Statistical Cesàro Summability in Intuitionistic Fuzzy n-Normed Linear Spaces Leading towards Tauberian Theorems

Abstract

The concept of summability is crucial in deriving formal solutions to partial differential equations. This paper explores the connection between the methods of statistical convergence of sequences and statistical Cesàro summability in intuitionistic fuzzy n-normed linear space (IFnNLS). While the existing literature covers Cesàro summability and its statistical variant in fuzzy, intuitionistic fuzzy, and classical normed spaces, this study stands out not only for its methodology but also for its comprehensive approach, encompassing a broader range of spaces and detailing the pathway from the statistical Cesàro summability method to statistical convergence. These results will lead us to Tauberian theorems in IFnNLS.

1. Introduction

Zadeh’s concept of fuzzy set theory [1] has been applied across diverse mathematical areas, including approximation theory [2], the theory of functions [3,4], and metric and topological spaces [5,6,7]. Moreover, this theory finds extensive use in quantum physics [8], chaos control [9], computer programming [10], population dynamics [11], and nonlinear dynamical systems [12].

Katsaras [13] originally introduced the concept of a fuzzy norm, which has since been refined by various authors from different perspectives [7,8,14,15,16]. The idea of an IFnNLS [17] naturally extends the intuitionistic fuzzy normed space introduced by Saadati and Park [18].

In this paper, our objective is to put forward the idea of statistical summability theory within an IFnNLS. To achieve this goal, we introduce the concepts of Cesàro and statistical Cesàro summability. These concepts pave the way for future investigations into associated Tauberian theorems in an IFnNLS context.

Our findings rely on the definition of sequence convergence in IFnNLS, for which a new and precise definition was proposed in prior works by Debnath and Sen [19,20]. The current results are built upon this refined definition.

For classical counterparts of the results discussed here, we refer to works such as Asama et al. [21], Dutta and Rhoades [22], Talo and Yavuz [23], and their respective literature reviews.

Very recently, some significant Tauberian theorems have been studied by Onder et al. [24] and Debnath [25].

Existing Research Gaps and Novelty of the Current Work

One key contribution of this study is that it will help us to establish Tauberian conditions that facilitate the transition from statistical Cesàro summability to statistical convergence of sequences within the framework of IFnN ${(\eta ,\gamma )}^n$. This research introduces novel techniques for proving associated theorems, which we hope will complement investigators in this field by both methodological approach and content.

The primary contributions of this research paper are to address the following inquiries:

1. What constitutes the sufficient and necessary condition for statistical Cesàro summability in regard to the IFnN ${(\eta ,\gamma )}^n$?

2. How can subsets of the sequence space included in an IFnNLS be identified such that a sequence statistically Cesàro summable with respect to the IFnN also converges in the same manner?

3. Can a proof be provided for the properties outlined in (1) and (2) regarding these related subsets?

These questions underscore the study’s focus on establishing rigorous conditions and techniques within summability theory, particularly concerning statistical Cesàro summability and convergence in IFnNLS.

2. Preliminaries

Researchers have already proven that a fuzzy metric and an intuitionistic fuzzy metric produce identical topologies [26]. In pursuit of significant and innovative results, mathematicians have slightly adjusted the definition of an intuitionistic fuzzy norm [27,28]. Building on this progress, Debnath and Sen have proposed a modified definition of an IFnNLS as follows [29,30]:

Definition 1.

The five-tuple $(V,\eta ,\gamma ,\ast ,\circ )$ is referred to as an IFnNLS, where V represents a vector space of dimension $d\ge n$ over the field $\mathbb{R}$ (the field of reals), ∗ denotes a continuous t-norm, ∘ denotes a continuous t-conorm, and η and γ are fuzzy sets defined on $V^n\times (0,\infty )$. In this context, η signifies the degree of membership and γ denotes the degree of non-membership for elements $(u_1,u_2,\dots ,u_n,t)\in V^n\times (0,\infty )$. The following conditions hold for every $(u_1,u_2,\dots ,u_n)\in V^n$ and $s,t>0$:

(i) 

$\eta (u_1,u_2,\dots ,u_n,r)+\gamma (u_1,u_2,\dots ,u_n,r)\le 1$;

(ii) 

$\eta (u_1,u_2,\dots ,u_n,r)=1$ and $\gamma (u_1,u_2,\dots ,u_n,r)=0$ for all positive r if and only if $u_1,u_2,\dots ,u_n$ are linearly dependent;

(iii) 

$\eta (u_1,u_2,\dots ,u_n,r)$ and $\gamma (u_1,u_2,\dots ,u_n,r)$ are invariant under any permutation of $u_1,u_2,\dots ,u_n$;

(iv) 

$\eta (u_1,u_2,\dots ,c{u}_n,r)=\eta (u_1,u_2,\dots ,u_n,{\displaystyle \frac{r}{|c|}})$ and $\gamma (u_1,u_2,\dots ,c{u}_n,r)=\gamma (u_1,u_2,\dots ,u_n,{\displaystyle \frac{r}{|c|}})$ if $c\ne 0,c\in F$;

(v) 

$\eta (u_1,u_2,\dots ,u_n,s)\ast \eta (u_1,u_2,\dots ,u_n^{{ }^{\prime }},r)\le \eta (u_1,u_2,\dots ,u_n+u_n^{{ }^{\prime }},s+r)$;

(vi) 

$\gamma (u_1,u_2,\dots ,u_n,s)\circ \gamma (u_1,u_2,\dots ,u_n^{{ }^{\prime }},r)\ge \gamma (u_1,u_2,\dots ,u_n+u_n^{{ }^{\prime }},s+r)$;

(vii) 

$\eta (u_1,u_2,\dots ,u_n,r):(0,\infty )\to [0,1]$ and $\gamma (u_1,u_2,\dots ,u_n,r):(0,\infty )\to [0,1]$ are continuous in r;

(viii) 

${lim}_{r\to \infty }\eta (u_1,u_2,\dots ,u_n,r)=1$ and ${lim}_{r\to 0}\eta (u_1,u_2,\dots ,u_n,r)=0$;

(ix) 

${lim}_{r\to \infty }\gamma (u_1,u_2,\dots ,u_n,r)=0$ and ${lim}_{r\to 0}\gamma (u_1,u_2,\dots ,u_n,r)=1$.

