λ - statistical convergence in fuzzy normed linear spaces

Abstract

In this paper, we have introduced λ- statistical convergence and condition of being λ- statistical Cauchy of real number sequences in fuzzy normed linear spaces. At the same time, in fuzzy normed spaces, we have introduced the concept of (V, λ) summability and (C, 1) summability, and then, we have studied the relation between these concepts and λ- statistical convergence.

Keywords

fuzzy normed linear space summability

1 Introduction

The concept of statistical convergence of real number sequences was introduced by Fast [7] and Steinhaus [18] and later reintroduced by Schoenberg [17] independently. The concept of statistical convergence have been studied in many branches of mathematics; for example, Fourier analysis, Banach spaces, Number theory, Measure theory. Many mathematicians such as Connor [3], Fridy [9], Colak [4], Šalát [15], Et and Cinar [6], and so on have studied the concept of statistical convergence in summability theory.

Let λ = (λ_n) be a non-decreasing sequence of positive real numbers tending to ∞ such that λ_n ≤ λ_n + 1, λ₁ = 1. The set of all such sequences will be denoted by Λ.

The concept of λ- statistical convergence was defined by Mursaleen [13] as follows.

A sequence x = (x_k) is said to be λ- statistically convergent or S_λ- convergent to L if for every ɛ > 0 $lim_{n \to \infty} \frac{1}{λ_{n}} ∥ {k \in I_{n} : ∥ x_{k} - L ∥ \geq ɛ} ∥ = 0,$ where I_n = [n - λ_n + 1, n]. In this case, we have written S_λ - lim x = L or x_k → L (S_λ) and $S_{λ} = {x : \exists L \in ℝ, S_{λ} - lim x = L} .$

Fuzzy sets and fuzzy numbers are also an uncertainty. So, many writers have studied the relationship of fuzzy numbers to probability ([22, 23]). Further, soft sets and rough sets concepts have been studied and new methods have been developed for decision-making using fuzzy sets, soft sets and rough sets concepts together ([24 –29]).

Fuzzy sets are considered with respect to a nonempty base set X of elements of interest. The essential idea is that each element x ∈ X is assigned a membership grade u (x) taking values in [0, 1], with u (x) =0 corresponding to nonmembership, 0 < u (x) <1 to partial membership, and u (x) =1 to full membership.

According to Zadeh [21], a fuzzy subset of X is a nonempty subset {(x, u (x)) : x ∈ X} of X × [0, 1] for some function u : X → [0, 1]. The function u itself is often used for the fuzzy set.

A fuzzy set u on $ℝ$ is called a fuzzy number if it has the following properties:

i) u is normal, that is, there exists an $x_{0} \in ℝ$ such that u (x₀) =1 ;

ii) u is fuzzy convex, that is, for $x, y \in ℝ$ and 0 ≤ λ ≤ 1, $u (λ x + (1 - λ) y) \geq min [u (x), u (y)];$

iii) u is upper semicontinuous;

iv) $suppu = cl {x \in ℝ : u (x) > 0},$ or denoted by [u] ₀, is compact.

Let $L (ℝ)$ be set of all fuzzy numbers. If $u \in L (ℝ)$ and u (t) = 0 for t < 0, then u is called a non-negative fuzzy number. We have written $L^{*} (ℝ)$ by the set of all non-negative fuzzy numbers. We can say that $u \in L^{*} (ℝ)$ iff $u_{α}^{-} \geq 0$ for each α ∈ [0, 1] . Clearly we have $\tilde{0} \in L (ℝ) .$ For $u \in L (ℝ),$ the α level set of u is defined by ${[u]}_{α} = {\begin{matrix} {x \in ℝ : u (x) \geq α}, & if α \in (0, 1] \\ suppu, & if α = 0 . \end{matrix}$ A partial order ⪯ on $L (ℝ)$ is defined by

u ⪯ v iff $u_{α}^{-} \leq v_{α}^{-}$ and $u_{α}^{+} \leq v_{α}^{+}$ for all α ∈ [0, 1].

