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 Open Access
Best proximity point theorems for αψproximal contractions in intuitionistic fuzzy metric spaces
 Abdul Latif^{1},
 Masoomeh Hezarjaribi^{2},
 Peyman Salimi^{3}Email author and
 Nawab Hussain^{1}
https://doi.org/10.1186/1029242X2014352
© Latif et al.; licensee Springer 2014
 Received: 28 February 2014
 Accepted: 5 September 2014
 Published: 16 September 2014
Abstract
The aim of this paper is to introduce and study certain new concepts of αψproximal contractions in an intuitionistic fuzzy metric space. Then we establish certain best proximity point theorems for such proximal contractions in intuitionistic fuzzy metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered intuitionistic fuzzy metric spaces. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain some recent fixed point theorems as special cases. Moreover, we discuss some illustrative examples to highlight the realized improvements.
MSC:47H10, 54H25.
Keywords
 αproximal admissible mapping
 fuzzy αψproximal contractions
 best proximity point
 intuitionistic fuzzy ordered metric space
1 Introduction
Many problems arising in different areas of mathematics, such as optimization, variational analysis, and differential equations, can be modeled as fixed point equations of the form $Tx=x$. If T is not a selfmapping, the equation $Tx=x$ could have no solutions and, in this case, it is of a certain interest to determine an element x that is in some sense closest to Tx. Fan’s best approximation theorem [1] asserts that if K is a nonempty compact convex subset of a Hausdorff locally convex topological vector space X and $T:K\to X$ is a continuous mapping, then there exists an element x satisfying the condition $d(x,Tx)=inf\{d(y,Tx):y\in K\}$, where d is a metric on X.
A best approximation theorem guarantees the existence of an approximate solution, a best proximity point theorem is contemplated for solving the problem to find an approximate solution which is optimal. Given the nonempty closed subsets A and B of X, when a nonselfmapping $T:A\to B$ has not a fixed point, it is quite natural to find an element ${x}^{\ast}$ such that $d({x}^{\ast},T{x}^{\ast})$ is minimum. Best proximity point theorems provide the existence of an element ${x}^{\ast}$ such that $d({x}^{\ast},T{x}^{\ast})=d(A,B):=inf\{d(x,y):x\in A\text{and}y\in B\}$; this element is called a best proximity point of T. Moreover, if the mapping under consideration is a selfmapping, we note that this best proximity theorem reduces to a fixed point. For more details, we refer to [2–6] and references therein.
The concept of fuzzy set was introduced by Zadeh [7] in 1965 and it is well known that there are many viewpoints of the notion of metric space in fuzzy topology. In 1975, Kramosil and Michálek [8] introduced the concept of a fuzzy metric space, which can be regarded as a generalization of the statistical (probabilistic) metric space. Clearly, this work provides an important basis for the construction of fixed point theory in fuzzy metric spaces. Afterwards, Grabiec [9] defined the completeness of the fuzzy metric space (now known as a Gcomplete fuzzy metric space) and extended the Banach contraction theorem to Gcomplete fuzzy metric spaces. Subsequently, George and Veeramani [10] modified the definition of the Cauchy sequence introduced by Grabiec. Meanwhile, they slightly modified the notion of a fuzzy metric space introduced by Kramosil and Michálek and then defined a Hausdorff and first countable topology. Since then, the notion of a complete fuzzy metric space presented by George and Veeramani (now known as an complete fuzzy metric space) has emerged as another characterization of completeness, and some fixed point theorems have also been constructed on the basis of this metric space. From the above analysis, we can see that there are many studies related to fixed point theory based on the above two kinds of complete fuzzy metric spaces; see [11–22] and the references therein. On the other hand the concept of intuitionistic fuzzy set was introduced by Atanassov [23] as generalization of fuzzy set. In 2004, Park introduced the notion of intuitionistic fuzzy metric space [24]. He showed that for each intuitionistic fuzzy metric space $(X,M,N,\ast ,\diamond )$, the topology generated by the intuitionistic fuzzy metric $(M,N)$ coincides with the topology generated by the fuzzy metric M. For more details on intuitionistic fuzzy metric space and related results we refer the reader to [24–31].
