Tauberian theorems for statistical Cesàro and statistical logarithmic summability of sequences in intuitionistic fuzzy normed spaces

Abstract

We define statistical Cesàro and statistical logarithmic summability methods of sequences in intuitionistic fuzzy normed spaces(IFNS) and give slowly oscillating type and Hardy type Tauberian conditions under which statistical Cesàro summability and statistical logarithmic summability imply convergence in IFNS. Besides, we obtain analogous results for the higher order summability methods as corollaries. Also, two theorems concerning the convergence of statistically convergent sequences in IFNS are proved in the paper.

Keywords

Intuitionistic fuzzy normed space tauberian theorem cesàro and logarithmic summability methods statistical convergence slow oscillation 03E72 40A05 40G05 40G15 40E05

1 Introduction and preliminaries

Events and problems that humankind encounter in real world are commonly complex and imprecise due to uncertainty of parameters and indefiniteness of the objects involved. Researchers proposed various theories of uncertainty to handle such events and problems involving incomplete information. Among them, Lotfi A. Zadeh proposed fuzzy logic as an extension Boolean logic with the introduction of the mathematical concept of fuzzy sets [1]. In [1], Zadeh extended classical sets to fuzzy sets by using gradual memberships taking real values in the interval [0, 1], instead of Boolean memberships which take only integer values {0, 1}. Following its introduction, fuzzy sets were utilized by many researchers in different fields of science to process non-categorical data. Besides, motivated by fuzzy sets, Atanassov [2, 3] defined intuitionistic fuzzy sets(IFS) by considering gradual non-memberships as well as Zadeh’s gradual memberships. Like fuzzy sets, IFS were also applied in many areas of science such as control systems, robotics, computer, medical diagnosis, education, etc. Theoretical basis of intuitionistic fuzzy set theory also developed in time. In particular, Park [4] defined IF-metric and it was pursued by IF-norm [5]. Convergence of sequences with respect to IF-norm was defined and various convergence methods in the statistical sense were proposed to achieve a limit where ordinary convergence fails [6 –15]. Also, there are recent studies applying some weighted mean summation methods in IFNS to achieve limit of sequences and providing Tauberian conditions which guarantee ordinary convergence [16, 17].

In some cases of sequences as in Example 2.2, Example 2.9, Example 3.2 and Example 3.9, a limit can not be achieved via statistical convergence or via weighted mean summation methods in IFNS. In such cases, we need to use some stronger methods of convergence to achieve a limit in IFNS, and to investigate whether ordinary convergence can be guaranteed with some extra conditions. For this aim, in this study we introduce statistical Cesàro and statistical logarithmic summability of sequences in IFNS, and obtain Tauberian conditions of slowly oscillating type and Hardy type under which ordinary convergence in IFNS follows from statistical Cesàro and statistical logarithmic summability. Besides, we show that statistical convergence of slowly oscillating sequences in IFNS implies ordinary convergence in the space. Examples in the paper provide also new types of sequences for statistical Cesàro and statistical logarithmic summation methods in classical normed spaces. Before to continue main results, we now give some preliminaries.

Definition 1.1. [18] The triplicate (N, μ, ν) is said to be an IFNS if N is a real vector space, and μ, ν are fuzzy sets on $N \times ℝ$ satisfying the following conditions for every u, w ∈ N and $t, s \in ℝ$ :

μ (u, t) =0 for t ≤ 0,

μ (u, t) =1 for all $t \in ℝ^{+}$ if and only if u = θ

$μ (cu, t) = μ (u, \frac{t}{| c |})$ for all $t \in ℝ^{+}$ and c ≠ 0,

μ (u + w, t + s) ≥ min {μ (u, t) , μ (w, s)},

$lim_{t \to \infty} μ (u, t) = 1$ and $lim_{t \to 0} μ (u, t) = 0$ ,

ν (u, t) =1 for t ≤ 0,

ν (u, t) =0 for all $t \in ℝ^{+}$ if and only if u = θ

$ν (cu, t) = ν (u, \frac{t}{| c |})$ for all $t \in ℝ^{+}$ and c ≠ 0,

max {ν (u, t) , ν (w, s)} ≥ ν (u + w, t + s),

$lim_{t \to \infty} ν (u, t) = 0$ and $lim_{t \to 0} ν (u, t) = 1$ .

We call (μ, ν) an IF-norm on N.

Example 1.2. Let (N, || · ||) be a normed space and μ₀, ν₀ be fuzzy sets on $N \times ℝ$ defined by $\begin{matrix} μ_{0} (u, t) = {\begin{matrix} 0, & t \leq 0, \\ \frac{t}{t + ‖ u ‖}, & t > 0, \end{matrix} \\ ν_{0} (u, t) = {\begin{matrix} 1, & t \leq 0, \\ \frac{‖ u ‖}{t + ‖ u ‖}, & t > 0 . \end{matrix} \end{matrix}$ Then (μ₀, ν₀) is IF-norm on N. We note that we will use this IF-norm in the examples of the paper.

Throughout the paper (N, μ, ν) will denote an IFNS.

Definition 1.3. [18] A sequence (u_k) in (N, μ, ν) is said to be convergent to a ∈ N and denoted by u_k → a if for every ɛ > 0 and t > 0 there exists $k_{0} \in ℕ$ such that μ (u_k - a, t) >1 - ɛ and ν (u_k - a, t) < ɛ for all k ≥ k₀.

Definition 1.4. [6] Let sequence (u_k) be in (N, μ, ν). We say that (u_k) is statistically convergent to a ∈ N with respect to fuzzy norm (μ, ν) provided that, for every ɛ > 0 and t > 0, $\begin{matrix} lim_{n \to \infty} \frac{1}{n} & | {k \leq n : μ (u_{k} - a, t) \leq 1 - ɛ or \\ ν (u_{k} - u_{n}, t) \geq ɛ} | = 0 . \end{matrix}$ In this case we write st_μ,ν - lim u = a.

