Applications of statistical convergence of order ( η,δ + γ ) in difference sequence spaces of fuzzy numbers

Abstract

In this article, we introduce and study some difference sequence spaces of fuzzy numbers by making use of λ-statistical convergence of order (η, δ + γ) . With the aid of MATLAB software, it appears that the statistical convergence of order (η, δ + γ) is well defined every time when (δ + γ) > η and this convergence fails when (δ + γ) < η. Moreover, we try to set up relations between (Δ^v, λ)-statistical convergence of order (η, δ + γ) and strongly (Δ^v, p, λ)-Cesàro summability of order (η, δ + γ) and give some compelling instances to show that the converse of these relations is not valid. In addition to the above results, we also graphically exhibits that if a sequence of fuzzy numbers is bounded and statistically convergent of order (η, δ + γ) in (Δ^v, λ), then it need not be strongly (Δ^v, p, λ)-Cesàro summable of order (η, δ + γ).

Keywords

Cesàro summability difference operator fuzzy numbers

1 Introduction

The hypothesis of statistical convergence was first presented by Fast [11] in 1951 and further by Schoenberg [33] in 1959, which brings the attention of many researchers from different fields of mathematics. Connor [9], Fridy [12], Šalát [31] linked the notion of statistical convergence with that of summability theory. In 1980 Gadjiev and Orhan [13] studied the order of statistical convergence for a sequence of numbers and later on, Çolak [8] introduced the statistical convergence of order β and strongly p-Cesàro summability of order β. After that, the concept of λ-statistical convergence of order β and strongly λp-summability of order β for the sequence of fuzzy numbers were given by Altinok [1]. Also, Altinok [2] gave the idea of statistical convergence of order β and strongly p-Cesàro summability of order β for generalized difference sequences of fuzzy numbers which was further enlarged to a statistical convergence of order (β, γ) by Altinok and Et in [4]. Recently, E. Gülle studied double Wijsman asymptotically statistically equivalence of order α in [14] and U. Ulusu and E. Gülle study the concept of double Wijsman asymptotically ideal statistically equivalence of order η in [38]. For a detailed study of statistical convergence, one can refer ([16 , 39]).

In 2000 Savas initiated the idea of λ-statistical convergence of sequence of fuzzy numbers. Let us consider Λ be a class of increasing sequence of positive real numbers λ = (λ_n) such that $λ_{1} \leq λ_{2} \leq λ_{3} \leq . . . and lim_{n \to \infty} λ_{n} = \infty,$ satisfying λ_n+1 ≤ λ_n + 1 where λ₁ = 1.

Kizmaz [21] in 1981 gave the concept of difference sequence spaces which was generalized by Et and Çolak [10], Altinok et al. [3] and Çolak [7]. For a literature in difference sequence spaces one can refer to ([5 , 19]). Let Ω (F) be the set of all sequences of fuzzy numbers, then Δ^v : Ω (F) → Ω (F) is defined by $(Δ^{0} Y)_{s} = Y_{s}, (Δ^{1} Y)_{s} = Δ^{1} Y_{s} = Y_{s} - Y_{s + 1}$ $(Δ^{v} Y)_{s} = Δ^{1} (Δ^{v - 1} Y)_{s}, (v \geq 2),$ for all $v \in ℕ .$ A sequence Y = (Y_s) of fuzzy numbers is called Δ^v-bounded if there exists fuzzy numbers a and b such that a ≤ Δ^vY_s ≤ b, for each $v \in ℕ$ and Δ^v-convergent to a fuzzy number Y₀ if for every ε ≥ 0 there exists a positive integer s₀ such that d (Δ^vY_s, Y₀) < ε, for s > s₀.

The idea of fuzzy numbers and its arithmetic operation were introduced by Zadeh [42] in 1965. Later, many other authors discussed various applications of the fuzzy sets such as fuzzy ordering, the fuzzy measure of fuzzy event, fuzzy mathematical programming, etc. Matloka [22] first studied the theory of a sequence of fuzzy numbers. Later on, Nanda [24] introduced the sequences of fuzzy numbers and studied that set of all convergent sequences of fuzzy numbers form complete metric space. Further, Savas and Mursaleen [32], Tripathy and Nanda [36], Hazarika and Savas [15] and many others discussed the theory of sequences of fuzzy numbers. Sakhre [30] studied some applications of fuzzy number system and used fuzzy counter-propagation network (FCPN) model to control different discrete time nonlinear systems with unknown distances and in [34] Singh et al. used the fuzzy numbers about an adaptive fuzzy model-based controller (AFMBC), which is studied and implemented for class of nonlinear discrete-time system with dead zone. To know more about fuzzy sequence spaces (see [17 , 41]).

A fuzzy number $Y : ℝ \to [0, 1]$ is a fuzzy set of real numbers which satisfies the given assertions:

(i) Normal, that is Y (u) =1 for some $u \in ℝ,$

(ii) Fuzzy convex, that is Y (u) ≥ min {Y (p) , Y (q)} where p < u < q,

(iii) Y is upper semi continuous,

(iv) The closure of $Y^{0} = {u \in ℝ : Y (u) > 0}$ is compact.

Also, for α ∈ (0, 1] , the α-level cut of a fuzzy number Y can be defined as $[Y]^{α} = {u \in ℝ : Y (u) \geq α} .$ Hence, Y is a fuzzy number iff [Y] ^α is a closed interval for each α ∈ (0, 1] and [Y] ¹ ≠ φ . The space of all fuzzy numbers is denoted by L (R).

