Pointwise ideal convergence and uniformly ideal convergence of sequences of fuzzy valued functions

Abstract

In this paper we introduce the concepts of ideal convergence, pointwise ideal convergence, and uniformly ideal convergence of sequences of fuzzy valued functions based on the concept of convergence of sequences of fuzzy numbers and obtain the relationship among pointwise ideal convergence and uniformly ideal convergence of sequences of fuzzy valued functions, and study their representations of sequences of α-level cuts.

Keywords

Fuzzy number fuzzy valued function ideal convergence pointwise ideal convergence uniformly ideal convergence

1 Introduction

In analysis, either in classical or fuzzy, the convergence sequences means that almost all elements of the sequence belong to an arbitrary small neighborhood of a point, called the limit point. The main purpose of the statistical convergence is to relax this condition and to demand validity of the convergence criterion only for a majority of the elements. As always, statistics are concerned only about the big quantities and the term “majority” is simply imply the concept “for almost all” in the classical analysis. As the set of real numbers can be embedded in the set of all fuzzy numbers and statistical convergence of reals can be considered as a special case of fuzzy numbers. The set of all fuzzy numbers is not partially ordered and does not carry the group structure, so most of the results in real sequences may not be valid in fuzzy setting. Therefore the theory is not a trivial extension of what has been known in real cases.

The idea of statistical convergence was introduced by Fast [12] and Steinhaus [28] and reintroduced by Schoenberg [27] and studied some basic properties of statistical convergence, and statistical convergence as a summability method. In 1937, Cartan [9] introduced the notion of the ideal convergence is the dual (equivelant) to the notion of filter convergence. The notion of the filter convergence is a generalization of the classical notion of convergence of a sequence and it has been an important tool in general topology and functional analysis. Now-a-days many authors to use an equivalent dual notion of the ideal convergence. Kostyrko et al. [19] and Nuray and Ruckle [24] independently studied about the notion of ideal convergence which is based on the structure of the admissible ideal I of subsets of natural numbers $ℕ .$ Later on it was further investigated by many authors, e.g. Šalát et al. [25], Hazarika and Mohiuddine [14], Hazarika [15 –17], Mursaleen and Mohiuddine [22], Tripathy and Sen [29], Tripathy and Dutta [30], and references therein. Balcerzak et al. [7] discussed the statistical convergence and ideal convergence for sequences of real valued functions. Jasinski and Recław [18] introduced the concept of ideal convergence of continuous functions. Considering the uncertainty of data and information in a specific modeling process, this uncertainty was usually represented by a fuzzy number [23]. Şavas [26] proved the characterization theorem for the sequence of fuzzy numbers. Aytar and Pehlvian [6] discussed the statistical convergence of sequences of fuzzy numbers and sequences of α-cuts. Kumar and Kumar [20] introduced the concept of ideal convergence of fuzzy numbers and prove some basic results in the new settings. Altin et al. [1] introduced the concept of pointwise statistical convergence sequences of fuzzy mappings and established some basic properties of fuzzy mappings. Ansastassiou and Duman [5] proved Korovkin’s approximation theorem for fuzzy positive linear operators and estimated the rates of statistical fuzzy convergence based on a non-negative regularly summable matrix of operators via the fuzzy modulus of continuity of fuzzy number valued functions. Çinar et al. [8] introduced the concepts of pointwise and uniformly statistical convergence of sequences of functions of order α and established some basic properties of sequences of functions. Et et al. [11] introduced the concept of pointwise statistical convergence of sequences of fuzzy mappings of order α and established some basic properties. For details on statistical convergence of order β for sequences of fuzzy numbers, we refer to [2 –4] and references therein. Recently, Gong et al. [13] studied statistical convergence, uniformly statistical convergence and equi-statistical convergence for sequences of fuzzy valued functions and established some basic properties of sequences of fuzzy valued functions based on sequences of α-level cuts. In this paper, based on the concept of ideal convergence of sequences of fuzzy numbers, we propose ideal convergence, pointwise ideal convergence, and uniformly ideal convergence for sequences of fuzzy valued functions and prove some classical results in this new settings, and their representations of sequences of α-level cuts are also dicsussed. If we take $I = I_{δ} = {A \subseteq ℕ : δ (A) = 0},$ where δ (A) denote the asymptotic density of the set A . Then I_δ is a non-trivial admissible ideal of $ℕ$ and the corresponding convergence coincide with statistical convergence. So, for I = I_δ the results of this article are reduced to the results discussed in [1, 13].

2 Preliminaries

Let E be a nonempty set. According to Zadeh (see [31]) a fuzzy subset of E is a nonempty subset ${(t, \bar{x} (t)) : t \in E}$ of E × J (= [0, 1]) for some function $\bar{x} : E \to J (= [0, 1]) .$ A function $\bar{x} : ℝ \to J (= [0, 1])$ is called a fuzzy number if the function $\bar{x}$ satisfies the following properties:

$\bar{x}$ is convex i.e. $\bar{x} (t) \geq \bar{x} (s) \land \bar{x} (r) = min {\bar{x} (s), \bar{x} (r)},$ where s < t < r .

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

$\bar{x}$ is upper semi-continuous i.e. for each $ε > 0, {\bar{x}}^{- 1} ([0, a + ε))$ for all a ∈ [0, 1] is open in the usual topology of $ℝ .$

$[\bar{x}]^{0} = cl ({t \in ℝ : \bar{x} (t) > 0})$ is compact, where cl is the closure operator.

We denote the set of all fuzzy numbers by $F (ℝ) .$ The set $ℝ$ of real numbers can be embedded in $F (ℝ)$ if we define $\bar{r} \in F (ℝ)$ by $\bar{r} (t) = {\begin{matrix} 1, & if t = r : \\ 0, & if t \neq r . \end{matrix}$

For 0 < α ≤ 1, α-cut of $\bar{x}$ is defined by $[\bar{x}]_{α} = {t \in ℝ : \bar{x} (t) \geq α} = [{\bar{x}}_{α}^{-}, {\bar{x}}_{α}^{+}]$ is a closed and bounded interval of $ℝ .$ As in [23], the Hausdorff distance between two fuzzy numbers $\bar{x}$ and $\bar{y}$ given by $D : F (ℝ) \times F (ℝ) ⟶ [0, + \infty)$ $\begin{matrix} D (\bar{x}, \bar{y}) & = & sup_{α \in [0, 1]} d ([\bar{x}]_{α}, [\bar{y}]_{α}) \\ = & sup_{α \in [0, 1]} max {| {\bar{x}}_{α}^{-} - {\bar{x}}_{α}^{+} |, | {\bar{y}}_{α}^{-} - {\bar{y}}_{α}^{+} |}, \end{matrix}$ where d is the Hausdorff metric. For any $\bar{x}, \bar{y}, \bar{z}, \bar{u} \in F (ℝ)$ we know that

$(F (ℝ), D)$ is a complete metric space.

$D (γ \bar{x}, γ \bar{y}) = | γ | D (\bar{x}, \bar{y}); γ \in ℝ .$

$D (\bar{x} + \bar{u}, \bar{y} + \bar{u}) = D (\bar{x}, \bar{y}) .$

$D (\bar{x} + \bar{z}, \bar{y} + \bar{u}) \leq D (\bar{x}, \bar{y}) + D (\bar{z}, \bar{u}) .$

Lemma 2.1. [23] Let $\bar{x} \in F (ℝ)$ and $[\bar{x}]_{α} = [{\bar{x}}_{α}^{-}, {\bar{x}}_{α}^{+}] .$ Then the following conditions are satisfied:

${\bar{x}}_{α}^{-}$ is a left continuous monotone-nondecreasing function on (0, 1] .

${\bar{x}}_{α}^{+}$ is a right continuous monotone-nonincreasing function on (0, 1] .

${\bar{x}}_{α}^{-}$ and ${\bar{x}}_{α}^{+}$ are right continuous at α = 0 .

${\bar{x}}_{1}^{-} \leq {\bar{x}}_{1}^{+} .$

For a subset M of

ℕ

the asymptotic density or density of M, denoted by δ (M) , is given by

δ (M) = lim_{n \to \infty} \frac{1}{n} | {k \leq n : k \in M} |,

if this limit exists, where | {k ≤ n : k ∈ M} | denotes the cardinality of the set {k ≤ n : k ∈ M} .