Theorem 1.5. [6] Let sequence (u_k) be in (N, μ, ν) and a ∈ N. Then, st_μ,ν - lim u = a if and only if st - lim μ (u_k - a, t) =1 and st - lim ν (u_k - a, t) =0 for each t.

Theorem 1.6. [6] Let sequence (u_k) be in (N, μ, ν) and a ∈ N. If (u_k) is convergent to a, then (u_k) is statistically convergent to a.

Definition 1.7. [19] A sequence (u_k) in (N, μ, ν) is called q-bounded if $lim_{t \to \infty} inf_{k \in ℕ} μ (u_{k}, t) = 1$ and $lim_{t \to \infty} sup_{k \in ℕ} ν (u_{k}, t) = 0$ .

Definition 1.8. [16] A sequence (u_k) is slowly oscillating if and only if for all t > 0 and for all ɛ ∈ (0, 1) there exist λ > 1 and $m_{0} \in ℕ$ , depending on t and ɛ, such that $μ (u_{k} - u_{n}, t) > 1 - ɛ and ν (u_{k} - u_{n}, t) < ɛ$ whenever m₀≤ n < k ≤ ⌊ λn ⌋.

Definition 1.9. [17] A sequence (u_k) is slowly oscillating with respect to logarithmic summability if and only if for all t > 0 and for all ɛ ∈ (0, 1) there exist λ > 1 and $m_{0} \in ℕ$ , depending on t and ɛ, such that $μ (u_{k} - u_{n}, t) > 1 - ɛ and ν (u_{k} - u_{n}, t) < ɛ$ whenever m₀≤ n < k ≤ ⌊ n^λ ⌋.

Theorem 1.10. [16] Let sequence (u_k) be in (N, μ, ν). If {k (u_k - u_k-1)} is q-bounded, then (u_k) is slowly oscillating.

Theorem 1.11. [17] Let sequence (u_k) be in (N, μ, ν). If {k ln k (u_k - u_k-1)} is q-bounded, then (u_k) is slowly oscillating with respect to logarithmic summability.

Theorem 1.12. [16] Let sequence (u_k) be in (N, μ, ν). If (u_k) is Cesàro summable to some a ∈ N and slowly oscillating, then (u_k) converges to a.

Theorem 1.13. [17] Let sequence (u_k) be in (N, μ, ν). If (u_k) is logarithmic summable to some a ∈ N and slowly oscillating with respect to logarithmic summability, then (u_k) converges to a.

2 Tauberian theorems for statistical Cesàro summability in IFNS

Now, we define statistical Cesàro summability method in IFNS. For some other studies concerning the concepts of Cesàro summability and statistical convergence see [16 , 20–23].

Definition 2.1. Let (u_k) be a sequence in (N, μ, ν). Cesàro means σ_k of (u_k) is defined by $\begin{matrix} σ_{k} = \frac{1}{k} \sum_{j = 1}^{k} u_{j} . \end{matrix}$ We say that (u_k) is statistically Cesàro summable to a ∈ N if st_μ,ν - lim σ = a.

In view of Theorem 1.6 and [16, Theorem 3.2], convergence implies statistical Cesàro summability in IFNS. But converse statement is not true in general by the next example.

Example 2.2. Let $\begin{matrix} u_{k} = {\begin{matrix} 1 + (- 1)^{k} + k^{2}, & k = n^{2} \\ 1 + (- 1)^{k} - (k - 1)^{2}, & k = n^{2} + 1 \\ 1 + (- 1)^{k}, & otherwise \end{matrix} \end{matrix}$ be in IF-normed space $(ℝ, μ_{0}, ν_{0})$ . Sequence (u_k) is neither convergent nor statistically convergent in $(ℝ, μ_{0}, ν_{0})$ . Besides, it is not Cesàro summable.

Let us apply statistical Cesàro summability to achieve a limit. Cesàro means (σ_k) of sequence (u_k) is $\begin{matrix} σ_{k} = {\begin{matrix} 1 + \frac{1}{k} \sum_{j = 1}^{k} (- 1)^{j} + k, & k = n^{2} \\ 1 + \frac{1}{k} \sum_{j = 1}^{k} (- 1)^{j}, & otherwise . \end{matrix} \end{matrix}$ Sequence (σ_k) is statistically convergent to 1 since for each t > 0 we have st - lim μ₀ (σ_k - 1, t) = 1 and st - lim ν₀ (σ_k - 1, t) = 0 where $\begin{matrix} μ_{0} (σ_{k} - 1, t) = {\begin{matrix} \frac{t}{t + | \frac{1}{k} \sum_{j = 1}^{k} (- 1)^{j} + k |}, & k = n^{2} \\ \frac{t}{t + | \frac{1}{k} \sum_{j = 1}^{k} (- 1)^{j} |}, & otherwise, \end{matrix} \\ ν_{0} (σ_{k} - 1, t) = {\begin{matrix} \frac{| \frac{1}{k} \sum_{j = 1}^{k} (- 1)^{j} + k |}{t + | \frac{1}{k} \sum_{j = 1}^{k} (- 1)^{j} + k |}, & k = n^{2} \\ \frac{| \frac{1}{k} \sum_{j = 1}^{k} (- 1)^{j} |}{t + | \frac{1}{k} \sum_{j = 1}^{k} (- 1)^{j} |}, & otherwise . \end{matrix} \end{matrix}$ Hence, sequence (u_k) is statistically Cesàro summable to 1 in $(ℝ, μ_{0}, ν_{0})$ .

