Approaches of linear operators in the intuitionistic fuzzy 2-Banach spaces

Approaches of linear operators in the intuitionistic fuzzy 2-Banach spaces Abstract The focus of this paper is to present some concepts of intuitionistic fuzzy T-convergence in intuitionistic fuzzy 2-Banach spaces. We will modify and correct the definition of regularity of 2-norms on 2-Banach spaces, which was given by Gürdal et al. in (2009, Nonlinear Analysis: Theory, Methods & Applications, 71, 1654–1661) to guarantee uniqueness of T-limit of T-convergence. Furthermore, we will introduce the concepts of intuitionistic fuzzy regularity and investigate consistency of intuitionistic fuzzy 2-norms on domain and range of the operator $$T_{n}$$. We will give some illustrative examples supporting our theoretical results. 1 Introduction Gähler defined the concept of 2-normed space in 1960. We recall some basic facts as follows: Definition 1.1 [6] A function $$\left \Vert ,\right \Vert :V\times V\rightarrow \mathbb{R} $$ is called a 2-norm on V if (i) $$\left \Vert z,w\right \Vert =0\mathit{ \Leftrightarrow }z$$and w are linearly dependent, (ii) $$\left \Vert z,w\right \Vert =\left \Vert w,z\right \Vert $$, (iii) $$\left \Vert z,w+r\right \Vert \leq \left \Vert z,r\right \Vert +\left \Vert w,r\right \Vert, $$ where V is a d-dimensional real linear space, $$2\leq d<\infty .$$ Also $$\left ( V,\left \Vert ,\right \Vert \right ) $$is called a 2-normed space when $$\left \Vert ,\right \Vert $$ is a 2-norm on V. Example 1.2 $$V= \mathbb{R} ^{2}$$ is a 2-normed space with $$\left \Vert z,w\right \Vert =\left \vert z_{1}w_{2}-z_{2}w_{1}\right \vert $$ for $$z=\left ( z_{1},z_{2}\right ) ,w=\left ( w_{1},w_{2}\right ) \in \mathbb{R} ^{2}$$. Definition 1.3 A sequence $$\left ( z_{n}\right ) $$ in V is called convergent to z if $$ \underset{n\rightarrow \infty }{\lim }\left \Vert z_{n}-z,w\right \Vert =0$$ for every w ∈ V. Definition 1.4 A sequence $$\left ( z_{n}\right ) $$ in V is called Cauchy if there exist two linearly independent elements w, r ∈ V such that $$ \underset{m,n\rightarrow \infty }{\lim }\left\Vert z_{m}-z_{n},w\right\Vert =0\text{ and }\underset{ m,n\rightarrow \infty }{\lim }\left\Vert z_{m}-z_{n},r\right\Vert =0 $$. Definition 1.5 A linear 2-normed space $$\left ( V,\left \Vert ,\right \Vert \right ) $$ is called a 2-Banach space if every Cauchy sequence in V is convergent. Many authors researched various topics on this concept which can be considered as two dimensional analogues of a normed space. Approximation theory is one of these topics. By the sequence of linear operators $$T_{n}$$, approximation was done between the a Banach space and sequences of Banach spaces (see [4], [7], [19]). Gürdal et al. [8] investigated approximation theory in 2-Banach space. To accomplish this goal, they defined linear operators $$ T_{n}:V\rightarrow W_{n}$$ where $$W_{n}$$ is a sequence of 2-Banach spaces for n ≥ 2 and examined the problem of approximation to the space V by $$ W_{n}.$$ They gave the definition of T-convergence to approximate to elements of V by elements of $$W_{n}$$ as the following: Definition 1.6 [8] A sequence $$\left ( w_{n}\right ) $$ is called T-convergent to z ∈ V if $$ \underset{n\rightarrow \infty }{\lim }\left\Vert w_{n}-T_{n}z,r\right\Vert _{W_{n}}=0 $$ for all $$r\in W_{n}$$ and is denoted by $$w_{n}\overset{T_{\left \Vert .,.\right \Vert }}{\rightarrow }z.$$ They showed that T-limit of generated sequence $$(w_{n})$$ from $$W_{n}$$ does not have to be unique. To resolve this problem, they defined regularity of 2-norms on $$W_{n},$$ which guarantees uniqueness of T-limit as the following: Definition 1.7 [8] 2-norms in $$W_{n}$$ are called to be regular, if (i) $$\underset{n\rightarrow \infty }{\lim } \left \Vert T_{n}z,r\right \Vert{{ }}_{W_{n}}=0$$implies$$\left \Vert z,r\right \Vert =0,$$ (ii) if z and r are linear independent, then z = 0. In Definition 1.7, (i) and (ii) contradict each other. Because,$$\ \left \Vert z,r\right \Vert =0$$ implies z and r linearly dependent. So, we can not consider (ii) when z and r linearly dependent. That is, it is not possible that (i) and (ii) are correct at the same time. The evaluation of (i) and (ii) at the same time leads to a contradiction. Now, we first give the corrected version of Definition 1.7. Definition 1.8 2-norms in $$W_{n}$$ are called to be regular, if $$ \underset{n\rightarrow \infty }{\lim }\left \Vert T_{n}z,r\right \Vert _{W_{n}}=0$$for all r$$\in W_{n}$$implies z = 0. On the other hand, fuzzy theory that was introduced by Zadeh [21] was generalized by Atanassov [1]. Current literature reveals that intuitionistic fuzzy analogues of concepts from functional analysis have a great importance and so a great effort have been devoted study in this direction. Some of them can be given as follows: [15], [16], [9], [10], [13], [2], [3], [11], [14], [17] and [18]. This paper concentrates to handle the idea of approximation between the a 2-Banach space and sequences of 2-Banach spaces with intuitionistic fuzzy logic. Firstly let’s give some basic concepts to make our paper clearer. 2 Preliminaries In the following, we give some of the basic concepts, which will be used in sequel of this paper to study approximation theory in the intuitionistic fuzzy 2-Banach spaces. Definition 2.1 [20] Let $$\ast :\left [ 0,1\right ] \times \left [ 0,1\right ] \rightarrow \left [ 0,1\right ] $$ be a continuous binary operation * is called a continuous t-norm if the following conditions are satisfied: (i) * is commutative and associative, (ii) x * 1 = x for all $$x\in \left [ 0,1\right ] $$, (iii) $$x\ast x^{\prime }\leq y\ast y^{\prime }$$ whenever x ≤ y and $$x^{\prime }\leq y^{\prime }$$ for each $$x,y,x^{\prime },y^{\prime }\in \left [ 0,1\right ].$$ For example, x * y = x.y is a continuous t-norm. Definition 2.2 [20] Let $$\diamond :\left [ 0,1\right ] \times \left [ 0,1\right ] \rightarrow \left [ 0,1\right ] $$ be a continuous binary operation. $$ \diamond $$ is called a t-conorm if it satisfies the following conditions: (i) $$ \diamond $$ is commutative and associative, (ii) x$$ \diamond $$ 0 = x for all $$x\in \left [ 0,1\right ] $$, (iii) $$x\diamond x^{\prime }\leq y\diamond y^{\prime }$$ whenever x ≤ y and $$x^{\prime }\leq y^{\prime }$$ for each $$x,y,x^{\prime },y^{\prime }\in \left [ 0,1\right ] .$$ For example, $$x\diamond y=\min \left \{ x+y,1\right \} $$ is a continuous t-norm. Definition 2.3 [15] Let V be a linear space, * be a continuous t-norm and ◊ be a continuous t-conorm. The five-tuple $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is called an intuitionistic fuzzy 2-normed space if $$\mu ,\upsilon $$ are fuzzy sets on V$$\times V\times \left ( 0,\infty \right ) $$ satisfying the following conditions for every z, w, r ∈ V and s, t > 0: (a) $$\mu \left ( w,z;s\right ) +\upsilon \left ( w,z;s\right ) \leq 1,$$ (b) $$\mu \left ( w,z;s\right )>0,$$ (c) $$\mu \left ( w,z;s\right ) =$$ 1 if and only if z and w are linearly dependent, (d) $$\mu \left ( \alpha w,z;s\right ) =\,$$$$\mu \left ( w,z;\frac{s}{\left \vert \alpha \right \vert }\right ) $$ for each $$\alpha \neq 0$$, (e) $$\mu \left ( w,z;t\right ) \ast $$$$\mu \left ( z,r;s\right ) \leq\, $$$$\mu \left ( w,z+r;t+s\right ),$$ (f) $$\mu \left ( w,z;.