A sequence x = (x_n) is statistically convergent to ℓ if $δ ({n \in ℕ : | x_{n} - ℓ | \geq ɛ}) = 0$ for every ɛ > 0 . In this case ℓ is called the statistical limit of x . Let S be a non-empty set. Then a non empty class I ⊆ P (S) is said to be an ideal on S if and only if (i) φ ∈ I . (ii) I is additive (iii) hereditary. An ideal I ⊆ P (S) is said to be non trivial if I ≠ φ and S ∉ I . A non-empty family of sets F ⊆ P (S) is said to be a filter on S if and only if (i) φ ∉ F (ii) for each A, B ∈ F we have A ∩ B ∈ F (iii) for each A ∈ F and B ⊃ A, implies B ∈ F . For each ideal I, there is a filter F (I) corresponding to I i.e. F (I) = {K ⊆ S : K^c ∈ I} , where K^c = S - K . We say that a non-trivial ideal I ⊆ P (S) is an admissible ideal on S if and only if it contains all singletons, i.e. if it contains {{x} : x ∈ S} . Recall that a sequence x = (x_k) of points in $ℝ$ is said to be I-convergent to the number ℓ (denoted by I - lim x_k =ℓ) if for every ɛ > 0, the set ${k \in ℕ, : | x_{k} - ℓ | \geq ɛ} \in I .$ Throughout the article we assume that I is an admissible ideal of $ℕ .$

3 Pointwise and uniformaly convergence of sequences of fuzzy valued functions

In this section, we define pointwise convergence and uniformly convergence of sequences of fuzzy valued functions, and prove some classical results in this settings. Suppose that a fuzzy valued function $\bar{f} : [a, b] ⟶ F (ℝ)$ and sequence of fuzzy valued functions ${\bar{f}}_{n} : [a, b] ⟶ F (ℝ), n \in ℕ .$

Definition 3.1. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to pointwise convergent to fuzzy valued function $\bar{f}$ if, for each x ∈ [a, b] and for each ɛ > 0 there exists a positive integer N (x, ɛ) such that $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ$ for all n > N . We write $lim_{n \to \infty} {\bar{f}}_{n} (x) = \bar{f} (x)$ or ${\bar{f}}_{n} (x) \overset{p}{⟶} \bar{f} (x)$ for each x ∈ [a, b] . The fuzzy valued function $\bar{f}$ is called the limit function of the sequences of fuzzy valued functions $({\bar{f}}_{n})$ and we say that $({\bar{f}}_{n})$ converges pointwise to $\bar{f}$ on [a, b] .

Lemma 3.1. Let $({\bar{f}}_{n})$ be a sequence of fuzzy valued functions and $\bar{f}$ be a fuzzy valued function defined on [a, b] . Then $lim_{n \to \infty} {\bar{f}}_{n} (x) = \bar{f} (x) for each x \in [a, b]$ (3.1) exists if and only if for all strictly increasing functions $n : ℕ \to ℕ$ the ordinary limit $lim_{r \to \infty} {\bar{f}}_{n (r)} (x) = \bar{f} (x) for each x \in [a, b] exists .$ (3.2)

Proof. It is obvious that (3.1) implies (3.2).

Conversely suppose that the limit in (3.1) does not exist. Then for every fuzzy valued function $\bar{f}$ there exists ɛ > 0 such that for each N, there exists integer $n \in ℕ$ for which $D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ for all x \in [a, b] .$ (3.3)

For N = 1 we consider the corresponding values of n for which (3.3) is true by n (1) . Put N₁ = n (1) +1 and consider values of n satisfying (3.3) by n (1) . Continuing in this way we obtain an infinite sequence of values {n (r)} for which (3.3) is true. Therefore limit of (3.2) does not exist. □

Definition 3.2. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to converge uniformly to $\bar{f}$ on [a, b] if, for every ɛ > 0, there exists an integer N (= N (ɛ)) such that $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all x \in [a, b], for all n > N .$

We denote this symbolically by ${\bar{f}}_{n} \overset{u}{⇉} \bar{f} .$

Theorem 3.1. Let $\bar{f}$ be fuzzy valued functions and $({\bar{f}}_{n})$ be a sequence of fuzzy valued functions on [a, b] . Then ${\bar{f}}_{n} \overset{u}{⇉} \bar{f}$ on [a, b] if and only if d_n $\overset{p}{⟶}$ 0, where $d_{n} = sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) .$

Proof. Suppose that ${\bar{f}}_{n} \overset{u}{⇉} \bar{f}$ on [a, b] . Then for every ɛ > 0 there exists an integer N (= N (ɛ)) such that $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all x \in [a, b] and for all n > N$ $\Rightarrow sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all n > N, as ɛ is arbitrary .$

Hence d_n $\overset{p}{⟶}$ 0 .

Next we suppose that d_n $\overset{p}{⟶}$ 0 . i.e. $sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) \overset{p}{⟶} 0 .$ Now for every ɛ > 0 we have s $\begin{matrix} sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ \\ \Rightarrow D ({\bar{f}}_{n} (x), \bar{f} (x)) \\ \leq sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all x \in [a, b] . \end{matrix}$

Thus ${\bar{f}}_{n} \overset{u}{⇉} \bar{f}$ on [a, b] . □

Remark 3.1. It is clear that uniform convergence implies pointwise convergence with the same limit on [a, b] . But converse is not necessarily true. For this we consider the following example.

Example 3.1. For any x ∈ [0, 1] we define ${\bar{f}}_{n} (x) = {\begin{matrix} \bar{(\frac{nx}{1 + n^{2} x^{2}})}, & for x \in [0, 1]; \\ \bar{0} & otherwise . \end{matrix}$

Then $({\bar{f}}_{n})$ converges pointwise to $\bar{f} = \bar{0} .$ But for all $n \in ℕ$ $\begin{matrix} sup_{x \in [0, 1]} D ({\bar{f}}_{n} (x), \bar{0}) = sup_{x \in [0, 1]} \frac{nx}{1 + n^{2} x^{2}} \\ = sup_{x \in [0, 1]} \frac{1}{\frac{1}{nx} + nx} = \frac{1}{2} \neq 0 . \end{matrix}$

Hence $({\bar{f}}_{n})$ is not uniformly convergent to $\bar{0}$ on [0, 1] .

Definition 3.3. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to uniformly Cauchy on [a, b] if, for every ɛ > 0, there exists an integer N (= N (ɛ)) such that $\begin{matrix} D ({\bar{f}}_{n} (x), {\bar{f}}_{k} (x)) < ɛ for all x \in [a, b], \\ for all n, k > N . \end{matrix}$

Theorem 3.2. Let $({\bar{f}}_{n})$ be fuzzy valued functions on [a, b] . Then $({\bar{f}}_{n})$ is uniformly convergent if and only if it is uniformly Cauchy.

Proof. Suppose that $({\bar{f}}_{n})$ is uniformly convergent to a fuzzy valued function $\bar{f}$ on [a, b] . Then for each ɛ > 0 there exists a positive integer N (= N (ɛ)) such that $\begin{matrix} D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all x \in [a, b] \\ and for all n > N . \end{matrix}$

Now, we have $\begin{matrix} D ({\bar{f}}_{n} (x), {\bar{f}}_{k} (x)) \leq D ({\bar{f}}_{n} (x), \bar{f} (x)) + D ({\bar{f}}_{k} (x), \bar{f} (x)) \\ < \frac{ɛ}{2} + \frac{ɛ}{2} = ɛ for all x \in [a, b], for all n, k > N . \end{matrix}$

Hence $({\bar{f}}_{n})$ is uniformly Cauchy on [a, b] .

Next we suppose that $({\bar{f}}_{n})$ is uniformly Cauchy on [a, b] . Then for each ɛ > 0 there exists $N (= N (ɛ)) \in ℕ$ such that

$\begin{matrix} D ({\bar{f}}_{n} (x), {\bar{f}}_{k} (x)) < ɛ for all x \in [a, b] \\ and for all n, k > N . \end{matrix}$ (3.4) Since sequence of fuzzy numbers $({\bar{f}}_{n} (x))$ is Cauchy sequence for every x ∈ [a, b] and by the completeness of $F (ℝ)$ we have $lim_{n \to \infty} {\bar{f}}_{n} (x) = \bar{f} (x) .$ In (3.4) we fixed n > N and taking limit k→ ∞ we have $\begin{matrix} D ({\bar{f}}_{n} (x), \bar{f} (x)) < \frac{ɛ}{2} < ɛ for all x \in [a, b] \\ and for all n > N . \end{matrix}$

Hence $({\bar{f}}_{n})$ is uniformly convergent to $\bar{f}$ on [a, b] . □

Theorem 3.3. Let $\bar{f}$ and $({\bar{f}}_{n})$ be fuzzy valued functions on [a, b] , and $({\bar{f}}_{n})$ be uniformly convergent to $\bar{f} .$ Suppose that x ∈ [a, b] and $lim_{t \to x} {\bar{f}}_{n} (t) = {\bar{a}}_{n}, n \in ℕ for all t \in [a, b] .$ (3.5)

Then sequence of fuzzy numbers $({\bar{a}}_{n})$ is convergent and $lim_{t \to x} \bar{f} (t) = {\bar{a}}_{n} for all t \in [a, b] .$ (3.6)

Proof. Suppose that $({\bar{f}}_{n})$ is uniformly convergent to $\bar{f}$ on [a, b] . Therefore $({\bar{f}}_{n})$ is uniformly Cauchy on [a, b] . Then for each ɛ > 0 there exists $N (= N (ɛ)) \in ℕ$ such that