Definition 2

([19,20]). Let $(V,\eta ,\gamma ,\ast ,\circ )$ be an IFnNLS. A sequence $v=v_k$ in V is considered convergent to $\varsigma \in V$ under the intuitionistic fuzzy n-norm (IFnN) ${(\eta ,\gamma )}^n$ if, for every $ϵ\in (0,1)$, $r>0$, and $u_1,u_2,\dots ,u_{n-1}\in V$, there exists a natural number $k_0$ such that $\eta (u_1,u_2,\dots ,u_{n-1},v_k-\varsigma ,r)>1-ϵ$ and $\gamma (u_1,u_2,\dots ,u_{n-1},v_k-\varsigma ,r)<ϵ$ for all $k\ge k_0$. This convergence is denoted by ${(\eta ,\gamma )}^n-limv=\varsigma$ or $v_k\stackrel{{(\eta ,\gamma )}^n}{\to }\varsigma$ as $k\to \infty$.

Proposition 1

([31]). In an IFnNLS V, ${(\eta ,\gamma )}^n-limv=\varsigma$ if and only if for every $r>0$ and $u_1,u_2,\dots ,u_{n-1}\in V$, the conditions $\eta (u_1,\dots ,u_{n-1},v_k-\varsigma ,r)\to 1$ and $\gamma (u_1,\dots ,u_{n-1},v_k-\varsigma ,r)\to 0$ hold as $k\to \infty$.

Definition 3

([19,20]). Let $(V,\eta ,\gamma ,\ast ,\circ )$ be an IFnNLS. A sequence $v=v_k$ in V is defined to be Cauchy with respect to the IFnN ${(\eta ,\gamma )}^n$ if, for every $ϵ\in (0,1)$, $r>0$ and $u_1,u_2,\dots ,u_{n-1}\in V$, there exists a natural number $k_0$ such that $\eta (u_1,u_2,\dots ,u_{n-1},v_k-v_m,r)>1-ϵ$ and $\gamma (u_1,u_2,\dots ,u_{n-1},v_k-v_m,r)<ϵ$ for all $k,m\ge k_0$.

Definition 4.

An IFnNLS V is said to be complete with respect to the IFnN ${(\eta ,\gamma )}^n$ if every Cauchy sequence in V converges.

Following Efe and Alaca [32], bounded sets in the context of IFnNLS are defined below.

Definition 5.

Let $(V,\eta ,\gamma ,\ast ,\circ )$ be an IFnNLS and B be any subset of V. The set B is said to be bounded if there exist $ϵ>0$ and $r_0>0$ such that $\eta (u_1,u_2,\dots ,u_n,r_0)>1-ϵ$ and $\gamma (u_1,u_2,\dots ,u_n,r_0)<ϵ$ for all $u_1,u_2,\dots ,u_n\in B$.

The set B is said to be p-bounded if ${lim}_{r\to \infty }{\Phi }_B\left(r\right)=1$ and ${lim}_{r\to \infty }{\Psi }_B\left(r\right)=0$, where

$${\Phi }_B\left(r\right)=inf\{\eta (u_1,u_2,\dots ,u_n,r):u_1,u_2,\dots ,u_n\in B\};$$

$${\Psi }_B\left(r\right)=sup\{\gamma (u_1,u_2,\dots ,u_n,r):u_1,u_2,\dots ,u_n\in B\}.$$

The next few definitions are related to the concept of statistical convergence.

Definition 6

([33]). Let P be a subset of $\mathbb{N}$. The natural density of P is defined by

$$\begin{array}{c}\hfill \delta \left(P\right)=\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\left|\{k\le n:k\in P\}\right|\end{array}$$

whenever the limit exists, where $\left|P\right|$ signifies the cardinality of the set P.

Definition 7

([33]). A sequence $x=\left\{x_k\right\}$ of numbers is considered statistically convergent to the number l if, for every $ϵ>0$,

$$\delta \left(\right\{k\in \mathbb{N}:|x_k-l|\ge ϵ\})=0.$$

In such instances, we denote this as $S-limx=l$.

Definition 8

([33]). The upper density of a subset P of the natural numbers $\mathbb{N}$ is defined as

$$\begin{array}{c}\hfill \overline{\delta }\left(P\right)=\underset{n}{lim\; sup}{\displaystyle \frac{1}{n}}\left|\{k\le n:k\in P\}\right|.\end{array}$$

Definition 9

([20]). Let $(X,\mu ,\nu ,\ast ,\circ )$ be an IFnNLS. A sequence $x=\left\{x_k\right\}$ in X is considered statistically convergent to $L\in X$ in regard to the IFnN ${(\mu ,\nu )}^n$ if, for every $ϵ\in (0,1)$, $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in X$,

$$\begin{array}{cc}\delta \left(\right\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)\le 1-ϵ\hfill & \\ & \hfill or \nu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)\ge ϵ\left\}\right)=0.\end{array}$$

It is denoted by $S_{{(\mu ,\nu )}^n}-limx=L$.

From some established properties of the natural density and Definition 9, the following lemma were obtained by Debnath and Sen [20].