Arithmetic operation ⊕, ⊖ , ⊙ and ø on $L (ℝ) \times L (ℝ)$ are defined by $\begin{matrix} (u \oplus v) (t) = sup_{s \in ℝ} {u (s) \land v (t - s)} t \in ℝ \\ (u ⊖ v) (t) = sup_{s \in ℝ} {u (s) \land v (s - t)} t \in ℝ \\ (u ⊙ v) (t) = sup_{\begin{matrix} s \in ℝ \\ s \neq 0 \end{matrix}} {u (s) \land v (t / s)} t \in ℝ \\ (u ø v) (t) = sup_{s \in ℝ} {u (st) \land v (s)} t \in ℝ . \end{matrix}$

For $k \in ℝ^{+},$ ku is defined as ku (t) = u (t/k) and $0 u (t) = \tilde{0},$ $t \in ℝ .$ Some arithmetic operations for α-level sets are defined as follows:

$u, v \in L (ℝ)$ and ${[u]}_{α} = [u_{α}^{-}, u_{α}^{+}]$ and ${[v]}_{α} = [v_{α}^{-}, v_{α}^{+}],$ α ∈ (0, 1] . Then $\begin{matrix} {[u \oplus v]}_{α} = [u_{α}^{-} + v_{α}^{-}, u_{α}^{+} + v_{α}^{+}] \\ {[u ⊖ v]}_{α} = [u_{α}^{-} - v_{α}^{+}, u_{α}^{+} - v_{α}^{-}] \\ {[u ⊙ v]}_{α} = [u_{α}^{-} . v_{α}^{-}, u_{α}^{+} . v_{α}^{+}] \\ {[\tilde{1} ø u]}_{α} = [\frac{1}{u_{α}^{+}}, \frac{1}{u_{α}^{-}}] u_{α}^{-} > 0 \end{matrix}$

For $u, v \in L (ℝ),$ the supremum metric on $L (ℝ)$ is defined as $D (u, v) = sup_{0 \leq α \leq 1} max {∥ u_{α}^{-} - v_{α}^{-} ∥, ∥ u_{α}^{+} - v_{α}^{+} ∥} .$ It is known that D is a metric on $L (ℝ),$ and $(L (ℝ), D)$ is a complete metric space.

A sequence x = (x_k) of fuzzy numbers is said to be convergent to the fuzzy number x₀ if for every ɛ > 0, there exists a positive integer k₀ such that D (x_k, x₀) < ɛ for k > k₀. And a sequence x = (x_k) of fuzzy numbers convergens to levelwise to x₀, iff $lim_{k \to \infty} {[x_{k}]}_{α} = {[x_{0}]}_{α}^{-}$ and $lim_{k \to \infty} {[x_{k}]}_{α} = {[x_{0}]}_{α}^{+}$ where ${[x_{k}]}_{α} = [{(x_{k})}_{α}^{-}, {(x_{k})}_{α}^{+}]$ and ${[x_{0}]}_{α} = [{(x_{0})}_{α}^{-}, {(x_{0})}_{α}^{+}]$ for every α ∈ (0, 1) . ([2 , 11]).

The statistical converge of fuzzy number defined Savas [16] is as follows; A sequence X = (X_k) of fuzzy numbers is said to be λ- statistically convergent to fuzzy numbers X₀ if every ɛ > 0 $lim_{n} \frac{1}{λ_{n}} ∥ {k \in I_{n} : d (X_{k}, X_{0}) \geq ɛ} ∥ = 0 .$

Later, many mathematicians such as Mohiuddine et al. [12], Altınok et al. [1] and so on studied statistical convergence of fuzzy numbers.

Let X be a vector space over $ℝ,$ let $| | . | | : X \to L^{*} (ℝ)$ and the mappings L ; R (respectively, left norm and right norm) : [0, 1] × [0, 1] → [0, 1] be symetric, nondecreasing in both arguments and satisfy L (0, 0) = 0 and R (1, 1) =1 . ([8, 19]).

The quadruple (X, || . ||, L, R) is called fuzzy normed linear space (briefly (X, || . ||) FNS) and || . || a fuzzy norm if the following axioms are satisfied

$∥ x ∥ = \tilde{0}$ iff x = θ,

||rx|| = ∥ r ∥ ⊙ ||x|| for x ∈ X, $r \in ℝ,$

For all x, y ∈ X

∥x + y ∥ (s + t) ≥ L (∥ x ∥ (s) , ∥ y ∥ (t)) , whenever $s \leq {∥ x ∥}_{1}^{-}, t \leq {∥ y ∥}_{1}^{-}$ and $s + t \leq {∥ x + y ∥}_{1}^{-},$

∥x + y ∥ (s + t) ≤ R (∥ x ∥ (s) , ∥ y ∥ (t)) , whenever $s \geq {∥ x ∥}_{1}^{-}, t \geq {∥ y ∥}_{1}^{-}$ and $s + t \geq {∥ x + y ∥}_{1}^{-} .$