$\begin{matrix} D ({\bar{f}}_{n} (t), {\bar{f}}_{k} (t)) < ɛ for all t \in [a, b] \\ and for all n, k > N . \end{matrix}$ (3.7)

Fixing n, k > N in (3.7) and taking limit t → x, then from (3.5) we have $D ({\bar{a}}_{n}, {\bar{a}}_{k}) < ɛ .$ Therefore sequence of fuzzy numbers $({\bar{a}}_{n})$ is a Cauchy sequence and it is convergent, say $lim_{n \to \infty} a_{n} = \bar{a} .$ Since $({\bar{f}}_{n})$ is uniformly convergent to $\bar{f}$ on [a, b] and $lim_{n \to \infty} a_{n} = \bar{a} .$ Then for every ɛ > 0, there exists a positive integer N₀ (= N₀ (ɛ)) such that $\begin{matrix} D ({\bar{f}}_{n} (t), {\bar{f}}_{k} (t)) < \frac{ɛ}{3} for all t \in [a, b] \\ and for all n, k > N_{0} \end{matrix}$ and $D ({\bar{a}}_{n}, \bar{a}) < \frac{ɛ}{3} for all n > N_{0} .$

For fixed n > N₀ and $lim_{t \to x} {\bar{f}}_{n} (t) = {\bar{a}}_{n}$ for all t ∈ [a, b] there exists a punctured neighborhood $\tilde{A} (x)$ of x such that $D ({\bar{f}}_{n} (t), {\bar{a}}_{n}) < \frac{ɛ}{3} for all t \in \tilde{A} (x) \cap [a, b] .$

Then for every n > N₀ and for each $t \in \tilde{A} (x) \cap [a, b],$ we have $\begin{matrix} D (\bar{f} (t), \bar{a}) \leq D (\bar{f} (t), {\bar{f}}_{n}) + D ({\bar{f}}_{n} (t), {\bar{a}}_{n}) + D ({\bar{a}}_{n}, \bar{a}) \\ < \frac{ɛ}{3} + \frac{ɛ}{3} + \frac{ɛ}{3} = ɛ . \end{matrix}$

Hence $lim_{t \to x} \bar{f} (t) = \bar{a} = lim_{n \to \infty} {\bar{a}}_{n} for all t \in [a, b] .$ □

Theorem 3.4. Let $({\bar{f}}_{n})$ be sequence of fuzzy valued continuous functions and $\bar{f}$ be a fuzzy valued function on [a, b] . If $({\bar{f}}_{n})$ is uniformly convergent to $\bar{f}$ on [a, b] , then $\bar{f}$ is continuous on [a, b] .

Proof. Let x ∈ [a, b] be arbitrary. Since each ${\bar{f}}_{n}$ is continuous on [a, b] , then for all $n \in ℕ$ we have $lim_{t \to x} {\bar{f}}_{n} (t) = {\bar{f}}_{n} (x) for all t \in [a, b] .$

Therefore by Theorem 3.3, $({\bar{f}}_{n})$ is convergent and for all t ∈ [a, b] we have $lim_{t \to x} \bar{f} (t) = lim_{n \to \infty} {\bar{f}}_{n} (x) = \bar{f} (x) for all n \in ℕ .$

Thus $\bar{f}$ is continuous at x . Since x ∈ [a, b] is arbitrary, and hence $\bar{f}$ is continuous on [a, b] . □

Theorem 3.5. (Dini’s type theorem) Let D be a compact subset of $ℝ$ and let $({\bar{f}}_{n})$ be a sequence of fuzzy valued continuous functions on D and pointwise converges to a fuzzy valued continuous function $\bar{f}$ on D . Suppose that $({\bar{f}}_{n})$ be monotonic on D i.e. ${\bar{f}}_{n + 1} (x) \leq {\bar{f}}_{n} (x), n = 1, 2, 3, . . .,$ for every x ∈ D or ${\bar{f}}_{n + 1} (x) \geq {\bar{f}}_{n} (x), n = 1, 2, 3, . . .,$ for every x ∈ D . Then $({\bar{f}}_{n})$ is uniformly convergent to $\bar{f}$ on D .

Proof. Let x ∈ D and $({\bar{f}}_{n})$ be monotonic decreasing on D and let ${\bar{g}}_{n} (x) = {\bar{f}}_{n} (x) - \bar{f} (x) .$ (If $({\bar{f}}_{n})$ be monotonic increasing on D and let ${\bar{g}}_{n} (x) = \bar{f} (x) - {\bar{f}}_{n} (x) .$ ) Then ${\bar{g}}_{n} (x)$ is continuous on D and $lim_{n \to \infty} {\bar{g}}_{n} (x) = \bar{0}$ for each x ∈ D, and $({\bar{g}}_{n})$ is monotonic decreasing on D . To show that $({\bar{g}}_{n})$ is uniformly convergent to $\bar{0}$ on D . Let ɛ > 0 . Since $lim_{n \to \infty} {\bar{g}}_{n} (x) = \bar{0}$ for each x ∈ D, there exists $n_{x} \in ℕ$ such that $\bar{0} \leq {\bar{g}}_{n_{x}} (t) < \frac{ɛ}{2} for every t \in D .$ (3.8)

Since ${\bar{g}}_{n_{x}} (t)$ is continuous at x ∈ D, by relation (3.8), there exists an open neighborhood $\bar{U} (x)$ of x such that $\bar{0} \leq {\bar{g}}_{n_{x}} (t) < ɛ for every t \in \bar{U} (x) \cap D \equiv \bar{A} (x) .$

Since $({\bar{g}}_{n} (t))$ is monotonic decreasing for every $t \in \bar{A} (x),$ we have ${\bar{g}}_{n} (t) < ɛ for all n > n_{x} .$

Since D is a compact subset of $ℝ,$ there exists finite set {x₁, x₂,. . . x_j} such that $D \subset \bar{A} (x_{1}) \cup \bar{A} (x_{2}) \cup . . . \cup \bar{A} (x_{j}) .$

We define N₁ = max {n_{x
₁}, n_{x
₂},. . . n_{x
_j}} . Then we have ${\bar{g}}_{n} (t) < ɛ for all n > N_{1} and for all t \in D .$

Hence $({\bar{g}}_{n})$ is uniformly convergent to $\bar{0}$ on D . □

4 Ideal convergence of sequences of fuzzy valued functions

In this section, we will give the definitions of pointwise ideal convergence and uniformly ideal convergence of sequences of fuzzy valued functions and discuss some fundamental results in this new settings.

Definition 4.1. [20] A sequence of fuzzy numbers $({\bar{x}}_{n})$ is said to be ideally convergent to a fuzzy numbers ${\bar{x}}_{0}$ if for any ɛ > 0 ${n \in ℕ : D ({\bar{x}}_{n}, {\bar{x}}_{0}) \geq ɛ} \in I .$ We write $I - lim {\bar{x}}_{n} = {\bar{x}}_{0}$ or ${\bar{x}}_{n} \overset{I}{⟶} {\bar{x}}_{0} .$

It is clear that $I - lim {\bar{x}}_{n} = {\bar{x}}_{0}$ if and only if for given ɛ > 0, there exists a subset $M \subset ℕ$ satisfying M ∈ I such that $D ({\bar{x}}_{n}, {\bar{x}}_{0}) < ɛ$ for all $n \in ℕ ∖ M .$

Definition 4.2. A sequnece fuzzy valued functions $({\bar{f}}_{n})$ is said to be ideally convergent to a fuzzy valued function $\bar{f}$ if for each x ∈ [a, b] and for each ɛ > 0, there exists a finite set $A \subset ℕ, A \in I$ such that $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all n \in ℕ ∖ A .$ In this case we write $I - lim_{n \to \infty} {\bar{f}}_{n} (x) = \bar{f} (x) .$

Theorem 4.1. Let $({\bar{f}}_{n}),$ $({\bar{g}}_{n})$ and $({\bar{h}}_{n})$ be three sequences of fuzzy valued functions and $\bar{f}$ be fuzzy valued function on [a, b] . If

$({\bar{f}}_{n}) \leq ({\bar{g}}_{n}) \leq ({\bar{h}}_{n})$ for every n ∈ K, where $ℕ \supseteq K$ with K∈ F ;

$I - lim {\bar{f}}_{n} (x) = \bar{f} (x)$ and $I - lim {\bar{h}}_{n} (x) = \bar{f} (x)$ for each x ∈ [a, b] ,

then

I - lim {\bar{g}}_{n} (x) = \bar{f} (x) .