In this section, we will give some Tauberian conditions for statistical Cesàro summability to imply convergence in IFNS. To this end, firstly we now show that statistical convergence of slowly oscillating sequences yields convergence in IFNS.

Theorem 2.3. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically convergent to some a ∈ N and slowly oscillating, then (u_k) is convergent to a.

Proof. Let st_μ,ν - lim u = a and sequence (u_k) be slowly oscillating. Then by Theorem 1.5, for every t > 0 we have st - lim μ (u_k - a, t) = 1 and st - lim ν (u_k - a, t) = 0. Our aim to show that $lim_{k \to \infty} μ (u_{k} - a, t) = 1$ and $lim_{k \to \infty} ν (u_{k} - a, t) = 0$ .

Fix t > 0. Since $st - lim μ (u_{k} - a, \frac{t}{2}) = 1$ , from the proof of [23, Lemma 6] there is a subsequence of integers 1≤ l₁ < l₂ < ⋯ such that for any λ > 1 inequality l_m < l_m+1 < λl_m holds for large enough m; and $\begin{matrix} lim_{m \to \infty} μ (u_{l_{m}} - a, \frac{t}{2}) = 1 . \end{matrix}$ So, for given ɛ > 0 there exists m₁ such that

$μ (u_{l_{m}} - a, \frac{t}{2}) > 1 - ɛ for m > m_{1} .$ (1) Besides, since (u_k) is slowly oscillating there exist m₂ and λ > 1 such that

$μ (u_{k} - u_{l_{m}}, \frac{t}{2}) > 1 - ɛ$ (2) for m₂ ≤ l_m < k ≤ λl_m. It follows from (2.1)–(2.2) that $\begin{matrix} μ (u_{k} - a, t) \\ \geq & min {μ (u_{k} - u_{l_{m}}, \frac{t}{2}), μ (u_{l_{m}} - a, \frac{t}{2})} > 1 - ɛ \end{matrix}$ holds for l_m < k ≤ l_m+1 where m > max {m₁, m₂}. By considering all m’s, we get $\begin{matrix} μ (u_{k} - a, t) > 1 - ɛ for k > l_{m_{3}}, \end{matrix}$ where m₃ = max {m₁, m₂}. Hence $lim_{k \to \infty} μ (u_{k} - a, t) = 1$ is proved. The proof of $lim_{k \to \infty} ν (u_{k} - a, t) = 0$ can be done similarly.□

We need next lemma to prove main theorem of this section.

Lemma 2.4. Let (u_k) be slowly oscillating sequence in (N, μ, ν). Let t be an arbitrary but fixed positive number. Then, for each ɛ > 0 followings hold: $\begin{matrix} μ & (u_{n} - u_{k}, {1 + \frac{ln (2 n / k)}{ln λ}} t) > 1 - ɛ, \\ ν & (u_{n} - u_{k}, {1 + \frac{ln (2 n / k)}{ln λ}} t) < ɛ \end{matrix}$ (3) and $\begin{matrix} μ & (\frac{1}{n} \sum_{k = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{k}), {1 + \frac{ln 2 + 1}{ln λ}} t) > 1 - ɛ, \\ ν & (\frac{1}{n} \sum_{k = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{k}), {1 + \frac{ln 2 + 1}{ln λ}} t) < ɛ \end{matrix}$ (4) where m₀ ≤ k ≤ n/λ, and m₀ = m₀ (t, ɛ) and λ = λ (t, ɛ) are from definition of slow oscillation.

Proof. Let (u_k) be slowly oscillating sequence in (N, μ, ν), and m₀ and λ > 1 be from Definition 1.8. Let t be an arbitrary but fixed positive number. Fix m₀ ≤ k ≤ n/λ. Consider the sequence(see [23, Proof of Lemma 8]) $\begin{matrix} n_{0} : = n, n_{p} : = 1 + ⌊ \frac{n_{p - 1}}{λ} ⌋, p = 1, 2, \dots, q + 1, \end{matrix}$ where q is determined by the condition $\begin{matrix} n_{q + 1} \leq k < n_{q} . \end{matrix}$ Then, we get $\begin{matrix} 1 - ɛ \\ < & min {μ (u_{n} - u_{n_{1}}, t), μ (u_{n_{1}} - u_{n_{2}}, t), \\ \dots, μ (u_{n_{q}} - u_{k}, t)} \\ \leq & μ (u_{n} - u_{k}, (q + 1) t) \\ \leq & μ (u_{n} - u_{k}, {1 + \frac{ln (2 n / k)}{ln λ}} t) \end{matrix}$ and $\begin{matrix} ɛ & > & max {ν (u_{n} - u_{n_{1}}, t), \dots, ν (u_{n_{q}} - u_{k}, t)} \\ \geq & ν (u_{n} - u_{k}, {1 + \frac{ln (2 n / k)}{ln λ}} t) \end{matrix}$ by virtue of the inequality $q \leq \frac{1}{ln λ} ln (\frac{2 n}{k})$ which was calculated in [23, Proof of Lemma 8]. This proves (2.3).