Let X and Y be two fuzzy numbers. Then Matloka [22] introduce the distance between X and Y which is given by $\begin{matrix} d (X, Y) = sup_{0 \leq α \leq 1} d_{H} ([X]^{α}, [Y]^{α}), \end{matrix}$ where d_H is Hausdorff metric defined by $\begin{matrix} d_{H} ([X]^{α}, [Y]^{α}) = max {| {\underline{X}}^{α} - {\underline{Y}}^{α} |, | {\bar{X}}^{α} - {\bar{Y}}^{α} |}, \end{matrix}$ where ${\underline{Y}}^{α}$ and ${\bar{Y}}^{α}$ are the lower and upper bound of α-level cut.

As defined by Nuray and Savas [25], let Y = (Y_s) be the sequence of fuzzy numbers, then (Y_s) is said to be statistical convergent to a fuzzy number Y₀, if $\begin{matrix} \frac{1}{n} | {s \leq n : d (Y_{s}, Y_{0}) \geq ε} | = 0, \end{matrix}$ for each ε > 0, where the vertical bar indicates cardinality of the enclosed set.

Throughout the paper, λ = (λ_n) will be the increasing sequence of positive numbers tending to infinity such that $λ_{n + 1} = λ_{n} + 1, λ_{1} = 1 .$ As defined by P. D. Srivastava and S. Ojha [35], a sequence Y = (Y_s) is said to be λ-statistical convergent of order η to a fuzzy number Y₀ if for every ε > 0 $\begin{matrix} lim_{n \to \infty} \frac{1}{(λ_{n})^{η}} | {s \in I_{n} : d (Y_{s}, Y_{0}) \geq ε} | = 0, \end{matrix}$ where I_n = [n - λ_n + 1, n] and a sequence Y = (Y_s) is said to be strongly p-Cesàro summable of order η if $\begin{matrix} lim_{n \to \infty} \frac{1}{(λ_{n})^{η}} \sum_{s \in I_{n}} [d (Y_{s}, Y_{0})]^{p} = 0, \end{matrix}$ where p is a positive real number.

In this article, our main motive is to present new results related to (Δ^v, λ)-statistical convergence and strongly (Δ^v, p, λ)-Cesàro summability of order (η, δ + γ). To achieve these advance results we have studied Altinok et al. [4] and extend it by making use of (Δ^v, λ)-statistical convergence and the parameters η, δ and γ. Further, these results are also present graphically which proves that if a sequence of fuzzy numbers is bounded and statistically convergent of order (η, δ + γ) in (Δ^v, λ), then it need not be strongly (Δ^v, p, λ)-Cesàro summable in general when 0 < η ≤ δ ≤ γ ≤ 1 such that 0 < η ≤ δ + γ ≤ 1.

2 Main results

Definition 2.1. Let 0 < η ≤ δ ≤ γ ≤ 1 such that 0 < η ≤ δ + γ ≤ 1. A sequence Y = (Y_s) of fuzzy numbers is said to be (Δ^v, λ)-statistical convergent of order (η, δ + γ) or $(S_{λ}^{(η, δ + γ)}, Δ^{v}) -$ convergent to a fuzzy number Y₀ if for every ε > 0, $lim_{n \to \infty} \frac{1}{(λ_{n})^{η}} | {s \in I_{n} : d (Δ^{v} Y_{s}, Y_{0}) \geq ε} |^{(δ + γ)} = 0,$ where I_n = [n - λ_n + 1, n], (λ_n) ∈ Λ. We may write $(S_{λ}^{(η, δ + γ)}, Δ^{v}) -$ convergent to fuzzy number Y₀ as $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim Y_{s} = Y_{0}$ .

Definition 2.2. Let 0 < η ≤ δ ≤ γ ≤ 1 such that 0 < η ≤ δ + γ ≤ 1 and 0< p < ∞. A sequence Y = (Y_s) of fuzzy numbers is said to be strongly (Δ^v, p, λ)-Cesàro summable of order (η, δ + γ) or $(ω_{λ}^{(η, δ + γ)}, Δ^{v}, p) -$ summable if there exists a fuzzy real number Y₀ such that $lim_{n \to \infty} \frac{1}{(λ_{n})^{η}} (\sum_{s \in I_{n}} [d (Δ^{v} Y_{s}, Y_{0})]^{p})^{(δ + γ)} = 0,$ or $(ω_{λ}^{(η, δ + γ)}, Δ^{v}, p) - lim Y_{s} = Y_{0}$ .

We know that the statistical convergence is well defined for order (η, δ + γ) every time (δ + γ) > η and statistical convergence fails for order (η, δ + γ) every time (δ + γ) < η as shown in the example below.