Proof. Let ɛ > 0 be given. Then for each x ∈ [a, b] , we have ${n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} \in I$ and ${n \in ℕ : D ({\bar{h}}_{n} (x), \bar{f} (x)) \geq ɛ} \in I .$

For each x ∈ [a, b] , we put $\begin{matrix} C & = & {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ} \\ D & = & {n \in ℕ : D ({\bar{h}}_{n} (x), \bar{f} (x)) < ɛ} . \end{matrix}$

Let $E = {n \in ℕ : D ({\bar{g}}_{n} (x), \bar{f} (x)) < ɛ} for each$ x ∈ [a, b] . It is clear that C ∩ D ∩ K contained in E, therefore we have E^c ⊆ C^c ∪ D^c ∪ K^c . This implies that E^c ∈ I . Hence $I - lim {\bar{g}}_{n} (x) = \bar{f} (x)$ for each x ∈ [a, b] . □

Definition 4.3. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to pointwise ideally convergent to $\bar{f}$ if ${\bar{f}}_{n} (x) \overset{pI}{⟶} \bar{f} (x)$ for each x ∈ [a, b] i.e for every ɛ > 0 there exists M_x ∈ I such that for all x ∈ [a, b] , $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all n \in ℕ ∖ M_{x} .$

In this case we write $pI - lim {\bar{f}}_{n} (x) = \bar{f} (x)$ or ${\bar{f}}_{n} \overset{pI}{⟶} \bar{f}$ on [a, b] . The function $\bar{f}$ is called the ideal limit function of the sequence $({\bar{f}}_{n}) .$

Theorem 4.2. Let $({\bar{f}}_{n})$ and $({\bar{g}}_{n})$ be two sequences of fuzzy valued functions defined on [a, b] . If

${\bar{f}}_{n} (x) \overset{pI}{⟶} \bar{f} (x)$ on [a, b] ,

${\bar{g}}_{n} (x) \overset{pI}{⟶} \bar{g} (x)$ on [a, b] .

Then

$({\bar{f}}_{n} (x) + {\bar{g}}_{n} (x)) \overset{pI}{⟶} \bar{f} (x) + \bar{g} (x)$ on [a, b] ,

$c {\bar{f}}_{n} (x) \overset{pI}{⟶} c \bar{f} (x)$ on [a, b] , where $c \in ℝ .$

Proof. (a) Since ${\bar{f}}_{n} (x) \overset{pI}{⟶} \bar{f} (x)$ and ${\bar{g}}_{n} (x) \overset{pI}{⟶} \bar{g} (x)$ on [a, b] . Therefore every ɛ > 0, there exists M_ɛ ∈ I such that for each x ∈ [a, b] and for all $n \in ℕ ∖ M_{ɛ},$ $D ({\bar{f}}_{n} (x), \bar{f} (x)) < \frac{ɛ}{2} and D ({\bar{g}}_{n} (x), \bar{g} (x)) < \frac{ɛ}{2} .$

Also, for each x ∈ [a, b] $\begin{matrix} D ({\bar{f}}_{n} (x) + {\bar{g}}_{n} (x), \bar{f} (x) + \bar{g} (x)) \\ \leq D ({\bar{f}}_{n} (x), \bar{f} (x)) + D ({\bar{g}}_{n} (x), \bar{g} (x)) . \end{matrix}$

Therefore, for each x ∈ [a, b] , $\begin{matrix} D ({\bar{f}}_{n} (x) + {\bar{g}}_{n} (x), \bar{f} (x) + \bar{g} (x)) < ɛ \\ for all n \in ℕ ∖ M_{ɛ} . \end{matrix}$

Hence $({\bar{f}}_{n} (x) + {\bar{g}}_{n} (x)) \overset{pI}{⟶} \bar{f} (x) + \bar{g} (x)$ on x ∈ [a, b] .

(b) Suppose ${\bar{f}}_{n} (x) \overset{pI}{⟶} \bar{f} (x)$ on [a, b]. Therefore every ɛ > 0, there exists M_ɛ ∈ I such that $\begin{matrix} D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for each x \in [a, b] \\ and for all n \in ℕ ∖ M_{ɛ} . \end{matrix}$

For c = 0, it is obvious.

For any non-zero $c \in ℝ,$ and for each x ∈ [a, b] , we have $D (c {\bar{f}}_{n} (x), c \bar{f} (x)) = | c | D ({\bar{f}}_{n} (x), \bar{f} (x)) .$

Thus every ɛ > 0, we have $\begin{matrix} D (c {\bar{f}}_{n} (x), c \bar{f} (x)) < | c | ɛ for each x \in [a, b] \\ and for all n \in ℕ ∖ M_{ɛ} . \end{matrix}$ Hence $c {\bar{f}}_{n} (x) \overset{pI}{⟶} c \bar{f} (x)$ on [a, b] , where $c \in ℝ .$ □

Let $({\bar{f}}_{n})$ be a sequence fuzzy valued function on [a, b] such that ${\bar{f}}_{n} (x) \overset{pI}{⟶} \bar{f} (x)$ on [a, b] , and [c, d] ⊂ [a, b] , then ${\bar{f}}_{n} (x) \overset{pI}{⟶} \bar{f} (x)$ on [c, d] .

Theorem 4.3. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is pointwise ideally convergent to $\bar{f}$ on [a, b] if and only if there exists a subset $K_{x} = {n} \subseteq ℕ, n = 1, 2, 3, . . .$ for each x ∈ [a, b] such that K_x ∈ F and $lim_{\begin{matrix} n \to \infty \end{matrix}; n \in K_{x}} {\bar{f}}_{n} (x) = \bar{f} (x)$ for each x ∈ [a, b] .

Proof. Suppose that $({\bar{f}}_{n})$ is pointwise ideally convergent to $\bar{f}$ on [a, b] . Then we have $pI - lim_{n} {\bar{f}}_{n} (x) = \bar{f} (x)$ on [a, b] . We put $\begin{matrix} K_{x, j} & = {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{j}} \\ L_{x, j} & = {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) < \frac{1}{j}} \end{matrix}$ for j = 1, 2, 3 . . . and for each x ∈ [a, b] . Then we have

K_x,j ∈ I,

L_x,1 ⊃ L_2,x ⊃ . . . ⊃ L_x,j ⊃ L_x,j+1 ⊃ . . .

L_x,j ∈ F, for j = 1, 2, 3 . . . and for each x ∈ [a, b] .

We show that for n ∈ L_x,j, $({\bar{f}}_{n})$ is convergent to $\bar{f}$ on [a, b] . Suppose that $({\bar{f}}_{n})$ is not convergent to $\bar{f}$ on [a, b] . Therefore for every ɛ > 0 such that $D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ$ for infinitely many terms and for some x ∈ [a, b] .

Let $L_{x, ɛ} = {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ}$ and $ɛ > \frac{1}{j}$ for j = 1, 2, 3 . . . and for each x ∈ [a, b] . Then L_x,ɛ ∈ I and by the relation (ii) we have L_x,j ⊂ L_x,ɛ . Therefore L_x,j ∈ I which contradicts (iii). Hence $({\bar{f}}_{n})$ is convergent to $\bar{f}$ on [a, b] .

Next, we suppose that there exists a subset $K_{x} = {n} \subset ℕ$ for each x ∈ [a, b] such that K_x ∈ F and $lim_{\begin{matrix} n \to \infty \end{matrix}; n \in K_{x}} {\bar{f}}_{n} (x) = \bar{f} (x)$ for each x ∈ [a, b] i.e. for every ɛ > 0, there exists an integer N (x, ɛ) such that for each x ∈ [a, b] $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all n > N .$

Therefore $\begin{matrix} K_{x, ɛ} = {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} \\ \subseteq ℕ ∖ {n_{N + 1}, n_{N + 2}, . . .} for each x \in [a, b] . \end{matrix}$

Thus we have K_x,ɛ ∈ I for each x ∈ [a, b] . Hence $({\bar{f}}_{n})$ is pointwise ideally convergent to $\bar{f}$ on [a, b] . □

If a sequence of fuzzy valued functions $({\bar{f}}_{n})$ is pointwise ideally convergent to $\bar{f}$ on [a, b] , then there exists a sequence of fuzzy valued functions $({\bar{g}}_{n})$ such that $lim_{n \to \infty} {\bar{g}}_{n} (x) = \bar{f} (x)$ on [a, b] and ${n \in ℕ : {\bar{f}}_{n} (x) = {\bar{g}}_{n} (x)} \in F$ for each x ∈ [a, b] .

Definition 4.4. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to uniformly ideally convergent to $\bar{f}$ on [a, b] if for every ɛ > 0, there exists M ∈ I such that for all $n \in ℕ ∖ M,$ $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all x \in [a, b] .$

In this case we write $uI - lim {\bar{f}}_{n} (x) = \bar{f} (x)$ on [a, b] or ${\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ on [a, b] .

Remark 4.1. From the above definition, we have ${\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ on [a, b] if for every ɛ > 0, ${n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} \in I for all x \in [a, b] .$

In view of the above definitions we state the following result without proof.

Theorem 4.4. If the sequence of fuzzy valued functions ${\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ on [a, b] , then ${\bar{f}}_{n} \overset{pI}{⟶} \bar{f}$ on [a, b] .