On the other hand by using (2.3) we have: $\begin{matrix} 1 - ɛ & < & min_{m_{0} \leq k \leq ⌊ n / λ ⌋} μ (u_{n} - u_{k}, {1 + \frac{ln (2 n / k)}{ln λ}} t) \\ \leq & μ (\sum_{k = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{k}), {\sum_{k = m_{0}}^{⌊ n / λ ⌋} (1 + \frac{ln (2 n / k)}{ln λ})} t) \\ \leq & μ (\sum_{k = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{k}), {\sum_{k = 1}^{n} (1 + \frac{ln (2 n / k)}{ln λ})} t) \\ \leq & μ (\sum_{k = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{k}), {1 + \frac{ln 2 + 1}{ln λ}} nt) \\ = & μ (\frac{1}{n} \sum_{k = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{k}), {1 + \frac{ln 2 + 1}{ln λ}} t) \end{matrix}$ and $\begin{matrix} ɛ & > & max_{m_{0} \leq k \leq ⌊ n / λ ⌋} ν (u_{n} - u_{k}, {1 + \frac{ln (2 n / k)}{ln λ}} t) \\ \geq & ν (\frac{1}{n} \sum_{k = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{k}), {1 + \frac{ln 2 + 1}{ln λ}} t) \end{matrix}$ by virtue of $\sum_{k = 2}^{n} ln k > \int_{1}^{n} ln (x) dx$ , which proves (2.4). □

Theorem 2.5. Let sequence (u_k) be in (N, μ, ν). If (u_k) is slowly oscillating, then sequence (σ_k) of Cesàro means is also slowly oscillating.

Proof. Let sequence (u_k) be slowly oscillating and $σ_{k} = \frac{1}{k} \sum_{j = 1}^{k} u_{j}$ . Fix t > 0. For given ɛ > 0 there exists $m_{0}, m_{1} \in ℕ$ and 1 < λ < 2 such that

μ (u_k - u_n, t/16) >1 - ɛ and ν (u_k - u_n, t/16) < ɛ whenever m₀≤ n < k ≤ ⌊ λn ⌋.

$\begin{matrix} μ (\frac{k - n}{kn} \sum_{j = 1}^{m_{0} - 1} (u_{n} - u_{j}), \frac{t}{4}) > 1 - ɛ \\ and ν (\frac{k - n}{kn} \sum_{j = 1}^{m_{0} - 1} (u_{n} - u_{j}), \frac{t}{4}) < ɛ \end{matrix}$ whenever m₁≤ n < k ≤ ⌊ λn ⌋, by virtue of inequalities in (2.3).

Then, for max {m₀, m₁}≤ n < k ≤ ⌊ λn ⌋, we get

\begin{matrix} μ (σ_{k} - σ_{n}, t) \\ = & μ (\frac{k - n}{kn} \sum_{j = 1}^{n} (u_{n} - u_{j}) + \frac{1}{k} \sum_{j = n + 1}^{k} (u_{j} - u_{n}), t) \\ \geq & min {μ (\frac{k - n}{kn} \sum_{j = 1}^{m_{0} - 1} (u_{n} - u_{j}), \frac{t}{4}), \\ μ (\frac{k - n}{kn} \sum_{j = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{j}), \frac{t}{4}), \\ μ (\frac{k - n}{kn} \sum_{j = ⌊ n / λ ⌋ + 1}^{n} (u_{n} - u_{j}), \frac{t}{4}), \\ μ (\frac{1}{k} \sum_{j = n + 1}^{k} (u_{j} - u_{n}), \frac{t}{4})} \\ \geq & min {μ (\frac{k - n}{kn} \sum_{j = 1}^{m_{0} - 1} (u_{n} - u_{j}), \frac{t}{4}), \\ μ (\frac{k - n}{kn} \sum_{j = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{j}), \frac{t}{4}), \\ min_{⌊ n / λ ⌋ \leq j \leq n} μ (u_{n} - u_{j}, \frac{t}{4}), \\ min_{n + 1 \leq j \leq k} μ (u_{j} - u_{n}, \frac{t}{4})} \\ > & min {μ (\frac{k - n}{kn} \sum_{j = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{j}), \frac{t}{4}), 1 - ɛ} \\ \geq & min {μ (\frac{1}{n} \sum_{j = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{j}), \frac{t}{4 (λ - 1)}), 1 - ɛ} \\ \geq & min {μ (\frac{1}{n} \sum_{j = m_{0}}^{⌊ n / λ ⌋} (u_{n} - u_{j}), {1 + \frac{ln 2 + 1}{ln λ}} \frac{t}{16}), 1 - ɛ} \\ \geq & 1 - ɛ \end{matrix}

by virtue of the facts that

\frac{k - n}{k} < λ - 1

and

\frac{λ - 1}{ln λ} (ln λ + ln 2 + 1) < 4

, and of Lemma 2.4. On the other hand, ν (σ_k - σ_n, t) < ɛ can be shown similarly. Hence, the proof is completed.□

Now we give the main theorem of this section.

Theorem 2.6. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically Cesàro summable to some a ∈ N and slowly oscillating, then (u_k) is convergent to a.

Proof. Let (u_k) be statistically Cesàro summable to some a ∈ N and slowly oscillating. Then, by Theorem 2.5 sequence (σ_k) of Cesàro means is also slowly oscillating. From Theorem 2.3, (σ_k) → a. This means that (u_k) is Cesàro summable to a. Since (u_k) is slowly oscillating, from Theorem 1.12 (u_k) is convergent to a.□

In view of Theorem 1.10 and Theorem 2.6 we get following theorem.

Theorem 2.7. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically Cesàro summable to a ∈ N and {k (u_k - u_k-1)} is q-bounded, then (u_k) converges to a.

In some cases of sequences as in Example 2.9, first order Cesàro means fails to converge both ordinarily and statistically in IFNS. To handle such sequences, we consider higher order Cesàro means and define statistical Hölder summability method in IFNS. In the sequel, we give corresponding Tauberian theorems as corollaries.