Example 2.3. Consider Y = (Y_s) be a sequence of fuzzy numbers defined as $Y_{s} (x) = {\begin{matrix} x + 5, for - 5 \leq x \leq - 4 \\ - x - 3 for - 4 \leq x \leq - 3 \\ 0, otherwise \end{matrix}} = Y^{'}$ if s is odd and $Y_{s} (x) = {\begin{matrix} x - 3, for 3 \leq x \leq 4 \\ - x + 5 for 4 \leq x \leq 5 \\ 0, otherwise \end{matrix}} = Y^{″},$ if s is even. Now, the α-level set of sequences for (Y_s) and (Δ^vY_s) are as follows $[Y_{s}]^{α} = {\begin{matrix} [- 5 + α, - 3 - α], if s is odd \\ [3 + α, 5 - α], if s is even \end{matrix}$ and $[Δ^{v} Y_{s}]^{α} = {\begin{matrix} [2^{v} (- 5 + α, - 3 - α)], if s is odd \\ [2^{v} (3 + α, 5 - α)], if s is even, \end{matrix}$ where $v \in ℕ$ . Thus, we have $lim_{n \to \infty} \frac{1}{λ_{n}^{η}} | {s \in I_{n} : d ([Δ^{v} Y_{s}]^{α}, [Y^{'}]^{α}) \geq ε} |^{(δ + γ)}$ $\leq \frac{[λ_{n}]^{(δ + γ)}}{2 [λ_{n}]^{η}} = 0$ and $lim_{n \to \infty} \frac{1}{λ_{n}^{η}} | {s \in I_{n} : d ([Δ^{v} Y_{s}]^{α}, [Y^{″}]^{α}) \geq ε} |^{(δ + γ)}$ $\leq \frac{[λ_{n}]^{(δ + γ)}}{2 [λ_{n}]^{η}} = 0,$ for δ + γ < η. Here [Y′] ^α = [2^v (-5 + α) , 2^v (-3 - α)] and [Y″] ^α = [2^v (3 + α) , 2^v (5 - α)]. Thus, it is clear from Figure 1 that Y = (Y_s) is (Δ^v, λ)- statistical convergent of order (η, δ + γ) both to fuzzy numbers Y′ and Y″, that is $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim Y_{s} = Y^{'}$ and $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim Y_{s} = Y^{″}$ . Thus Y = (Y_s) is not statistically convergent i.e. limit of Y = (Y_s) is not unique.

Fig. 1

Y = (Y_s) is (Δ^v, λ)- statistical convergence of order (η, δ + γ), both to fuzzy number Y′ and Y″ .

Theorem 2.4. Let 0 < η ≤ δ ≤ γ ≤ 1 such that 0 < η ≤ δ + γ ≤ 1 and suppose that X = (X_s), Y = (Y_s) be two fuzzy sequences and $a \in ℂ$ , if

(i) $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim Y_{s} = Y_{0}$ , then $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim {aY}_{s} = {aY}_{0} .$

(ii) If $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim X_{s} = X_{0}$ and $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim Y_{s} = Y_{0},$ then $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim (X_{s} + Y_{s}) = X_{0} + Y_{0} .$

Proof. (i) This proof is easily obtained from the equality $\frac{1}{λ_{n}^{η}} | {s \in I_{n} : d (a Δ^{v} Y_{s}, {aY}_{0}) \geq ε} |^{(δ + γ)}$ $\begin{matrix} = & \frac{1}{λ_{n}^{η}} | {s \in I_{n} : d (Δ^{v} Y_{s}, Y_{0}) \geq \frac{ε}{| a |}} |^{(δ + γ)} . \end{matrix}$ (ii) This can also be obtained easily from the inequality $\frac{1}{λ_{n}^{η}} | {s \in I_{n} : d (Δ^{v} (X_{s} + Y_{s}), X_{0} + Y_{0}) \geq ε} |^{(δ + γ)}$

$\begin{matrix} \leq & \frac{1}{λ_{n}^{η}} | {s \in I_{n} : d (Δ^{v} X_{s}, X_{0}) + d (Δ^{v} Y_{s}, Y_{0}) \geq ε} |^{(δ + γ)} \\ \leq & \frac{1}{λ_{n}^{η}} | {s \in I_{n} : d (Δ^{v} X_{s}, X_{0}) \geq \frac{ε}{2}} |^{(δ + γ)} \\ + & \frac{1}{λ_{n}^{η}} | {s \in I_{n} : d (Δ^{v} Y_{s}, Y_{0}) \geq \frac{ε}{2}} |^{(δ + γ)} . \end{matrix}$

Remark 2.5. Here, it is easy to see that (Δ^v, λ)-conv-ergent sequence of fuzzy numbers is (Δ^v, λ)-statistical convergent of order (η, δ + γ). But the converse need not be true as shown in the following example.

Example 2.6. Let us take a fuzzy sequence Y = (Y_s) such that

If s = j⁴ $Y_{s} (x) = {\begin{matrix} \frac{2}{5} x - \frac{2}{5}, for 1 \leq x \leq \frac{7}{2} \\ - \frac{2}{5} x + \frac{12}{5}, for \frac{7}{2} \leq x \leq 6 \\ 0, otherwise \end{matrix}$ and if s ≠ j⁴ $Y_{s} (x) = {\begin{matrix} \frac{2}{5} x + \frac{12}{5}, for - 6 \leq x \leq \frac{- 7}{2} \\ - \frac{2}{5} x - \frac{2}{5}, for \frac{- 7}{2} \leq x \leq - 1 \\ 0, otherwise . \end{matrix}$ For j = 1, 2, 3, . . ., the α- level set of sequences (Y_s) and (ΔY_s) are as follows $[Y_{s}]^{α} = {\begin{matrix} [1 + \frac{5}{2} α, 6 - \frac{5}{2} α], if s = j^{4} \\ [- 6 + \frac{5}{2} α, - 1 - \frac{5}{2} α], if s \neq j^{4} \end{matrix}$ and $[Δ Y_{s}]^{α} = {\begin{matrix} [2 + 5 α, 12 - 5 α], if s = j^{4} \\ [- 12 + 5 α, - 2 - 5 α], if s + 1 = j^{4} \\ [- 5 + 5 α, 5 - 5 α], otherwise . \end{matrix}$ For instance, if we take (λ_n) = n and proceed similarly by taking difference v times for $v \in ℕ$ . It is easily seen that for (δ + γ) <4η, Y = (Y_s) is (Δ^v, λ)-statistical convergent of order (η, δ + γ) to fuzzy number Y₀, where Y₀ = [2^v-1 (-5 + 5α) , 2^v-1 (5 -5α)]. But it is not (Δ^v, λ)-convergent.

