Triple sets of -summable sequences of fuzzy numbers defined by an Orlicz function

Abstract

In this paper we introduce the ${\mathit{\chi }}^3$ fuzzy numbers defined by an Orlicz function and study some of their properties and inclusion results.

Keywords:

• gai sequence
• analytic sequence
• Orlicz function
• triple sequences
• completeness
• solid space
• symmetric space
• Orthogonal polynomials and special functions

AMS Subject Classifications:

• 40A05
• 40C05
• 40D05

Public Interest Statement

In this paper, we introduced the ${\mathit{\chi }}^3$ fuzzy numbers defined by an Orlicz function and study some of their properties with inclusion results. Furthermore we provided an example of triple sequence of gai which is not symmetric, not solid, not monotone and not convergent free.

Our result unifies the results of several author’s in the case of classical Orlicz spaces. One can extend our results for more general spaces.

1. Introduction

A triple sequence (real or complex) can be defined as a function $x:\mathbb{N}\times \mathbb{N}\times \mathbb{N}\to \mathbb{R}\left(\mathbb{C}\right),$ where $\mathbb{N},\mathbb{R}$ and $\mathbb{C}$ denote the set of natural numbers, real numbers and complex numbers, respectively.

Some initial work on double series is found in Apostol (1978), Alzer, Karayannakis, and Srivastava (2006), Bor, Srivastava, and Sulaiman (2012), Choi and Srivastava (1991), Liu and Srivastava (2006 and double sequence spaces are found in Hardy (1917), Deepmala Subramanian, and Mishra (in press), Deepmala, Mishra, and Subramanian (2016) and many others. Later on some initial work on triple sequence spaces is found in Sahiner, Gurdal, and Duden (2007), Esi (2014), Esi and Necdet Catalbas (2014),Esi and Savas (2015), Subramanian and Esi (2015) and many others.

A sequence $x=(x_{mnk})$ is said to be triple analytic if ${\text{sup}}_{m,n,k}{\left|x_{mnk}\right|}^{\frac{1}{m+n+k}}<\infty .$ The vector space of all triple analytic sequences are usually denoted by ${\mathrm{\Lambda }}^3$.

A sequence $x=(x_{mnk})$ is called triple entire sequence if ${\left|x_{mnk}\right|}^{\frac{1}{m+n+k}}\to 0$ as $m,n,k\to \infty .$

A sequence $x=(x_{mnk})$ is called triple chi sequence if ${\left((m+n+k)!\left|x_{mnk}\right|\right)}^{\frac{1}{m+n+k}}\to 0$ as $m,n,k\to \infty .$ The triple gai sequences will be denoted by ${\mathit{\chi }}^3$.

This paper deals with introducing the ${\mathit{\chi }}^3$-fuzzy number defined by an Orlicz function and study some topological properties, inclusion relations and give some examples. Some interesting results may be seen in Alzer et al. (2006), Bor et al. (2012), Choi and Srivastava (1991), Liu and Srivastava (2006).

2. Definitions and preliminaries

Definition 2.1

An Orlicz function (see Kamthan & Gupta, 1981) is a function $M:\left[0,\infty \right)\to \left[0,\infty \right)$ which is continuous, non-decreasing and convex with $M\left(0\right)=0,M\left(x\right)>0,$ for $x>0$ and $M\left(x\right)\to \infty$ as $x\to \infty .$ If convexity of Orlicz function M is replaced by $M\left(x+y\right)\le M\left(x\right)+M\left(y\right),$ then this function is called modulus function. Lindenstrauss and Tzafriri (1971) used the idea of Orlicz function to construct Orlicz sequence space.

Throughout a triple sequence is denoted by $〈X_{mnk}〉,$ a triple infinite array of fuzzy real numbers.

Let D denote the set of all closed and bounded intervals $X=\left[a_1,a_2,a_3\right]$ on the real line $\mathbb{R}.$ For $X=\left[a_1,a_2,a_3\right]\in D$ and $Y=\left[b_1,b_2,b_3\right]\in D,$ define$$\begin{array}{c}\hfill d\left(X,Y\right)=max\left(\left|a_1-b_1\right|,\left|a_2-b_2\right|,\left|a_3-b_3\right|\right)\end{array}$$

It is known that $\left(D,d\right)$ is a complete metric space.