Let (X, || . ||_C) be an ordinary normed linear space. Then a fuzzy norm || . || on X can be obtained $| | x | | (t) = {\begin{matrix} 0 & if, 0 \leq t \leq a {| | x | |}_{C} or t \geq b {| | x | |}_{C} \\ \frac{t}{(1 - a) {| | x | |}_{C}} - \frac{a}{1 - a} & a {| | x | |}_{C} \leq t \leq {| | x | |}_{C} \\ \frac{- t}{(b - 1) {| | x | |}_{C}} + \frac{b}{b - 1} & {| | x | |}_{C} \leq t \leq b {| | x | |}_{C} \end{matrix}$ (1) where ||x||_C is the ordinary norm of x (≠ θ), 0 < a < 1 and 1 < b < ∞ . For x = θ, define $| | x | | = \tilde{0} .$

Hence, (X, || . ||) is a fuzzy normed linear space. [8] Sençimen [14] has defined convergence in fuzzy normed spaces by taking advantage of Kaleva [10] and Felbin [8], as follows;

Let (X, || . ||) be an FNS. A sequence ${(x_{n})}_{n = 1}^{\infty}$ in X is convergent to x ∈ X with respect to the fuzzy norm on X and we denote by $x_{n} \overset{FN}{\to} x,$ provided that $(D) - lim_{n \to \infty} | | x_{n} - x | | = \tilde{0};$ i.e. for every ɛ > 0 there is an $N (ɛ) \in ℕ$ such that $D (| | x_{n} - x | |, \tilde{0}) < ɛ$ for all $n \in ℕ .$ This means that for every ɛ > 0 there is an $N (ɛ) \in ℕ$ such that $sup_{α \in [0, 1]} {| | x_{n} - x | |}_{α}^{+} = {| | x_{n} - x | |}_{0}^{+} < ɛ$ for all n ≥ N (ɛ) .

2 Main result

In this section, we have defined λ- statistical convergence sequences, λ- statistically Cauchy sequence, strongly λ- summability, and strongly Cesaro summability in fuzzy normed spaces. And then we have studied the relation between λ- statistically convergence and λ- summability in fuzzy normed spaces.

2.1 Definition

Let (X, ∥ · ∥) be an FNS and λ ∈ Λ. A sequence x = (x_k) in X is said to be λ-statistically convergent to L ∈ X with respect to fuzzy norm on X or FS_λ-convergent

if for each ɛ > 0 $lim_{n \to \infty} \frac{1}{λ_{n}} ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ = 0$ So, we have written $x_{k} \overset{{FS}_{λ}}{\to} L$ or x_k → L (FS_λ) or $S_{λ} \overset{FN}{-} lim x_{k} = L$ where I_n = [n - λ_n + 1, n] . This implies that for each ɛ > 0, the set $K (ɛ) = {k \in I_{n} : {| | x_{k} - L | |}_{0}^{+} \geq ɛ}$ has a natural density zero, namely, for each ɛ > 0, $| | x_{k} - L | |_{0}^{+} < ɛ$ for a.a.k.

In this case, we have written $S_{λ} \overset{FN}{-} lim x_{k} = L$ . The set of all statistically convergent sequence w.r.t fuzzy norm on X will be denoted by FS_λ.

The element L ∈ X is the FS_λ- limit of (x_k). In terms of neigborhoods, we have had $x_{k} \overset{{FS}_{λ}}{\to} L$ provided that for each ɛ > 0, x_k ∈ ℵ _x (ɛ, 0) for a.a.k.

A useful interpretation of the above definition is the following: $x_{k} \overset{{FS}_{λ}}{\to} L iff S_{λ} \overset{FN}{-} lim {| | x_{k} - L | |}_{0}^{+} = 0$

Note that $S_{λ} \overset{FN}{-} lim {| | x_{k} - L | |}_{0}^{+} = 0$ implies that $S_{λ} \overset{FN}{-} lim {| | x_{k} - L | |}_{α}^{-} = S_{λ} \overset{FN}{-} lim {| | x_{k} - L | |}_{α}^{+} = 0$ for each α ∈ [0, 1] since $0 \leq {| | x_{k} - L | |}_{α}^{-} \leq {| | x_{k} - L | |}_{α}^{+} \leq {| | x_{k} - L | |}_{0}^{+}$ holds for every k ∈ I_n and for each α ∈ [0, 1] .

In this case, we have written throughout the paper (x_k) is FS_λ-convergent to L ∈ X means that (x_k) is λ-statistically convergent to L ∈ X w.r.t the fuzzy norm on X. Since the natural density of a finite set is zero, every convergent sequence is statistically convergent on FNS but the converse is not true in general as can be seen in the following example.