Definition 2.8. Let (u_k) be in (N, μ, ν). m-th order Hölder means $H_{k}^{m}$ of (u_k) is defined by $\begin{matrix} H_{k}^{m} = \frac{1}{k} \sum_{j = 1}^{k} H_{j}^{m - 1}, \end{matrix}$ where $H_{k}^{0} = u_{k}$ . Sequence (u_k) is said to be (H, m) summable to a ∈ N if $lim_{k \to \infty} H_{k}^{m} = a$ and it is said to be statistically (H, m) summable to a if st_μ,ν - lim H^m = a

Example 2.9. Let $\begin{matrix} u_{k} = \\ {\begin{matrix} (- 1)^{k} k^{2} + k^{4}, & k = n^{2} \\ (- 1)^{k} k^{2} - (k - 1)^{4} - (k - 1)^{3} k - (k - 1)^{2} k^{2}, & k = n^{2} + 1 \\ (- 1)^{k} k^{2} + 3 (k - 2)^{2} (k - 1)^{2}, & k = n^{2} + 2 \\ (- 1)^{k} k^{2} - (k - 3)^{2} (k - 2) (k - 1), & k = n^{2} + 3 \\ (- 1)^{k} k^{2}, & otherwise \end{matrix} \end{matrix}$ be in IF-normed space $(ℝ, μ_{0}, ν_{0})$ where n ≥ 2. Sequence (u_k) is neither convergent nor statistically convergent in $(ℝ, μ_{0}, ν_{0})$ . Furthermore, it is neither Cesàro summable nor statistically Cesàro summable.

Let us apply statistical (H, 3) summability to achieve a limit. Hölder means $(H_{k}^{1})$ , $(H_{k}^{2})$ and $(H_{k}^{3})$ of sequence (u_k) are $\begin{matrix} H_{k}^{1} = {\begin{matrix} σ_{k}^{(1)} + k^{3}, & k = n^{2} \\ σ_{k}^{(1)} - (k - 1)^{2} (2 k - 1), & k = n^{2} + 1 \\ σ_{k}^{(1)} + (k - 2)^{2} (k - 1), & k = n^{2} + 2 \\ σ_{k}^{(1)}, & otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} H_{k}^{2} = {\begin{matrix} σ_{k}^{(2)} + k^{2}, & k = n^{2} \\ σ_{k}^{(2)} - (k - 1)^{2}, & k = n^{2} + 1 \\ σ_{k}^{(2)}, & otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} H_{k}^{3} = {\begin{matrix} σ_{k}^{(3)} + k, & k = n^{2} \\ σ_{k}^{(3)}, & otherwise \end{matrix} \end{matrix}$ where sequence ${σ_{k}^{(m)}}$ denotes m-fold Cesàro means of sequence { (-1) ^kk²} and n ≥ 2. Sequence $(H_{k}^{3})$ is statistically convergent to 0 since for each t > 0 we have $st - lim μ_{0} (H_{k}^{3}, t) = 1$ and $st - lim ν_{0} (H_{k}^{3}, t) = 0$ where $\begin{matrix} μ_{0} (H_{k}^{3}, t) = {\begin{matrix} \frac{t}{t + | σ_{k}^{(3)} + k |}, & k = n^{2} \\ \frac{t}{t + | σ_{k}^{(3)} |}, & otherwise, \end{matrix} \\ ν_{0} (H_{k}^{3}, t) = {\begin{matrix} \frac{| σ_{k}^{(3)} + k |}{t + | σ_{k}^{(3)} + k |}, & k = n^{2} \\ \frac{| σ_{k}^{(3)} |}{t + | σ_{k}^{(3)} |}, & otherwise . \end{matrix} \end{matrix}$ in view of the fact that $lim_{k \to \infty} σ_{k}^{(3)} = 0$ . Hence, sequence (u_k) is statistically (H, 3) summable to 0 in $(ℝ, μ_{0}, ν_{0})$ .

In view of Theorem 1.12, Theorem 2.5 and Theorem 2.6 we get following Tauberian theorem.

Theorem 2.10. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically (H, m) summable to some a ∈ N and slowly oscillating, then (u_k) is convergent to a.

Also, in view of theorem above and Theorem 1.10 we get following theorem.

Theorem 2.11. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically (H, m) summable to some a ∈ N and {k (u_k - u_k-1)} is q-bounded, then (u_k) is convergent to a.

3 Tauberian theorems for statistical logarithmic summability in IFNS

We define statistical logarithmic summability method in IFNS as the following.

Definition 3.1. Let sequence (u_k) be in (N, μ, ν). Logarithmic mean τ_k of (u_k) is defined by $\begin{matrix} τ_{k} = \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} \frac{u_{j}}{j} where ℓ_{k} = \sum_{j = 1}^{k} \frac{1}{j} \cdot \end{matrix}$ (u_k) is said to be statistically logarithmic summable to a ∈ N if st_μ,ν - lim τ = a.

In view of Theorem 1.6 and [17, Theorem 2.2], convergence implies statistical logarithmic summability in IFNS. But converse statement is not true in general by the next example.

Example 3.2. Consider the vector space C [0, 1] equipped with the norm $‖ f ‖ = max_{x \in [0, 1]} | f (x) |$ . Let $\begin{matrix} f_{k} (x) = {\begin{matrix} (- x)^{k} k + k {(ℓ_{k})}^{2}, & k = n^{2} \\ (- x)^{k} k - k {(ℓ_{k - 1})}^{2}, & k = n^{2} + 1 \\ (- x)^{k} k, & otherwise \end{matrix} \end{matrix}$ be in IF-normed space (C [0, 1] , μ₀, ν₀) where $ℓ_{k} = \sum_{j = 1}^{k} 1 / j$ . Sequence (f_k) is neither convergent nor statistically convergent in (C [0, 1] , μ₀, ν₀). Besides, (f_k) is neither Cesàro summable nor logarithmic summable. Furthermore, (f_k) is not statistically Cesàro summable which we have defined in previous section.