\right ) :\left ( 0,\infty \right ) \rightarrow \left [ 0,1 \right ] $$ is continuous, (g) $$\underset{s\rightarrow \infty }{lim}$$$$\mu \left ( w,z;s\right ) =1$$ and $$ \underset{s\rightarrow 0}{lim}$$$$\mu \left ( w,z;s\right ) =0,$$ (h) $$\mu \left ( w,z;s\right ) =\mu \left ( z,w;s\right ),$$ (i) $$\upsilon \left ( w,z;s\right ) $$ < 1, (j) $$\upsilon \left ( w,z;s\right ) =$$ 0 if and only if z and w are linearly dependent, (k) $$\upsilon \left ( \alpha w,z;s\right ) =\,$$$$\upsilon \left ( w,z;\frac{s}{ \left \vert \alpha \right \vert }\right ) $$ for each $$\alpha \neq 0$$, (l) $$\upsilon \left ( w,z;s\right ) \Diamond $$$$\upsilon \left ( z,r;s\right ) \geq \upsilon \left ( w,z+r;s+s\right ),$$ (m) $$\upsilon \left ( w,z;.\right ) :\left ( 0,\infty \right ) \rightarrow \left [ 0,1\right ] $$ is continuous, (n) $$\underset{s\rightarrow \infty }{\lim }\upsilon \left ( w,z;s\right ) =0$$ and $$\underset{s\rightarrow 0}{\lim }\upsilon \left ( w,z;s\right ) =1,$$ (o) $$\upsilon \left ( w,z;s\right ) =\upsilon \left ( z,w;s\right )\!.$$ Also, $$\left ( \mu ,\upsilon \right ) $$ is called an intuitionistic fuzzy 2-norm on V. We will abbreviate an intuitionistic fuzzy 2-normed space and an intuitionistic fuzzy 2-norm as IF-2-NS and IF-2-N, respectively. Example 2.4 Let $$\ ( \mathbb{R} ^{n},\left \Vert .,.\right \Vert )$$ be a 2-normed space, let x * y = xy and $$x\Diamond y=\min \left \{ x,y\right \} $$ for all $$x,y\in \left [ 0,1\right ] $$. For all $$z\in \mathbb{R} ^{n}$$ and every s > 0, we consider $$ \mu \left( z,w;s\right) =\frac{s}{s+\left\Vert w,z\right\Vert }\ \textrm{and}\ \upsilon \left( z,w;s\right) =\frac{\left\Vert w,z\right\Vert }{s+\left\Vert w,z\right\Vert }. $$ Then $$\left ( \mathbb{R} ^{n},\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is an IF-2-NS. Definition 2.5 [15] A sequence $$\left ( z_{n}\right ) $$ in $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is called Cauchy if for each $$\epsilon>0 $$ and each s > 0, there exists $$n_{0}$$ ∈ $$ \mathbb{N} $$ such that $$\mu \left ( z_{n}-z_{m},r;s\right )>1-\epsilon $$ and $$\upsilon \left ( z_{n}-z_{m},r;s\right ) $$$$<\epsilon $$ for all $$n,m\geq n_{0}$$ and for all r ∈ V. Definition 2.6 [15] A sequence $$\left ( z_{n}\right ) $$ in $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is called convergent to L ∈ V with respect to IF-2-N if, for every $$\epsilon>0$$ and s > 0, there exists $$k\in \mathbb{N} $$ such that $$\mu \left ( z_{n}-L,r;s\right )>1-\epsilon $$ and $$\upsilon \left ( z_{n}-L,r;s\right ) $$$$<\epsilon $$ for all $$k\geq k_{0}$$ and for all r ∈ V. We will show convergence of $$\left ( z_{n}\right ) $$ to L in $$ \left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ as $$\textrm{IF-2-N}-\lim z_{n}=L$$. Definition 2.7 [15] $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is called a complete IF-2-NS if every Cauchy sequence in $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is convergent with respect to IF-2-N. Now, we can give main results in the following section. 3 Intuitionistic fuzzy T-convergence in the intuitionistic fuzzy 2-Banach Spaces Let $$\left ( W_{n}\right ) $$ be a sequence of IF-2-NSs. Denote IF-2-N on $$W_{n}$$ by $$\left(\mu_{W_{n}}, \upsilon_{W_{n}}\right)$$. Take notice linear operators $$T_{n}:V\rightarrow W_{n}$$ where $$T_{n}(V)=W_{n}$$ for n = 2, 3, 4, .... We use the members of $$W_{n}$$ to approach to the members of V. Constitute sequence $$(w_{n})$$ by the element $$w_{n}\in W_{n}$$. Definition 3.1 Sequence $$(w_{n})$$ is called intuitionistic fuzzy T-convergent to z ∈ V, if $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( w_{n}-T_{n}z,r;s\right) =1 \textrm{ and}\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( w_{n}-T_{n}z,r;s\right) =0 $$ for all $$r\in W_{n}$$ and s > 0. We will denote this by $$w_{n}\overset{\mathrm T_{\mathrm{IF-2-N}}}{\rightarrow }z.$$ If $$W_{n}=V$$ for n = 2, 3, ... then intuitionistic fuzzy T-convergence is equivalent to intuitionistic fuzzy convergence in the IN-2-NS V. Example 3.2 Let $$V=\ell _{2}=\left \{ z=(z_{k}):\overset{\infty }{\underset{k=1}{\sum }} \left \vert z_{k}\right \vert{{ }}^{2}<\infty \right \} $$. We define operator $$ T_{n} $$ as $$T_{n}z=\left ( z_{k+1}\right )_{k=1}^{n}$$ for $$z=\left ( z_{k}\right ) \in \ell _{2}$$ where n = 2, 3, 4, .... In this case $$ T_{n}(V)=W_{n}= \mathbb{R} ^{n}.$$ Let’s consider the 2-normed space $$( \mathbb{R}^{n},\left \Vert .,.\right \Vert )$$ where $$\left \Vert z,w\right \Vert $$ is the area of the parallelogram spanned by z and w given by the formula $$ \left\Vert z,w\right\Vert =\left\{ \left( \overset{n}{\underset{k=1}{\sum }} {z_{k}^{2}}\right) \left( \overset{n}{\underset{k=1}{\sum }}{w_{k}^{2}}\right) -\left( \overset{n}{\underset{k=1}{\sum }}z_{k}w_{k}\right)^{2}\right\}^{ \frac{1}{2}}. $$ Let x * y = xy and $$x\Diamond y=min\left \{ x,y\right \} $$ for all $$ x,y\in \left [ 0,1\right ] $$ . $$\left ( \mathbb{R} ^{n},\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is an IF-2-NS as stated in Example 2.4 with $$ \mu_{\mathbb{R}^{n}}\left( z,w;s\right) =\frac{s}{s+\left\Vert w,z\right\Vert }\ \textrm{and}\ \upsilon_{\mathbb{R}^{n}} \left( z,w;s\right) =\frac{\left\Vert w,z\right\Vert }{s+\left\Vert w,z\right\Vert } $$ for all $$z\in \mathbb{R} ^{n}$$ and every s > 0. Consider $$w_{k}=\left ( \frac{1}{k},\frac{1}{k^{2}} ,\ldots ,\frac{1}{k^{n}}\right ) \in T_{n}V= \mathbb{R} ^{n}$$ where $$k\in \mathbb{N} .$$ For every $$(u,0,0,\ldots ,0,...)\in \mathbb{R} ^{n},$$ we get \begin{eqnarray} &&\underset{n\rightarrow \infty }{\lim }\mu_{\mathbb{R}^{n}} \left( \left( \frac{1}{n},\frac{1 }{n^{2}},\ldots,\frac{1}{n^{n}}\right) -T_{n}(u,0,0,\ldots,0,...),r;s\right) \nonumber \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{\mathbb{R}^{n}} \left( \left( \frac{1}{n},\frac{ 1}{n^{2}},\ldots,\frac{1}{n^{n}}\right) -\overset{n\text{-times}}{\overbrace{ (0,0,\ldots,0,0)}},r;s\right) \nonumber \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{\mathbb{R}^{n}} \left( \left( \frac{1}{n},\frac{ 1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r;s\right) \nonumber \\ \nonumber &=&\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\Vert \left( \frac{1 }{n},\frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r\right\Vert } \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\{ \left( \overset{n }{\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{\underset{ i=1}{\sum }}{r_{i}^{2}}\right) -\left( \overset{n}{\underset{i=1}{\sum }}\frac{ 1}{n^{i}}r_{i}\right)^{2}\right\}^{\frac{1}{2}}} \nonumber \\ &\geq &\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\{ \left( \overset{n}{\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{ \underset{i=1}{\sum }}{r_{i}^{2}}\right) \right\}^{\frac{1}{2}}}=1 \end{eqnarray} (3.1) and \begin{eqnarray} &&\underset{n\rightarrow \infty }{\lim }\upsilon_{\mathbb{R}^{n}} \left( \left( \frac{1}{n}, \frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right) -T_{n}(u,0,0,\ldots,0,...),r;s\right) \nonumber \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{\mathbb{R}^{n}} \left( \left( \frac{1}{n}, \frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right) -\overset{n\text{-times}}{ \overbrace{(0,0,\ldots,0,0)}},r;s\right) \nonumber \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{\mathbb{R}^{n}} \left( \left( \frac{1}{n}, \frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r;s\right) \nonumber \\ \nonumber &=&\underset{n\rightarrow \infty }{\lim }\frac{\left\Vert \left( \frac{1}{n}, \frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r\right\Vert }{s+\left\Vert \left( \frac{1}{n},\frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r\right\Vert }\\ &=&\underset{n\rightarrow \infty }{\lim }\frac{\left\{ \left( \overset{n}{ \underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{\underset{ i=1}{\sum }}{r_{i}^{2}}\right) -\left( \overset{n}{\underset{i=1}{\sum }}\frac{ 1}{n^{i}}r_{i}\right)^{2}\right\}^{\frac{1}{2}}}{s+\left\{ \left( \overset{ n}{\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{\underset{i=1}{\sum }}{r_{i}^{2}}\right) -\left( \overset{n}{\underset{i=1}{\sum }} \frac{1}{n^{i}}r_{i}\right)^{2}\right\}^{\frac{1}{2}}} \nonumber \\ &\leq& \underset{n\rightarrow \infty }{\lim }\frac{\left\{ \left( \overset{n} {\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{\underset{ i=1}{\sum }}{r_{i}^{2}}\right) \right\}^{\frac{1}{2}}}{s+\left\{ \left( \overset{n}{\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{ \underset{i=1}{\sum }}{r_{i}^{2}}\right) -\left( \overset{n}{\underset{i=1}{ \sum }}\frac{1}{n^{i}}r_{i}\right)^{2}\right\}^{\frac{1}{2}}}=0. \end{eqnarray} (3.2) Equations (3.1) and (3.2) say that $$w_{n}\overset{\mathrm T_{\mathrm{IF-2-N}}}{\rightarrow }(u,0,0,\ldots ,0,...).$$ As can be seen, $$\left ( w_{n}\right ) $$ has infinitely many intuitionistic fuzzy T-limits. This means that approximation of V to $$T_{n}V$$ is very bad. Now, we define a property guaranteeing uniqueness of limit of intuitionistic fuzzy T-convergence. Definition 3.3 The IF-2-Ns in $$W_{n}$$ are called to be intuitionistic fuzzy regular if $$\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left ( T_{n}z,r;s\right ) =1$$ and $$\underset{n\rightarrow \infty }{\lim } \upsilon_{W_{n}} \left ( T_{n}z,r;s\right ) =0$$ imply z = 0 for all $$r\in W_{n}$$ and s > 0. Theorem 3.4 IF-2-Ns in $$W_{n}$$ are regular if and only if intuitionistic fuzzy T-limit is unique. Proof. ⟹ : Suppose that $$\left ( w_{n}\right ) $$ is T-convergent to $$z^{\prime}$$ and $$z^{\prime\prime}.$$ In this case, $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left(w_{n}-T_{n}z^{\prime },r;s\right) =1\textrm{ and}\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\left( w_{n}-T_{n}z,r;s\right) =0 $$ $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( w_{n}-T_{n}z^{\prime \prime },r;s\right) =1\textrm{ and}\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( w_{n}-T_{n}z,r;s\right) =0 $$ for all $$r\in w_{n}$$ and s > 0. Using these arguments, we obtain \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( T_{n}(z^{\prime }-z^{\prime \prime }),r;s\right) &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( T_{n}z^{\prime }-T_{n}z^{\prime \prime },r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( T_{n}z^{\prime }-w_{n}+w_{n}-T_{n}z^{\prime \prime },r;s\right) \\ &\geq &\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( T_{n}z^{\prime }-w_{n},r;\frac{s}{2}\right) \ast \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( w_{n}-T_{n}z^{\prime \prime },r;\frac{s}{2}\right) \\ &=&1\ast 1=1 \end{eqnarray*} and \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( T_{n}\big(z^{\prime }-z^{\prime \prime }\big),r;s\right) &=&\underset{n\rightarrow \infty }{\lim } \upsilon_{W_{n}} \left( T_{n}z^{\prime }-T_{n}z^{\prime \prime },r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( T_{n}z^{\prime }-w_{n}+w_{n}-T_{n}z^{\prime \prime },r;s\right) \\ &\leq &\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( T_{n}z^{\prime }-w_{n},r;\frac{s}{2}\right) \Diamond \underset{n\rightarrow \infty }{\lim } \upsilon_{W_{n}} \left( w_{n}-T_{n}z^{\prime \prime },r;\frac{s}{2}\right) \\ &=&0\Diamond 0=0. \end{eqnarray*} Since the IF-2-Ns in $$W_{n}$$ are regular, $$z^{\prime }-z^{\prime \prime }$$ is equal to 0. That is, $$z^{\prime }=z^{\prime \prime }.$$ ⇐: Now, suppose that intuitionistic fuzzy T-limit is unique. In this case, since $$\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}}\! \left (T_{n}z,r;s\right )=1$$ and $$\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\! \left ( T_{n}z,r;s\right ) =0$$ for all $$r\in W_{n},$$ we have $$z_{n}\overset{ \text T_{\mathrm{IF-2-N}}}{\rightarrow }0.$$ From the linearity of T, we get z = 0. It means that IF-2-Ns of $$W_{n}$$ are regular. Theorem 3.5 Let $$z_{n},w_{n}$$$$\in W_{n}$$; z, w ∈ V and $$\alpha $$ be a scalar. If $$z_{n}\overset{\text T_{\mathrm{IF-2-N}}}{\rightarrow }z$$ and $$w_{n}\overset{\text T_{\mathrm{IF-2-N}}}{\rightarrow }w,$$ then (a) $$\left ( \alpha z_{n}\right ) \overset{\text T_{\mathrm{IF-2-N}}}{\rightarrow }\left ( \alpha z\right ) $$ for each $$\alpha ,$$ (b) $$\left ( z_{n}+w_{n}\right ) \overset{\text T_{\mathrm{IF-2-N}}}{\rightarrow }\left ( z+w\right )\!.$$ Proof. (a) Using $$z_{n}\overset{\text T_{\mathrm{IF-2-N}}}{ \rightarrow }z,$$ we obtain \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \alpha z_{n}-T_{n}\left( \alpha z\right),r;s\right) &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \alpha z_{n}-\alpha T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \alpha \left( z_{n}-T_{n}z\right),r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( z_{n}-T_{n}z,r;\frac{s}{ \left\vert \alpha \right\vert }\right) \\ &=&1 \end{eqnarray*} and \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( \alpha z_{n}-T_{n}\left( \alpha z\right),r;s\right) &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( \alpha z_{n}-\alpha T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( \alpha \left( z_{n}-T_{n}z\right),r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( z_{n}-T_{n}z,r; \frac{s}{\left\vert \alpha \right\vert }\right) \\ &=&0. \end{eqnarray*} (b) Using by $$z_{n}\overset{\text T_{\mathrm{IF-2-N}}}{ \rightarrow }z$$ and $$w_{n}\overset{\text T_{\mathrm{IF-2-N}}}{ \rightarrow }w,$$ we obtain \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \left( z_{n}+w_{n}\right) -T_{n}(z+w),r;s\right)\!\!\!\!\! &=&\!\!\!\!\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \left( z_{n}+w_{n}\right) -\left( T_{n}z+T_{n}w\right),r;s\right) \\ &\geq&\!\!\!\!\!\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( z_{n}-T_{n}z,r;\frac{ s}{2}\right)\ast \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( w_{n}-T_{n}w,r;\frac{s}{2}\right) \\ &=&\!\!\!\!\!1\ast 1=1 \end{eqnarray*} and \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( \left( z_{n}+w_{n}\right) -T_{n}(z+w),r;s\right)\!\!\!\!\! &=&\!\!\!\!\!\underset{n\rightarrow \infty }{ \lim }\upsilon_{W_{n}} \left( \left( z_{n}+w_{n}\right) -\left( T_{n}z+T_{n}w\right) ,r;s\right) \\ &\leq&\!\!\!\!\!\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( z_{n}-T_{n}z,r; \frac{s}{2}\right) \Diamond \underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( w_{n}-T_{n}w,r;\frac{s}{2}\right) \\ &=&\!\!\!\!