Remark 2.7. For 0 < η ≤ δ ≤ γ ≤ 1 such that 0 < η ≤ δ + γ ≤ 1 . If $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim Y_{s} = Y_{0}$ , then $(S_{λ}^{(η, δ)}, Δ^{v}) - lim Y_{s} = Y_{0}$ , that is $(S_{λ}^{(η, δ)}, Δ^{v}) \subset (S_{λ}^{(η, δ + γ)}, Δ^{v})$ . But the converse does not hold as shown below.

Example 2.8. Let us take (λ_n) = n and $η = \frac{1}{4}$

If s = j⁴ $Y_{s} (x) = {\begin{matrix} x, for 0 \leq x \leq 1 \\ - x + 2, for 1 \leq x \leq 2 \\ 0, otherwise \end{matrix}$ and if s ≠ j⁴ $Y_{s} (x) = {\begin{matrix} x - 2, for 2 \leq x \leq 3 \\ - x + 4, for 3 \leq x \leq 4 \\ 0, otherwise . \end{matrix}$ Now, the α-level set of sequences for (Y_s) and (ΔY_s) are as follows $[Y_{s}]^{α} = {\begin{matrix} [α, 2 - α], if s = j^{4} \\ [2 + α, 4 - α], if s \neq j^{4} \end{matrix}$ and $[Δ Y_{s}]^{α} = {\begin{matrix} [- 4 + 2 α, - 2 α], if s = j^{4} \\ [2 α, 4 - 2 α], if s + 1 = j^{4} \\ [- 2 + 2 α, 2 - 2 α], otherwise . \end{matrix}$ After some routine operation, when we take the difference v-times, we see that for $δ = \frac{1}{3}$ , $(S_{λ}^{(η, δ)}, Δ^{v}) - lim Y_{s} = Y_{0}$ where [Y₀] ^α = [2^v (-1 + α) , 2^v (1 - α)], but for $δ = \frac{1}{3}$ and $γ = \frac{2}{3}$ , $(S_{λ}^{(η, δ + γ)}, Δ^{v}) - lim Y_{s} \neq Y_{0} .$

Theorem 2.9. Let 0 < η₁ ≤ η₂ ≤ δ₁ ≤ δ₂ ≤ γ₁ ≤ γ₂ ≤ 1 such that 0 < η₁ ≤ η₂ ≤ δ₁ + γ₁ ≤ δ₂ + γ₂ ≤ 1 . If $(S_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}) - lim Y_{s} = Y_{0}$ , then $(S_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}) - lim Y_{s} = Y_{0} .$

Proof. Suppose that $(S_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}) - lim Y_{s} = Y_{0}$ and take δ₁ + γ₁ ≤ δ₂ + γ₂ and η₁ ≤ η₂, then $\frac{1}{λ_{n}^{η_{2}}} | {s \in I_{n} : d (Δ^{v} Y_{s}, Y_{0}) \geq ε} |^{(δ_{1} + γ_{1})}$ $\begin{matrix} \leq \frac{1}{λ_{n}^{η_{1}}} | {s \in I_{n} : d (Δ^{v} Y_{s}, Y_{0}) \geq ε} |^{(δ_{2} + γ_{2})} \end{matrix}$ for every ε > 0, which implies that $(S_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v} Y) - lim Y_{s} = Y_{0} .$ The converse of this result does not exists. For this, if s = j³ $Y_{s} (x) = {\begin{matrix} x - s + 1, for s - 1 \leq x \leq s \\ - x + s + 1, for s \leq x \leq s + 1 \\ 0, otherwise \end{matrix}$ and if s ≠ j³ $Y_{s} (x) = {\begin{matrix} x - 1, for 1 \leq x \leq 2 \\ - x + 3, for 2 \leq x \leq 3 \\ 0, otherwise . \end{matrix}$ The α- level set of sequences of (Y_s) and (ΔY_s) are as follows $[Y_{s}]^{α} = {\begin{matrix} [s - 1 + α, s + 1 - α], if s = j^{3} \\ [1 + α, 3 - α], if s \neq j^{3} \end{matrix}$ and

$[Δ Y_{s}]^{α} = {\begin{matrix} [s - 4 + 2 α, s - 2 α], if s = j^{3} \\ [- s - 1 + 2 α, - s + 3 - 2 α], if s + 1 = j^{3} \\ [- 2 + 2 α, 2 - 2 α], otherwise . \end{matrix}$

Here, for j > 1 and λ_n = n we see that, if we take difference v times, we get $(S_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}) - lim Y_{s} = Y_{0}$ , where [Y₀] ^α = [2^v (-1 + α) , 2^v (1 - α)] for $δ_{1} = \frac{1}{5}$ , $γ_{1} = \frac{3}{10}$ and $\frac{1}{6} < η_{2} \leq \frac{1}{5}$ , but $(S_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}) - lim Y_{s} \neq Y_{0}$ for $δ_{2} = \frac{1}{5}$ , $γ_{2} = \frac{3}{10}$ and $0 < η_{1} \leq \frac{1}{6} .$