A fuzzy real number X is a fuzzy set on $\mathbb{R},$ that is, a mapping $X:\mathbb{R}\times \mathbb{R}\times \mathbb{R}\to I\times I\times I\left(=\left[0,1\right]\right)$ associating each real number t with its grade of membership $X\left(t\right).$

The $\mathit{\alpha }$-level set ${\left[X\right]}^{\mathit{\alpha }},$ of the fuzzy real number X,  for $0<\mathit{\alpha }\le 1;$ is defined by$$\begin{array}{c}\hfill {\left[X\right]}^{\mathit{\alpha }}=\left\{t\in \mathbb{R}:X\left(t\right)\ge \mathit{\alpha }\right\}.\end{array}$$

The 0-level set is th closure of the strong 0-cut that is, $cl\left\{t\in \mathbb{R}:X\left(t\right)>0\right\}.$

A fuzzy real number X is called convex if $X\left(t\right)\ge X\left(s\right)\wedge X\left(r\right)\wedge X\left(v\right)=min\left\{X\left(s\right),X\left(r\right),X\left(v\right)\right\},$ where $s<t<r<v.$ If there exists $t_0\in \mathbb{R}$ such that $X\left(t_0\right)=1$ then, the fuzzy real number X is called normal.

A fuzzy real number X is said to be upper-semi continuous if, for each $\mathit{ϵ}>0,X^{-1}\left(\left[0,a+\mathit{ϵ}\right)\right)$ is open in the usual topology of $\mathbb{R}$ for all $a\in I.$

The set of all upper-semi continuous, normal, convex fuzzy real numbers is denoted by $L\left(\mathbb{R}\right).$

The absolute value, $\left|X\right|$ of $X\in L\left(\mathbb{R}\right)$ is defined by$$\begin{array}{c}\hfill \left|X\right|\left(t\right)=\left\{\begin{array}{cc}max\left\{X\left(t\right),X\left(-t\right)\right\},\hfill & \text{if}t\ge 0;\hfill \\ 0,\hfill & \text{if}t<0\hfill \end{array}\right.\end{array}$$

Let $\overline{d}:L\left(\mathbb{R}\right)\times L\left(\mathbb{R}\right)\times L\left(\mathbb{R}\right)\to \mathbb{R}\times \mathbb{R}\times \mathbb{R}$ be defined by$$\begin{array}{c}\hfill \overline{d}\left(X,Y\right)=\underset{0\le \mathit{\alpha }\le 1}{sup}d\left({\left[X\right]}^{\mathit{\alpha }},{\left[Y\right]}^{\mathit{\alpha }}\right).\end{array}$$

Then, $\overline{d}$ defines a metric on $L\left(\mathbb{R}\right)$ and it is well-known that $\left(L\left(\mathbb{R}\right),\overline{d}\right)$ is a complete metric space.

A sequence $〈X_{mnk}〉\subset L\left(\mathbb{R}\right)$ is said to be null if $\overline{d}\left(X_{mnk},\overline{0}\right)=0.$

A triple sequence $〈X_{mnk}〉$ of fuzzy real numbers is said to be gai in Pringsheim’s sense to a fuzzy number 0 if ${lim}_{m,n,k\to \infty }{\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k}=0.$

A triple sequence $〈X_{mnk}〉$ is said to $\mathit{\chi }$ regularly if it converges in the Prinsheim’s sense and the following limts zero:$$\begin{array}{c}\hfill \underset{m,n,k\to \infty }{lim}{\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k}=0\text{for}\text{each}m,n,k\in \mathbb{N}.\end{array}$$

A fuzzy real-valued double sequence space $E^F$ is said to be solid if $〈Y_{mnk}〉\in E^F$ whenever $〈X_{mnk}〉\in E^F$ and $\left|Y_{mnk}\right|\le \left|X_{mnk}\right|$ for all $m,n,k\in \mathbb{N}.$

Let $K=\left\{\left(m_i,n_i,k_i\right):i\in \mathbb{N};m_1<m_2<m_3\cdots \text{and}{n}_1<n_2<n_3\cdots \text{and}{k}_1<k_2<k_3<\cdots \right\}\subseteq \mathbb{N}\times \mathbb{N}\times \mathbb{N}$ and $E^F$ be a triple sequence space. A K-step space of $E^F$ is a sequence space ${\mathit{\lambda }}_K^E=\left\{〈X_{m_i{n}_i{k}_i}〉\in w^{3F}:〈X_{mnk}〉\in E^F\right\}.$