Let us apply statistical logarithmic summability to achieve a limit. Logarithmic means (τ_k) of sequence (f_k) is $\begin{matrix} τ_{k} (x) = {\begin{matrix} \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} (- x)^{j} + ℓ_{k}, & k = n^{2} \\ \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} (- x)^{j}, & otherwise . \end{matrix} \end{matrix}$ Hence, (f_k) is statistically logarithmic summable to 0 since for each t > 0 we have st - lim μ₀ (τ_k, t) = 1 and st - lim ν₀ (τ_k, t) = 0 where $\begin{matrix} μ_{0} (τ_{k}, t) = {\begin{matrix} \frac{t}{t + ‖ \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} (- x)^{j} + ℓ_{k} ‖}, & k = n^{2} \\ \frac{t}{t + ‖ \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} (- x)^{j} ‖}, & otherwise, \end{matrix} \\ ν_{0} (τ_{k}, t) = {\begin{matrix} \frac{‖ \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} (- x)^{j} + ℓ_{k} ‖}{t + ‖ \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} (- x)^{j} + ℓ_{k} ‖}, & k = n^{2} \\ \frac{‖ \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} (- x)^{j} ‖}{t + ‖ \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} (- x)^{j} ‖}, & otherwise . \end{matrix} \end{matrix}$

In this section, we will give some Tauberian conditions for statistical logarithmic summability to imply convergence in IFNS. To this end, firstly we now show that statistical convergence of slowly oscillating sequences with respect to logarithmic summability implies convergence in IFNS.

Theorem 3.3. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically convergent to some a ∈ N and slowly oscillating with respect to logarithmic summability, then (u_k) is convergent to a.

Proof. Let st_μ,ν - lim u = a and sequence (u_k) be slowly oscillating with respect to logarithmic summability. Then by Theorem 1.5, for every t > 0 we have st - lim μ (u_k - a, t) = 1 and st - lim ν (u_k - a, t) = 0. Our aim to show that $lim_{k \to \infty} μ (u_{k} - a, t) = 1$ and $lim_{k \to \infty} ν (u_{k} - a, t) = 0$ .

Fix t > 0. Since $st - lim μ (u_{k} - a, \frac{t}{2}) = 1$ , from the proof of [24, Lemma 10] there is a subsequence of integers 1≤ j₁ < j₂ < ⋯ such that for any λ > 1 inequality $j_{m} < j_{m + 1} < j_{m}^{λ}$ holds for large enough m; and $\begin{matrix} lim_{m \to \infty} μ (u_{j_{m}} - a, \frac{t}{2}) = 1 . \end{matrix}$ So, for ɛ > 0 there exists m₁ so that

$μ (u_{j_{m}} - a, \frac{t}{2}) > 1 - ɛ for m > m_{1} .$ (5) Besides, since (u_k) is slowly oscillating with respect to logarithmic summability there exist m₂ and λ > 1 such that

$μ (u_{k} - u_{j_{m}}, \frac{t}{2}) > 1 - ɛ$ (6) for $m_{2} \leq j_{m} < k \leq j_{m}^{λ}$ . It follows from (3.1)–(3.2) that $\begin{matrix} μ (u_{k} - a, t) \\ \geq & min {μ (u_{k} - u_{j_{m}}, \frac{t}{2}), μ (u_{j_{m}} - a, \frac{t}{2})} \\ > & 1 - ɛ \end{matrix}$ holds for j_m < k ≤ j_m+1 where m > max {m₁, m₂}. By considering all m’s, we get $\begin{matrix} μ (u_{k} - a, t) > 1 - ɛ for k > j_{m_{3}}, \end{matrix}$ where m₃ = max {m₁, m₂}. Hence $lim_{k \to \infty} μ (u_{k} - a, t) = 1$ is proved. The proof of $lim_{k \to \infty} ν (u_{k} - a, t) = 0$ can be done similarly. □

We need the next lemma to prove main theorem of this section.

Lemma 3.4. Let sequence (u_k) be slowly oscillating with respect to logarithmic summability in (N, μ, ν). Let t be an arbitrary but fixed positive number. Then, for each ɛ > 0 followings hold: $\begin{matrix} μ & (u_{n} - u_{k}, {1 + \frac{1}{ln λ} ln (\frac{2 ln n}{ln k})} t) > 1 - ɛ, \\ ν & (u_{n} - u_{k}, {1 + \frac{1}{ln λ} ln (\frac{2 ln n}{ln k})} t) < ɛ \end{matrix}$ (7) and $\begin{matrix} μ & (\frac{1}{ℓ_{n}} \sum_{k = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{k}}{k}), {1 + \frac{ln 2 + 2}{ln λ}} t) > 1 - ɛ, \\ ν & (\frac{1}{ℓ_{n}} \sum_{k = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{k}}{k}), {1 + \frac{ln 2 + 2}{ln λ}} t) < ɛ \end{matrix}$ (8) where 1 < m₀ ≤ k ≤ n^1/λ, and m₀ = m₀ (t, ɛ) and λ = λ (t, ɛ) are from definition of slow oscillation with respect to logarithmic summability.