Corollary 2.10. Let 0 < η₁ ≤ η₂ ≤ δ₁ ≤ δ₂ ≤ γ₁ ≤ γ₂ ≤ 1 such that 0 < η₁ ≤ η₂ ≤ δ₁ + γ₁ ≤ δ₂ + γ₂ ≤ 1. Then

(i) If δ₁ + γ₁ = δ₂ + γ₂ and η₁ = η₂, then $(S_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}) = (S_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}) .$

(ii) If δ₁ + γ₁ = δ₂ + γ₂ = 1 and η₁ = η₂, then $(S_{λ}^{η_{1}}, Δ^{v}) = (S_{λ}^{η_{2}}, Δ^{v}) .$

(iii) If δ₁ + γ₁ = δ₂ + γ₂ = 1 and η₂ = 1, then $(S_{λ}^{η_{1}}, Δ^{v}) \subseteq (S_{λ}, Δ^{v}) .$

(iv) δ₂ + γ₂ = 1, then $(S_{λ}^{η_{1}}, Δ^{v}) \subseteq (S_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}) .$

Theorem 2.11. Let 0 < η₁ ≤ η₂ ≤ δ₁ ≤ δ₂ ≤ γ₁ ≤ γ₂ ≤ 1 such that 0 < η₁ ≤ η₂ ≤ δ₁ + γ₁ ≤ δ₂ + γ₂ ≤ 1 . If $(w_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}, p) - lim Y_{s} = Y_{0}$ , then $(w_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}, p) - lim Y_{s} = Y_{0} .$

Proof. Let $(w_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}, p) - lim Y_{s} = Y_{0}$ . Let 0 < η ≤ δ ≤ γ ≤ 1 such that 0 < η ≤ δ + γ ≤ 1 and 0< p < ∞. Then from the inequality $\frac{1}{n^{η_{2}}} (\sum_{s = 1}^{n} [d (Δ^{v} Y_{s}, Y_{0})]^{p})^{(δ_{1} + γ_{1})}$ $\begin{matrix} \leq & \frac{1}{n^{η_{1}}} (\sum_{s = 1}^{n} [d (Δ^{v} Y_{s}, Y_{0})]^{p})^{(δ_{2} + γ_{2})}, \end{matrix}$ we have $(w_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}, p) - lim Y_{s} = Y_{0} .$ But the converse does not exists as shown below.

Example 2.12. Let us consider (λ_n) = n and the fuzzy sequence (Y_s) defined as $Y_{s} (x) = {\begin{matrix} \frac{2}{3} x, for 0 \leq x \leq \frac{3}{2} \\ - \frac{2}{3} x + 2, for \frac{3}{2} \leq x \leq 3 \\ 0, otherwise \end{matrix}$ if s = j³ and $Y_{s} (x) = {\begin{matrix} \frac{3}{2} x - \frac{8}{3}, for 4 \leq x \leq \frac{11}{2} \\ - \frac{2}{3} x + \frac{14}{3}, for \frac{11}{2} \leq x \leq 7 \\ 0, otherwise . \end{matrix}$ if s ≠ j³. Here, we find respectively the α- level set of sequences (Y_s) and (ΔY_s) as follows $[Y_{s}]^{α} = {\begin{matrix} [\frac{3}{2} α, 3 - \frac{3}{2} α], for s = j^{3} \\ [4 + \frac{3}{2} α, 7 - \frac{3}{2} α], for s \neq j^{3} \end{matrix}$ and $[Δ Y_{s}]^{α} = {\begin{matrix} [- 7 + 3 α, - 1 - 3 α], for s = j^{3} \\ [1 + 3 α, 7 - 3 α], for s + 1 = j^{3} \\ [- 3 + 3 α, 3 - 3 α], otherwise . \end{matrix}$ For j = 1, 2, 3, . . . , if we take difference v-times, we get the inequality $\frac{1}{n^{η_{2}}} (\sum_{s = 1}^{n} [d ([Δ^{v} Y_{s}]^{α}, [Y_{0}]^{α})]^{p})^{(δ_{1} + γ_{1})}$ $\leq \frac{2^{v + 2} (\sqrt[3]{n})^{(δ_{1} + γ_{1})}}{n^{η_{2}}},$ (1) by using addition of fuzzy real numbers by α-level set. For p=1, where [Y₀] ^α = [2^v-1 (-3 + 3α) , 2^v-1 (3 -3α)] . On the other hand $\frac{2^{v + 2} ({\sqrt[3]{n}}^{(δ_{2} + γ_{2})} - 1) - (2^{v + 2} - 4)}{n^{η_{1}}}$ $\leq \frac{1}{n^{η_{1}}} (\sum_{s = 1}^{n} [d ([Δ^{v} Y_{s}]^{α}, [Y_{0}]^{α})]^{p})^{(δ_{2} + γ_{2})},$ (2) for p = 1 . From equation (1), it is clear that Y_s is $(w_{λ}^{(η, δ + γ)}, Δ^{v}, p) -$ summable of order (η₂, δ₁ + γ₁) for $δ_{1} = \frac{1}{4}$ , $γ_{1} = \frac{7}{20}$ and $\frac{1}{5} < η_{2} \leq \frac{1}{4}$ . From equation (2), Y_s is not $(w_{λ}^{(η_{,} δ + γ)}, Δ^{v}, p) -$ summable of order (η₁, δ₂ + γ₂) to fuzzy number Y₀ for $0 < η_{1} \leq \frac{1}{5}$ and $δ_{2} = \frac{1}{4}$ , $γ_{2} = \frac{7}{20}$ . Hence, the converse is not valid.

