2}}$ for each ɛ > 0 $\begin{matrix} \frac{p (λ x)}{p (x) - p (λ x)} σ (λ x) - \frac{2 \bar{ɛ}}{3} + μ \\ ⪯ \frac{p (λ x)}{p (x) - p (λ x)} σ (x) + σ (x) \\ = \frac{p (λ x)}{p (x) - p (λ x)} σ (λ x) \\ + \frac{1}{p (x) - p (λ x)} \int_{λ x}^{x} s (t) dp (t) \\ ⪯ \frac{p (λ x)}{p (x) - p (λ x)} σ (λ x) + s (x) + \frac{\bar{ɛ}}{3} . \end{matrix}$

Therefore, by Lemma 2.6, we get $μ - \bar{ɛ} ⪯ s (x) .$ (15)

Combining (10) and (15), for any given ɛ > 0 we obtain $μ - \bar{ɛ} ⪯ s (x) ⪯ \bar{ɛ} + μ .$

By Lemma 2.5, we conclude that $D (s (x), μ) \leq ɛ .$

This implies that $lim_{x \to \infty} s (x) = lim_{x \to \infty} \int_{0}^{x} f (t) dt = μ$ . □

Lemma 3.5. Let f : [0, ∞) → E¹ be continuous and p (x) be a nondecreasing real valued differentiable function on [0, ∞) such that the condition $\frac{p (λ x)}{p (x)} \leq H whenever λ > 1$ (16) for some H ≥ 0 is satisfied. If the function $f (x) \in C ([0, \infty), E^{1})$ satisfies the condition $\frac{p (x)}{p^{'} (x)} f (x) ⪰ υ whenever x > x_{0}$ (17) for some constant fuzzy number $υ ⪯ \bar{0}$ and x₀ ≥ 0, then s (x) is slowly decreasing.

Proof. Assume that the fuzzy-number-valued function f (x) satisfies (17) under the given conditions. Then, for all α ∈ [0, 1] and x > x₀ we an write that $\frac{p (x)}{p^{'} (x)} f_{α}^{-} (x) \geq υ_{α}^{-} \geq υ_{0}^{-}$ and $\frac{p (x)}{p^{'} (x)} f_{α}^{+} (x) \geq υ_{α}^{+} \geq υ_{1}^{+} \geq υ_{0}^{-} .$

On behalf of simplicity let take $υ_{0}^{-} = - C$ whenever C ≥ 0. Therefore, for all α ∈ [0, 1] and x > x₀ we have $f_{α}^{-} (x) \geq - C (\frac{p^{'} (x)}{p (x)})$ (18) and $f_{α}^{+} (x) \geq - C (\frac{p^{'} (x)}{p (x)}) .$ (19)

For all x ≤ t and α ∈ [0, 1], by using (18) and (19) we get $\begin{matrix} s_{α}^{-} (t) - s_{α}^{-} (x) & = & \int_{x}^{t} f_{α}^{-} (u) du \\ \geq & - C \int_{x}^{t} \frac{p^{'} (u)}{p (u)} du \\ = & - C log \frac{p (t)}{p (x)} . \end{matrix}$

Hence, for all ɛ > 0 and 1 < λ, by using (16) we obtain for all x ≤ t ≤ λx $\begin{matrix} s_{α}^{-} (t) - s_{α}^{-} (x) & \geq & - C log \frac{p (t)}{p (x)} \\ \geq & - C log \frac{p (λ x)}{p (x)} \\ \geq & - C log H . \end{matrix}$

If we choose $H : = e^{\frac{ɛ}{C}}$ , then we conclude that $s_{α}^{-} (t) - s_{α}^{-} (x) \geq - ɛ .$ (20)

In a similar way, for all x ≤ t and α ∈ [0, 1], by using (18) and (19) we get $s_{α}^{+} (t) - s_{α}^{+} (x) \geq - C log \frac{p (t)}{p (x)}$

Thus, for all ɛ > 0 and 1 < λ, by using (16) we obtain for all x ≤ t ≤ λx $\begin{matrix} s_{α}^{+} (t) - s_{α}^{+} (x) & \geq & - C log \frac{p (t)}{p (x)} \\ \geq & - C log \frac{p (λ x)}{p (x)} \\ \geq & - C log H . \end{matrix}$

If we choose $H : = e^{\frac{ɛ}{C}}$ , then we conclude that $s_{α}^{+} (t) - s_{α}^{+} (x) \geq - ɛ .$ (21)

Therefore, combining (20) and (21), it can be seen that $s (t) ⪰ s (x) - \bar{ɛ}$ . This implies that s (x) is slowly decreasing. □

Remark 3.6. Lemma 3.5 remains valid if the condition (16) is replaced by $\frac{p (x)}{p (λ x)} \leq \tilde{H} whenever 0 < λ < 1$ (22) for some $\tilde{H} \geq 0$ . In other words, if $f (x) \in C ([0, \infty), E^{1})$ satisfies the condition (17) and (22), then the fuzzy-number-valued function s (x) is slowly decreasing.

By Theorem 3.4 and Lemma 3.5, we have the following corollary.

Corollary 3.7. Let f : [0, ∞) → E¹ be continuous and the condition (3) is satisfied. If $\int_{0}^{\infty} f (t) dt$ is $(\bar{N}, p)$ integrable to μ ∈ E¹ and the conditions (16) and (17) are satisfied, then $\int_{0}^{\infty} f (t) dt$ converges to μ.