Proof. Let sequence (u_k) be slowly oscillating with respect to logarithmic summability in (N, μ, ν), and 1 < m₀ and λ > 1 be from Definition 1.9. Let t be an arbitrary but fixed positive number. Fix m₀ ≤ k ≤ n^1/λ. Consider the sequence(see [24, Proof of Lemma 5]) $\begin{matrix} n_{0} : = n, n_{p} : = 1 + ⌊ n_{p - 1}^{1 / λ} ⌋, p = 1, 2, \dots, q + 1, \end{matrix}$ where q is determined by the condition $\begin{matrix} n_{q + 1} \leq k < n_{q} . \end{matrix}$ Then, we get $\begin{matrix} 1 - ɛ \\ < & min {μ (u_{n} - u_{n_{1}}, t), μ (u_{n_{1}} - u_{n_{2}}, t), \\ \dots, μ (u_{n_{q}} - u_{k}, t)} \\ \leq & μ (u_{n} - u_{k}, (q + 1) t) \\ \leq & μ (u_{n} - u_{k}, {1 + \frac{1}{ln λ} ln (\frac{2 ln n}{ln k})} t) \end{matrix}$ and $\begin{matrix} ɛ & > & max {ν (u_{n} - u_{n_{1}}, t), \dots, ν (u_{n_{q}} - u_{k}, t)} \\ \geq & ν (u_{n} - u_{k}, {1 + \frac{1}{ln λ} ln (\frac{2 ln n}{ln k})} t) \end{matrix}$ by virtue of the inequality $q \leq \frac{1}{ln λ} ln (\frac{2 ln n}{ln k})$ which was calculated in [24, Proof of Lemma 5]. This proves (3.3).

On the other hand by using (3.3) we have: $\begin{matrix} 1 - ɛ \\ < & min_{m_{0} \leq k \leq ⌊ n^{1 / λ} ⌋} μ (u_{n} - u_{k}, {1 + \frac{1}{ln λ} ln (\frac{2 ln n}{ln k})} t) \\ \leq & μ (\sum_{k = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{k}}{k}), {\sum_{k = m_{0}}^{⌊ n^{1 / λ} ⌋} \frac{1}{k} (1 + \frac{1}{ln λ} ln (\frac{2 ln n}{ln k}))} t) \\ \leq & μ (\sum_{k = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{k}}{k}), {\sum_{k = 2}^{n} \frac{1}{k} (1 + \frac{1}{ln λ} ln (\frac{2 ln n}{ln k}))} t) \\ \leq & μ (\sum_{k = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{k}}{k}), {1 + \frac{ln 2 + 2}{ln λ}} ℓ_{n} t) \\ = & μ (\frac{1}{ℓ_{n}} \sum_{k = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{k}}{k}), {1 + \frac{ln 2 + 2}{ln λ}} t) \end{matrix}$ and $\begin{matrix} ɛ \\ > & max_{m_{0} \leq k \leq ⌊ n^{1 / λ} ⌋} ν (u_{n} - u_{k}, {1 + \frac{1}{ln λ} ln (\frac{2 ln n}{ln k})} t) \\ \geq & ν (\frac{1}{ℓ_{n}} \sum_{k = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{k}}{k}), {1 + \frac{ln 2 + 2}{ln λ}} t) \end{matrix}$ which proves (3.4). □

Theorem 3.5. Let sequence (u_k) be in (N, μ, ν). If (u_k) is slowly oscillating with respect to logarithmic summability, then sequence (τ_k) of logarithmic means is also slowly oscillating with respect to logarithmic summability.

Proof. Let sequence (u_k) be slowly oscillating with respect to logarithmic summability and $τ_{k} = \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} \frac{u_{j}}{j}$ . Fix t > 0. For given ɛ > 0 there exists $1 < m_{0}, m_{1} \in ℕ$ and 1 < λ < 2 such that

μ (u_k - u_n, t/20) >1 - ɛ and ν (u_k - u_n, t/20) < ɛ whenever m₀≤ n < k ≤ ⌊ n^λ ⌋.

$\begin{matrix} μ & ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = 1}^{m_{0} - 1} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4}) > 1 - ɛ, \\ ν & ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = 1}^{m_{0} - 1} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4}) < ɛ \end{matrix}$ whenever m₁≤ n < k ≤ ⌊ n^λ ⌋, by virtue of inequalities in (3.3).

Then, for max {m₀, m₁}≤ n < k ≤ ⌊ n^λ ⌋, we get

\begin{matrix} μ (τ_{k} - τ_{n}, t) \\ = & μ ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = 1}^{n} (\frac{u_{n} - u_{j}}{j}) + \frac{1}{ℓ_{k}} \sum_{j = n + 1}^{k} (\frac{u_{j} - u_{n}}{j}), t) \\ \geq & min {μ ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = 1}^{m_{0} - 1} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4}), \\ μ ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4}), \\ μ ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = ⌊ n^{1 / λ} ⌋ + 1}^{n} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4}), \\ μ (\frac{1}{ℓ_{k}} \sum_{j = n + 1}^{k} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4})} \\ \geq & min {μ ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = 1}^{m_{0} - 1} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4}), \\ μ ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4}), \\ min_{⌊ n^{1 / λ} ⌋ \leq j \leq n} μ (u_{n} - u_{j}, \frac{t}{4}), min_{n + 1 \leq j \leq k} μ (u_{j} - u_{n}, \frac{t}{4})} \\ > & min {μ ((\frac{1}{ℓ_{n}} - \frac{1}{ℓ_{k}}) \sum_{j = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4}), 1 - ɛ} \\ \geq & min {μ (\frac{1}{ℓ_{n}} \sum_{j = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{j}}{j}), \frac{t}{4 (λ - 1)}), 1 - ɛ} \\ \geq & min {μ (\frac{1}{ℓ_{n}} \sum_{j = m_{0}}^{⌊ n^{1 / λ} ⌋} (\frac{u_{n} - u_{j}}{j}), {1 + \frac{ln 2 + 2}{ln λ}} \frac{t}{20}), 1 - ɛ} \\ \geq & 1 - ɛ \end{matrix}

by virtue of the facts that

\frac{ℓ_{k} - ℓ_{n}}{ℓ_{k}} \leq \frac{ln k - ln n}{ln k} < λ - 1

and

\frac{λ - 1}{ln λ} (ln λ + ln 2 + 2) < 5

, and of Lemma 3.4. On the other hand, ν (τ_k - τ_n, t) < ɛ can be shown similarly. Hence, the proof is completed.□

Now we give the main theorem of this section.