A canonical pre-image of a sequence $〈X_{m_i{n}_i{k}_i}〉\in E^F$ is a sequence $〈Y_{mnk}〉$ defined as follows:$$\begin{array}{c}\hfill Y_{mnk}=\left\{\begin{array}{cc}{X}_{mnk},\hfill & \text{if}\left(m,n,k\right)\in K,\hfill \\ \overline{0},\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

A canonical pre-image of a step space ${\mathit{\lambda }}_K^E$ is a set of canonical pre-images of all elements in ${\mathit{\lambda }}_K^E.$

A sequence set $E^F$ is said to be monotone if $E^F$ contains the canonical pre-images of all its step spaces.

A sequence set $E^F$ is said to be symmetric if $〈X_{{\mathit{\pi }}_{\left(m\right)},{\mathit{\pi }}_{\left(n\right)},{\mathit{\pi }}_{\left(k\right)}}〉\in E^F$ whenever $〈X_{mnk}〉\in E^F,$ where $\mathit{\pi }$ is a permutation of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}.$

A fuzzy real-valued sequence set $E^F$ is said to be convergent free if $〈Y_{mnk}〉\in E^F$ whenever $〈X_{mnk}〉\in E^F$ and $X_{mnk}=\overline{0}$ implies $Y_{mnk}=\overline{0}.$

We define the following classes of sequences:$$\begin{array}{cc}\hfill {\mathrm{\Lambda }}_f^{3F}& =\left\{〈X_{mnk}〉:\underset{mnk}{sup}f\left(\overline{d}\left(X_{mnk}^{1/m+n+k},\overline{0}\right)\right)<\infty ,X_{mnk}\in L\left(\mathbb{R}\right)\right\}.\hfill \\ \hfill {\mathit{\chi }}_f^{3F}& =\left\{〈X_{mnk}〉:\underset{mnk\to \infty }{lim}f\left(\overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\right)=0\right\}.\hfill \end{array}$$

Also, we define the classes of sequences ${\mathit{\chi }}_f^{3F^R}$ as follows :

A sequence $〈X_{mnk}〉\in {\mathit{\chi }}_f^{3F^R}$ if $〈x_{mnk}〉\in {\mathit{\chi }}_f^{3F}$ and the following limits hold$$\begin{array}{cc}\hfill \underset{m\to \infty }{lim}f\left(\overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\right)& =0\text{for}\text{each}m\in \mathbb{N}.\hfill \\ \hfill \underset{n\to \infty }{lim}f\left(\overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\right)=0\text{for}\text{each}n\in \mathbb{N}.\\ \hfill \underset{k\to \infty }{lim}f\left(\overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\right)=0\text{for}\text{each}k\in \mathbb{N}.\end{array}$$

3. Main results

Theorem 3.1

Let $N_1=min\left\{n_0:{sup}_{mnk\ge n_0}f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}-Y_{mnk}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}}<\infty \right\}$

$N_2=min\left\{n_0:{sup}_{mnk\ge n_0}{P}_{mnk}<\infty \right\}$ and $N=max\left(N_1,N_2,N_3\right).$

(i)

${\mathit{\chi }}_{f_p}^{3F^R}$ is not a paranormed space with (3.1) $$\begin{array}{c}\hfill g\left(X\right)=\underset{N\to \infty }{lim}\underset{mnk\ge N}{sup}f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}-Y_{mnk}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}\end{array}$$(3.1) if and only if $\mathit{\mu }>0,$ where $\mathit{\mu }={lim}_{N\to \infty }in{f}_{mnk\ge N}{P}_{mnk}$ and $M=max\left(1,{sup}_{mnk\ge N}{P}_{mnk}\right)$

(ii)

${\mathit{\chi }}_{f_p}^{3F^R}$ is complete with the paranorm (3.1).