Lemma 1.

Let $(X,\mu ,\nu ,\ast ,\circ )$ be an IFnNLS. Then, for every $ϵ>0$, $t>0$ and $y_1,y_2,\dots ,y_{n-1}\in X$, the following statements are equivalent:

(i) 

$S_{{(\mu ,\nu )}^n}-lim{x}_k=L$.

(ii) 

$\delta \left(\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)\le 1-ϵ\}\right)=\delta \left(\{k\in \mathbb{N}:\nu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)\ge ϵ\}\right)=0$.

(iii) 

$\delta \left(\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)>1-ϵ and \nu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)<ϵ\}\right)=1$.

(iv) 

$\delta \left(\{k\in \mathbb{N}:\mu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)>1-ϵ\}\right)=\delta \left(\{k\in \mathbb{N}:\nu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)<ϵ\}\right)=1$.

(v) 

$S-lim\mu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)=1andS-lim\nu (y_1,y_2,\dots ,y_{n-1},x_k-L,t)=0$.

The following lemmas will be used in the sequel.

Lemma 2

([34]). For every $\delta >0$, we define $<\delta >=\delta -\left[\delta \right]$, where $[·]$ denotes the greatest integer function. The following are true:

(i) 

If $\delta >1$, then ${\delta }_n>n$ for each $n\in \mathbb{N}\setminus \left\{0\right\}$ with $n\ge {\displaystyle \frac{1}{<\delta >}}$.

(ii) 

If $0<\delta <1$, then ${\delta }_n<n$ for each $n\in \mathbb{N}\setminus \left\{0\right\}$, where ${\delta }_n=\left[n\delta \right]$.

Lemma 3

([34]). The following statements are true:

(i) 

If $\delta >1$, then for each $n\in \mathbb{N}\setminus \left\{0\right\}$ with $n\ge {\displaystyle \frac{3\delta -1}{\delta (\delta -1)}}$, we have

${\displaystyle \frac{\delta }{\delta -1}}<{\displaystyle \frac{{\delta }_n+1}{{\delta }_n-n}}<{\displaystyle \frac{2\delta }{\delta -1}}$.

(ii) 

If $0<\delta <1$, then for each $n\in \mathbb{N}\setminus \left\{0\right\}$ with $n>{\displaystyle \frac{1}{\delta }}$, we have

$0<{\displaystyle \frac{{\delta }_n+1}{n-{\delta }_n}}<{\displaystyle \frac{2\delta }{1-\delta }}$.

3. Statistical Cesàro summability in IFnNLS

First, we introduce the notion of Cesàro summability.

Definition 10

([25]). Let $\left\{v_n\right\}$ be a sequence in an IFnNLS $(V,\eta ,\gamma ,\ast ,\circ )$. The arithmetic means ${\chi }_n$ of $v_n$ are defined as

$${\chi }_n={\displaystyle \frac{1}{n+1}}\sum _{k=0}^n{v}_k.$$

$\left\{v_n\right\}$ is said to be Cesàro summable to $v\in V$ if ${(\eta ,\gamma )}^n-{lim}_{m\to \infty }{\chi }_m=v$.

Further, $\left\{v_n\right\}$ is said to be statistically Cesàro summable to $v\in V$ if $S_{{(\eta ,\gamma )}^n}-{lim}_{m\to \infty }{\chi }_m=v$.

The next theorem indicates the regularity of the statistical Cesàro summability method in an IFnNLS under p-boundedness of sequence.

Theorem 1.

Let $\left\{v_n\right\}$ be a p-bounded sequence in an IFnNLS $(V,\eta ,\gamma ,\ast ,\circ )$. If $\left\{v_n\right\}$ converges statistically to $v\in V$, then $\left\{v_n\right\}$ is also statistically Cesàro summable to v with respect to IFnN ${(\eta ,\gamma )}^n$.

Proof. 

Let $\left\{v_n\right\}$ converges statistically to $v\in V$ and consider it to be p-bounded.

Fix $u_1,u_2,\dots ,u_{n-1}\in V$. Then, for a given $ϵ>0$, there exist $M,M^{\prime }>0$ such that

$$\begin{array}{cc}\underset{n\in \mathbb{N}}{inf}\eta (u_1,u_2,\dots ,u_{n-1},v_n,r)>1-ϵ\mathrm{and}\hfill & \\ & \hfill \underset{n\in \mathbb{N}}{sup}\gamma (u_1,u_2,\dots ,u_{n-1},v_n,r)<ϵ,\mathrm{for}\mathrm{all}r>2M.\end{array}$$

Clearly, ${inf}_{n\in \mathbb{N}}\eta (u_1,u_2,\dots ,u_{n-1},v,{\displaystyle \frac{r}{2}})>1-ϵ$ and ${sup}_{n\in \mathbb{N}}\gamma (u_1,u_2,\dots ,u_{n-1},v,{\displaystyle \frac{r}{2}})<ϵ$ for all $r>2M^{\prime }$. This, in turn, implies the following inequalities:

$$\begin{array}{cc}\hfill \underset{n\in \mathbb{N}}{inf}& \eta (u_1,u_2,\dots ,u_{n-1},v_n-v,r)\hfill \\ \hfill & \ge min\{\underset{n\in \mathbb{N}}{inf}\eta (u_1,u_2,\dots ,u_{n-1},v_n,{\displaystyle \frac{r}{2}}),\underset{n\in \mathbb{N}}{inf}\eta (u_1,u_2,\dots ,u_{n-1},v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & >1-ϵ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \underset{n\in \mathbb{N}}{sup}& \gamma (u_1,u_2,\dots ,u_{n-1},v_n-v,r)\hfill \\ \hfill & \le max\{\underset{n\in \mathbb{N}}{sup}\gamma (u_1,u_2,\dots ,u_{n-1},v_n,{\displaystyle \frac{r}{2}}),\underset{n\in \mathbb{N}}{sup}\gamma (u_1,u_2,\dots ,u_{n-1},v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & <ϵ\hfill \end{array}$$

for all $r>min\{2M,2M^{\prime }\}$.