2 Mathematical preliminaries
 (1)
∗ is commutative and associative;
 (2)
∗ is continuous;
 (3)
$a\ast 1=a$ for all $a\in [0,1]$
 (4)
$a\ast b\le c\ast d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$.
Examples of tnorm are $a\ast b=min\{a,b\}$ and $a\ast b=ab$.
 (a)
⋄ is commutative and associative;
 (b)
⋄ is continuous;
 (c)
$a\diamond 0=a$ for all $a\in [0,1]$;
 (d)
$a\diamond b\le c\phantom{\rule{0.2em}{0ex}}\mathrm{\u25ca}\phantom{\rule{0.2em}{0ex}}d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in [0,1]$.
Examples of a tconorm are $a\diamond b=max\{a,b\}$ and $a\diamond b=min\{1,a+b\}$.
 (i)
$M(x,y,t)+N(x,y,t)\le 1$;
 (ii)
$M(x,y,0)=0$;
 (iii)
$M(x,y,t)=1$ for all $t>0$ if and only if $x=y$;
 (iv)
$M(x,y,t)=M(y,x,t)$;
 (v)
$M(x,y,t)\ast M(y,z,s)\le M(x,z,t+s)$;
 (vi)
$M(x,y,\cdot ):(0,\mathrm{\infty})\to [0,1]$ is left continuous;
 (vii)
${lim}_{t\to \mathrm{\infty}}M(x,y,t)=1$;
 (viii)
$N(x,y,0)=1$;
 (ix)
$N(x,y,t)=0$ if and only if $x=y$;
 (x)
$N(x,y,t)=N(y,x,t)$;
 (xi)
$N(x,y,t)\diamond N(y,z,s)\ge N(x,z,t+s)$;
 (xii)
$N(x,y,\cdot ):(0,\mathrm{\infty})\to [0,1]$ is right continuous;
 (xiii)
${lim}_{t\to \mathrm{\infty}}N(x,y,t)=0$.
Then $(M,N)$ is called an intuitionistic fuzzy metric on X. The functions $M(x,y,t)$ and $N(x,y,t)$ denote the degree of nearness and the degree of nonnearness between x and y with respect to t, respectively.
Remark 1 Note that, if $(M,N)$ is an intuitionistic fuzzy metric on X and $\{{x}_{n}\}$ be a sequence in X such that ${lim}_{m,n\to \mathrm{\infty}}M({x}_{n},{x}_{m},t)=1$, then ${lim}_{m,n\to \mathrm{\infty}}N({x}_{n},{x}_{m},t)=0$. Indeed, from (i) of Definition 3 we know that $M(x,y,t)+N(x,y,t)\le 1$ for all $x,y\in X$ and all $t>0$.

a sequence $\{{x}_{n}\}$ is said to be Cauchy sequence whenever ${lim}_{m,n\to \mathrm{\infty}}M({x}_{n},{x}_{m},t)=1$ and ${lim}_{m,n\to \mathrm{\infty}}N({x}_{n},{x}_{m},t)=0$ for all $t>0$;

a sequence $\{{x}_{n}\}$ is said to converge $x\in X$, if ${lim}_{m,n\to \mathrm{\infty}}M({x}_{n},x,t)=1$ and ${lim}_{m,n\to \mathrm{\infty}}N({x}_{n},x,t)=0$ for all $t>0$;

$(X,M,N,\ast ,\diamond )$ is called complete whenever every Cauchy sequence is convergent in X.