Proof

 

(i)

Necessity: Let ${\mathit{\chi }}_{f_p}^{3F^R}$ be a paranormed space with (3.1) and suppose that $\mathit{\mu }=0.$ Then $\mathit{\alpha }={\text{inf}}_{mnk\ge N}{P}_{mnk}=0$ for all $N\in \mathbb{N}$ and $g〈\mathit{\lambda }X〉={lim}_{N\to \infty }{sup}_{mnk\ge N}{\left|\mathit{\lambda }\right|}^{P_{mnk/M}}=1$ for all $\mathit{\lambda }\in (0,1],$ where $X=〈\mathit{\alpha }〉\in {\mathit{\chi }}_{f_p}^{3F^R}$ whence $\mathit{\lambda }\to 0$ does not imply $\mathit{\lambda }X\to \mathit{\theta },$ when X is fixed. But this contradicts to (3.1) to be a paranorm. Sufficiency: Let $\mathit{\mu }>0.$ It is trivial that $g\left(\mathit{\theta }\right)=0,g\left(-X\right)=g\left(X\right)$ and $g〈X+Y+Z,\overline{0}〉\le g〈X,\overline{0}〉+g〈Y,\overline{0}〉+g〈Z,\overline{0}〉.$ Since $\mathit{\mu }>0$ there exists a positive number $\mathit{\beta }$ such that $P_{mnk}>\mathit{\beta }$ for sufficiently large positive integer m, n, k. Hence for any $\mathit{\lambda }\in \mathbb{C},$ we may write ${\left|\mathit{\lambda }\right|}^{P_{mnk}}\le max\left({\left|\mathit{\lambda }\right|}^M,{\left|\mathit{\lambda }\right|}^{\mathit{\beta }}\right)$ for suffciently large positive integers $m,n,k\ge N.$ Therefore, we obtain $g〈\mathit{\lambda }X,\overline{0}〉\le max\left(\left|\mathit{\lambda }\right|,{\left|\mathit{\lambda }\right|}^{\mathit{\beta }/M}\right)g〈X〉.$ Using this, one can prove that $\mathit{\lambda }X\to \mathit{\theta },$ whenever X is fixed and $\mathit{\lambda }\to 0$ or $\mathit{\lambda }\to 0$ and $X\to \mathit{\theta },$ or $\mathit{\lambda }$ is fixed and $X\to \mathit{\theta }.$ Because a paranormed space is a vector space. ${\mathit{\chi }}_{f_p}^{3F^R}$ is a set of sequences of fuzzy numbers. But the set $w^F=\left\{〈X_{mnk}〉:X_{mnk}\in L\left(R\right)\right\}$ of all sequences of fuzzy numbers is not a vector space. That is why, in order to say that ${\mathit{\chi }}_{f_p}^{3F^R}$ is a vector subspace (that is a sequence space) it is not sufficient to show that ${\mathit{\chi }}_{f_p}^{3F^R}$ is closed under addition and scalar multiplication. Consequently since $w^F$ is not a vector space, then ${\mathit{\chi }}_{f_p}^{3F^R}$ is not a vector subspace so that it is not a sequence space. Therefore it cannot be a paranormed space.

Proof

 

(ii)