\!0\Diamond 0=0. \end{eqnarray*} Definition 3.6 For all z, r ∈ V and $$w\in W_{n}$$ if $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}}\left( T_{n}z,w;s\right) =\mu_{V}\left( z,r;s\right) \textrm{ and}\underset{ n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\left( T_{n}z,w;s\right) =\upsilon_{V}\left( z,r;s\right) $$ then IF-2-Ns in $$W_{n}$$ are called consistent with intuitionistic fuzzy norm $$\left ( \mu_{V} ,\upsilon_{V} \right )$$ of the space V. Theorem 3.7 If IF-2-Ns in $$W_{n}$$ are consistent with $$\left ( \mu_{V} ,\upsilon_{V} \right )$$ of the space V, then IF-2-Ns in $$W_{n}$$ are intuitionistic fuzzy regular. Proof. Since IF-2-Ns in $$W_{n}$$ are consistent with $$\left ( \mu_{V} ,\upsilon_{V} \right )$$ of the space V, $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}}\left( T_{n}z,w;s\right) =\mu_{V}\left( z,r;s\right) =1\textrm{ and}\underset{ n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\left( T_{n}z,r;s\right) =\upsilon_{V}\left( z,r;s\right) =0. $$ For each z ∈ V and all $$w\in W_{n},r\in V,$$$$\mu_{V}\left ( z,r;s\right ) =1$$ and $$\upsilon_{V}\left ( z,r;s\right ) =0$$ if and only if z and r are linear dependent. Since each z is linear dependent with any r, z is equal to 0. Thus, we can get the intuitionistic fuzzy regularity of the space $$W_{n}.$$ Example 3.8 Consider $$V=C\left [ 0,n\right ] =\{ z(s):[ 0,n] \ \rightarrow \mathbb{R} :z(s)$$ is continuous on [0, n]}. Let $$T_{n}\left ( z(s)\right ) =z({s_{k}^{n}})_{k=1}^{n}$$ for $$0\leq{s_{1}^{n}}<{s_{2}^{n}}<...<{s_{n}^{n}}\leq n.$$ In this case \begin{eqnarray*} T_{2}\left( z(s)\right) &=&z\big({s_{k}^{2}}\big)_{k=1}^{2}=\left( z\big({s_{1}^{2}}\big),z\big({s_{2}^{2}}\big)\right) \\ T_{3}\left( z(s)\right) &=&z\big({s_{k}^{3}}\big)_{k=1}^{3}=\left( z\big({s_{1}^{3}}\big),z\big({s_{2}^{3}}\big),z\big({s_{3}^{3}}\big)\right) \\ &&\vdots \\ T_{n}\left( z(s)\right) &=&z\big({s_{k}^{n}}\big)_{k=1}^{n}=\left( z\big({s_{1}^{n}}\big),z\big({s_{2}^{n}}\big),z\big({s_{3}^{n}}\big),\ldots,z\big({s_{n}^{n}}\big)\right) . \end{eqnarray*} That is $$W_{n}=\mathbb{R} ^{n}.$$ Since $$\left \Vert z,w\right \Vert =\left \vert \begin{array}{cc} {\int \limits _{0}^{n}}z(s)z(s)\,\mathrm{d}s & {\int \limits _{0}^{n}}z(s)w(s)\,\mathrm{d}s \\{\int \limits _{0}^{n}}z(s)w(s)\,\mathrm{d}s & {\int \limits _{0}^{n}}w(s)w(s)\,\mathrm{d}s \end{array} \right \vert ^{\frac{1}{2}}$$ is a 2-norm on $$V\!\!=\!\!C\left [ 0,n\right ],\ \left ( \mu_{V},\upsilon_{V}\right ) $$ is a IF-2-N on $$ V=C\left [ 0,n\right ] $$ with x * y = xy and $$x\Diamond y=min\left \{ x,y\right \} $$ for all $$x,y\in \left [ 0,1\right ] $$ where $$\mu_{V}\left ( z,w;s\right ) =\frac{s}{s+\left \Vert z,w\right \Vert }$$ and $$\upsilon_{V}\left ( z,w;s\right ) =\frac{\left \Vert z,w\right \Vert }{s+\left \Vert z,w\right \Vert }.$$ Also, take into consideration $$\left ( \mathbb{R} ^{n},\mu ,\upsilon ,\ast ,\Diamond \right ) $$ in Example 3.2. $$ {s_{1}^{n}},{s_{2}^{n}},\ldots ,{s_{n}^{n}}$$ is uniform partition of $$\left [ 0,n\right ] .$$ That is $$\triangle s_{k}=s_{k+1}-s_{k}=$$ 1 and so $$\overset{n}{\underset{k=1}{\sum }}\left ( z\big ({s_{k}^{n}}\big )\right ) \triangle s_{k}=\overset{n}{\underset{k=1}{\sum }}\left ( z\big ({s_{k}^{n}}\big )\right ) .$$ Since \begin{eqnarray*} &&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}}\left( T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{\mathbb{R}^{n}}\left( T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\{ \left( \overset{n }{\underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\right) \left( \overset{n}{\underset{k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right)^{2}\right) -\left( \overset{n}{\underset{k=1}{\sum }}z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\right) ^{2}\right\}^{\frac{1}{2}}} \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\{ \left( \overset{n }{\underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) \left( \overset{n}{\underset{k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) -\left( \overset{n}{\underset{ k=1}{\sum }}z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\triangle s_{k}\right)^{2}\right\}^{ \frac{1}{2}}} \\ &=&\frac{s}{s+\left\{ {\int\limits_{0}^{n}}z(s)z(s)\,\mathrm{d}s{\int\limits_{0}^{n}}r(s)r(s)\,\mathrm{d}s-\left( {\int\limits_{0}^{n}}z(s)r(s)\, \mathrm{d}s\right)^{2}\right\}^{\frac{1}{2}}} \\ &=&\mu_{V}\left( z,w;s\right) \end{eqnarray*} and \begin{eqnarray*} &&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\left( T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{\mathbb{R}^{n}}\left( T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{\left\{ \left( \overset{n}{ \underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\right) \left( \overset{ n}{\underset{k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right) \right) -\left( \overset{ n}{\underset{k=1}{\sum }}z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\right)^{2}\right\}^{ \frac{1}{2}}}{s+\left\{ \left( \overset{n}{\underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\right) \left( \overset{n}{\underset{k=1}{\sum }} \left( r\big({s_{k}^{n}}\big)\right) \right) -\left( \overset{n}{\underset{k=1}{\sum }} z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\right)^{2}\right\}^{\frac{1}{2}}} \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{\left\{ \left( \overset{n}{ \underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) \left( \overset{n}{\underset{k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right) ^{2}\triangle s_{k}\right) -\left( \overset{n}{\underset{k=1}{\sum }} z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\triangle s_{k}\right)^{2}\right\}^{\frac{1}{2}}}{ s+\left\{ \left( \overset{n}{\underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) \left( \overset{n}{\underset{ k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) -\left( \overset{n}{\underset{k=1}{\sum }}z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\triangle s_{k}\right)^{2}\right\}^{\frac{1}{2}}} \\ &=&\frac{\left\{ {\int\limits_{0}^{l}}z(s)z(s)\,\mathrm{d}s{\int\limits_{0}^{l}}w(s)w(s)\,\mathrm{d}s-\left( {\int\limits_{0}^{l}}z(s)w(s)\,\mathrm{d}s\right)^{2}\right\}^{\frac{1}{2}}}{s+\left\{ {\int\limits_{0}^{l}}z(s)z(s)\,\mathrm{d}s{\int\limits_{0}^{l}}w(s)w(s)\,\mathrm{d}s-\left( {\int\limits_{0}^{l}}z(s)w(s)\,\mathrm{d}s\right)^{2}\right\}^{\frac{1}{2}}} \\ &=&\upsilon_{V}\left( z,w;s\right)\!, \end{eqnarray*} IF-2-Ns in $$ \mathbb{R} ^{n}$$ are consistent with IF-2-N of the space $$V=C \left [ 0,n\right ]\!.$$ Funding This work is supported by The Scientific and Technological Research Council of Turkey under the Project number 110T699. 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Gürsoy . $$\lambda $$- statistical convergence of sequence of functions in intuitionistic fuzzy normed spaces. Journal of Function Spaces and Applications , 1 – 14 , 2012 . [13] V. Karakaya , N. Şimşek , F. Gürsoy and M. Ertürk . Lacunary statistical convergence of sequences of functions in intuitionistic fuzzy normed space . Journal of Intelligent & Fuzzy Systems , 26 , 1289 – 1299 , 2014 . [14] M. Mursaleen and Q. Danish Lohani . Baire’s and Cantor’s theorems in intuitionistic fuzzy 2-metric spaces . Chaos, Solitons & Fractals , 42 , 2254 – 2259 , 2009 . Google Scholar CrossRef Search ADS [15] M. Mursaleen and Q. Danish Lohani . Intuitionistic fuzzy 2-normed space and some related concepts . Chaos, Solitons & Fractals , 42 , 2009 , 224 – 234 . Google Scholar CrossRef Search ADS [16] M. Mursaleen , V. Karakaya and S. Mohiuddine . Schauder basis, separability, and approximation property in intuitionistic fuzzy normed space . Abstract and Applied Analysis, 14 , 2010 . [17] M. Mursaleen and S. Mohiuddine . Statistical convergence of double sequences in intuitionistic fuzzy normed spaces . Chaos, Solitons & Fractals , 41 , 2414 – 2421 , 2009 . Google Scholar CrossRef Search ADS [18] P. P. Murthy , R. Mishra and V. N. Mishra . Tripled coincidence theorems for compatible maps in fuzzy metric spaces. Electronic Journal of Mathematical Analysis and Applications , 4 , 96 – 106 , 2016 . [19] M. A. Özarslan , O. Duman and O. Dogru . Rates of A-statistical convergence of approximating operators . Calcolo , 422 , 93 – 104 , 2005 . Google Scholar CrossRef Search ADS [20] B. Schweizer and A. Skaler . Statistical metric spaces . Pacific Journal of Mathematics , 10 , 21 , 1960 . [21] L. A. Zadeh . Fuzzy sets. Information and Control , 8 , 338 – 353 . © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. 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Approaches of linear operators in the intuitionistic fuzzy 2-Banach spaces