Corollary 2.13. Let 0 < η₁ ≤ η₂ ≤ δ₁ ≤ δ₂ ≤ γ₁ ≤ γ₂ ≤ 1 such that 0 < η₁ ≤ η₂ ≤ δ₁ + γ₁ ≤ δ₂ + γ₂ ≤ 1 and 0< p < ∞. Then the following hold

(i) If η₁ = η₂ and δ₁ + γ₁ = δ₂ + γ₂, then $(w_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}, p) = (w_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}, p) .$

(ii) If η₁ = η₂ and δ₁ + γ₁ = δ₂ + γ₂ = 1, then $(w_{λ}^{η_{1}}, Δ^{v}, p,) = (w_{λ}^{η_{2}}, Δ^{v}, p) .$

(iii) If 0 < η₁ ≤ 1, then $(w_{λ}^{η_{1}}, Δ^{v}, p) \subseteq (w_{λ}, Δ^{v}, p) .$

(iv) If 0 < δ₁ + γ₁ ≤ 1, then $(w_{λ}^{η_{1}}, Δ^{v}, p) \subseteq (w_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}, p) .$

Theorem 2.14. Let 0 < η₁ ≤ η₂ ≤ δ₁ ≤ δ₂ ≤ γ₁ ≤ γ₂ ≤ 1 such that 0 < η₁ ≤ η₂ ≤ δ₁ + γ₁ ≤ δ₂ + γ₂ ≤ 1 and 0< p < ∞. If $(w_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}, p) - lim Y_{s} = Y_{0},$ then $(S_{λ}^{(η_{2}, δ_{1} + γ_{1})}, Δ^{v}, p) - lim Y_{s} = Y_{0} .$

Proof. Let $(w_{λ}^{(η_{1}, δ_{2} + γ_{2})}, Δ^{v}, p) - lim Y_{s} = Y_{0}$ and given ε > 0. Then $(\sum_{s = 1}^{n} [d (Δ^{v} Y_{s}, Y_{0})]^{p})^{(δ_{2} + γ_{2})}$ $\begin{matrix} \geq & (\sum_{s = 1}^{n} [d (Δ^{v} Y_{s}, Y_{0})]^{p})^{(δ_{1} + γ_{1})} \\ \geq & ε^{p (δ_{1} + γ_{1})} | {s \in I_{n} : d (Δ^{v} Y_{s}, Y_{0}) \geq ε} |^{(δ_{1} + γ_{1})} \end{matrix}$ and $\frac{1}{ε^{p (δ_{1} + γ_{1})} n^{η_{1}}} (\sum_{s = 1}^{n} [d (Δ^{v} Y_{s}, Y_{0})]^{p})^{(δ_{2} + γ_{2})}$ $\geq \frac{1}{n^{η_{2}}} | {s \in I_{n} : d (Δ^{v} Y_{s}, Y_{0}) \geq ε} |^{(δ_{1} + γ_{1})}$

To prove that converse is not true. Consider the sequence Y = (Y_s) from Theorem 2.6, we have $lim_{n \to \infty} \frac{1}{n^{η_{2}}} | {s \in I_{n} : d (Δ^{v} Y_{s}, Y_{0}) \geq ε} |^{(δ_{1} + γ_{1})}$ $\begin{matrix} \leq & \frac{n^{\frac{δ_{1} + γ_{1}}{3}}}{n^{η_{2}}} \\ = & lim_{n \to \infty} \frac{1}{n^{η_{2} - (\frac{δ_{1} + γ_{1}}{3})}} \\ = & 0, \end{matrix}$ when $δ_{1} = \frac{2}{15}$ , $γ_{1} = \frac{1}{5}$ and $\frac{1}{9} < η_{2} \leq \frac{2}{15},$ but $\frac{1}{n^{η_{1}}} (\sum_{s = 1}^{n} [d (Δ^{v} X_{s}, X_{0})]^{p})^{(δ_{2} + γ_{2})} \to \infty,$ when $η_{1} \in (0, \frac{1}{9}]$ and $δ_{2} = \frac{2}{15}$ , $γ_{2} = \frac{1}{5} .$ Therefore, sequence Y = (Y_s) is not strongly (Δ^v, p, λ)-Cesàro summable of order (η₁, δ₂ + γ₂) .

Remark 2.15. We know that if sequence (Y_s) of fuzzy numbers is (Δ^v, λ)-bounded and (Δ^v, λ)-statistical convergent, then the sequence is strongly $(ω_{λ}^{(η, δ + γ)}, Δ^{v}, p) -$ summable. But this is not possible in our definition. We will show the contradiction via example.

Example 2.16. Let us take (λ_n) = n and s ≠ j⁴, the fuzzy sequence (Y_s) is defined as

$Y_{s} (x) = {\begin{matrix} {xs}^{2} + 1, for - \frac{1}{s^{2}} \leq x \leq 0 \\ - {xs}^{2} + 1, for 0 \leq x \leq \frac{1}{s^{2}} \\ 0, otherwise \end{matrix}$ and if s = j⁴, the fuzzy sequence is defined as

$Y_{s} (x) = {\begin{matrix} x - 4, for 4 \leq x \leq 5 \\ - x + 6, for 5 \leq x \leq 6 \\ 0, otherwise . \end{matrix}$ Then for j = 1, 2, 3, . . . we find respectively the α-level set of sequences (Y_s) and (ΔY_s) as