Definition 5 [28]
Let $(X,M,N,\ast ,\diamond )$ be an intuitionistic fuzzy metric space. We say the mapping $T:X\to X$ is tuniformly continuous if for each $0<\u03f5<1$, there exists $0<\delta <1$, such that $M(x,y,t)\ge 1\delta $ and $N(x,y,t)\le \delta $ implies $M(Tx,Ty,t)\ge 1\u03f5$ and $N(Tx,Ty,t)\le \u03f5$ for all $x,y\in X$ and for all $t>0$.
Lemma 1 [32]
Let $(X,M,N,\ast ,\diamond )$ be an intuitionistic fuzzy metric space and T be a tuniformly continuous mapping on X. If ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, then $T{x}_{n}\to Tx$ as $n\to \mathrm{\infty}$.
Lemma 2 [32]
Let $(X,M,N,\ast ,\diamond )$ be an intuitionistic fuzzy metric space. If ${x}_{n}\to x$ and ${y}_{n}\to y$ as $n\to \mathrm{\infty}$, then $M({x}_{n},{y}_{n},t)\to M(x,y,t)$ and $N({x}_{n},{y}_{n},t)\to N(x,y,t)$, $n\to \mathrm{\infty}$, for all $t>0$.
for all $x,y,z\in X$ and all $t>0$.
On the other hand, Samet et al. [34] defined the notion of αadmissible mappings as follows.
Salimi et al. [35] generalized the notion of αadmissible mappings in the following ways.
Definition 8 [35]
Note that if we take $\eta (x,y)=1$ then this definition reduces to Definition 7. Also, if we take, $\alpha (x,y)=1$ then we say that T is an ηsubadmissible mapping.
Definition 9 [5]
for all ${x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A$, where $\alpha ,\eta :A\times A\to [0,\mathrm{\infty})$. Also, if we take $\eta (x,y)=1$ for all $x,y\in A$ then we say T is an αproximal admissible mapping.
Clearly, if $A=B$, T is αproximal admissible implies that T is αadmissible.
3 Main results
In [34] the authors consider the family Ψ of nondecreasing functions $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ such that ${\sum}_{n=1}^{+\mathrm{\infty}}{\psi}^{n}(t)<+\mathrm{\infty}$ for each $t>0$, where ${\psi}^{n}$ is the n th iterate of ψ.
where $M(A,B,t)=sup\{M(x,y,t):x\in A,y\in B\}$.
for all $t>0$.
holds for all $t>0$, where $\psi \in \mathrm{\Psi}$.
 (i)
T is an αproximal admissible mapping and $T({A}_{0}(t))\subseteq {B}_{0}(t)$ for all $t>0$;
 (ii)
T is a αψproximal contractive mapping;
 (iii)
for any sequence $\{{y}_{n}\}$ in ${B}_{0}(t)$ and $x\in A$ satisfying $M(x,{y}_{n},t)\to M(A,B,t)$ as $n\to +\mathrm{\infty}$, then $x\in {A}_{0}(t)$ for all $t>0$;
 (iv)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}(t)$ such that$M({x}_{1},T{x}_{0},t)=M(A,B,t)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1},t)\ge t\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t0.$
 (v)
Moreover, if $M(x,Tx,t)=M(A,B,t)$, $M(y,Ty,t)=M(A,B,t)$ implies $\alpha (x,y,t)\ge t$ for all $t>0$, then T has a unique best proximity point.
for all $n\in \mathbb{N}\cup \{0\}$ and all $t>0$.
which is a contradiction. Hence, $M({x}^{\ast},w,{t}_{0})=1$ for all $t>0$. i.e., ${x}^{\ast}=w$. Thus T has unique best proximity point. □
 (i)
T is an αproximal admissible mapping and $T({A}_{0}(t))\subseteq {B}_{0}(t)$ for all $t>0$;
 (ii)
T is a αψproximal contractive mapping such that ψ is continuous;
 (iii)
for any sequence $\{{y}_{n}\}$ in ${B}_{0}(t)$ and $x\in A$ satisfying $M(x,{y}_{n},t)\to M(A,B,t)$ as $n\to +\mathrm{\infty}$, then $x\in {A}_{0}(t)$ for all $t>0$;
 (iv)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}(t)$ such that$M({x}_{1},T{x}_{0},t)=M(A,B,t)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1},t)\ge t;$
 (v)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1},t)\ge t$ for all $t>0$ and n with ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\alpha ({x}_{n},x,t)\ge t$ for all $t>0$ and all n.