A special case of Theorem 3.4 and Lemma 3.5 can be obtained by choosing p (x) = x as follows.

Corollary 3.8. If the function $f (x) \in C ([0, \infty), E^{1})$ satisfies the condition $xf (x) ⪰ υ whenever x > x_{0}$ (23) for some constant fuzzy number $υ ⪯ \bar{0}$ and x₀ ≥ 0, then s (x) is slowly decreasing.

Taking advantage of Corollary 3.8, we construct an example illustrating that conditions in Theorem 3.4 are satisfied and so the mentioned theorem is applicable.

Example 3.9. Consider the fuzzy-number-valued function f on the unbounded interval [0, ∞) defined by $\begin{matrix} (f (u)) (t) \\ = {\begin{matrix} {te}^{u} (3 u + 5)^{- 1} & if t \in [0, \frac{(3 u + 5)}{e^{u}}] \\ 2 - {te}^{u} (3 u + 5)^{- 1} & if t \in [\frac{(3 u + 5)}{e^{u}}, \frac{2 (3 u + 5)}{e^{u}}] \\ 0 & otherwise \end{matrix} \end{matrix}$ for all u ≥ 0. Since the fuzzy-number-valued function f with the parametric representation $[f (u)]_{α} = [f_{α}^{-} (u), f_{α}^{+} (u)]$ for each u ∈ [0, ∞) is continuous, it is integrable on the same interval. Using the components of the parametric representation of f, we find that $\begin{matrix} s_{α}^{-} (t) & = & \int_{0}^{t} f_{α}^{-} (u) du = α \int_{0}^{t} (3 u + 5) e^{- u} du \\ = & α ((- 3 t - 5) e^{- t} - 3 e^{- t} + 8) \end{matrix}$ and $\begin{matrix} s_{α}^{+} (t) & = & \int_{0}^{t} f_{α}^{+} (u) du = (2 - α) \int_{0}^{t} (3 u + 5) e^{- u} du \\ = & (2 - α) ((- 3 t - 5) e^{- t} - 3 e^{- t} + 8) . \end{matrix}$

Condition (3) is clearly satisfied. Choosing p (x) = x, we obtain the components of parametric representation of weighted mean of the function s (t) as follows: $\begin{matrix} σ_{α}^{-} (x) & = & \frac{1}{x} \int_{0}^{x} s_{α}^{-} (t) dt \\ = & α (3 e^{- x} + 8) + \frac{11 α}{x} (e^{- x} - 1) \end{matrix}$ and $\begin{matrix} σ_{α}^{+} (x) & = & \frac{1}{x} \int_{0}^{x} s_{α}^{+} (t) dt \\ = & (2 - α) (3 e^{- x} + 8) + \frac{11 (2 - α)}{x} (e^{- x} - 1) . \end{matrix}$

We claim that if we take [μ] _α = [8α, 8 (2 - α)], then the improper integral $\int_{0}^{\infty} f (u) du$ is integrable by the weighted mean method determined by the function p (x) = x to the fuzzy number μ, which corresponds to the α-level set [μ] _α, defined by $μ (t) = {\begin{matrix} t / 8 & if t \in [0, 8] \\ 2 - t / 8 & if t \in [8, 16] \\ 0 & otherwise . \end{matrix}$

It easily follows that $σ_{α}^{-} (x)$ and $σ_{α}^{+} (x)$ converge to 8α and 8 (2 - α) as x→ ∞, respectively. Hence, we obtain $lim_{x \to \infty} D (σ (x), μ) = 0$ , that is, σ (x) converges to μ. In other words, the improper integral $\int_{0}^{\infty} f (u) du$ is integrable by the weighted mean method determined by the function p (x) = x to the fuzzy number μ. In addition to this, for each α ∈ [0, 1] and u ≥ 0, we see that ${uf}_{α}^{-} (u) = α u (3 u + 5) e^{- u} \geq 0$ (24) and ${uf}_{α}^{+} (u) = (2 - α) u (3 u + 5) e^{- u} \geq 0 .$ (25)

It follows from (24) and (25) that $uf (u) ⪰ \bar{0}$ . By Corollary 3.8, s (t) is slowly decreasing. Since all conditions of Theorem 3.4 are satisfied, the improper integral $\int_{0}^{\infty} f (u) du$ converges to the fuzzy number μ.

Remark 3.10. The condition (211) is a Tauberian condition for the $(\bar{N}, p)$ integrability method as well.

Corollary 3.11. (Yavuz et al. (2016)) Let f : [0, ∞) → E¹ be continuous.If $\int_{0}^{\infty} f (t) dt$ is Cesàro integrable to μ ∈ E¹ and the condition (211) is satisfied, then $\int_{0}^{\infty} f (t) dt$ converges to μ.

Corollary 3.12. (Yavuz et al. (2016)) Let f : [0, ∞) → E¹ be continuous.If $\int_{0}^{\infty} f (t) dt$ is Cesàro integrable to μ ∈ E¹ and s (x) is slowly decreasing, then $\int_{0}^{\infty} f (t) dt$ converges to μ.

4 Conclusion

In this paper we have introduced the weighted mean method of improper integrals of fuzzy-number-valued functions and obtained a Tauberian theorem for this method. This work can be extended for the improper integrals of fuzzy-number-valued functions of several variables and Tauberian conditions can be improved.

Footnotes

Acknowledgments

The authors would like to thank the referee for very helpful suggestions which improved the manuscript.

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