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Abstract

Abstract The focus of this paper is to present some concepts of intuitionistic fuzzy T-convergence in intuitionistic fuzzy 2-Banach spaces. We will modify and correct the definition of regularity of 2-norms on 2-Banach spaces, which was given by Gürdal et al. in (2009, Nonlinear Analysis: Theory, Methods & Applications, 71, 1654–1661) to guarantee uniqueness of T-limit of T-convergence. Furthermore, we will introduce the concepts of intuitionistic fuzzy regularity and investigate consistency of intuitionistic fuzzy 2-norms on domain and range of the operator $$T_{n}$$. We will give some illustrative examples supporting our theoretical results. 1 Introduction Gähler defined the concept of 2-normed space in 1960. We recall some basic facts as follows: Definition 1.1 [6] A function $$\left \Vert ,\right \Vert :V\times V\rightarrow \mathbb{R} $$ is called a 2-norm on V if (i) $$\left \Vert z,w\right \Vert =0\mathit{ \Leftrightarrow }z$$and w are linearly dependent, (ii) $$\left \Vert z,w\right \Vert =\left \Vert w,z\right \Vert $$, (iii) $$\left \Vert z,w+r\right \Vert \leq \left \Vert z,r\right \Vert +\left \Vert w,r\right \Vert, $$ where V is a d-dimensional real linear space, $$2\leq d<\infty .$$ Also $$\left ( V,\left \Vert ,\right \Vert \right ) $$is called a 2-normed space when $$\left \Vert ,\right \Vert $$ is a 2-norm on V. Example 1.2 $$V= \mathbb{R} ^{2}$$ is a 2-normed space with $$\left \Vert z,w\right \Vert =\left \vert z_{1}w_{2}-z_{2}w_{1}\right \vert $$ for $$z=\left ( z_{1},z_{2}\right ) ,w=\left ( w_{1},w_{2}\right ) \in \mathbb{R} ^{2}$$. Definition 1.3 A sequence $$\left ( z_{n}\right ) $$ in V is called convergent to z if $$ \underset{n\rightarrow \infty }{\lim }\left \Vert z_{n}-z,w\right \Vert =0$$ for every w ∈ V. Definition 1.4 A sequence $$\left ( z_{n}\right ) $$ in V is called Cauchy if there exist two linearly independent elements w, r ∈ V such that $$ \underset{m,n\rightarrow \infty }{\lim }\left\Vert z_{m}-z_{n},w\right\Vert =0\text{ and }\underset{ m,n\rightarrow \infty }{\lim }\left\Vert z_{m}-z_{n},r\right\Vert =0 $$. Definition 1.5 A linear 2-normed space $$\left ( V,\left \Vert ,\right \Vert \right ) $$ is called a 2-Banach space if every Cauchy sequence in V is convergent. Many authors researched various topics on this concept which can be considered as two dimensional analogues of a normed space. Approximation theory is one of these topics. By the sequence of linear operators $$T_{n}$$, approximation was done between the a Banach space and sequences of Banach spaces (see [4], [7], [19]). Gürdal et al. [8] investigated approximation theory in 2-Banach space. To accomplish this goal, they defined linear operators $$ T_{n}:V\rightarrow W_{n}$$ where $$W_{n}$$ is a sequence of 2-Banach spaces for n ≥ 2 and examined the problem of approximation to the space V by $$ W_{n}.$$ They gave the definition of T-convergence to approximate to elements of V by elements of $$W_{n}$$ as the following: Definition 1.6 [8] A sequence $$\left ( w_{n}\right ) $$ is called T-convergent to z ∈ V if $$ \underset{n\rightarrow \infty }{\lim }\left\Vert w_{n}-T_{n}z,r\right\Vert _{W_{n}}=0 $$ for all $$r\in W_{n}$$ and is denoted by $$w_{n}\overset{T_{\left \Vert .,.\right \Vert }}{\rightarrow }z.$$ They showed that T-limit of generated sequence $$(w_{n})$$ from $$W_{n}$$ does not have to be unique. To resolve this problem, they defined regularity of 2-norms on $$W_{n},$$ which guarantees uniqueness of T-limit as the following: Definition 1.7 [8] 2-norms in $$W_{n}$$ are called to be regular, if (i) $$\underset{n\rightarrow \infty }{\lim } \left \Vert T_{n}z,r\right \Vert{{ }}_{W_{n}}=0$$implies$$\left \Vert z,r\right \Vert =0,$$ (ii) if z and r are linear independent, then z = 0. In Definition 1.7, (i) and (ii) contradict each other. Because,$$\ \left \Vert z,r\right \Vert =0$$ implies z and r linearly dependent. So, we can not consider (ii) when z and r linearly dependent. That is, it is not possible that (i) and (ii) are correct at the same time. The evaluation of (i) and (ii) at the same time leads to a contradiction. Now, we first give the corrected version of Definition 1.7. Definition 1.8 2-norms in $$W_{n}$$ are called to be regular, if $$ \underset{n\rightarrow \infty }{\lim }\left \Vert T_{n}z,r\right \Vert _{W_{n}}=0$$for all r$$\in W_{n}$$implies z = 0. On the other hand, fuzzy theory that was introduced by Zadeh [21] was generalized by Atanassov [1]. Current literature reveals that intuitionistic fuzzy analogues of concepts from functional analysis have a great importance and so a great effort have been devoted study in this direction. Some of them can be given as follows: [15], [16], [9], [10], [13], [2], [3], [11], [14], [17] and [18]. This paper concentrates to handle the idea of approximation between the a 2-Banach space and sequences of 2-Banach spaces with intuitionistic fuzzy logic. Firstly let’s give some basic concepts to make our paper clearer. 2 Preliminaries In the following, we give some of the basic concepts, which will be used in sequel of this paper to study approximation theory in the intuitionistic fuzzy 2-Banach spaces. Definition 2.1 [20] Let $$\ast :\left [ 0,1\right ] \times \left [ 0,1\right ] \rightarrow \left [ 0,1\right ] $$ be a continuous binary operation * is called a continuous t-norm if the following conditions are satisfied: (i) * is commutative and associative, (ii) x * 1 = x for all $$x\in \left [ 0,1\right ] $$, (iii) $$x\ast x^{\prime }\leq y\ast y^{\prime }$$ whenever x ≤ y and $$x^{\prime }\leq y^{\prime }$$ for each $$x,y,x^{\prime },y^{\prime }\in \left [ 0,1\right ].$$ For example, x * y = x.y is a continuous t-norm. Definition 2.2 [20] Let $$\diamond :\left [ 0,1\right ] \times \left [ 0,1\right ] \rightarrow \left [ 0,1\right ] $$ be a continuous binary operation. $$ \diamond $$ is called a t-conorm if it satisfies the following conditions: (i) $$ \diamond $$ is commutative and associative, (ii) x$$ \diamond $$ 0 = x for all $$x\in \left [ 0,1\right ] $$, (iii) $$x\diamond x^{\prime }\leq y\diamond y^{\prime }$$ whenever x ≤ y and $$x^{\prime }\leq y^{\prime }$$ for each $$x,y,x^{\prime },y^{\prime }\in \left [ 0,1\right ] .