Theorem 4.5. Let $\bar{f}$ be fuzzy valued functions and $({\bar{f}}_{n})$ be a sequences of fuzzy valued functions on [a, b] . Then ${\bar{f}}_{n} \overset{{uS}_{2}}{⇉} \bar{f}$ on [a, b] if and only if c_n $\overset{pI}{⟶}$ 0, where $c_{n} = sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) .$

Proof. Assume that c_n $\overset{pI}{⟶}$ 0, i.e. $sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) \overset{pI}{⟶} 0 .$ Then for every ɛ > 0 for each x ∈ [a, b] , we have $\begin{matrix} D ({\bar{f}}_{n} (x), \bar{f} (x)) \leq sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) . \\ i . e {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ for all x \in [a, b]} \\ \subseteq {n \in ℕ : sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

Hence ${\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ on [a, b] .

Next suppose that ${\bar{f}}_{n} \overset{{uS}_{2}}{⇉} \bar{f}$ on [a, b] . For every ɛ > 0, consider the sets $\begin{matrix} A & = {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ for all x \in [a, b]} \\ B & = {n \in ℕ : sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

If $n \in ℕ ∖ A,$ then we have $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all x \in [a, b]$

$\Rightarrow sup_{x \in [a, b]} D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ as ɛ is arbitrary$

$\Rightarrow n \in ℕ ∖ B .$

Therefore we have $ℕ ∖ A \subseteq ℕ ∖ B \Rightarrow B \subseteq A \in I .$ Hence c_n $\overset{pI}{⟶}$ 0 . □

If $({\bar{f}}_{n})$ is a sequence of fuzzy valued functions and $\bar{f}$ is a fuzzy valued function on [a, b] then

${\bar{f}}_{n} \overset{u}{⇉} \bar{f}$ $\Rightarrow {\bar{f}}_{n} \overset{p}{⇉} \bar{f}$ $\Rightarrow {\bar{f}}_{n} \overset{pI}{⇉} \bar{f}$ on [a, b] ,

${\bar{f}}_{n} \overset{u}{⇉} \bar{f}$ $\Rightarrow {\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ $\Rightarrow {\bar{f}}_{n} \overset{pI}{⇉} \bar{f}$ on [a, b] .

Definition 4.5. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to be uniformly ideally Cauchy if for each ɛ > 0 there exists positive integer N = N (ɛ) such that for each x ∈ [a, b] ${n \in ℕ : D ({\bar{f}}_{n} (x), {\bar{f}}_{N} (x)) \geq ɛ} \in I .$

Theorem 4.6. Let $({\bar{f}}_{n})$ be a sequence of fuzzy valued functions defined on [a, b] . Then $({\bar{f}}_{n})$ is pointwise ideally convergent on [a, b] if and only if $({\bar{f}}_{n})$ is uniformly ideally Cauchy on [a, b] .

Proof. Suppose that $({\bar{f}}_{n})$ is pointwise ideally convergent to $\bar{f}$ on [a, b] . Then for every ɛ > 0 there exists M_x ∈ I such that $\begin{matrix} D ({\bar{f}}_{n} (x), \bar{f} (x)) < \frac{ɛ}{2} for all x \in [a, b] \\ and for all n \in ℕ ∖ M_{x} . \end{matrix}$

If we choose $N \in ℕ,$ so that $D ({\bar{f}}_{N} (x), \bar{f} (x)) < \frac{ɛ}{2} for all x \in [a, b] .$

Now, we have $\begin{matrix} D ({\bar{f}}_{n} (x), {\bar{f}}_{N} (x)) \leq D ({\bar{f}}_{n} (x), \bar{f} (x)) + D ({\bar{f}}_{N} (x), \bar{f} (x)) \\ < \frac{ɛ}{2} + \frac{ɛ}{2} = ɛ for all x \in [a, b] . \end{matrix}$ Therefore $({\bar{f}}_{n})$ is uniformly ideally Cauchy on [a, b] .

Next, we suppose that $({\bar{f}}_{n})$ is uniformly ideally Cauchy on [a, b] . then we choose $N \in ℕ$ such that the band $J = [{\bar{f}}_{N} (x) - 1, {\bar{f}}_{N} (x) + 1]$ contains ${\bar{f}}_{n} (x)$ for all $n \in ℕ$ and for each x ∈ [a, b] . Now choose L such that $J_{0} = [{\bar{f}}_{L} (x) - \frac{1}{2}, {\bar{f}}_{L} (x) + \frac{1}{2}]$ contains ${\bar{f}}_{n} (x)$ for all $n \in ℕ$ and for each x ∈ [a, b] . We assert that J₁ = J ∩ J₀ contains ${\bar{f}}_{n} (x)$ for all $n \in ℕ$ and for each x ∈ [a, b] , so $\begin{matrix} {n \in ℕ : {\bar{f}}_{n} (x) \notin J \cap J_{0}} \\ = {n \in ℕ : {\bar{f}}_{n} (x) \notin J} \cup {n \in ℕ : {\bar{f}}_{n} (x) \notin J_{0}} . \end{matrix}$

Therefore, for each x ∈ [a, b] we have $\begin{matrix} {n \in ℕ : {\bar{f}}_{n} (x) \notin J \cap J_{0}}) \\ \subseteq {n \in ℕ : {\bar{f}}_{n} (x) \notin J}) \cup {n \in ℕ : {\bar{f}}_{n} (x) \notin J_{0}} \in I . \end{matrix}$

Thus J₁ is a closed band of height less than equal to 1 that contains ${\bar{f}}_{n} (x)$ for all $n \in ℕ$ and for each x ∈ [a, b] . We proceed by choosing N (2) so that $J_{2} = [{\bar{f}}_{N (2)} (x) - \frac{1}{4}, {\bar{f}}_{N (2)} (x) + \frac{1}{4}]$ contains ${\bar{f}}_{n} (x)$ for all $n \in ℕ$ and for each x ∈ [a, b] . By the preceding argument J₃ = J₁ ∩ J₂ contains ${\bar{f}}_{n} (x)$ for all $n \in ℕ$ and for each x ∈ [a, b] and J₃ has height less than equal to $\frac{1}{2} .$ Continuing inductively we construct a sequence (J_i) of closed band such that for each i, J_i ⊇ J_i+1, the height of J_i is not greater than 2^1-i, and ${\bar{f}}_{n} (x) \in J_{i}$ for all $n \in ℕ$ and for each x ∈ [a, b] . Thus there exists a fuzzy valued function $\bar{f} (x)$ on [a, b] such that $\bar{f} (x) = ⋂_{i = 1}^{\infty} J_{i} .$ We show that ${\bar{f}}_{n} (x)$ is ideally convergent to $\bar{f} (x)$ on [a, b] . Let ɛ > 0 be given. Then there exists j such that ɛ > 2^1-j . Then from the above argument it follows that ${\bar{f}}_{n} (x) \in J_{i}$ for all $n \in ℕ$ and for each x ∈ [a, b] . Then we have for each x ∈ [a, b] ${n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ}) \subseteq {n \in ℕ : {\bar{f}}_{n} (x) \notin J_{i}} \in I .$

Hence completes the proof. □

Theorem 4.7. Let $({\bar{f}}_{n})$ be a sequence of fuzzy valued functions defined on [a, b] . Then the followings are equivalent:

$({\bar{f}}_{n})$ is pointwise deally convergent to $\bar{f}$ on [a, b] ;

$({\bar{f}}_{n})$ is uniformly ideally Cauchy on [a, b] ;

there exists a subsequence $({\bar{g}}_{n})$ of $({\bar{f}}_{n})$ such that $lim_{n \to \infty} {\bar{g}}_{n} (x) = \bar{f} (x)$ on [a, b] .

Proof. The proof of the result follows from Theorems 4.3 and 4.6

Theorem 4.8. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is ideally convergent to a fuzzy valued function $\bar{f}$ if and only if $[{\bar{f}}_{n}]_{α}$ is uniformly ideally convergent to $[\bar{f}]_{α}$ with respect to α .

Proof. Suppose that $({\bar{f}}_{n})$ is ideally convergent to a fuzzy valued function $\bar{f}$ on [a, b] . For any x ∈ [a, b] and for all α ∈ [0, 1] , and for every ɛ > 0 there exists M_x ∈ I such that $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all n \in ℕ ∖ M_{x} .$ $\Rightarrow sup_{α \in [0, 1]} max {| ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ for all n \in ℕ ∖ M_{x} .$

But $[{\bar{f}}_{n} (x)]_{α} = [({\bar{f}}_{n} (x))_{α}^{-}, ({\bar{f}}_{n} (x))_{α}^{+}]$ and $[\bar{f} (x)]_{α} = [(\bar{f} (x))_{α}^{-}, (\bar{f} (x))_{α}^{+}],$ therefore we have, for each $n \in ℕ ∖ M_{x}$ and for all α ∈ [0, 1] $\begin{matrix} | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | < ɛ and \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ . \end{matrix}$

Hence for all α ∈ [0, 1] , $[{\bar{f}}_{n} (x)]_{α}$ is uniformly ideally convergent to $[\bar{f} (x)]_{α}$ with respect to α .