2.2 Example

Let $(ℝ^{m}, | | \cdot | |)$ be an FNS and $x = (x_{1}, x_{2}, . . ., x_{m}) \in ℝ^{m}$ be a fixed non-zero vector where the fuzzy norm on $ℝ^{m}$ is defined as in (1) such that ${| | x | |}_{C} = {(x_{1}^{2} + x_{2}^{2} + . . . + x_{m}^{2})}^{1 / 2}$ and λ = (λ_n) = (n) ∈ Λ.

Now, we have defined a sequence (x_k) in $ℝ^{m}$ as $x_{k} = {\begin{matrix} x if k = t^{2}, \\ θ if k \neq t^{2}, \end{matrix}$ where $t \in ℕ .$

Then we have seen that for any ɛ satisfying $0 < ɛ \leq b {(x_{1}^{2} + x_{2}^{2} + . . . + x_{m}^{2})}^{1 / 2},$ we have had $K (ɛ) = {k \in I_{n} : {| | x_{k} - θ | |}_{0}^{+} \geq ɛ} = {1, 4, 9, . . .}$ and hence δ (K (ɛ)) = 0.

If we choose $ɛ > b {(x_{1}^{2} + x_{2}^{2} + . . . + x_{m}^{2})}^{1 / 2}$ , then we get K (ɛ) =∅ and hence δ (∅) = 0 . Thus, $x_{k} \overset{{FS}_{λ}}{\to} θ$ but (x_k) is not convergent since the set {1, 4, 9, 16, . . .} has infinitely many elements.

Moreover, since the subsequence (x_{k
²}) statistically converges to x, we have seen that a subsequence of a statistically convergent sequence need not statistically convergent to the statistical limit of the sequence in an FNS.

In the following proposition, some basic proporties of the statisticall limit is summarized.

2.3 Proposition

Let (x_k) and (y_k) be sequences in an FNS (X, || · ||) such that $x_{k} \overset{{FS}_{λ}}{\to} x$ and $y_{k} \overset{{FS}_{λ}}{\to} y$ where x, y ∈ X. Then we have

$(x_{k} + y_{k}) \overset{{FS}_{λ}}{\to} x + y$ ,

${tx}_{k} \overset{{FS}_{λ}}{\to} tx$ $(t \in ℝ)$ ,

$S_{λ} \overset{FN}{\to} lim | | x_{k} | | = | | x | |$ .

2.4 Definition

Let (X, || · ||) be an FNS. A sequence (x_k) in X is λ-statistically Cauchy with respect to the fuzzy norm on X provided that for every ɛ > 0, there exist a number $N = N (ɛ) \in ℕ$ such that $lim_{n \to \infty} \frac{1}{λ_{n}} ∥ {k \in I_{n} : | | x_{k} - x_{N} | |_{0}^{+} \geq ɛ} ∥ = 0 .$

In the sequel, (x_k) is FS_λ-Cauchy means that (x_k) is λ-statistically Cauchy with respect to the fuzzy norm on X.

2.5 Example

Example 2.2 is a statistical Cauchy sequence.

2.6 Proposition

In an FNS (X, || · ||) , every FS_λ- convergent sequence is also an FS_λ- Cauchy sequence.

Proof. Let $x_{k} \overset{{FS}_{λ}}{\to} L$ and ɛ > 0. Then we have ${| | x_{k} - L | |}_{0}^{+} < ɛ / 2$ for a.a.k. Choose $N \in ℕ$ such that ${| | x_{N} - x | |}_{0}^{+} < ɛ / 2 .$ Now, $| | \cdot | |_{0}^{+}$ being a norm in the usual sense, we get $\begin{matrix} {| | x_{k} - x_{N} | |}_{0}^{+} & = & {| | (x_{k} - x) + (x - x_{N}) | |}_{0}^{+} \\ \leq & {| | x_{k} - x | |}_{0}^{+} + {| | x_{N} - x | |}_{0}^{+} \\ < & ɛ / 2 + ɛ / 2 < ɛ \end{matrix}$ for a.a.k. This shows that (x_k) is FS_λ- Cauchy.

In the following definition, we have introduced and studied the concepts of strongly λ-summability w.r.t fuzzy norm on X and found its relation with λ-statistically convergent w.r.t fuzzy norm on X. Before giving the promised relations we have given definitions of λ- summability w.r.t fuzzy norm on X.