$$ For example, $$x\diamond y=\min \left \{ x+y,1\right \} $$ is a continuous t-norm. Definition 2.3 [15] Let V be a linear space, * be a continuous t-norm and ◊ be a continuous t-conorm. The five-tuple $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is called an intuitionistic fuzzy 2-normed space if $$\mu ,\upsilon $$ are fuzzy sets on V$$\times V\times \left ( 0,\infty \right ) $$ satisfying the following conditions for every z, w, r ∈ V and s, t > 0: (a) $$\mu \left ( w,z;s\right ) +\upsilon \left ( w,z;s\right ) \leq 1,$$ (b) $$\mu \left ( w,z;s\right )>0,$$ (c) $$\mu \left ( w,z;s\right ) =$$ 1 if and only if z and w are linearly dependent, (d) $$\mu \left ( \alpha w,z;s\right ) =\,$$$$\mu \left ( w,z;\frac{s}{\left \vert \alpha \right \vert }\right ) $$ for each $$\alpha \neq 0$$, (e) $$\mu \left ( w,z;t\right ) \ast $$$$\mu \left ( z,r;s\right ) \leq\, $$$$\mu \left ( w,z+r;t+s\right ),$$ (f) $$\mu \left ( w,z;.\right ) :\left ( 0,\infty \right ) \rightarrow \left [ 0,1 \right ] $$ is continuous, (g) $$\underset{s\rightarrow \infty }{lim}$$$$\mu \left ( w,z;s\right ) =1$$ and $$ \underset{s\rightarrow 0}{lim}$$$$\mu \left ( w,z;s\right ) =0,$$ (h) $$\mu \left ( w,z;s\right ) =\mu \left ( z,w;s\right ),$$ (i) $$\upsilon \left ( w,z;s\right ) $$ < 1, (j) $$\upsilon \left ( w,z;s\right ) =$$ 0 if and only if z and w are linearly dependent, (k) $$\upsilon \left ( \alpha w,z;s\right ) =\,$$$$\upsilon \left ( w,z;\frac{s}{ \left \vert \alpha \right \vert }\right ) $$ for each $$\alpha \neq 0$$, (l) $$\upsilon \left ( w,z;s\right ) \Diamond $$$$\upsilon \left ( z,r;s\right ) \geq \upsilon \left ( w,z+r;s+s\right ),$$ (m) $$\upsilon \left ( w,z;.\right ) :\left ( 0,\infty \right ) \rightarrow \left [ 0,1\right ] $$ is continuous, (n) $$\underset{s\rightarrow \infty }{\lim }\upsilon \left ( w,z;s\right ) =0$$ and $$\underset{s\rightarrow 0}{\lim }\upsilon \left ( w,z;s\right ) =1,$$ (o) $$\upsilon \left ( w,z;s\right ) =\upsilon \left ( z,w;s\right )\!.$$ Also, $$\left ( \mu ,\upsilon \right ) $$ is called an intuitionistic fuzzy 2-norm on V. We will abbreviate an intuitionistic fuzzy 2-normed space and an intuitionistic fuzzy 2-norm as IF-2-NS and IF-2-N, respectively. Example 2.4 Let $$\ ( \mathbb{R} ^{n},\left \Vert .,.\right \Vert )$$ be a 2-normed space, let x * y = xy and $$x\Diamond y=\min \left \{ x,y\right \} $$ for all $$x,y\in \left [ 0,1\right ] $$. For all $$z\in \mathbb{R} ^{n}$$ and every s > 0, we consider $$ \mu \left( z,w;s\right) =\frac{s}{s+\left\Vert w,z\right\Vert }\ \textrm{and}\ \upsilon \left( z,w;s\right) =\frac{\left\Vert w,z\right\Vert }{s+\left\Vert w,z\right\Vert }. $$ Then $$\left ( \mathbb{R} ^{n},\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is an IF-2-NS. Definition 2.5 [15] A sequence $$\left ( z_{n}\right ) $$ in $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is called Cauchy if for each $$\epsilon>0 $$ and each s > 0, there exists $$n_{0}$$ ∈ $$ \mathbb{N} $$ such that $$\mu \left ( z_{n}-z_{m},r;s\right )>1-\epsilon $$ and $$\upsilon \left ( z_{n}-z_{m},r;s\right ) $$$$<\epsilon $$ for all $$n,m\geq n_{0}$$ and for all r ∈ V. Definition 2.6 [15] A sequence $$\left ( z_{n}\right ) $$ in $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is called convergent to L ∈ V with respect to IF-2-N if, for every $$\epsilon>0$$ and s > 0, there exists $$k\in \mathbb{N} $$ such that $$\mu \left ( z_{n}-L,r;s\right )>1-\epsilon $$ and $$\upsilon \left ( z_{n}-L,r;s\right ) $$$$<\epsilon $$ for all $$k\geq k_{0}$$ and for all r ∈ V. We will show convergence of $$\left ( z_{n}\right ) $$ to L in $$ \left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ as $$\textrm{IF-2-N}-\lim z_{n}=L$$. Definition 2.7 [15] $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is called a complete IF-2-NS if every Cauchy sequence in $$\left ( V,\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is convergent with respect to IF-2-N. Now, we can give main results in the following section. 3 Intuitionistic fuzzy T-convergence in the intuitionistic fuzzy 2-Banach Spaces Let $$\left ( W_{n}\right ) $$ be a sequence of IF-2-NSs. Denote IF-2-N on $$W_{n}$$ by $$\left(\mu_{W_{n}}, \upsilon_{W_{n}}\right)$$. Take notice linear operators $$T_{n}:V\rightarrow W_{n}$$ where $$T_{n}(V)=W_{n}$$ for n = 2, 3, 4, .... We use the members of $$W_{n}$$ to approach to the members of V. Constitute sequence $$(w_{n})$$ by the element $$w_{n}\in W_{n}$$. Definition 3.1 Sequence $$(w_{n})$$ is called intuitionistic fuzzy T-convergent to z ∈ V, if $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( w_{n}-T_{n}z,r;s\right) =1 \textrm{ and}\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( w_{n}-T_{n}z,r;s\right) =0 $$ for all $$r\in W_{n}$$ and s > 0. We will denote this by $$w_{n}\overset{\mathrm T_{\mathrm{IF-2-N}}}{\rightarrow }z.$$ If $$W_{n}=V$$ for n = 2, 3, ... then intuitionistic fuzzy T-convergence is equivalent to intuitionistic fuzzy convergence in the IN-2-NS V. Example 3.2 Let $$V=\ell _{2}=\left \{ z=(z_{k}):\overset{\infty }{\underset{k=1}{\sum }} \left \vert z_{k}\right \vert{{ }}^{2}<\infty \right \} $$. We define operator $$ T_{n} $$ as $$T_{n}z=\left ( z_{k+1}\right )_{k=1}^{n}$$ for $$z=\left ( z_{k}\right ) \in \ell _{2}$$ where n = 2, 3, 4, .... In this case $$ T_{n}(V)=W_{n}= \mathbb{R} ^{n}.$$ Let’s consider the 2-normed space $$( \mathbb{R}^{n},\left \Vert .,.\right \Vert )$$ where $$\left \Vert z,w\right \Vert $$ is the area of the parallelogram spanned by z and w given by the formula $$ \left\Vert z,w\right\Vert =\left\{ \left( \overset{n}{\underset{k=1}{\sum }} {z_{k}^{2}}\right) \left( \overset{n}{\underset{k=1}{\sum }}{w_{k}^{2}}\right) -\left( \overset{n}{\underset{k=1}{\sum }}z_{k}w_{k}\right)^{2}\right\}^{ \frac{1}{2}}. $$ Let x * y = xy and $$x\Diamond y=min\left \{ x,y\right \} $$ for all $$ x,y\in \left [ 0,1\right ] $$ . $$\left ( \mathbb{R} ^{n},\mu ,\upsilon ,\ast ,\Diamond \right ) $$ is an IF-2-NS as stated in Example 2.4 with $$ \mu_{\mathbb{R}^{n}}\left( z,w;s\right) =\frac{s}{s+\left\Vert w,z\right\Vert }\ \textrm{and}\ \upsilon_{\mathbb{R}^{n}} \left( z,w;s\right) =\frac{\left\Vert w,z\right\Vert }{s+\left\Vert w,z\right\Vert } $$ for all $$z\in \mathbb{R} ^{n}$$ and every s > 0. Consider $$w_{k}=\left ( \frac{1}{k},\frac{1}{k^{2}} ,\ldots ,\frac{1}{k^{n}}\right ) \in T_{n}V= \mathbb{R} ^{n}$$ where $$k\in \mathbb{N} .$$ For every $$(u,0,0,\ldots ,0,...)