Since $v_n$ is statistically convergent to v, using Lemma 1, we have that

$$\delta \left(N_{\eta }(ϵ,r)\right)=\delta \left(N_{\gamma }(ϵ,r)\right)=0$$

for all $r>0$, where

$$N_{\eta }(ϵ,r)=\{n\in \mathbb{N}:\eta (u_1,u_2,\dots ,u_{n-1},v_n-v,r)\le 1-ϵ\}$$

and

$$N_{\gamma }(ϵ,r)=\{n\in \mathbb{N}:\gamma (u_1,u_2,\dots ,u_{n-1},v_n-v,r)\ge ϵ\}.$$

Define the sets $B=\{k\in \mathbb{N}:k\in N_{\eta }(ϵ,r)\}$, $C=\{k\in \mathbb{N}:k\in N_{\eta }^c(ϵ,r)\}$, and $B^{\prime }=\{k\in \mathbb{N}:k\in N_{\gamma }(ϵ,r)\}$, $C^{\prime }=\{k\in \mathbb{N}:k\in N_{\gamma }^c(ϵ,r)\}$ such that $|B|+|C|=n+1=|B^{\prime }|+|C^{\prime }|$, where $|·|$ denotes the cardinality of a set.

Thus, we may conclude that $B\cap C=\varphi =B^{\prime }\cap C^{\prime }$.

In view of the above information, we conclude that there exists a number $n_0\in \mathbb{N}$ such that

$$\begin{array}{cc}\hfill \eta (u_1& ,u_2,\dots ,u_{n-1},{\chi }_n-v,r)\hfill \\ \hfill & =\eta (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{1}{n+1}}\sum _{k=0}^n(v_k-v),r)\hfill \\ \hfill & =\eta (u_1,u_2,\dots ,u_{n-1},\sum _{k\in N_{\eta }}(v_k-v)+\sum _{k\in N_{\eta }^c}(v_k-v),(n+1)r)\hfill \\ \hfill & \ge min\left\{\eta \right(u_1,u_2,\dots ,u_{n-1},\sum _{k\in N_{\eta }}(v_k-v),|B|r),\eta (u_1,u_2,\dots ,u_{n-1},\sum _{k\in N_{\eta }^c}(v_k-v),|C|r\left)\right\}\hfill \\ \hfill & \ge min\{\underset{k\in N_{\eta }}{min}\eta (u_1,u_2,\dots ,u_{n-1},(v_k-v),r),\underset{k\in N_{\eta }^c}{min}\eta (u_1,u_2,\dots ,u_{n-1},(v_k-v),r)\}\hfill \\ \hfill & \ge min\{\underset{k\in N_{\eta }}{inf}\eta (u_1,u_2,\dots ,u_{n-1},(v_k-v),r),\underset{k\in N_{\eta }^c}{min}\eta (u_1,u_2,\dots ,u_{n-1},(v_k-v),r)\}\hfill \\ \hfill & \ge min\{1-ϵ,1-ϵ\}\hfill \\ \hfill & =1-ϵ\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \gamma (u_1,& u_2,\dots ,u_{n-1},{\chi }_n-v,r)\hfill \\ \hfill & \le max\{\underset{k\in N_{\eta }}{max}\eta (u_1,u_2,\dots ,u_{n-1},(v_k-v),r),\underset{k\in N_{\eta }^c}{max}\eta (u_1,u_2,\dots ,u_{n-1},(v_k-v),r)\}\hfill \\ \hfill & \le max\{\underset{k\in N_{\eta }}{sup}\eta (u_1,u_2,\dots ,u_{n-1},(v_k-v),r),\underset{k\in N_{\eta }^c}{max}\eta (u_1,u_2,\dots ,u_{n-1},(v_k-v),r)\}\hfill \\ \hfill & \le ϵ\hfill \end{array}$$

for all $r>min\{2M,2M^{\prime }\}>0$ and $n\ge n_0$. This implies that the set

$$S=\{n\in \mathbb{N}:\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,r)\le 1-ϵ\mathrm{or}\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,r)\ge ϵ\}$$

contains finitely many terms at the maximum. Because a finite subset of the natural numbers has zero density, we observe that $\delta \left(S\right)=0$, which indicates that the sequence $v_n$ is statistically Cesàro summable to v in regard to the IFnN ${(\eta ,\gamma )}^n$. □

Our next example exhibits that the converse of Theorem 1 need not hold true. This example is an improvement over the one constructed by Talo and Yavuz [23].

Example 1.

Let $V={\mathbb{R}}^n$ with

$$\parallel u_1,u_2,\dots ,u_n\parallel =abs\left(\left|\begin{array}{ccc}{u}_{11}& \cdots & u_{1n}\\ ⋮& \ddots & ⋮\\ u_{n1}& \cdots & u_{nn}\end{array}\right|\right),$$

where $u_i=(u_{i1},u_{i2},\dots ,u_{in})\in {\mathbb{R}}^n$ for each $i=1,2,\dots ,n$ and let $a\ast b=ab$, $a\circ b=min\{a+b,1\}$ for all $a,b\in [0,1]$. Now, for all $w_1,w_2,\dots ,w_n\in {\mathbb{R}}^n$ and $r>0$, let us define $\eta (w_1,w_2,\dots ,w_n,r)={\displaystyle \frac{r}{r+\parallel w_1,w_2,\dots ,w_n\parallel }}$ and $\gamma (w_1,w_2,\dots ,w_n,r)={\displaystyle \frac{\parallel w_1,w_2,\dots ,w_n\parallel }{r+\parallel w_1,w_2,\dots ,w_n\parallel }}$. Then $({\mathbb{R}}^n,\eta ,\gamma ,\ast ,\circ )$ is an IFnNLS.