2.7 Definition

Let (X, ∥ · ∥) be an FNS and λ = (λ_n) be a non-decreasing sequence of positive numbers tending to ∞ and λ_n+1 ≤ λ_n + 1, λ₁ = 1 and x = (x_k) be a sequence in X . The sequence x is said to be strongly λ - summable with respect to fuzzy norm on X if there is a L ∈ X such that $lim_{n \to \infty} \frac{1}{λ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) = 0$ where I_n = [n - λ_n + 1, n].

In this case, we have written [V, λ] _FN - lim x_k = L. The set of all strongly (V, λ) summable to fuzzy norm on X has been denoted by [V, λ] _FN.

In that case, we have said that X is strongly λ- summable to L with respect to fuzzy norm on X. If λ_n = n, then strongly λ- summable reduces to strongly Cesaro summable w.r.t fuzzy norm on X defined as follows: $lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} D (| | x_{k} - L | |, \tilde{0}) = 0$

In this case, we have written [C, 1] _FN - lim x_k = L. The set of all strongly (C, 1) summable to fuzzy norm on X has been denoted by [C, 1] _FN.

So, we have written $\begin{matrix} {[V, λ]}_{FN} = {x = (x_{k}) : lim_{n \to \infty} \frac{1}{λ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) = 0 for some L} \\ {[C, 1]}_{FN} = {x = (x_{k}) : lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} D (| | x_{k} - L | |, \tilde{0}) = 0 for some L} \end{matrix}$

2.8 Example

Let $(ℝ, ∥ \cdot ∥)$ be an FNS as defined in Example 2.2 and λ = (λ_n) be a non-decreasing sequence of positive numbers tending to ∞ and λ_n+1 ≤ λ_n + 1, λ₁ = 1 and $x = (x_{k}) = {\begin{matrix} 1 \\ 0 \end{matrix} \begin{matrix} n - \sqrt{λ_{n}} + 1 < k \leq n \\ otherwise \end{matrix}$ be a sequence in $ℝ .$ In this case, we have written [V, λ] _FN - lim x_k = 0. In that case, we have said that x is strongly λ- summable to 0 with respect to fuzzy norm on $ℝ$ .

2.9 Theorem

If a sequence x = (x_k) is [V, λ] _FN- summable to L, then it is FS_λ- convergent to L.

Proof. Let ɛ > 0. Since $\begin{matrix} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) & \geq & \sum_{\underset{D (| | x_{k} - L | |, \tilde{0}) \geq ɛ}{k \in I_{n}}} D (| | x_{k} - L | |, \tilde{0}) \\ \geq & ɛ . ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ . \end{matrix}$ This implies that if [V, λ] _FN- summable to L, then x is FS_λ- convergent to L.

2.10 Theorem

If a bounded x = (x_k) is FS_λ- convergent to L, then it is [V, λ] _FN- summable to L, and hence x is [C, 1] _FN- summable to L.

Proof. Suppose that x = (x_k) is bounded and FS_λ- convergent to L. Since x is bounded we have written $D (| | x_{k} - L | |, \tilde{0}) < M$ for all k.

Given ɛ > 0, we have had $\begin{matrix} \frac{1}{λ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) = \frac{1}{λ_{n}} \sum_{\underset{D (| | x_{k} - L | |, \tilde{0}) \geq ɛ}{k \in I_{n}}} D (| | x_{k} - L | |, \tilde{0}) \\ + \frac{1}{λ_{n}} \sum_{\underset{D (| | x_{k} - L | |, \tilde{0}) < ɛ}{k \in I_{n}}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq \frac{M}{λ_{n}} ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ + ɛ \end{matrix}$ which implies that x is [V, λ] _FN-summable to L.

Further, we have had $\begin{matrix} \frac{1}{n} \sum_{k = 1}^{n} D (| | x_{k} - L | |, \tilde{0}) = \frac{1}{n} \sum_{k = 1}^{n - λ_{n}} D (| | x_{k} - L | |, \tilde{0}) + \frac{1}{n} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq \frac{1}{λ_{n}} \sum_{k = 1}^{n - λ_{n}} D (| | x_{k} - L | |, \tilde{0}) + \frac{1}{λ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq \frac{2}{λ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) . \end{matrix}$ Hence, x is [C, 1] _FN- summable to L since x is [V, λ] _FN- summable to L.

2.11 Theorem

If a sequence x = (x_k) is statistically convergent to L w.r.t fuzzy norm on X and $lim inf_{n} (\frac{λ_{n}}{n}) > 0$ then it is FS_λ- convergent to L.