Next, we suppose that for all α ∈ [0, 1] , $[{\bar{f}}_{n}]_{α}$ is uniformly ideally convergent to $[\bar{f}]_{α}$ with respect to α . Then for every ɛ > 0 and for any x ∈ [a, b] , we consider $\begin{matrix} A & = {n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \geq ɛ} \\ B & = {n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \geq ɛ} \\ C & = {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

Now for any n ∈ A^c ∩ B^c, for any x ∈ [a, b] and α ∈ [0, 1] , we have $\begin{matrix} | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | < ɛ and \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | < ɛ . \end{matrix}$

Therefore $\begin{matrix} D ({\bar{f}}_{n} (x), \bar{f} (x)) \\ = sup_{α \in [0, 1]} max {| ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ . \end{matrix}$ i.e. n ∈ C^c . So C ⊆ A ∪ B ∈ I . Hence $({\bar{f}}_{n})$ is ideally convergent to a fuzzy valued function $\bar{f}$ on [a, b] . □

Definition 4.6. Two sequence of fuzzy valued functions $({\bar{f}}_{n})$ and $({\bar{g}}_{n})$ on [a, b] are said to be almost all n relatively to I, if ${n \in ℕ : {\bar{g}}_{n} (x) \neq {\bar{f}}_{n} (x)} \in I$ for each [a, b] .

Theorem 4.9. Let $({\bar{f}}_{n})$ be sequence of fuzzy valued functions, then the following statements are equivalent:

$({\bar{f}}_{n})$ is ideally convergent to $\bar{f};$

there exist sequences of fuzzy valued functions $({\bar{g}}_{n})$ and $({\bar{h}}_{n})$ such that ${\bar{f}}_{n} (x) = {\bar{g}}_{n} (x) + {\bar{h}}_{n} (x),$ $lim_{n \to \infty} {\bar{g}}_{n} (x) = \bar{f} (x)$ and ${\bar{h}}_{n} (x) \overset{pI}{⟶} \bar{0}$ for x ∈ [a, b] ,

there exists a subsequence $K = {n_{l}} \subset ℕ$ such that K ∈ F and $D ({\bar{f}}_{n_{l}} (x), \bar{f} (x)) ⟶ \bar{0}$ as l → ∞ .

Proof. (i) ⇒ (ii) : Suppose that $({\bar{f}}_{n})$ is ideally convergent to $\bar{f}$ on [a, b] . For every ɛ > 0 and for a given x ∈ [a, b] we have ${n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} \in I .$

Let (n_t) be a nondecreasing sequence with $n_{t} \in ℕ$ (t ≥ 1) such that ${n_{t} \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{t}} \in I .$

For a given x ∈ [a, b] , if 0 < n ≤ n₁ we define ${\bar{g}}_{n} (x) = {\bar{f}}_{n} (x) and {\bar{h}}_{n} (x) = {\bar{f}}_{n} (x) + \bar{f} (x) .$ and for n_t < n ≤ n_t+1 ${\bar{g}}_{n} (x) = {\begin{matrix} {\bar{f}}_{n} (x), & if D ({\bar{f}}_{n} (x), \bar{f} (x)) < \frac{1}{t} \\ \bar{f} (x), & if D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{t} \end{matrix}$ and ${\bar{h}}_{n} (x) + \bar{f} (x) = {\begin{matrix} \bar{f} (x), & if D ({\bar{f}}_{n} (x), \bar{f} (x)) < \frac{1}{t} \\ {\bar{f}}_{n} (x), & if D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{t} \end{matrix}$

Then we have ${\bar{f}}_{n} (x) = {\bar{g}}_{n} (x) + {\bar{h}}_{n} (x) .$

Now for every ɛ > 0 and for a given x ∈ [a, b] choose $t \in ℕ$ such that $\frac{1}{t} < ɛ,$ if $D ({\bar{f}}_{n} (x), \bar{f} (x)) < \frac{1}{t}$ then we have $D ({\bar{g}}_{n} (x), \bar{f} (x)) = D ({\bar{f}}_{n} (x), \bar{f} (x)) < \frac{1}{t} < ɛ$ and if $D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{t},$ we have $D ({\bar{g}}_{n} (x), \bar{f} (x)) = 0 .$

Therefore $D ({\bar{g}}_{n} (x), \bar{f} (x)) < ɛ .$

Hence $lim_{n \to \infty} {\bar{g}}_{n} (x) = \bar{f} (x)$ for a given x ∈ [a, b] .

Again if n_t < n ≤ n_t+1, we have $\begin{matrix} {n \in ℕ : D ({\bar{h}}_{n} (x) + \bar{f} (x), \bar{f} (x)) \geq \frac{1}{t}} \\ \subset {n_{t} \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{t}} . \\ \Rightarrow {n \in ℕ : D ({\bar{h}}_{n} (x) + \bar{f} (x), \bar{f} (x)) \geq \frac{1}{t}} \\ \subset {n_{t} \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{t}} \in I . \end{matrix}$

Hence ${\bar{h}}_{n} (x) + \bar{f} (x) \overset{I}{⟶} \bar{f} (x)$ for a given x ∈ [a, b] .

For every ɛ > 0 and for a given x ∈ [a, b] , there exists M_x ∈ I such that for all α ∈ [0, 1] , $| ({\bar{h}}_{n} (x))_{α}^{-} + (\bar{f} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | < ɛ$ and $| ({\bar{h}}_{n} (x))_{α}^{+} + (\bar{f} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | < ɛ .$ $\Rightarrow | ({\bar{h}}_{n} (x))_{α}^{-} | < ɛ and | {\bar{h}}_{n} (x))_{α}^{+} | ɛ for all α \in [0, 1] .$ $\Rightarrow sup_{α \in [0, 1]} max {| ({\bar{h}}_{n} (x))_{α}^{-} - 0 |, | {\bar{h}}_{n} (x))_{α}^{+} - 0 |} < ɛ .$

Hence ${\bar{h}}_{n} (x) \overset{pI}{⟶} \bar{0}$ for x ∈ [a, b] .

(ii) ⇒ (iii) : Since ${\bar{h}}_{n} (x) \overset{pI}{⟶} \bar{0},$ so for a given x ∈ [a, b] and for every ɛ > 0, there exists M_x ∈ I such that $D ({\bar{h}}_{n} (x), \bar{0}) < ɛ$ for all $n \in ℕ ∖ M_{x} .$ i.e. ${n \in ℕ : D ({\bar{h}}_{n} (x), \bar{0}) \geq ɛ} \in I .$ We define $K = {n_{l}} \subseteq ℕ ∖ M_{x}$ as a subsequence of $ℕ$ such that ${\bar{h}}_{n} (x) = \bar{0}$ for n ∈ K . Therefore we have $K = ℕ ∖ {n \in ℕ : D ({\bar{h}}_{n} (x), \bar{0}) \geq ɛ} \in F .$ Again, since ${\bar{f}}_{n_{l}} (x) = {\bar{g}}_{n_{l}} (x)$ for $l \in ℕ$ and $D ({\bar{g}}_{n} (x), \bar{f} (x)) ⟶ 0,$ it follows that $D ({\bar{f}}_{n_{l}} (x), \bar{f} (x)) ⟶ 0 .$ (iii) ⇒ (i) : For any x ∈ [a, b] , suppose that (iii) holds, then there exists a subsequence K = {n_l} of $ℕ$ with K ∈ F such that for every ɛ > 0, $D ({\bar{f}}_{n_{l}} (x), \bar{f} (x)) < ɛ$ for all n_l ∈ K . Then we have ${n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} \subset ℕ ∖ K \in I .$

Hence $({\bar{f}}_{n})$ is ideally convergent to $\bar{f}$ on [a, b] . □

Let $({\bar{f}}_{n})$ be sequence of fuzzy valued functions, then the following statements are equivalent:

$({\bar{f}}_{n})$ is ideally convergent to $\bar{f};$

there exists a sequence of fuzzy valued functions $({\bar{g}}_{n})$ such that ${\bar{g}}_{n} (x) = {\bar{f}}_{n} (x)$ a . a . n . r . I for each x ∈ [a, b] and $lim_{n \to \infty} {\bar{g}}_{n} (x) = \bar{f} (x) .$

there exists a subsequence $K = {n_{l}} \subset ℕ$ such that K ∈ F and $D ({\bar{f}}_{n_{l}} (x), \bar{f} (x)) ⟶ \bar{0}$ as l → ∞ .

Theorem 4.10. Let the sequence of fuzzy valued functions ${\bar{f}}_{n} \overset{pI}{⟶} \bar{f}$ on [a, b] , where $({\bar{f}}_{n})$ are equi-continuous on [a, b] , then $\bar{f}$ is continuous and ${\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ on [a, b] .