\in \mathbb{R} ^{n},$$ we get \begin{eqnarray} &&\underset{n\rightarrow \infty }{\lim }\mu_{\mathbb{R}^{n}} \left( \left( \frac{1}{n},\frac{1 }{n^{2}},\ldots,\frac{1}{n^{n}}\right) -T_{n}(u,0,0,\ldots,0,...),r;s\right) \nonumber \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{\mathbb{R}^{n}} \left( \left( \frac{1}{n},\frac{ 1}{n^{2}},\ldots,\frac{1}{n^{n}}\right) -\overset{n\text{-times}}{\overbrace{ (0,0,\ldots,0,0)}},r;s\right) \nonumber \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{\mathbb{R}^{n}} \left( \left( \frac{1}{n},\frac{ 1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r;s\right) \nonumber \\ \nonumber &=&\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\Vert \left( \frac{1 }{n},\frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r\right\Vert } \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\{ \left( \overset{n }{\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{\underset{ i=1}{\sum }}{r_{i}^{2}}\right) -\left( \overset{n}{\underset{i=1}{\sum }}\frac{ 1}{n^{i}}r_{i}\right)^{2}\right\}^{\frac{1}{2}}} \nonumber \\ &\geq &\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\{ \left( \overset{n}{\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{ \underset{i=1}{\sum }}{r_{i}^{2}}\right) \right\}^{\frac{1}{2}}}=1 \end{eqnarray} (3.1) and \begin{eqnarray} &&\underset{n\rightarrow \infty }{\lim }\upsilon_{\mathbb{R}^{n}} \left( \left( \frac{1}{n}, \frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right) -T_{n}(u,0,0,\ldots,0,...),r;s\right) \nonumber \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{\mathbb{R}^{n}} \left( \left( \frac{1}{n}, \frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right) -\overset{n\text{-times}}{ \overbrace{(0,0,\ldots,0,0)}},r;s\right) \nonumber \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{\mathbb{R}^{n}} \left( \left( \frac{1}{n}, \frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r;s\right) \nonumber \\ \nonumber &=&\underset{n\rightarrow \infty }{\lim }\frac{\left\Vert \left( \frac{1}{n}, \frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r\right\Vert }{s+\left\Vert \left( \frac{1}{n},\frac{1}{n^{2}},\ldots,\frac{1}{n^{n}}\right),r\right\Vert }\\ &=&\underset{n\rightarrow \infty }{\lim }\frac{\left\{ \left( \overset{n}{ \underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{\underset{ i=1}{\sum }}{r_{i}^{2}}\right) -\left( \overset{n}{\underset{i=1}{\sum }}\frac{ 1}{n^{i}}r_{i}\right)^{2}\right\}^{\frac{1}{2}}}{s+\left\{ \left( \overset{ n}{\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{\underset{i=1}{\sum }}{r_{i}^{2}}\right) -\left( \overset{n}{\underset{i=1}{\sum }} \frac{1}{n^{i}}r_{i}\right)^{2}\right\}^{\frac{1}{2}}} \nonumber \\ &\leq& \underset{n\rightarrow \infty }{\lim }\frac{\left\{ \left( \overset{n} {\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{\underset{ i=1}{\sum }}{r_{i}^{2}}\right) \right\}^{\frac{1}{2}}}{s+\left\{ \left( \overset{n}{\underset{i=1}{\sum }}\frac{1}{n^{2i}}\right) \left( \overset{n}{ \underset{i=1}{\sum }}{r_{i}^{2}}\right) -\left( \overset{n}{\underset{i=1}{ \sum }}\frac{1}{n^{i}}r_{i}\right)^{2}\right\}^{\frac{1}{2}}}=0. \end{eqnarray} (3.2) Equations (3.1) and (3.2) say that $$w_{n}\overset{\mathrm T_{\mathrm{IF-2-N}}}{\rightarrow }(u,0,0,\ldots ,0,...).$$ As can be seen, $$\left ( w_{n}\right ) $$ has infinitely many intuitionistic fuzzy T-limits. This means that approximation of V to $$T_{n}V$$ is very bad. Now, we define a property guaranteeing uniqueness of limit of intuitionistic fuzzy T-convergence. Definition 3.3 The IF-2-Ns in $$W_{n}$$ are called to be intuitionistic fuzzy regular if $$\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left ( T_{n}z,r;s\right ) =1$$ and $$\underset{n\rightarrow \infty }{\lim } \upsilon_{W_{n}} \left ( T_{n}z,r;s\right ) =0$$ imply z = 0 for all $$r\in W_{n}$$ and s > 0. Theorem 3.4 IF-2-Ns in $$W_{n}$$ are regular if and only if intuitionistic fuzzy T-limit is unique. Proof. ⟹ : Suppose that $$\left ( w_{n}\right ) $$ is T-convergent to $$z^{\prime}$$ and $$z^{\prime\prime}.$$ In this case, $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left(w_{n}-T_{n}z^{\prime },r;s\right) =1\textrm{ and}\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\left( w_{n}-T_{n}z,r;s\right) =0 $$ $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( w_{n}-T_{n}z^{\prime \prime },r;s\right) =1\textrm{ and}\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( w_{n}-T_{n}z,r;s\right) =0 $$ for all $$r\in w_{n}$$ and s > 0. Using these arguments, we obtain \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( T_{n}(z^{\prime }-z^{\prime \prime }),r;s\right) &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( T_{n}z^{\prime }-T_{n}z^{\prime \prime },r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( T_{n}z^{\prime }-w_{n}+w_{n}-T_{n}z^{\prime \prime },r;s\right) \\ &\geq &\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( T_{n}z^{\prime }-w_{n},r;\frac{s}{2}\right) \ast \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( w_{n}-T_{n}z^{\prime \prime },r;\frac{s}{2}\right) \\ &=&1\ast 1=1 \end{eqnarray*} and \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( T_{n}\big(z^{\prime }-z^{\prime \prime }\big),r;s\right) &=&\underset{n\rightarrow \infty }{\lim } \upsilon_{W_{n}} \left( T_{n}z^{\prime }-T_{n}z^{\prime \prime },r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( T_{n}z^{\prime }-w_{n}+w_{n}-T_{n}z^{\prime \prime },r;s\right) \\ &\leq &\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( T_{n}z^{\prime }-w_{n},r;\frac{s}{2}\right) \Diamond \underset{n\rightarrow \infty }{\lim } \upsilon_{W_{n}} \left( w_{n}-T_{n}z^{\prime \prime },r;\frac{s}{2}\right) \\ &=&0\Diamond 0=0. \end{eqnarray*} Since the IF-2-Ns in $$W_{n}$$ are regular, $$z^{\prime }-z^{\prime \prime }$$ is equal to 0. That is, $$z^{\prime }=z^{\prime \prime }.$$ ⇐: Now, suppose that intuitionistic fuzzy T-limit is unique. In this case, since $$\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}}\! \left (T_{n}z,r;s\right )=1$$ and $$\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\! \left ( T_{n}z,r;s\right ) =0$$ for all $$r\in W_{n},$$ we have $$z_{n}\overset{ \text T_{\mathrm{IF-2-N}}}{\rightarrow }0.$$ From the linearity of T, we get z = 0. It means that IF-2-Ns of $$W_{n}$$ are regular. Theorem 3.5 Let $$z_{n},w_{n}$$$$\in W_{n}$$; z, w ∈ V and $$\alpha $$ be a scalar. If $$z_{n}\overset{\text T_{\mathrm{IF-2-N}}}{\rightarrow }z$$ and $$w_{n}\overset{\text T_{\mathrm{IF-2-N}}}{\rightarrow }w,$$ then (a) $$\left ( \alpha z_{n}\right ) \overset{\text T_{\mathrm{IF-2-N}}}{\rightarrow }\left ( \alpha z\right ) $$ for each $$\alpha ,$$ (b) $$\left ( z_{n}+w_{n}\right ) \overset{\text T_{\mathrm{IF-2-N}}}{\rightarrow }\left ( z+w\right )\!.$$ Proof. (a) Using $$z_{n}\overset{\text T_{\mathrm{IF-2-N}}}{ \rightarrow }z,$$ we obtain \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \alpha z_{n}-T_{n}\left( \alpha z\right),r;s\right) &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \alpha z_{n}-\alpha T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \alpha \left( z_{n}-T_{n}z\right),r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( z_{n}-T_{n}z,r;\frac{s}{ \left\vert \alpha \right\vert }\right) \\ &=&1 \end{eqnarray*} and \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( \alpha z_{n}-T_{n}\left( \alpha z\right),r;s\right) &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( \alpha z_{n}-\alpha T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( \alpha \left( z_{n}-T_{n}z\right),r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( z_{n}-T_{n}z,r; \frac{s}{\left\vert \alpha \right\vert }\right) \\ &=&0. \end{eqnarray*} (b) Using by $$z_{n}\overset{\text T_{\mathrm{IF-2-N}}}{ \rightarrow }z$$ and $$w_{n}\overset{\text T_{\mathrm{IF-2-N}}}{ \rightarrow }w,$$ we obtain \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \left( z_{n}+w_{n}\right) -T_{n}(z+w),r;s\right)\!\!\!\!\! &=&\!\!\!\!\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( \left( z_{n}+w_{n}\right) -\left( T_{n}z+T_{n}w\right),r;s\right) \\ &\geq&\!\!\!\!\!\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( z_{n}-T_{n}z,r;\frac{ s}{2}\right)\ast \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}} \left( w_{n}-T_{n}w,r;\frac{s}{2}\right) \\ &=&\!\!\!\!\!1\ast 1=1 \end{eqnarray*} and \begin{eqnarray*} \underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( \left( z_{n}+w_{n}\right) -T_{n}(z+w),r;s\right)\!