Let $〈X^{k\ell }〉$ be a Cauchy sequence in ${\mathit{\chi }}_{f_p}^{3F^R},$ where $X^{k\ell }={〈X_{mnk}^{k\ell }〉}_{m,n,k\in \mathbb{N}}$. Then for every $\mathit{ϵ}>0\left(0<\mathit{ϵ}<1\right)$ there exists a positive integer $s_0$ such that (3.2) $$\begin{array}{c}\hfill g〈X^{k\ell }-X^{rt}〉=\underset{N\to \infty }{lim}\underset{mnk\ge N}{sup}f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}^{k\ell }-X_{mnk}^{rt}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}<\frac{\mathit{ϵ}}{2}\end{array}$$(3.2) for all $k,\ell ,r,t>s_0.$ By (3.2) there exists a positive integer $n_0$ such that (3.3) $$\begin{array}{c}\hfill \underset{mnk\ge N}{sup}f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}^{k\ell }-X_{mnk}^{rst}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}<\frac{\mathit{ϵ}}{2}\end{array}$$(3.3) for all $k,\ell ,r,t>s_0$ and for $N>n_0.$ Hence we obtain (3.4) $$\begin{array}{c}\hfill f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}^{k\ell }-X_{mnk}^{rt}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}<\frac{\mathit{ϵ}}{2}<1\end{array}$$(3.4) so that (3.5) $$\begin{array}{c}\hfill f\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}^{k\ell }-X_{mnk}^{rt}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)<f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}^{k\ell }-X_{mnk}^{rt}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}<\frac{\mathit{ϵ}}{2}\end{array}$$(3.5) for all $k,\ell ,r,t>s_0.$ This implies that ${〈X_{mnk}^{k\ell }〉}_{k\ell \in \mathbb{N}}$ is a Cauchy sequence in $\mathbb{C}$ for each fixed $m,n,k\ge n_0.$ Hence the sequence ${〈X_{mnk}^{k\ell }〉}_{k\ell \in \mathbb{N}}$ is convergent to $X_{mnk}$ say, (3.6) $$\begin{array}{c}\hfill \underset{k\ell \to \infty }{lim}{X}_{mnk}^{k\ell }=X_{mnk}\text{for each fixed}m,n,k>n_0.\end{array}$$(3.6) Getting $X_{mnk},$ we define $X=〈X_{mnk}〉.$ From (3.2) we obtain (3.7) $$\begin{array}{c}\hfill g〈X^{k\ell }-X〉=\underset{N\to \infty }{lim}\underset{mnk\ge N}{sup}f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}^{k\ell }-X_{mnk}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}<\frac{\mathit{ϵ}}{2}\end{array}$$(3.7) as $r,t\to \infty ,$ for all $k,\ell ,r,t>s_0.$ by (3.6). This implies that ${lim}_{k\ell \to \infty }{X}^{k\ell }=X.$ Now we show that $X=〈X_{mnk}〉\in {\mathit{\chi }}_{f_p}^{3F^R}.$ Since $X^{k\ell }\in {\mathit{\chi }}_{f_p}^{3F^R}$ for each $\left(k,1\right)\in N\times N\times N$ for every $\mathit{ϵ}>0\left(0<\mathit{ϵ}<1\right)$ there exists a positive integer $n_1\in N$ such that (3.8) $$\begin{array}{c}\hfill f{\left(\overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}<\frac{\mathit{ϵ}}{2}\text{for every}m,n,k>n_1.\end{array}$$(3.8) By (3.6) and (3.7) we obtain $$\begin{array}{cc}\hfill f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}& \le f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}^{k\ell }\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}\hfill \\ \hfill & +f{\left(\overline{d}\left({\left(\left(m+n+k\right)!\left(X_{mnk}^{k\ell }-X_{mnk}\right)\right)}^{1/m+n+k},\overline{0}\right)\right)}^{P_{mnk}/M}\hfill \\ \hfill & <\frac{\mathit{ϵ}}{2}+\frac{\mathit{ϵ}}{2}=\mathit{ϵ}\text{for}k,\ell >\hfill \end{array}$$$max\left(s_0,s_1\right)$ and $m,n,k>max\left(n_0,n_1\right)$. This implies that $X\in {\mathit{\chi }}_{f_p}^{3F^R}$.

Proposition 3.2

The class of sequences ${\mathrm{\Lambda }}_f^{3F}$ is symmetric but the classes of sequences ${\mathit{\chi }}_f^{3F}$ and ${\mathit{\chi }}_f^{3F^R}$ are not symmetric.

Proof

Obviously the class of sequences ${\mathrm{\Lambda }}_f^{3F}$ is symmetric. For the other classes of sequences, consider the following example

Example

Consider the class of sequences ${\mathit{\chi }}_f^{3F}.$ Let $f\left(X\right)=X$ and consider the sequence $〈X_{mnk}〉$ be defined by$$\begin{array}{c}\hfill X_{1nk}\left(t\right)=\left\{\begin{array}{cc}\frac{{\left(-t+1\right)}^{1+n+k}}{\left(1+n+k\right)!},\hfill & \text{for}t=-1,\hfill \\ \frac{{\left(t-1\right)}^{1+n+k}}{\left(1+n+k\right)!},\hfill & \text{for}t=1,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

and for $m>1,$$$\begin{array}{c}\hfill X_{mnk}\left(t\right)=\left\{\begin{array}{cc}\frac{{\left(t+2\right)}^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}t=-2,\hfill \\ \frac{{\left(-t-1\right)}^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}t=-1,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