 (vi)
Moreover, if $M(x,Tx,t)=M(A,B,t)$, $M(y,Ty,t)=M(A,B,t)$ implies $\alpha (x,y,t)\ge t$ for all $t>0$, then T has a unique best proximity point.
and so, by condition (iii), ${x}^{\ast}\in {A}_{0}(t)$. Since $T({A}_{0}(t))\subseteq {B}_{0}(t)$, then there exists $z\in {A}_{0}(t)$ such that $M(z,T{x}^{\ast},t)=M(A,B,t)$. Also from (iv) we have $\alpha ({x}_{n},{x}^{\ast},t)\ge t$ for all $n\in \mathbb{N}\cup \{0\}$.
which implies $\frac{1}{2M({x}^{\ast},z,{t}_{0})}<\frac{1}{2}$, i.e., $M({x}^{\ast},z,{t}_{0})>1$, which is a contradiction. Hence, $M({x}^{\ast},z,t)=1$ for all $t>0$. So, ${x}^{\ast}=z$. Therefore, T has a best proximity point. □
that is, T is an αψproximal contractive mapping. Moreover, if $\{{x}_{n}\}$ is a sequence such that $\alpha ({x}_{n},{x}_{n+1},t)\ge t$ for all $n\in \mathbb{N}\cup \{0\}$ and $t>0$ such that ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\{{x}_{n}\}\subseteq [2,1]$ and hence $x\in [2,1]$. Consequently, $\alpha ({x}_{n},x,t)\ge t$ for all $n\in \mathbb{N}\cup \{0\}$ and all $t>0$. Therefore all the conditions of Theorem 2 hold and T has a unique best proximity point. Here $z=1$ is the best proximity point of T.
 (i)
T is an αproximal admissible mapping and $T({A}_{0}(t))\subseteq {B}_{0}(t)$ for all $t>0$;
 (ii)for $x,y,u,v\in A$,$\begin{array}{c}\begin{array}{l}\alpha (x,y,t)\ge t,\\ M(u,Tx,t)=M(A,B,t),\\ M(v,Ty,t)=M(A,B,t)\end{array}\}\hfill \\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\frac{1}{M(u,v,t)}1\le \left(\frac{\frac{1}{M(x,v,t)}1+\frac{1}{M(y,u,t)}1}{\frac{1}{M(x,v,t)}1+\frac{1}{M(y,u,t)}1+\frac{1}{t}}\right)(\frac{1}{M(x,y,t)}1)\hfill \end{array}$(3.9)
 (iii)
for any sequence $\{{y}_{n}\}$ in ${B}_{0}(t)$ and $x\in A$ satisfying $M(x,{y}_{n},t)\to M(A,B,t)$ as $n\to +\mathrm{\infty}$, then $x\in {A}_{0}(t)$ for all $t>0$;
 (iv)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}(t)$ such that$M({x}_{1},T{x}_{0},t)=M(A,B,t)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1},t)\ge t\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t0.$
 (v)
Moreover, if $M(x,Tx,t)=M(A,B,t)$, $M(y,Ty,t)=M(A,B,t)$ implies $\alpha (x,y,t)\ge t$ for all $t>0$, then T has a unique best proximity point.
which is a contradiction. Hence, $w={x}^{\ast}$. That is, T has a unique best proximity point. □
 (i)
T is an αproximal admissible mapping and $T({A}_{0}(t))\subseteq {B}_{0}(t)$ for all $t>0$;
 (ii)
(3.9) holds for all $t>0$;
 (iii)
for any sequence $\{{y}_{n}\}$ in ${B}_{0}(t)$ and $x\in A$ satisfying $M(x,{y}_{n},t)\to M(A,B,t)$ as $n\to +\mathrm{\infty}$, then $x\in {A}_{0}(t)$ for all $t>0$;
 (iv)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}(t)$ such that$M({x}_{1},T{x}_{0},t)=M(A,B,t)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1},t)\ge t;$
 (v)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1},t)\ge t$ for all n and all $t>0$ such that ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\alpha ({x}_{n},x,t)\ge t$ for all n and all $t>0$.