Proof. For given ɛ > 0, we have had ${k \leq n : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} \supset {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} .$ Therefore, $\begin{matrix} \frac{1}{n} ∥ {k \leq n : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \geq \frac{1}{n} ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \\ \geq \frac{λ_{n}}{n} \cdot \frac{1}{λ_{n}} ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ . \end{matrix}$ Taking the limit as n→ ∞ and using $lim inf_{n} (\frac{λ_{n}}{n}) > 0$ , we have got that x is FS_λ- convergent to L.

Throughout the paper, unless stated otherwise, by “for all $n \in ℕ_{n_{0}}$ ” we have meant “for all $n \in ℕ$ except finite numbers of positive integers” where $ℕ_{n_{0}} = {n_{0}, n_{0} + 1, n_{0} + 2, . . .}$ for some $n_{0} \in ℕ = {1, 2, 3, . . .}$ .

2.12 Theorem

Let λ = (λ_n) and μ = (μ_n) be two sequences in Λ such that λ_n ≤ μ_n for all $n \in ℕ_{n_{0}}$ .

If $lim_{n \to \infty} inf \frac{λ_{n}}{μ_{n}} > 0,$ (2) then FS_μ ⊆ FS_λ.

If $lim_{n \to \infty} \frac{λ_{n}}{μ_{n}} = 1,$ (3) then FS_λ ⊆ FS_μ.

Proof. (i) Suppose that λ_n ≤ μ_n for all $n \in ℕ_{n_{0}}$ and let (2) satisfied. Then I_n ⊂ J_n and so that for ɛ > 0 we may write $∥ {k \in J_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \geq ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥$ and, therefore, we have had $\frac{1}{μ_{n}} ∥ {k \in J_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \geq \frac{λ_{n}}{μ_{n}} \cdot \frac{1}{λ_{n}} ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥$ for all $n_{0} \in ℕ_{n_{0}}$ where J_n = [n - μ_n + 1, n].

Now, taking the limit as n→ ∞ in the last inequality and using (2), we have got FS_μ ⊂ FS_λ.

(ii) Let (x_k) ∈ FS_λ and (3) be satisfied. Since I_n ⊂ J_n, for ɛ > 0, we may write

$\begin{matrix} \frac{1}{μ_{n}} ∥ {k \in J_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \\ = \frac{1}{μ_{n}} ∥ {n - μ_{n} + 1 \leq k \leq n - λ_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \\ + \frac{1}{μ_{n}} ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \\ \leq \frac{μ_{n} - λ_{n}}{μ_{n}} + \frac{1}{λ_{n}} ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \\ \leq (1 - \frac{λ_{n}}{μ_{n}}) + \frac{1}{λ_{n}} ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \end{matrix}$

for all $n \in ℕ_{n_{0}}$ . Since $lim_{n \to \infty} \frac{λ_{n}}{μ_{n}} = 1$ by (3) the first term and since x = (x_k) ∈ FS_λ, the second term of right hand side of above inequality has tended to 0 as n → ∞ . This implies that $\frac{1}{μ_{n}} ∥ {k \in J_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \to 0$ as n → ∞ . Therefore x = (x_k) ∈ FS_μ and so FS_λ ⊆ FS_μ.

2.13 Corollary

Let λ = (λ_n) and μ = (μ_n) be two sequences in Λ such that λ_n ≤ μ_n for all $n \in ℕ_{n_{0}}$ . If (3) holds, then FS_λ = FS_μ.

If we take μ = (μ_n) = (n) in Corollary 2.13, we have the following result.

2.14 Corollary

Let λ = (λ_n) ∈ Λ. If $lim_{n} \frac{λ_{n}}{n} = 1$ , then we have FS_λ = FS.

2.15 Theorem

Let λ = (λ_n) , μ = (μ_n) ∈ Λ, and suppose that λ_n ≤ μ_n for all $n \in ℕ_{n_{0}}$ .

If (2), holds then [V, μ] _FN ⊆ [V, λ] _FN,

If (3), holds then ℓ_∞ ∩ [V, λ] _FN ⊆ [V, μ] _FN.

Proof. (i) Suppose that λ_n ≤ μ_n for all $n \in ℕ_{n_{0}}$ . Then I_n ⊂ J_n and so that we may write $\frac{1}{μ_{n}} \sum_{k \in J_{n}} D (| | x_{k} - L | |, \tilde{0}) \geq \frac{1}{μ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0})$ for all $n \in ℕ_{n_{0}}$ . This gives that $\frac{1}{μ_{n}} \sum_{k \in J_{n}} D (| | x_{k} - L | |, \tilde{0}) \geq \frac{λ_{n}}{μ_{n}} \cdot \frac{1}{λ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) .$ Then we have obtained [V, μ] _FN ⊆ [V, λ] _FN. Taking the limit as n→ ∞ in the last inequality and using (2).