Proof. First we prove that $\bar{f}$ is continuous on [a, b] . Let ɛ > 0 and let x₀ be any point in [a, b] . Since $({\bar{f}}_{n})$ are equi-continuous on [a, b] , there exists δ > 0 such that $\begin{matrix} D ({\bar{f}}_{n} (x), {\bar{f}}_{n} (x_{0})) < \frac{ɛ}{3} where \\ x_{0} - δ < x < x_{0} + δ for all n \in ℕ . \end{matrix}$

Also since ${\bar{f}}_{n} \overset{pI}{⟶} \bar{f}$ on x ∈ [a, b] , for any $n \in ℕ$ we have $\begin{matrix} {n \in ℕ : D ({\bar{f}}_{n} (x_{0}), {\bar{f}}_{n} (x_{0})) \geq \frac{ɛ}{3}} \\ \cup {n \in ℕ : D ({\bar{f}}_{n} (x), {\bar{f}}_{n} (x)) \geq \frac{ɛ}{3}} \in I . \end{matrix}$

Therefore there exists $n \in ℕ$ such that $D ({\bar{f}}_{n} (x_{0}), {\bar{f}}_{n} (x_{0})) < \frac{ɛ}{3} and D ({\bar{f}}_{n} (x), {\bar{f}}_{n} (x)) < \frac{ɛ}{3} .$

Then $\begin{matrix} D (\bar{f} (x), \bar{f} (x_{0})) \\ \leq D (\bar{f} (x_{0}), {\bar{f}}_{n} (x_{0})) + D ({\bar{f}}_{n} (x_{0}), {\bar{f}}_{n} (x)) \\ + D ({\bar{f}}_{n} (x), {\bar{f}}_{n} (x)) \\ < \frac{ɛ}{3} + \frac{ɛ}{3} + \frac{ɛ}{3} = ɛ . \end{matrix}$

Since x₀ is arbitrary, therefore $\bar{f}$ is continuous on [a, b] .

Next we prove that ${\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ on [a, b] . Since $\bar{f}$ is continuous on [a, b] , it implies that it is uniformly continuous on [a, b] and $({\bar{f}}_{n})$ are equi-continuous on [a, b] , then for any x, y ∈ [a, b] and for ɛ > 0 there exists δ > 0 such that $D ({\bar{f}}_{n} (x), {\bar{f}}_{n} (y)) < \frac{ɛ}{3} and D (\bar{f} (x), \bar{f} (y)) < \frac{ɛ}{3}$ whenever y - δ < x < y + δ .

Since [a, b] is compact, we can choose a finite cover (x₁ - δ, x₁ + δ) , (x₂ - δ, x₂ + δ) , (x₃ - δ, x₃ + δ) ,. . . (x_i - δ, x_i + δ) from the covers of [a, b] . Since ${\bar{f}}_{n} \overset{pI}{⟶} \bar{f}$ on x ∈ [a, b] , there exists a set M_{x
_k} ∈ I such that for k ∈ {1, 2,. . . i} , $D ({\bar{f}}_{n} (x_{k}), \bar{f} (x_{k})) < \frac{ɛ}{3} for all n \notin M_{x_{k}} .$

Then for any n ∉ M_{x
_k} and for x ∈ (x_k - δ, x_k + δ) , for some k ∈ {1, 2,. . . , i} we have $\begin{matrix} D ({\bar{f}}_{n} (x), \bar{f} (x)) \\ \leq D ({\bar{f}}_{n} (x), {\bar{f}}_{n} (x_{k})) + D ({\bar{f}}_{n} (x_{k}), \bar{f} (x_{k})) \\ + D (\bar{f} (x_{k}), \bar{f} (x)) \\ < \frac{ɛ}{3} + \frac{ɛ}{3} + \frac{ɛ}{3} = ɛ \end{matrix}$ which implies that ${\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ on [a, b] . □

Theorem 4.11. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is uniformly ideally convergent to a fuzzy valued function $\bar{f}$ on [a, b] if and only if $[{\bar{f}}_{n} (x)]_{α}$ is uniformly ideally convergent to $[\bar{f} (x)]_{α}$ with respect to α and x .

Proof. Let ɛ > 0 be given. Suppose that ${\bar{f}}_{n} \overset{uI}{⇉} \bar{f}$ on [a, b] . There exists M ∈ I such that $\begin{matrix} D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ for all n \in ℕ ∖ M and \\ x \in [a, b] . \end{matrix}$ i.e. for all $n \in ℕ ∖ M$ and x ∈ [a, b] , $\begin{matrix} sup_{α \in [0, 1]} max {| ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ . \end{matrix}$

Therefore for all α ∈ [0, 1] and for all x ∈ [a, b] we have $\begin{matrix} | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \leq D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \leq D ({\bar{f}}_{n} (x), \bar{f} (x)) < ɛ . \end{matrix}$ Since $[{\bar{f}}_{n} (x))]_{α} = [({\bar{f}}_{n} (x))_{α}^{-}, ({\bar{f}}_{n} (x))_{α}^{+}]$ and $[\bar{f} (x))]_{α} = [(\bar{f} (x))_{α}^{-}, (\bar{f} (x))_{α}^{+}] .$ Then for every ɛ > 0 and for all x ∈ [a, b] , we have $\begin{matrix} {n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \geq ɛ} \\ \subseteq {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} \end{matrix}$ and $\begin{matrix} {n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \geq ɛ} \\ \subseteq {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

This implies that $[{\bar{f}}_{n} (x)]_{α}$ is uniformly ideally convergent to $[\bar{f} (x)]_{α}$ with respect to α and x .

Next, for α ∈ [0, 1] and for all x ∈ [a, b] we suppose that $[{\bar{f}}_{n} (x)]_{α}$ is uniformly ideally convergent to $[\bar{f} (x)]_{α}$ with respect to α and x .

Then for any ɛ > 0 and for all x ∈ [a, b] , consider the sets $\begin{matrix} P & = {n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \geq ɛ} \\ Q & = {n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \geq ɛ} \\ R & = {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ɛ} . \end{matrix}$

Thus we have P, Q ∈ I such that for every ɛ > 0, for α ∈ [0, 1] , and for any x ∈ [0, 1] , $\begin{matrix} | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | < ɛ for all n \in ℕ ∖ P \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | < ɛ for all n \in ℕ ∖ Q . \end{matrix}$

Therefore for any x ∈ [a, b] , we have $D ({\bar{f}}_{n} (x), \bar{f} (x))$ $sup_{α \in [0, 1]} max {| ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} < ɛ$ $\Rightarrow n \in ℕ ∖ R$ $\Rightarrow (ℕ ∖ P) \cap (ℕ ∖ Q) \subseteq (ℕ ∖ R) \Rightarrow R \subseteq P \cup Q \in I .$

This completes the proof. □

5 Applications

The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets, introduced by L.A. Zadeh [31] in 1965. This theory brought a paradigmatic change in mathematics. But there exists diffculty, how to set the membership function in each particular case. But there are also cases where these theories failed to give satisfactory results, possibly due to inadequacy of the parameterization tool in them. As a necessary supplement to the existing mathematical tools for handling uncertainty, Molodtsov [21] introduced the theory of soft sets as a new mathematical tool to deal with uncertainties while modelling the problems in engineering, physics, computer science, economics, social sciences, medical sciences, smoothness of functions, game theory, operations research, Riemann integration, Perron integration, probability, and theory of measurement. We refer to the readers [32, 33] for new roughset theory.

Integral equations are very useful for solving many problems in several applied fields like mathematical economics and optimal control theory. The concept of integration of fuzzy functions was introduced by Dubois and Prade [10]. Wavelet theory is a relatively new and an emerging area in mathematical research. Also, wavelets are the suitable and powerful tool for approximating functions based on wavelet basis functions. In [34] Ziari et al. constructed fuzzy Haar wavelet and applied it on solving linear fuzzy Fredholm integral equations of the second kind (in short FFIE₂). By using fuzzy wavelet like operator, we propose a numerical approach for approximating the solution of linear fuzzy Fredholm integral equations of the second kind $\bar{f} (x) = \bar{g} (x) \oplus λ \int_{a}^{b} K (x, t) \otimes \bar{f} (t) dt,$ (5.9) where K (x, t) is an arbitrary kernel function over the square [a, b] × [a, b] and $\bar{g}$ is a fuzzy valued function. We should mention that we do not suppose any kind of orthogonality condition of the scaling function as φ and the operators act on fuzzy valued continuous function over reals. We denote $UCF (ℝ)$ the set of all uniformly continuous fuzzy valuedfunctions.