$[Y_{s}]^{α} = {\begin{matrix} [\frac{α - 1}{s^{2}}, \frac{1 - α}{s^{2}}], for s \neq j^{4} \\ [4 + α, 6 - α], for s = j^{4} \end{matrix}$ and

[ΔY_s] ^α $= {\begin{matrix} [\frac{(4 (s + 1)^{2} - 1) + ((s + 1)^{2} + 1) α}{(s + 1)^{2}}, \\ \frac{(6 (s + 1)^{2} + 1) - ((s + 1)^{2} + 1) α}{(s + 1)^{2}}], for s = j^{4} \\ [\frac{(- 6 s^{2} - 1) + (s^{2} + 1) α}{s^{2}}, \\ \frac{(- 4 s^{2} + 1) - (s^{2} + 1) α}{s^{2}}], for s + 1 = j^{4} \\ [\frac{- s^{2} - (s + 1)^{2} + (s^{2} + (s + 1)^{2}) α}{s^{2} (s + 1)^{2}}, \\ \frac{s^{2} + (s + 1)^{2} - (s^{2} + (s + 1)^{2}) α}{s^{2} (s + 1)^{2}}], otherwise . \end{matrix}$ Here, we know that (Y_s) is (Δ^v, λ)- bounded and statistical convergent for $δ = \frac{1}{5}$ and $γ = \frac{3}{10}$ and $\frac{1}{8} < η \leq \frac{1}{5}$ as shown in the Figure 2. But (Y_s) is not strongly (Δ^v, p, λ)-Cesàro summable. For this, let us suppose that $P_{s} = {s \leq n : s = j^{4}, j \in ℕ}$ and $Q_{s} = {s \leq n : s + 1 = j^{4}, j \in ℕ}$ . Now, we have

$\frac{1}{n^{η}} (\sum_{s = 1}^{n} [d ([Δ Y_{s}]^{α}, [Y_{0}]^{α})]^{p})^{(δ + γ)}$ $\begin{matrix} = & \frac{1}{n^{η}} (\sum_{s \in P_{s}} [d ([Δ Y_{s}]^{α}, [Y_{0}]^{α})]^{p})^{(δ + γ)} \\ + & \frac{1}{n^{η}} (\sum_{s \in Q_{s}} [d ([Δ Y_{s}]^{α}, [Y_{0}]^{α})]^{p})^{(δ + γ)} \\ + & \frac{1}{n^{η}} (\sum_{s \notin P_{s} s \notin Q_{s}} [d ([Δ Y_{s}]^{α}, [Y_{0}]^{α})]^{p})^{(δ + γ)} \\ = & \frac{1}{n^{η}} (\sum_{s \in P_{s}} 6 + \frac{1}{(1 + s)^{2}})^{(δ + γ)} \\ + & \frac{1}{n^{η}} (\sum_{s \in Q_{s}} 6 + \frac{1}{s^{2}})^{(δ + γ)} \\ + & \frac{1}{n^{η}} (\sum_{s \notin P_{s}, s \notin Q_{s}} \frac{1}{s^{2}} + \frac{1}{(s + 1)^{2}})^{(δ + γ)} \\ \geq & \frac{1}{n^{η}} (\sqrt[4]{n})^{(δ + γ)} \end{matrix}$

for v = 1 and p = 1. Thus, the sequence (Y_s) is not strongly $(ω_{λ}^{(η, δ + γ)}, Δ^{v}, p) -$ summable for $δ = \frac{1}{5}$ , $γ = \frac{3}{10}$ and $0 < η \leq \frac{1}{8} .$

Fig. 2

The blue lines graph in (i), (ii) and (iii) shows for particular values of s i.e. when s=15, s=2, s=1 respectively. The red lines in the graph (i), (ii) and (iii) shows that when s+1 is a fourth power i.e. s=80,255..., when neither s nor s+1 is a fourth power i.e. 3,4,5,..., and when s is a fourth power i.e. 16,81,256,..., respectively.

References

Altınok

, On λ statistical convergence of order β of sequences of fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 20 (2012), 303–314.

Altınok

, Statistical convergence of order β for generalized difference sequences of fuzzy numbers, Journal of Intelligent & Fuzzy Systems 26(2) (2014), 847–856.

Altınok

, Çolak

and Et

, λ–difference sequence spaces of fuzzy numbers, Fuzzy Sets and Systems 160(21) (2009), 3128–3139.

Altınok

and Et

, Statistical convergence of order (β;γ) for sequences of fuzzy numbers, Soft Computing 23 (2019), 6017–6022.

Başarır

and Kara

E.E.

, On some difference sequence spaces of weighted means and compact operators, Annals of Functional Analysis 2(2) (2011), 114–129.

Başarır

, Konca

and Kara

E.E.

, Some generalized difference statistically convergent sequence spaces in 2-normed space, Journal of Inequalities and Applications 177 (2013), 10.

Çolak

, Altınok

and Et

, Generalized difference sequences of fuzzy numbers, Chaos Solitons & Fractals 40(3) (2009), 1106–1117.

Çolak

, Statistical convergence of order α, Modern Methods in Analysis and its Applications (Anamaya Publ. New Delhi, India), (2010), 121–129.

Connor

J.S.

, The statistical and strong p- Cesàro convergence of sequences, Analysis 8 (1988), 47–63.

10.

and Çolak

, On some generalized difference sequence spaces, Soochow Journal of Mathematics 21(4) (1995), 377–386.

11.

Fast

, Sur la convergence statistique, Colloquium Mathematicae 2 (1951), 241–244.

12.

Fridy

J.A.