Consider the sequence $\left\{v_k\right\}=(z_k,0,0,\dots ,0)\in {\mathbb{R}}^n$, where

$$z_k=\left\{\begin{array}{c}1+{(-1)}^k+k^2,\mathrm{if}k=m^2\hfill \\ 1+{(-1)}^k-{(k-1)}^2,\mathrm{if}k=m^2+1\hfill \\ 1+{(-1)}^k,\mathrm{otherwise},\hfill \end{array}\right.$$

for $m\in \mathbb{N}$.

It can be noted that $v_k$ statistically Cesáro sums to 1 under the IFnN ${(\eta ,\gamma )}^n$, but it is neither p-bounded nor statistically convergent under the same IFnN ${(\eta ,\gamma )}^n$.

4. Additional Results Leading to Tauberian Theorems

The next lemma proves the additivity and homogeneity of the statistical limit in an IFnNLS. We omit the proof of the same as it can be obtained in an exactly similar manner as in [19,25,35,36].

Lemma 4.

Let $(V,\eta ,\gamma ,\ast ,\circ )$ be an IFnNLS and $x=\left\{x_k\right\},y=\left\{y_k\right\}$ be sequences in V. Then, the following are true:

(i) 

If the ${(\eta ,\gamma )}^n$-statistical limit of x is ξ, and the ${(\eta ,\gamma )}^n$-statistical limit of y is ρ, then the ${(\eta ,\gamma )}^n$-statistical limit of the sum $(x+y)$ is $\xi +\rho$.

(ii) 

If the ${(\eta ,\gamma )}^n$-statistical limit of x is ξ, and α is any real number, then the ${(\eta ,\gamma )}^n$-statistical limit of $\alpha x$ is $\alpha \xi$.

Theorem 2.

Let $(V,\eta ,\gamma ,\ast ,\circ )$ be an IFnNLS and $\left\{v_n\right\}$ be a sequence in V. If $\left\{v_n\right\}$ is a statistically Cesàro summable to v with respect to IFnN ${(\eta ,\gamma )}^n$, then ${\chi }_{{\lambda }_n}$ is statistically convergent to v for each $\lambda >0$, i.e.,

$$S_{{(\mu ,\nu )}^n}-\underset{n\to \infty }{lim}{\chi }_{{\lambda }_n}=v,$$

where ${\lambda }_n=\left[\lambda n\right]$ such that $[·]$ is the greatest integer function.

Proof. 

Suppose that $S_{{(\mu ,\nu )}^n}-{lim}_{n\to \infty }{\chi }_n=v$. Then, for a sufficiently large N, given $ϵ>0$ and fixed $u_1,u_2,\dots ,u_{n-1}\in V$, define the following sets:

$$\begin{array}{c}\hfill K_{\eta ,\chi }(ϵ,r)=\{k\le {\lambda }_N:\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_k-v,r)\le 1-ϵ\},\end{array}$$

$$\begin{array}{c}\hfill K_{\gamma ,\chi }(ϵ,r)=\{k\le {\lambda }_N:\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_k-v,r)\ge ϵ\},\end{array}$$

$$\begin{array}{c}\hfill K_{\eta ,{\chi }_{\lambda }}(ϵ,r)=\{k\le {\lambda }_N:\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_k}-v,r)\le 1-ϵ\},\end{array}$$

$$\begin{array}{c}\hfill K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)=\{k\le {\lambda }_N:\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_k}-v,r)\ge ϵ\}.\end{array}$$

Now, we discuss the following cases.

Case I: $\lambda >1$.

It is easy to see that $K_{\eta ,{\chi }_{\lambda }}(ϵ,r)\subseteq K_{\eta ,\chi }(ϵ,r)$ and $K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)\subseteq K_{\gamma ,\chi }(ϵ,r)$ for any $r>0$. This implies the following:

$$\begin{array}{cc}\hfill {\displaystyle \frac{|K_{\eta ,{\chi }_{\lambda }}(ϵ,r)|}{N+1}}& ={\displaystyle \frac{\lambda |K_{\eta ,{\chi }_{\lambda }}(ϵ,r)|}{\lambda N+\lambda }}\hfill \\ \hfill & \le {\displaystyle \frac{\lambda |K_{\eta ,{\chi }_{\lambda }}(ϵ,r)|}{{\lambda }_N+1}}\hfill \\ \hfill & \le {\displaystyle \frac{\lambda |K_{\eta ,\chi }(ϵ,r)|}{{\lambda }_N+1}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{|K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)|}{N+1}}& ={\displaystyle \frac{\lambda |K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)|}{\lambda N+\lambda }}\hfill \\ \hfill & \le {\displaystyle \frac{\lambda |K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)|}{{\lambda }_N+1}}\hfill \\ \hfill & \le {\displaystyle \frac{\lambda |K_{\gamma ,\chi }(ϵ,r)|}{{\lambda }_N+1}}.\hfill \end{array}$$

Using the above inequalities, respectively, we can determine that

$$\delta \left(K_{\eta ,{\chi }_{\lambda }}(ϵ,r)\right)\le \lambda \delta \left(K_{\eta ,\chi }(ϵ,r)\right)$$

and

$$\delta \left(K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)\right)\le \lambda \delta \left(K_{\gamma ,\chi }(ϵ,r)\right).$$

Since by hypothesis, $\left\{{\chi }_n\right\}$ is statistically convergent to $v\in V$, we obtain from Lemma 1 that

$$\delta \left(K_{\eta ,\chi }(ϵ,r)\right)=\delta \left(K_{\gamma ,\chi }(ϵ,r)\right)=0$$

for any $r>0$. Thus, for any $r>0$, we have that

$$\delta \left(K_{\eta ,{\chi }_{\lambda }}(ϵ,r)\right)=\delta \left(K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)\right)=0.$$

Therefore, using Lemma 1, we can prove that $S_{{(\mu ,\nu )}^n}-{lim}_{n\to \infty }{\chi }_{{\lambda }_n}=v$.