Let $〈Y_{mnk}〉$ be a rearrangement of $〈X_{mnk}〉$ defined by$$\begin{array}{c}\hfill Y_{nnn}\left(t\right)=\left\{\begin{array}{cc}\frac{{\left(-t+1\right)}^{3n}}{\left(3n\right)!},\hfill & \text{for}t=-1,\hfill \\ \frac{{\left(t-1\right)}^{3n}}{\left(3n\right)!},\hfill & \text{for}t=1,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

and for $m\ne n\ne k,$$$\begin{array}{c}\hfill Y_{mnk}\left(t\right)=\left\{\begin{array}{cc}\frac{{\left(t+2\right)}^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}t=-2,\hfill \\ \frac{{\left(-t-1\right)}^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}t=-1,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

Then, $〈X_{mnk}〉\in {\mathit{\chi }}_f^{3F}$ but $〈Y_{mnk}〉\notin {\mathit{\chi }}_f^{3F}.$ Hence, ${\mathit{\chi }}_p^{3F}$ is not symmetric. Similarly other sequences are also not symmetric.

Proposition 3.3

The classes of sequences ${\mathrm{\Lambda }}_f^{3F},{\mathit{\chi }}_f^{3F}$ and ${\mathit{\chi }}_f^{3F^R}$ are solid.

Proof

Consider the class of sequences ${\mathit{\chi }}_f^{3F}$. Let $〈X_{mnk}〉$ and $〈Y_{mnk}〉\in {\mathit{\chi }}_f^{3F}$ be such that $\overline{d}\left({\left(\left(m+n+k\right)!Y_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\le \overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k},\overline{0}\right).$ As f is non-decreasing, we have$$\begin{array}{c}\hfill \underset{mnk\to \infty }{lim}f\left(\overline{d}\left({\left(\left(m+n+k\right)!Y_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\right)\le \underset{mnk\to \infty }{lim}f\left(\overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\right)\end{array}$$

Hence, the class of sequence ${\mathit{\chi }}_f^{3F}$ is solid. Simlarly it can be shown that the other classes of sequences are also solid.

Proposition 3.4

The classes of sequences ${\mathit{\chi }}_f^{3F}$ and ${\mathit{\chi }}_f^{3F^R}$ are not monotone and hence not solid.

Proof

The result follows from the following example.

Example

Consider the class of sequences ${\mathit{\chi }}_f^{3F}$ and $f\left(X\right)=X$. Let $J=\left\{\left(m,n,k\right):m\ge n\ge k\right\}\subseteq N\times N\times N.$ Let $〈X_{mnk}〉$ be defined by$$\begin{array}{c}\hfill X_{mnk}\left(t\right)=\left\{\begin{array}{cc}\frac{{\left(t+3\right)}^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}-3<t\le -2,\hfill \\ \frac{{\left(mt\right)}^{m+n+k}}{{\left(3m-1\right)}^{m+n+k}\left(m+n+k\right)!}+\frac{{\left(3m\right)}^{m+n+k}}{{\left(3m-1\right)}^{m+n+k}\left(m+n+k\right)!},\hfill & \text{for}-2\le t\le -1+\frac{1}{m},\hfill \\ \overline{0},\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

for all $m,n,k\in N.$

Then $〈X_{mnk}〉\in {\mathit{\chi }}_f^{3F}.$ Let $〈Y_{mnk}〉$ be the canonical pre-image of ${〈X_{mnk}〉}_J$ for the subsequence J of $N\times N\times N.$ Then$$\begin{array}{c}\hfill Y_{mnk}=\left\{\begin{array}{cc}{X}_{mnk},\hfill & \text{for}\left(m,n,k\right)\in J,\hfill \\ \overline{0},\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

Then, $〈Y_{mnk}〉\notin {\mathit{\chi }}_f^{3F}.$ Hence ${\mathit{\chi }}_f^{3F}$ is not monotone. Similarly, it can be shown that the other classes of sequences are also not monotone. Hence, the classes of sequences ${\mathit{\chi }}_f^{3F}$ and ${\mathit{\chi }}_f^{3F^R}$ are not solid.