, On statistical convergence, Analysis 5(4) (1985), 301–313.

13.

Gadjiev

A.D.

and Orhan

, Some approximation theorems via statistical convergence, Rocky Mountain Journal of Mathematics 32(1) (2002), 129–138.

14.

Gülle

, Double Wijsman asymptotically statistical equivalence of order α, Journal of Intelligent and Fuzzy Systems 38(2) (2020), 2081–2087.

15.

Hazarika

and Savaş

, Some-convergent λ–summable difference sequence spaces of fuzzy real numbers defined by a sequence of Orlicz functions, Mathematical and Computer Modelling 54 (2011), 2986–2998.

16.

Ilkhan

and Kara

E.E.

, On statistical convergence in quasimetric spaces, Demonstratio Mathematica 52 (2019), 225–236.

17.

Ilkhan

and Kara

E.E.

, A new type of statistical Cauchy sequence and its relation to Bourbaki completeness, Cogent Mathematics & Statistics 5 1487500 (2018), 9.

18.

Kara

E.E.

and Demiriz

, Some new paranormed difference sequence spaces derived by Fibonacci numbers, Miskolc Mathematical Notes 16(2) (2015), 907–923.

19.

Kara

E.E.

and Ilkhan

, On some Banach sequence spaces derived by a new band matrix, British Journal of Mathematics & Computer Science 9(2) (2015), 141–159.

20.

Khan

V.A.

, Kara

E.E.

, Altaf

, Khan

and Ahmad

, Intuitionistic fuzzy I-convergent Fibonacci difference sequence spaces, Journal of Inequalities and Applications 202 (2019), 7.

21.

Kizmaz

, On certain sequence spaces, Canadian Mathematical Bulletin 24(2) (1981), 169–176.

22.

Matloka

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

23.

Mursaleen

and Mohiuddine

S.A.

, Statistical convergence of double sequences in intuitionistic fuzzy normed spaces, Chaos Solitons and Fractals 41(5) (2009), 2414–2421.

24.

Nanda

, On sequences of fuzzy numbers, Fuzzy Sets and Systems 33 (1989), 123–126.

25.

Nuray

and Savaş

, Statistical convergence of sequences of fuzzy numbers, Mathematica Slovaca 45(3) (1995), 269–273.

26.

Onasanya

B.O.

and Hoskova-Mayerova

, Some Topological and Algebraic Properties of alpha-level Subsets’ Topology of a Fuzzy Subset, Analele Stiintifice ale Universitatii Ovidius Constanta 26(3) (2018), 213–227, DOI: 10.2478/auom-2018-0042

27.

Raj

and Pandoh

, On some spaces of Cesàro sequences of fuzzy numbers associated with λ–convergence and Orlicz function, Mathematica 11(1) (2019), 156–171.

28.

Raj

, Sharma

and Choudhary

, Applications of Tauberian theorem in Orlicz spaces of double difference sequences of fuzzy numbers, Journal of Intelligent & Fuzzy Systems 35(2) (2018), 2513–2524.

29.

Raj

, Jamwal

and Sharma

S.K.

, Multiplier generalized double sequence spaces of fuzzy numbers defined by sequence of Orlicz functions, International Journal of Pure and Applied Mathematics 78(4) (2012), 509–522.

30.

Sakhre

, Singh

U.P.

and Jain

, FCPN approach for uncertain nonlinear dynamical system with unknown disturbance, International Journal of Fuzzy Systems 19 (2017), 452–469. htpps://org/10.1007/s40815-016-0145-5.

31.

Šalát

, On statistically Convergent sequences of real numbers, Mathematica Slovaca 30(2) (1980), 139–150.

32.

Savaş

and Mursaleen

, On statistically convergent double sequences of fuzzy numbers, Information Sciences 162 (2004), 183–192.

33.

Schoenberg

I.J.

, The integrability of certain functions and related summability methods, The American Mathematical Monthly 66(5) (1959), 361–775.

34.

Singh

U.P.

, Jain

, Gupta

R.K.

and Tiwari

, AFMBC for a class of nonlinear discrete-time systems with dead zone, International Journal of Fuzzy Systems 21 (2019), 1073–1084. htpps://org/10.1007/s40815-019-00621-1.

35.

Srivastava

P.D.

and Ojha

, λ–Statistical convergence of fuzzy numbers and fuzzy functions of order θ, Soft Computing 18 (2014), 1027–1032.

36.

Tripathy

B.C.

and Nanda

, Absolute value of fuzzy real number and fuzzy sequence spaces, The Journal of Fuzzy Mathematics 8 (2000), 883–892.

37.

Turan

, Kara

E.E.

and Ilkhan

, Quasi statistical convergence in cone metric spaces, Facta Univ (NIS) Ser Math Inform 33(4) (2018), 613–626.

38.

Ulusu

and Gülle

, Wijsman asymptotical I2-statistically equivalent double set sequences of order η, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69(1) (2020), 854–862.

39.

Ulusu

and Gülle

, Some statistical convergence types for double set sequences of order α, Facta Universitatis, Series: Mathematics and Informatics 35(3) (2020), 595–603.

40.

Yavuz

, Tauberian theorems for statistical summability methods of sequences of fuzzy numbers, Soft Computing 23 (2019), 5659–5665.

41.

Yavuz

, Euler summability method of sequences of fuzzy numbers and a Tauberian theorem, Journal of Intelligent & Fuzzy Systems 32(1) (2017), 937–943.

42.

Zadeh

L.A.

, Fuzzy set, Information and Control 8 (1965), 338–353.