Case II: $\lambda \in (0,1)$.

To complete the proof, first, we prove that the term ${\chi }_n$ does not appear more than $1+{\displaystyle \frac{1}{\lambda }}$ times in the sequence ${\chi }_{{\lambda }_n}$. Suppose that for some $i,j\in \mathbb{N}$, we have

$$n={\lambda }_i={\lambda }_{i+1}=\cdots ={\lambda }_{i+j-1}<{\lambda }_{i+j},$$

or equivalently,

$$n\le \lambda i<\lambda (i+1)<\cdots <\lambda (i+j-1)<n+1\le \lambda (i+j).$$

So, we have

$$n+\lambda (j-1)<\lambda i+\lambda (j-1)=\lambda (i+j-1)<n+1,$$

which means that $\lambda (j-1)<1$, i.e., $j<1+{\displaystyle \frac{1}{\lambda }}$. From this point of view, we obtain for each $ϵ>0$ and $r>0$ that

$$\begin{array}{cc}\hfill {\displaystyle \frac{|K_{\eta ,{\chi }_{\lambda }}(ϵ,r)|}{N+1}}& \le (1+{\displaystyle \frac{1}{\lambda }}){\displaystyle \frac{{\lambda }_N+1}{N+1}}{\displaystyle \frac{|K_{\eta ,\chi }(ϵ,r)|}{{\lambda }_N+1}}\hfill \\ \hfill & \le 2(\lambda +1){\displaystyle \frac{|K_{\eta ,\chi }(ϵ,r)|}{{\lambda }_N+1}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{|K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)|}{N+1}}& \le (1+{\displaystyle \frac{1}{\lambda }}){\displaystyle \frac{{\lambda }_N+1}{N+1}}{\displaystyle \frac{|K_{\gamma ,\chi }(ϵ,r)|}{{\lambda }_N+1}}\hfill \\ \hfill & \le 2(\lambda +1){\displaystyle \frac{|K_{\gamma ,\chi }(ϵ,r)|}{{\lambda }_N+1}}\hfill \end{array}$$

for which N is sufficiently large, such that ${\displaystyle \frac{({\lambda }_n+1)}{N+1}}\le 2\lambda$.

These consequently imply that

$$\delta \left(K_{\eta ,{\chi }_{\lambda }}(ϵ,r)\right)\le 2(\lambda +1)\delta \left(K_{\eta ,\chi }(ϵ,r)\right),$$

$$\delta \left(K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)\right)\le 2(\lambda +1)\delta \left(K_{\gamma ,\chi }(ϵ,r)\right),$$

respectively.

Since $\left\{{\chi }_n\right\}$ is statistically convergent to v, we obtain from Lemma 1 that

$$\delta \left(K_{\eta ,\chi }(ϵ,r)\right)=\delta \left(K_{\gamma ,\chi }(ϵ,r)\right)=0$$

for any $r>0$. Thus, for any $r>0$, we have that

$$\delta \left(K_{\eta ,{\chi }_{\lambda }}(ϵ,r)\right)=\delta \left(K_{\gamma ,{\chi }_{\lambda }}(ϵ,r)\right)=0.$$

Therefore, using Lemma 1, we have proven that $S_{{(\mu ,\nu )}^n}-{lim}_{n\to \infty }{\chi }_{{\lambda }_n}=v$ in this case as well. □

Theorem 3.

Let $(V,\eta ,\gamma ,\ast ,\circ )$ be an IFnNLS and $\left\{v_n\right\}$ be a sequence in V. If $\left\{v_n\right\}$ is a statistically Cesàro summable to v with respect to IFnN ${(\eta ,\gamma )}^n$. Then,

$$S_{{(\mu ,\nu )}^n}-\underset{n\to \infty }{lim}{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}{v}_k=v,$$

for each $\lambda >1$ and

$$S_{{(\mu ,\nu )}^n}-\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^n{v}_k=v,$$

for each $0<\lambda <1$.

Proof. 

Assume that $S_{{(\mu ,\nu )}^n}-{lim}_{n\to \infty }{\chi }_n=v$. For a given $ϵ>0$, choose $t_1,t_2>0$ such that $min\{1-t_1,1-t_2\}>1-ϵ$ and $max\{r_1,r_2\}<ϵ$. Then, for sufficiently large N and any $r>0$, define the following sets:

$$\begin{array}{c}\hfill K_{\eta ,\chi }(t_1,r)=\{k\le N:\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_k-v,r)\le 1-t_1\},\end{array}$$

$$\begin{array}{c}\hfill K_{\gamma ,\chi }(t_1,r)=\{k\le N:\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_k-v,r)\ge t_1\},\end{array}$$

$$\begin{array}{c}\hfill K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r)=\{k\le N:\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_k}-{\chi }_k,r)\le 1-r_2\},\end{array}$$

$$\begin{array}{c}\hfill K_{\gamma ,\chi ,{\chi }_{\lambda }}(t_2,r)=\{k\le N:\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_k}-{\chi }_k,r)\ge r_2\},\end{array}$$

Now, we discuss the following cases.