(ii) Let x = (x_k) ∈ ℓ _∞ ∩ [V, λ] _FN and suppose that (3) holds. Since x = (x_k) ∈ ℓ _∞, then there exist some M > 0 such that $D (| | x_{k} - L | |, \tilde{0}) \leq M$ for all k. Now, since λ_n ≤ μ_n and so that $\frac{1}{μ_{n}} \leq \frac{1}{λ_{n}}$ , and I_n ⊂ J_n for all $n \in ℕ_{n_{0}}$ , we may write $\begin{matrix} \frac{1}{μ_{n}} \sum_{k \in J_{n}} D (| | x_{k} - L | |, \tilde{0}) = \frac{1}{μ_{n}} \sum_{k \in J_{n} ∖ I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ + \frac{1}{μ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq \frac{μ_{n} - λ_{n}}{μ_{n}} \cdot M + \frac{1}{μ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq (1 - \frac{λ_{n}}{μ_{n}}) \cdot M + \frac{1}{λ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \end{matrix}$ for every $n \in ℕ_{n_{0}}$ . Since $lim_{n} \frac{λ_{n}}{μ_{n}} = 1$ by (3) the first term and since x = (x_k) ∈ [V, λ] _FN, the second term of right hand side of above inequality has tended to 0 as n→ ∞ (Note that $1 - \frac{λ_{n}}{μ_{n}} \geq 0$ for all $n \in ℕ_{n_{0}}$ ).

This iplies that ℓ_∞ ∩ [V, λ] _FN ⊆ [V, μ] _FN and so that ℓ_∞ ∩ [V, λ] _FN ⊆ ℓ _∞ ∩ [V, μ] _FN.

From Theorem 2.15, we have has the following result.

2.16 Corollary

Let λ, μ ∈ Λ such that λ_n ≤ μ_n for all $n \in ℕ_{n_{0}}$ . If (3), holds then ℓ_∞ ∩ [V, λ] _FN = ℓ _∞ ∩ [V, μ] _FN

2.17 Theorem

Let λ, μ ∈ Λ such that λ_n ≤ μ_n for all $n \in ℕ_{n_{0}}$ .

If (2), holds then $x_{k} \to L ({[V, μ]}_{FN}) \Rightarrow x_{k} \to L ({FS}_{λ}),$ and the inclusion [V, μ] _FN ⊂ FS_λ is strict for some λ, μ ∈ Λ.

If (x_k) ∈ ℓ _∞ and x_k → L (FS_λ), then x_k → L ([V, μ] _FN) whenever (3) holds.

Proof. (i) Let ɛ > 0 be given and let x_k → L ([V, μ] _FN). Now, for every ɛ > 0, we may write $\begin{matrix} [b] l \sum_{k \in J_{n}} D (| | x_{k} - L | |, \tilde{0}) \geq \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ \geq \sum_{\underset{D (| | x_{k} - L | |, \tilde{0}) \geq ɛ}{k \in I_{n}}} D (| | x_{k} - L | |, \tilde{0}) \\ \geq ɛ . ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ \end{matrix}$ and so that $\frac{1}{μ_{n}} \sum_{k \in J_{n}} D (| | x_{k} - L | |, \tilde{0}) \geq \frac{λ_{n}}{μ_{n}} \cdot \frac{1}{λ_{n}} \cdot ɛ \cdot ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥$ for all $n \in ℕ_{n_{0}}$ . Then taking the limit as n→ ∞ in the last inequality and using (2), we have obtained that x_k → L (FS_λ) whenever x_k → L ([V, μ] _FN).

To show that the inclusion [V, μ] _FN ⊂ FS_λ is strict for some λ, μ ∈ Λ, we have taken $λ_{n} = \sqrt{n}, μ_{n} = n$ for all $n \in ℕ$ . Define x = (x_k) as $x_{k} = {\begin{matrix} k & ; k = n^{2} \\ 0 & ; k \neq n^{2} \end{matrix}$

Then clearly x_k → 0 (FS_λ) same as Example 2.2. On the other hand, we have known that $1 + 2^{2} + 3^{2} + \dots + n^{2} = \frac{n . (n + 1) . (2 n + 1)}{6}$ is provided for every $n \in ℕ$ . Considering this equality, we can write $\begin{matrix} \frac{1}{μ_{n}} \sum_{j \in J_{n}} D (| | x_{k} - 0 | |, \tilde{0}) = \frac{1}{n} \sum_{k = 1}^{n} {| | x_{k} | |}_{0}^{+} \\ = \frac{1}{n} (1 + 0 + 0 + 4 + \dots + {[\sqrt{n}]}^{2}) \\ = \frac{1}{n} (\frac{([\sqrt{n}]) . ([\sqrt{n}] + 1) . (2 [\sqrt{n}] + 1)}{6}) . \end{matrix}$