Proposition 3.5

(i) ${\mathit{\chi }}_{f_1}^{3F}\bigcap {\mathit{\chi }}_{f_2}^{3F}\bigcap {\mathit{\chi }}_{f_3}^{3F}\subseteq {\mathit{\chi }}_{f_1+f_2+f_3}^{3F},$ (ii) ${\mathit{\chi }}_{f_1}^{3F^R}\bigcap {\mathit{\chi }}_{f_2}^{3F^R}\bigcap {\mathit{\chi }}_{f_3}^{3F^R}\subseteq {\mathit{\chi }}_{f_1+f_2+f_3}^{3F^R}$

Proof

It is easy, so omitted.

Proposition 3.6

Let $f,f_1$ and $f_2$ be three Orlicz functions, then, (i) ${\mathit{\chi }}_{f_1}^{3F}\subseteq {\mathit{\chi }}_{f\circ f_1\circ f_2}^{3F}$, (ii) ${\mathit{\chi }}_{f_1}^{3F^R}\subseteq {\mathit{\chi }}_{f\circ f_1\circ f_2}^{3F^R}$, (iii)${\mathrm{\Lambda }}_{f_1}^{3F}\subseteq {\mathrm{\Lambda }}_{f\circ f_1\circ f_2}^{3F}$

Proof

We prove the result for the case ${\mathit{\chi }}_{f_1}^{3F}\subseteq {\mathit{\chi }}_{f\circ f_1\circ f_2}^{3F},$ the other cases are similar. Let $\mathit{ϵ}>0$ be given. As f is continuous and non-decreasing, so there exists $\mathit{\eta }>0,$ such that $f\left(\mathit{\eta }\right)=\mathit{ϵ}.$ Let $〈X_{mnk}〉\in {\mathit{\chi }}_{f_1}^{3F}.$ Then, there exist $m_0,n_0,k_0\in \mathbb{N},$ such that$$\begin{array}{cc}\hfill f_1\left(\overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+K},\overline{0}\right)\right)& <\mathit{\eta },\text{for all}m\ge m_0,n\ge n_0,k\ge k_0\hfill \\ \hfill & \Rightarrow f\circ f_1\circ f_2\left(\overline{d}\left({\left(\left(m+n+k\right)!X_{mnk}\right)}^{1/m+n+k},\overline{0}\right)\right)<\mathit{ϵ},\hfill \\ \hfill & \text{for all}m\ge m_0,n\ge n_0,k\ge k_0.\hfill \end{array}$$

Hence, $〈X_{mnk}〉\in {\mathit{\chi }}_{f\circ f_1\circ f_2}^{3F}.$ Thus, ${\mathit{\chi }}_{f_1}^{3F}\subseteq {\mathit{\chi }}_{f\circ f_1\circ f_2}^{3F}.$

Proposition 3.7

(i) ${\mathit{\chi }}_f^{3F}\subseteq {\mathrm{\Lambda }}_f^{3F}$, (ii) ${\mathit{\chi }}_f^{3F^R}\subseteq {\mathrm{\Lambda }}_f^{3F},$ the inclusions are strict.

Proof

The inclusion (i) ${\mathit{\chi }}_f^{3F}\subseteq {\mathrm{\Lambda }}_f^{3F}$ (ii) ${\mathit{\chi }}_f^{3F^R}\subseteq {\mathrm{\Lambda }}_f^{3F}$ is obvious. For establishing that the inclusions are proper, consider the following example.

Example

We prove the result for the case ${\mathit{\chi }}_f^{3F}\subseteq {\mathrm{\Lambda }}_f^{3F},$ the other case similar. Let $f\left(X\right)=X.$ Let the sequence $〈X_{mnk}〉$ be defined by for $m>n>k,$$$\begin{array}{c}\hfill X_{mnk}\left(t\right)=\left\{\begin{array}{cc}\frac{{\left(mt-m-1\right)}^{m+n+k}{\left(m-1\right)}^{-\left(m+n+k\right)}}{\left(m+n+k\right)!},\hfill & \text{for}1+\frac{1}{m}\le t\le 2,\hfill \\ \frac{{\left(3-t\right)}^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}2<t\le 3,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

and for $m<n<k$$$\begin{array}{c}\hfill X_{mnk}\left(t\right)=\left\{\begin{array}{cc}\frac{{\left(mt-1\right)}^{m+n+k}{\left(m-1\right)}^{-\left(m+n+k\right)}}{\left(m+n+k\right)!},\hfill & \text{for}\frac{1}{m}\le t\le 1,\hfill \\ \frac{{\left(-t+2\right)}^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}1\le t\le 2,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