Case I: $\lambda >1$. For given $ϵ>0$ and any $r>0$, define the following sets:

$$\begin{array}{c}\hfill K_{\eta ,\tau }(ϵ,r)=\{k\le N:\eta (u_1,u_2,\dots ,u_{n-1},{\tau }_n\left(w\right)-v,r)\le 1-ϵ\},\end{array}$$

$$\begin{array}{c}\hfill K_{\gamma ,\tau }(ϵ,r)=\{k\le N:\gamma (u_1,u_2,\dots ,u_{n-1},{\tau }_n\left(w\right)-v,r)\ge ϵ\},\end{array}$$

where ${\tau }_n\left(w\right)={\displaystyle \frac{1}{{\lambda }_n-n}}{\sum }_{k=n+1}^{{\lambda }_n}{v}_k$ for all $n\in \mathbb{N}$.

For any $\lambda >1$ and sufficiently large $n\in \mathbb{N}\setminus \left\{0\right\}$ such that $n<{\lambda }_n$ with $n\ge {\displaystyle \frac{3\lambda -1}{\lambda (\lambda -1)}}$, we obtain from Lemma 3 for any $r>0$ and $u_1,u_2,\dots ,u_{n-1}\in V$,

$$\begin{array}{cc}\hfill \eta & (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}{v}_k-v,r)\hfill \\ \hfill & =\eta (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}{\displaystyle \frac{1}{{\lambda }_n+1}}\sum _{k=0}^{{\lambda }_n}{v}_k-{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=0}^n{v}_k-v,r)\hfill \\ \hfill & =\eta (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}{\chi }_{{\lambda }_n}-{\displaystyle \frac{{\lambda }_n+1-{\lambda }_n+n}{{\lambda }_n-n}}{\chi }_n-v,r)\hfill \\ \hfill & \ge min\{\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,{\displaystyle \frac{r}{2\frac{{\lambda }_n+1}{{\lambda }_n-n}}}),\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & \ge min\{\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,{\displaystyle \frac{(\lambda -1)r}{4\lambda }}),\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & =min\{\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,r_0),\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & >min\{1-t_2,1-t_1\}\hfill \\ \hfill & >1-ϵ\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \gamma & (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}{v}_k-v,r)\hfill \\ \hfill & =\gamma (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}{\displaystyle \frac{1}{{\lambda }_n+1}}\sum _{k=0}^{{\lambda }_n}{v}_k-{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=0}^n{v}_k-v,r)\hfill \\ \hfill & =\gamma (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{{\lambda }_n+1}{{\lambda }_n-n}}{\chi }_{{\lambda }_n}-{\displaystyle \frac{{\lambda }_n+1-{\lambda }_n+n}{{\lambda }_n-n}}{\chi }_n-v,r)\hfill \\ \hfill & \le max\{\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,{\displaystyle \frac{r}{2\frac{{\lambda }_n+1}{{\lambda }_n-n}}}),\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & \le max\{\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,{\displaystyle \frac{(\lambda -1)r}{4\lambda }}),\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & =max\{\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,r_0),\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & <max\{t_2,t_1\}\hfill \\ \hfill & <ϵ,\hfill \end{array}$$

where $r_0={\displaystyle \frac{r(\lambda -1)}{4\lambda }}>0$. Hence, we have for any $r>0$,

$$K_{\eta ,\chi }^c(t_1,r)\cup K_{\eta ,\chi ,{\chi }_{\lambda }}^c(t_2,r)\subseteq K_{\eta ,\tau }^c(ϵ,r)$$

$$K_{\gamma ,\chi }^c(t_1,r)\cup K_{\gamma ,\chi ,{\chi }_{\lambda }}^c(t_2,r)\subseteq K_{\gamma ,\tau }^c(ϵ,r)$$

or equivalently,

$$\begin{array}{c}\hfill K_{\eta ,\tau }(ϵ,r)\subseteq K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r)\cap K_{\eta ,\tau }(t_1,r)\\ \hfill K_{\gamma ,\tau }(ϵ,r)\subseteq K_{\gamma ,\chi ,{\chi }_{\lambda }}(t_2,r)\cap K_{\gamma ,\tau }(t_1,r).\end{array}$$

(1)

If we take the asymptotic densities of both sides of (1), we obtain for each $r>0$ that

$$\begin{array}{cc}\hfill 0& \le \delta \left(K_{\eta ,\tau }(ϵ,r)\right)\hfill \\ \hfill & \le \delta (K_{\eta ,\chi }(t_1,r)\cup K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r))\hfill \\ \hfill & =\delta \left(K_{\eta ,\chi }(t_1,r)\right)+\delta \left(K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r)\right)-\delta (K_{\eta ,\chi }(t_1,r)\cap K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r))\hfill \\ \hfill & \le \delta \left(K_{\eta ,\chi }(t_1,r)\right)+\delta \left(K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r)\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill 0& \le \delta \left(K_{\gamma ,\tau }(ϵ,r)\right)\hfill \\ \hfill & \le \delta \left(K_{\gamma ,\chi }(t_1,r)\right)+\delta \left(K_{\gamma ,\chi ,{\chi }_{\lambda }}(t_2,r)\right).\hfill \end{array}$$

Since $\left\{{\chi }_n\right\}$ is statistically convergent to $v\in V$, we have that

$$\delta \left(K_{\eta ,\chi }(t_1,r)\right)=\delta \left(K_{\gamma ,\chi }(t_1,r)\right)=0$$

for any $r>0$. Thus, $\left\{{\chi }_{{\lambda }_n}\right\}$ is statistically convergent to v as well.