Theorem 3.6. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically logarithmic summable to some a ∈ N and slowly oscillating with respect to logarithmic summability, then (u_k) is convergent to a.

Proof. Let (u_k) be statistically logarithmic summable to some a ∈ N and slowly oscillating with respect to logarithmic summability. Then, by Theorem 3.5 sequence (τ_k) of logarithmic means is also slowly oscillating with respect to logarithmic summability. From Theorem 3.3, (τ_k) → a. This means that (u_k) is logarithmic summable to a. Since (u_k) is slowly oscillating with respect to logarithmic summability, from Theorem 1.13 (u_k) is convergent to a.□

In view of Theorem 1.11 and Theorem 3.6 we get following theorem.

Theorem 3.7. Let sequence (u_k) be in (N, μ, ν). If (u_k) is logarithmic summable to a ∈ N and {k ln k (u_k - u_k-1)} is q-bounded, then (u_k) converges to a.

For the case of higher order logarithmic summability methods we can give following Tauberian theorems as corollaries.

Definition 3.8. Let (u_k) be in (N, μ, ν). m-th order logarithmic means $L_{k}^{m}$ of (u_k) is defined by $\begin{matrix} L_{k}^{m} = \frac{1}{ℓ_{k}} \sum_{j = 1}^{k} \frac{L_{j}^{m - 1}}{j}, \end{matrix}$ where $L_{k}^{0} = u_{k}$ . We say that sequence (u_k) is statistically (L, m) summable to a ∈ N if sequence $(L_{k}^{m})$ is statistically convergent to a.

Example 3.9. Consider the vector space C [0, 1] equipped with the norm $‖ f ‖ = max_{x \in [0, 1]} | f (x) |$ . Let $\begin{matrix} f_{k} (x) = \\ {\begin{matrix} (- x)^{k} k^{2} + k^{2} {(ℓ_{k})}^{3}, & k = n^{2} \\ (- x)^{k} k^{2} - k {(ℓ_{k - 1})}^{2} ((k - 1) ℓ_{k - 1} + k ℓ_{k}), & k = n^{2} + 1 \\ (- x)^{k} k^{2} + k (k - 1) ℓ_{k - 1} {(ℓ_{k - 2})}^{2}, & k = n^{2} + 2 \\ (- x)^{k} k^{2}, & otherwise \end{matrix} \end{matrix}$ be in IF-normed space (C [0, 1] , μ₀, ν₀) where $ℓ_{k} = \sum_{j = 1}^{k} 1 / j$ . Sequence (f_k) is neither convergent nor statistically convergent in (C [0, 1] , μ₀, ν₀). Besides, (f_k) is neither statistical Cesàro summable nor statistical logarithmic summable.

Let us apply statistical (L, 2) summability to achieve a limit. Logarithmic means $(L_{k}^{1})$ and $(L_{k}^{2})$ of sequence (f_k) is $\begin{matrix} L_{k}^{1} (x) = {\begin{matrix} τ_{k}^{(1)} (x) + k (ℓ_{k})^{2}, & k = n^{2} \\ τ_{k}^{(1)} (x) - k (ℓ_{k - 1})^{2}, & k = n^{2} + 1 \\ τ_{k}^{(1)} (x), & otherwise, \end{matrix} \end{matrix}$ $\begin{matrix} L_{k}^{2} (x) = {\begin{matrix} τ_{k}^{(2)} (x) + ℓ_{k}, & k = n^{2} \\ τ_{k}^{(2)} (x), & otherwise \end{matrix} \end{matrix}$ where sequence ${τ_{k}^{(m)}}$ denotes m-fold logarithmic means of sequence { (- x) ^kk²}. Hence, (f_k) is statistically (L, 2) summable to 0 since for each t > 0 we have $st - lim μ_{0} (L_{k}^{2}, t) = 1$ and $st - lim ν_{0} (L_{k}^{2}, t) = 0$ where $\begin{matrix} μ_{0} (L_{k}^{2}, t) = {\begin{matrix} \frac{t}{t + ‖ τ_{k}^{(2)} (x) + ℓ_{k} ‖}, & k = n^{2} \\ \frac{t}{t + ‖ τ_{k}^{(2)} (x) ‖}, & otherwise, \end{matrix} \\ ν_{0} (L_{k}^{2}, t) = {\begin{matrix} \frac{‖ τ_{k}^{(2)} (x) ‖}{t + ‖ τ_{k}^{(2)} (x) + ℓ_{k} ‖}, & k = n^{2} \\ \frac{‖ τ_{k}^{(2)} (x) ‖}{t + ‖ τ_{k}^{(2)} (x) ‖}, & otherwise . \end{matrix} \end{matrix}$

In view of Theorem 1.13, Theorem 3.5 and Theorem 3.6 we get following Tauberian theorem.

Theorem 3.10. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically (L, m) summable to some a ∈ N and slowly oscillating with respect to logarithmic summability, then (u_k) is convergent to a.

Also, in view of theorem above and Theorem 1.11 we get following theorem.

Theorem 3.11. Let sequence (u_k) be in (N, μ, ν). If (u_k) is statistically (L, m) summable to some a ∈ N and {k ln k (u_k - u_k-1)} is q-bounded, then (u_k) is convergent to a.

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