Then, $〈X_{mnk}〉\in {\mathrm{\Lambda }}_f^{3F}$ but $〈X_{mnk}〉\notin {\mathit{\chi }}_f^{3F}.$

Proposition 3.8

The classes of sequences ${\mathrm{\Lambda }}_f^{3F},{\mathit{\chi }}_f^{3F}$ and ${\mathit{\chi }}_f^{3F^R}$ are not convergent free.

Proof

The result follows from the following example.

Example

Consider the classes of sequences ${\mathit{\chi }}_f^{3F}.$ Let $f\left(X\right)=X$ and consider the sequence $〈X_{mnk}〉$ defined by ${\left(\left(1+n+k\right)!X_{1nk}\right)}^{1/1+n+k}=\overline{0},$ and for other values,$$\begin{array}{c}\hfill X_{mnk}\left(t\right)=\left\{\begin{array}{cc}\frac{1^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}0\le t\le 1,\hfill \\ \frac{{\left(-mt\right)}^{m+n+k}{\left(m+1\right)}^{-\left(m+n+k\right)}+{\left(2m+1\right)}^{m+n+k}{\left(1+m\right)}^{-\left(m+n+k\right)}}{\left(m+n+k\right)!},\hfill & \text{for}1<t\le 2+\frac{1}{m},\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

Let the sequence $〈Y_{mnk}〉$ be defined by ${\left(\left(1+n+k\right)!Y_{1nk}\right)}^{1/1+n+k}=\overline{0},$ and for other values,$$\begin{array}{c}\hfill Y_{mnk}\left(t\right)=\left\{\begin{array}{cc}\frac{1^{m+n+k}}{\left(m+n+k\right)!},\hfill & \text{for}0\le t\le 1,\hfill \\ \frac{{\left(m-t\right)}^{m+n+k}{\left(m-1\right)}^{-\left(m+n+k\right)}}{\left(m+n+k\right)!},\hfill & \text{for}1<t\le m,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

Then, $〈X_{mnk}〉\in {\mathit{\chi }}_f^{3F}$ but $〈Y_{mnk}〉\notin {\mathit{\chi }}_f^{3F}.$ Hence, the classes of sequences ${\mathit{\chi }}_f^{3F}$ are not convergent free. Similarly, the other spaces are also not convergent free.

4. Conclusion

The ${\mathit{\chi }}^3$ fuzzy numbers defined by an Orlicz function and discuss inclusion relation. Furthermore, the given example of triple sequence of gai is not symmetric, not solid, not monotone and not convergent free.

Acknowledgements

The authors are extremely grateful to the anonymous learned referee(s) for their keen reading, valuable suggestion and constructive comments for the improvement of the manuscript. The authors are thankful to the editor(s) and reviewers of Cogent Mathematics.The third author NS wish to thank the Department of Science and Technology, Government of India for the financial sanction towards this work under FIST program SR/FST/MSI-107/2015.The research of the second author Deepmala is supported by the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India under SERB National Post-Doctoral fellowship scheme File Number: PDF/2015/000799.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Vandana

Vandana is a research scholar at School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur-492010, (C.G.) India. Her research interests are in the areas of applied mathematics including Optimization, Mathematical Programming, Inventory control, Supply Chain Management, Operation Research, etc. She is member of several scientific committees, advisory boards as well as member of editorial board of a number of scientific journals.

Deepmala

Deepmala is Visiting Scientist at SQC & OR Unit at Indian Statistical Institute, Kolkata, India. Her research interests are in the areas of Optimization, Mathematical Programming, Fixed Point Theory and Applications, Operator theory, Approximation Theory etc. She is member of several scientific committees and also member of editorial board of a number of scientific journals.

N. Subramanian

N Subramanian received PhD degree in Mathematics from Alagappa University at Karaikudi,Tamil Nadu,India and also getting Doctor of Science (D.Sc) degree in Mathematics from Berhampur University, Berhampur, India. His research interests are in the areas of summability through functional analysis of applied mathematics and pure mathematics.