The above argument implies that $S_{{(\mu ,\nu )}^n}-{lim}_{n\to \infty }({\chi }_{{\lambda }_n}-{\chi }_n)=0$. Consequently, we have

$$\delta \left(K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r)\right)=\delta \left(K_{\gamma ,\chi ,{\chi }_{\lambda }}(t_2,r)\right)=0$$

for any $r>0$.

From the last four inequalities, we can conclude that

$$\delta \left(K_{\eta ,\tau }(ϵ,r)\right)=\delta \left(K_{\gamma ,\tau }(ϵ,r)\right)=0.$$

Thus, we have proven that

$$S_{{(\mu ,\nu )}^n}-\underset{n\to \infty }{lim}{\displaystyle \frac{1}{{\lambda }_n-n}}\sum _{k=n+1}^{{\lambda }_n}{v}_k=v,$$

for each $\lambda >1$.

Case 2: $\lambda \in (0,1)$.

For given $ϵ>0$ and any $r>0$, define the following sets:

$$\begin{array}{c}\hfill K_{\eta ,\tau }(ϵ,r)=\{k\le N:\eta (u_1,u_2,\dots ,u_{n-1},{\tau }_k\left(w\right)-v,r)\le 1-ϵ\},\end{array}$$

$$\begin{array}{c}\hfill K_{\gamma ,\tau }(ϵ,r)=\{k\le N:\gamma (u_1,u_2,\dots ,u_{n-1},{\tau }_k\left(w\right)-v,r)\ge ϵ\},\end{array}$$

where ${\tau }_k\left(w\right)={\displaystyle \frac{1}{n-{\lambda }_n}}{\sum }_{k={\lambda }_n+1}^{{\lambda }_n}{v}_k$ for all $n\in \mathbb{N}$.

For any $0<\lambda <1$ and sufficiently large $n\in \mathbb{N}\setminus \left\{0\right\}$ such that $n>{\lambda }_n$ with $n>{\displaystyle \frac{1}{\lambda }}$, we obtain from Lemma 3 for any $r>0$ and $u_1,u_2,\dots ,u_{n-1}\in V$, that

$$\begin{array}{cc}\hfill \eta & (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^{{\lambda }_n}{v}_k-v,r)\hfill \\ \hfill & \ge min\{\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,{\displaystyle \frac{r}{2\frac{{\lambda }_n+1}{n-{\lambda }_n}}}),\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & \ge min\{\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,r_1),\eta (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & >min\{1-t_2,1-t_1\}\hfill \\ \hfill & >1-ϵ\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \gamma & (u_1,u_2,\dots ,u_{n-1},{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{k={\lambda }_n+1}^{{\lambda }_n}{v}_k-v,r)\hfill \\ \hfill & \le max\{\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,{\displaystyle \frac{r}{2\frac{{\lambda }_n+1}{n-{\lambda }_n}}}),\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & \le max\{\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_{{\lambda }_n}-{\chi }_n,r_1),\gamma (u_1,u_2,\dots ,u_{n-1},{\chi }_n-v,{\displaystyle \frac{r}{2}})\}\hfill \\ \hfill & <max\{t_2,t_1\}\hfill \\ \hfill & <ϵ,\hfill \end{array}$$

where $r_1={\displaystyle \frac{(1-\lambda )r}{4\lambda }}>0$.

Hence, for any $t>0$, we have

$$\begin{array}{c}\hfill K_{\eta ,\tau }(ϵ,r)\subseteq K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r)\cup K_{\eta ,\tau }(t_1,r)\\ \hfill K_{\gamma ,\tau }(ϵ,r)\subseteq K_{\gamma ,\chi ,{\chi }_{\lambda }}(t_2,r)\cup K_{\gamma ,\tau }(t_1,r).\end{array}$$

(2)

If we take asymptotic densities of both sides of (2), then for any $r>0$ we have

$$\begin{array}{cc}\hfill 0& \le \delta \left(K_{\eta ,\tau }(ϵ,r)\right)\hfill \\ \hfill & \le \delta \left(K_{\eta ,\chi }(t_1,r)\right)+\delta \left(K_{\eta ,\chi ,{\chi }_{\lambda }}(t_2,r)\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill 0& \le \delta \left(K_{\gamma ,\tau }(ϵ,r)\right)\hfill \\ \hfill & \le \delta \left(K_{\gamma ,\chi }(t_1,r)\right)+\delta \left(K_{\gamma ,\chi ,{\chi }_{\lambda }}(t_2,r)\right).\hfill \end{array}$$

Since $\left\{{\chi }_n\right\}$ is statistically convergent to $v\in V$, we have that $\left\{{\chi }_{{\lambda }_n}\right\}$ is statistically convergent to v as well.

The above argument implies that $S_{{(\mu ,\nu )}^n}-{lim}_{n\to \infty }({\chi }_{{\lambda }_n}-{\chi }_n)=0$. Consequently, we have that

$$\delta \left(K_{\eta ,\tau }(ϵ,r)\right)=\delta \left(K_{\gamma ,\tau }(ϵ,r)\right)=0.$$

Thus, we have proven that

$$S_{{(\mu ,\nu )}^n}-\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n-{\lambda }_n}}\sum _{{\lambda }_n+1}^n{v}_k=v,$$

for each $\lambda \in (0,1)$. □

5. Conclusions and Future Work

Summability theory holds a pivotal position in the study of partial differential equations. In this study, we introduced the concept of Cesàro summability in an IFnNLS, a highly general mathematical framework with both algebraic and analytic characteristics. Consequently, our findings on Cesàro summability extend and generalize numerous established theorems. A significant future direction is to demonstrate Tauberian theorems using our current results in an IFnNLS, highlighting their importance.