\!\!\!\! &=&\!\!\!\!\!\underset{n\rightarrow \infty }{ \lim }\upsilon_{W_{n}} \left( \left( z_{n}+w_{n}\right) -\left( T_{n}z+T_{n}w\right) ,r;s\right) \\ &\leq&\!\!\!\!\!\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( z_{n}-T_{n}z,r; \frac{s}{2}\right) \Diamond \underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}} \left( w_{n}-T_{n}w,r;\frac{s}{2}\right) \\ &=&\!\!\!\!\!0\Diamond 0=0. \end{eqnarray*} Definition 3.6 For all z, r ∈ V and $$w\in W_{n}$$ if $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}}\left( T_{n}z,w;s\right) =\mu_{V}\left( z,r;s\right) \textrm{ and}\underset{ n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\left( T_{n}z,w;s\right) =\upsilon_{V}\left( z,r;s\right) $$ then IF-2-Ns in $$W_{n}$$ are called consistent with intuitionistic fuzzy norm $$\left ( \mu_{V} ,\upsilon_{V} \right )$$ of the space V. Theorem 3.7 If IF-2-Ns in $$W_{n}$$ are consistent with $$\left ( \mu_{V} ,\upsilon_{V} \right )$$ of the space V, then IF-2-Ns in $$W_{n}$$ are intuitionistic fuzzy regular. Proof. Since IF-2-Ns in $$W_{n}$$ are consistent with $$\left ( \mu_{V} ,\upsilon_{V} \right )$$ of the space V, $$ \underset{n\rightarrow \infty }{\lim }\mu_{W_{n}}\left( T_{n}z,w;s\right) =\mu_{V}\left( z,r;s\right) =1\textrm{ and}\underset{ n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\left( T_{n}z,r;s\right) =\upsilon_{V}\left( z,r;s\right) =0. $$ For each z ∈ V and all $$w\in W_{n},r\in V,$$$$\mu_{V}\left ( z,r;s\right ) =1$$ and $$\upsilon_{V}\left ( z,r;s\right ) =0$$ if and only if z and r are linear dependent. Since each z is linear dependent with any r, z is equal to 0. Thus, we can get the intuitionistic fuzzy regularity of the space $$W_{n}.$$ Example 3.8 Consider $$V=C\left [ 0,n\right ] =\{ z(s):[ 0,n] \ \rightarrow \mathbb{R} :z(s)$$ is continuous on [0, n]}. Let $$T_{n}\left ( z(s)\right ) =z({s_{k}^{n}})_{k=1}^{n}$$ for $$0\leq{s_{1}^{n}}<{s_{2}^{n}}<...<{s_{n}^{n}}\leq n.$$ In this case \begin{eqnarray*} T_{2}\left( z(s)\right) &=&z\big({s_{k}^{2}}\big)_{k=1}^{2}=\left( z\big({s_{1}^{2}}\big),z\big({s_{2}^{2}}\big)\right) \\ T_{3}\left( z(s)\right) &=&z\big({s_{k}^{3}}\big)_{k=1}^{3}=\left( z\big({s_{1}^{3}}\big),z\big({s_{2}^{3}}\big),z\big({s_{3}^{3}}\big)\right) \\ &&\vdots \\ T_{n}\left( z(s)\right) &=&z\big({s_{k}^{n}}\big)_{k=1}^{n}=\left( z\big({s_{1}^{n}}\big),z\big({s_{2}^{n}}\big),z\big({s_{3}^{n}}\big),\ldots,z\big({s_{n}^{n}}\big)\right) . \end{eqnarray*} That is $$W_{n}=\mathbb{R} ^{n}.$$ Since $$\left \Vert z,w\right \Vert =\left \vert \begin{array}{cc} {\int \limits _{0}^{n}}z(s)z(s)\,\mathrm{d}s & {\int \limits _{0}^{n}}z(s)w(s)\,\mathrm{d}s \\{\int \limits _{0}^{n}}z(s)w(s)\,\mathrm{d}s & {\int \limits _{0}^{n}}w(s)w(s)\,\mathrm{d}s \end{array} \right \vert ^{\frac{1}{2}}$$ is a 2-norm on $$V\!\!=\!\!C\left [ 0,n\right ],\ \left ( \mu_{V},\upsilon_{V}\right ) $$ is a IF-2-N on $$ V=C\left [ 0,n\right ] $$ with x * y = xy and $$x\Diamond y=min\left \{ x,y\right \} $$ for all $$x,y\in \left [ 0,1\right ] $$ where $$\mu_{V}\left ( z,w;s\right ) =\frac{s}{s+\left \Vert z,w\right \Vert }$$ and $$\upsilon_{V}\left ( z,w;s\right ) =\frac{\left \Vert z,w\right \Vert }{s+\left \Vert z,w\right \Vert }.$$ Also, take into consideration $$\left ( \mathbb{R} ^{n},\mu ,\upsilon ,\ast ,\Diamond \right ) $$ in Example 3.2. $$ {s_{1}^{n}},{s_{2}^{n}},\ldots ,{s_{n}^{n}}$$ is uniform partition of $$\left [ 0,n\right ] .$$ That is $$\triangle s_{k}=s_{k+1}-s_{k}=$$ 1 and so $$\overset{n}{\underset{k=1}{\sum }}\left ( z\big ({s_{k}^{n}}\big )\right ) \triangle s_{k}=\overset{n}{\underset{k=1}{\sum }}\left ( z\big ({s_{k}^{n}}\big )\right ) .$$ Since \begin{eqnarray*} &&\underset{n\rightarrow \infty }{\lim }\mu_{W_{n}}\left( T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\mu_{\mathbb{R}^{n}}\left( T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\{ \left( \overset{n }{\underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\right) \left( \overset{n}{\underset{k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right)^{2}\right) -\left( \overset{n}{\underset{k=1}{\sum }}z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\right) ^{2}\right\}^{\frac{1}{2}}} \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{s}{s+\left\{ \left( \overset{n }{\underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) \left( \overset{n}{\underset{k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) -\left( \overset{n}{\underset{ k=1}{\sum }}z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\triangle s_{k}\right)^{2}\right\}^{ \frac{1}{2}}} \\ &=&\frac{s}{s+\left\{ {\int\limits_{0}^{n}}z(s)z(s)\,\mathrm{d}s{\int\limits_{0}^{n}}r(s)r(s)\,\mathrm{d}s-\left( {\int\limits_{0}^{n}}z(s)r(s)\, \mathrm{d}s\right)^{2}\right\}^{\frac{1}{2}}} \\ &=&\mu_{V}\left( z,w;s\right) \end{eqnarray*} and \begin{eqnarray*} &&\underset{n\rightarrow \infty }{\lim }\upsilon_{W_{n}}\left( T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\upsilon_{\mathbb{R}^{n}}\left( T_{n}z,r;s\right) \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{\left\{ \left( \overset{n}{ \underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\right) \left( \overset{ n}{\underset{k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right) \right) -\left( \overset{ n}{\underset{k=1}{\sum }}z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\right)^{2}\right\}^{ \frac{1}{2}}}{s+\left\{ \left( \overset{n}{\underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\right) \left( \overset{n}{\underset{k=1}{\sum }} \left( r\big({s_{k}^{n}}\big)\right) \right) -\left( \overset{n}{\underset{k=1}{\sum }} z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\right)^{2}\right\}^{\frac{1}{2}}} \\ &=&\underset{n\rightarrow \infty }{\lim }\frac{\left\{ \left( \overset{n}{ \underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) \left( \overset{n}{\underset{k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right) ^{2}\triangle s_{k}\right) -\left( \overset{n}{\underset{k=1}{\sum }} z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\triangle s_{k}\right)^{2}\right\}^{\frac{1}{2}}}{ s+\left\{ \left( \overset{n}{\underset{k=1}{\sum }}\left( z\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) \left( \overset{n}{\underset{ k=1}{\sum }}\left( r\big({s_{k}^{n}}\big)\right)^{2}\triangle s_{k}\right) -\left( \overset{n}{\underset{k=1}{\sum }}z\big({s_{k}^{n}}\big)r\big({s_{k}^{n}}\big)\triangle s_{k}\right)^{2}\right\}^{\frac{1}{2}}} \\ &=&\frac{\left\{ {\int\limits_{0}^{l}}z(s)z(s)\,\mathrm{d}s{\int\limits_{0}^{l}}w(s)w(s)\,\mathrm{d}s-\left( {\int\limits_{0}^{l}}z(s)w(s)\,\mathrm{d}s\right)^{2}\right\}^{\frac{1}{2}}}{s+\left\{ {\int\limits_{0}^{l}}z(s)z(s)\,\mathrm{d}s{\int\limits_{0}^{l}}w(s)w(s)\,\mathrm{d}s-\left( {\int\limits_{0}^{l}}z(s)w(s)\,\mathrm{d}s\right)^{2}\right\}^{\frac{1}{2}}} \\ &=&\upsilon_{V}\left( z,w;s\right)\!, \end{eqnarray*} IF-2-Ns in $$ \mathbb{R} ^{n}$$ are consistent with IF-2-N of the space $$V=C \left [ 0,n\right ]\!.$$ Funding This work is supported by The Scientific and Technological Research Council of Turkey under the Project number 110T699. 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Logic Journal of the IGPLOxford University Press

Published: Apr 12, 2018

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