Since we have had $\sqrt{n} < [\sqrt{n}] + 1$ and $\sqrt{n} < 2 [\sqrt{n}] + 1$ , we can write $\frac{1}{n} > \frac{1}{([\sqrt{n}] + 1) . (2 [\sqrt{n}] + 1)}$ and so that $\frac{1}{μ_{n}} \sum_{j \in J_{n}} D (| | x_{k} - 0 | |, \tilde{0}) > \frac{[\sqrt{n}]}{6} \to \infty$ as n→ ∞. Therefore x ∉ [V, μ] _FN.

(ii) Suppose that x_k → L (FS_λ) and x = (x_k) ∈ ℓ _∞. Then there exist some M > 0 such that $D (| | x_{k} - L | |, \tilde{0}) \leq M$ for all k. Since $\frac{1}{μ_{n}} \leq \frac{1}{λ_{n}}$ , then for every ɛ > 0 we may write $\begin{matrix} \frac{1}{μ_{n}} \sum_{k \in J_{n}} D (| | x_{k} - L | |, \tilde{0}) = \frac{1}{μ_{n}} \sum_{k \in J_{n} ∖ I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ + \frac{1}{μ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq \frac{μ_{n} - λ_{n}}{μ_{n}} \cdot M + \frac{1}{μ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq (1 - \frac{λ_{n}}{μ_{n}}) \cdot M + \frac{1}{λ_{n}} \sum_{k \in I_{n}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq (1 - \frac{λ_{n}}{μ_{n}}) \cdot M + \frac{1}{λ_{n}} \sum_{\underset{D (| | x_{k} - L | |, \tilde{0}) \geq ɛ}{k \in I_{n}}} D (| | x_{k} - L | |, \tilde{0}) \\ + \frac{1}{λ_{n}} \sum_{\underset{D (| | x_{k} - L | |, \tilde{0}) < ɛ}{k \in I_{n}}} D (| | x_{k} - L | |, \tilde{0}) \\ \leq (1 - \frac{λ_{n}}{μ_{n}}) \cdot M + \frac{M}{λ_{n}} \cdot ∥ {k \in I_{n} : D (| | x_{k} - L | |, \tilde{0}) \geq ɛ} ∥ + ɛ \end{matrix}$ for all $n \in ℕ$ . Using (3), we have obtained that x_k → L ([V, μ] _FN) whenever x_k → L (FS_λ). Hence, we have had ℓ_∞ ∩ FS_λ ⊆ [V, μ] _FN.

If we take μ_n = n for all $n \in ℕ_{n_{0}}$ in Theorem 2.17, then we have the following results. Because $lim_{n \to \infty} \frac{λ_{n}}{μ_{n}} = 1$ implies that $lim_{n \to \infty} inf \frac{λ_{n}}{μ_{n}} = 1 > 0$ , that is (3) ⇒ (2).

2.18 Corollary

Let $lim_{n \to \infty} \frac{λ_{n}}{μ_{n}} = 1$ . Then

If (x_k) ∈ ℓ _∞ and x_k → L (FS_λ) then x_k → L ([C, 1] _FN)

If x_k → L ([C, 1] _FN) then x_k → L (FS_λ)

3 Conclusion

In this study, definitions of (V, λ) summabilty and (C, 1) summability were given in fuzzy normed spaces. It has been given the relation between [V, λ] _FN and [C, 1] _FN. Also, it has been given statistical convergent equal to λ- statistical convergent for which conditions. Moreover, the requirements for $x_{k} \to L ({[V, μ]}_{FN}) \Rightarrow x_{k} \to L ({FS}_{λ}) .$

Finally, the following similarities have been shown:

If we take μ_n = λ_n for all $n \in ℕ_{n_{0}}$ in Theorem 2.17, we obtain Theorem 2.1 (i) and (ii) in [13]. (Note that $lim_{n \to \infty} inf \frac{λ_{n}}{μ_{n}} = lim_{n \to \infty} \frac{λ_{n}}{μ_{n}} = 1 > 0$ and so that (2) and (3) satisfied in this case).

If we take μ_n = λ_n = n for all $n \in ℕ_{n_{0}}$ in Theorem 2.17, we obtain Theorem 2.1 with q = 1 in [3] (Note that (2) and (3) are satisfied also in this case).

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