Let $\bar{f} : ℝ \to F (ℝ)$ be a bounded fuzzy valued function. Then the function $ω_{ℝ} (\bar{f}) : ℝ^{+} \cup {0} \to ℝ^{+}$ defined by $\begin{matrix} ω_{ℝ} (\bar{f}, δ) : = sup {D (\bar{f} (x), \bar{g} (x)) : \\ x, y \in ℝ, | x - y | < δ}, \end{matrix}$ where $ℝ^{+}$ is the set of positive real numbers, is called the modulus of continuity of $\bar{f}$ on $ℝ .$ The properties of the modulus of continuity are as follows. (iii)

$D (\bar{f} (x), \bar{g} (x)) \leq ω_{[a, b]} (\bar{f}, | x - y |)$ for any x, y ∈ [a, b] ,

$ω_{[a, b]} (\bar{f}, δ)$ is an increasing function of δ,

$ω_{[a, b]} (\bar{f}, 0) = 0,$

$ω_{[a, b]} (\bar{f}, δ_{1} + δ_{2}) \leq ω_{[a, b]} (\bar{f}, δ_{1}) + ω_{[a, b]} (\bar{f}, δ_{2})$ for any δ₁, δ₂ ≥ 0,

$ω_{[a, b]} (\bar{f}, n δ) \leq n ω_{[a, b]} (\bar{f}, δ),$ for any $δ \geq 0, n \in ℕ,$

$ω_{[a, b]} (\bar{f}, λ δ) \leq (λ + 1) ω_{[a, b]} (\bar{f}, δ)$ for any δ, λ ≥ 0, where $\bar{f} : [a, b] \to F ℝ .$

Let $\bar{f} \in UCF (ℝ)$ and the scaling function φ a real valued bounded function with suppφ (x) ⊆ [- β, β] , 0 < β < + ∞ , φ (x) ≥0 such that $\sum_{j = - \infty}^{+ \infty} φ (x - j) = 1 .$

For $k \in ℤ^{+}, x \in ℝ,$ put $(\bar{B_{k} f}) (x) = \sum_{j = - \infty}^{+ \infty} \bar{f} (\frac{j}{2^{k}}) \otimes φ (2^{k} x - j)$ (5.10)

which is a fuzzy-wavelet like operator. Then $\begin{matrix} D ((\bar{B_{k} f}) (x), \bar{f} (x)) \leq ω_{k} (\bar{f}, \frac{β}{2^{k}}) \\ for all x \in ℝ, k \in ℤ^{+} . \end{matrix}$ If $\bar{f} \in UCF (ℝ)$ then $ω_{k} (\bar{f}, \frac{β}{2^{k}}) \to 0 as k \to + \infty and lim_{k \to + \infty} B_{k} \bar{f} = \bar{f}$ pointwise and uniformly with rates.

Suppose $φ (x) = {\begin{matrix} 1, & if - \frac{1}{2} \leq x \leq \frac{1}{2} \\ 0, & otherwise \end{matrix}$

Consider the linear fuzzy Fredholm integral equations of the second kind $\begin{matrix} {\bar{g}}^{-} (x, α) = α x + \frac{3}{26} - \frac{3 α}{26} - \frac{1}{13} x^{2} - \frac{1}{13} x^{2} α, \\ {\bar{g}}^{+} (x, α) = 2 x - α x + \frac{3 α}{26} + \frac{1}{13} x^{2} α - \frac{3}{26} - \frac{3}{13} x^{2} \\ K (x, t) = \frac{x^{2} + t^{2} - 2}{13}, 0 \leq x, t \leq 2 . \end{matrix}$

Also let a = 0, b = 1 . The exact solution in this case is given by ${\bar{f}}^{-} (x, α) = α x, {\bar{f}}^{+} (x, α) = (2 - α) x .$ We consider the techniques using by Ziari [34] to compute the numerical results with the exact solution in Table 1 and Table 2, and represented it in the Fig. 1 and Fig. 2, respectively.

6 Conclusions

In this paper, the concepts of ideal convergence and uniformly ideal convergence of sequences of fuzzy valued functions have been introduced. Also, we obtained the relationship among ideal convergence and uniformly ideal convergence of sequences of fuzzy valued functions, and studied their representations of sequences of α-level cuts. The study of lacunary ideal convergence, uniformly lacunary ideal convergence sequences of fuzzy valued functions based on the concept of convergnce of sequences of fuzzy numbers may be suggested as some important future work in this new setting.

Footnotes

Acknowledgments

The author thank Prof. Jianming Zhan and the referees for their valuable comments and helpful suggestions.

References

Altin

, Et

and Tripathy

B.C.

, On pointwise statistical convergence sequences of fuzzy mappings, J Fuzzy Math 15(2) (2007), 425–433.

Altinok

, Statistical convergence of order β for generalized difference sequences of fuzzy numbers, J Intell Fuzzy Systems 26 (2014), 847–856.

Altinok

, On λ-Statistical convergence of order β of sequences of fuzzy numbers, Internat J Uncertain Fuzziness Knowledge-Based Systems 20(2) (2012), 303–312.

Altinok

, Altin

and Işık

, Statistical convergence and strong p-Cesàro summability of order β of sequences of fuzzy numbers, Iranian J Fuzzy Systems 9(2) (2013), 63–73.

Anastassiou

G.A.

and Duman

, Statistical fuzzy approximation by fuzzy positive linear operators, Comput Math Appl 55(3) (2008), 573–580.

Aytar

and Pehlvian

, Statistical convergence of sequences of fuzzy numbers and sequences of α-cuts, Inter J General Systems 37(2) (2008), 231–237.

Balcerzak

, Dems

and Komisarski

, Statistical convergence and ideal convergence for sequences of functions, J Math Anal Appl 328 (2007), 715–729.

Çinar

, Karakaş

and Et

, On pointwise and uniform statistical convergence of order α for sequences of functions, Fixed Point Theory Appl 2013 (2013), 2013:33: 11.

Cartan

, Filters et ultrafilters, C R Acad Sci Paris 205 (1937), 777–779.

10.

Dubois

and Prade

, Towards fuzzy differential calculus, Fuzzy Sets Systems 8(3) (1982), 1–7.

11.

, Tripathy

B.C.

and Dutta

A.J.

, On pointwise statistical convergence of order α of sequences of fuzzy mappings, Kuwait J Sci 41(3) (2014), 17–30.

12.

Fast

, Sur la convergence statistique, Coll Math 2 (1951), 241–244.

13.

Gong

, Zhang

and Zhu

, The statistical convergence for sequences of fuzzy-number-valued functions, Inform Sci 295 (2015), 182–195.

14.

Hazarika

and Mohiuddine

S.A.

, Ideal convergence of random variables, J Func Spaces Appl 2013 (2013), 7. Article ID 148249.

15.

Hazarika

, Lacunary difference ideal convergent sequence spaces of fuzzy numbers, J Intell Fuzzy Systems 25(1) (2013), 157–166.

16.

Hazarika

, On σ-uniform density and ideal convergent sequences of fuzzy real numbers, J Intell Fuzzy Systems 26(2) (2014), 793–799.

17.

Hazarika

, On σ-bounded variation and ideal convergence of fuzzy sequences, J Intell Fuzzy Systems 28(1) (2015), 411–420.

18.

Jasinski

and Reclaw

, Ideal convergence of continuous functions, Topology Appl 153 (2006), 3511–3518.

19.

Kostyrko

, Šalát

and Wilczynśki

, On I-convergence, Real Analysis Exchange 26(2) (2000–2001), 669–686.

20.

Kumar

and Kumar

, On the ideal convergence of sequences of fuzzy numbers, Inform Sci 178 (2008), 4670–4678.

21.

Molodstov

D.A.

, Soft set theory-first result, Comput Math Appl 37 (1999), 19–31.

22.

Mursaleen

and Mohiuddine

S.A.

, On ideal convergence in probabilistic normed spaces, Math Slovaca 62(1) (2012), 49–62.

23.

Negoita

C.V.

and Ralescu

, Applications of fuzzy sets to systems analysis, Wiley, New York, 1975.

24.

Nuray

and Ruckle

W.H.

, Generalized statistical convergence and convergence free spaces, J Math Anal Appl 245 (2000), 513–527.

25.

Šalát

, Tripathy

B.C.

and Ziman

, On some properties of I-convergence, Tatra Mt Math Publ 28 (2004), 279–286.

26.

Şavas

, On statistically convergent sequences of fuzzy numbers, Inform Sci 137(1-4) (2001), 277–282.

27.

Schoenberg

I.J.

, The integrability of certain functions and related summability methods, Amer Math Monthly 66 (1959), 361–375.

28.

Steinhaus

, Sur la convergence ordinaire et la convergence asymptotique, Colloq Math 2 (1951), 73–74.

29.

Tripathy

B.C.

and Sen

, On fuzzy I-convergent difference sequence spaces, J Intell Fuzzy Systems 25(3) (2013), 643–647.

30.

Tripathy

B.C.

and Dutta

A.J.

, Lacunary I-convergent sequences of fuzzy real numbers, Proyecciones J Math 34(3) (2015), 205–218.

31.

Zadeh

L.A.

, Fuzzy sets, Inform Control 8 (1965), 338–353.

32.

Zhan

, Liu

and Davvaz

, A new rough set theory: Rough soft hemirings, J Intell Fuzzy Systems 28 (2015), 1687–1697.

33.

Zhan

, Yu

and Fotea

V.-E.

, Characterizations of two kinds of hemirings based on probability spaces, Soft Computing 20 (2016), 637–648.

34.

Ziari

, Ezzati

and Abbasbandy

, Numerical solution of linear fuzzy Fredholm integral equations of the second kind using Haar wavelet, Commun Comput Inform Sci 299(3) (2012), 79–89.