 (vi)
Moreover, if $M(x,Tx,t)=M(A,B,t)$, $M(y,Ty,t)=M(A,B,t)$ imply $\alpha (x,y,t)\ge t$ for all $t>0$, then T has a unique best proximity point.
Taking the limit as $n\to \mathrm{\infty}$ in the above inequality we get $\frac{1}{M({x}^{\ast},z,t)}1=0$, i.e., ${x}^{\ast}=z$. Therefore ${x}^{\ast}$ is a best proximity point of T. Uniqueness follows similarly as in Theorem 3. □
4 Best proximity point results in partially ordered intuitionistic fuzzy metric space
Fixed point theorems for monotone operators in partially ordered metric spaces are widely investigated and have found various applications in differential and integral equations (see [36–40] and references therein). The aim of this section is to deduce certain new best proximity results in the context of partially ordered intuitionistic fuzzy metric spaces.
holds for all $t>0$.
 (i)
T is proximally orderpreserving and $T({A}_{0}(t))\subseteq {B}_{0}(t)$ for all $t>0$;
 (ii)for $x,y,u,v\in A$,$\begin{array}{c}\begin{array}{l}x\u2aafy,\\ M(u,Tx,t)=M(A,B,t),\\ M(v,Ty,t)=M(A,B,t)\end{array}\}\hfill \\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\frac{1}{M(u,v,t)}1\le \psi ({\mathcal{M}}^{T}(x,y,u,v,t){\mathcal{N}}^{T}(x,y,u,v,t))\hfill \end{array}$(4.1)
 (iii)
for any sequence $\{{y}_{n}\}$ in ${B}_{0}(t)$ and $x\in A$ satisfying $M(x,{y}_{n},t)\to M(A,B,t)$ as $n\to +\mathrm{\infty}$, then $x\in {A}_{0}(t)$ for all $t>0$;
 (iv)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}(t)$ such that$M({x}_{1},T{x}_{0},t)=M(A,B,t)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t0\mathit{\text{and}}{x}_{0}\u2aaf{x}_{1}.$
Then there exists ${x}^{\ast}\in A$ such that $M({x}^{\ast},T{x}^{\ast},t)=M(A,B,t)$, for all $t>0$, that is, T has a best proximity point ${x}^{\ast}\in A$.
Therefore all conditions of Theorem 1 hold and T has a best proximity point. □
 (i)
T is proximally orderpreserving and $T({A}_{0}(t))\subseteq {B}_{0}(t)$ for all $t>0$;
 (ii)
(4.1) holds for all $t>0$;
 (iii)
for any sequence $\{{y}_{n}\}$ in ${B}_{0}(t)$ and $x\in A$ satisfying $M(x,{y}_{n},t)\to M(A,B,t)$ as $n\to +\mathrm{\infty}$, then $x\in {A}_{0}(t)$ for all $t>0$;
 (iv)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}(t)$ such that$M({x}_{1},T{x}_{0},t)=M(A,B,t)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t0\mathit{\text{and}}{x}_{0}\u2aaf{x}_{1};$
 (v)
if $\{{x}_{n}\}$ is an increasing sequence in X such that ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all n.
Then there exists ${x}^{\ast}\in A$ such that $M({x}^{\ast},T{x}^{\ast},t)=M(A,B,t)$, for all $t>0$, that is, T has a best proximity point ${x}^{\ast}\in A$.
Proof Define $\alpha :A\times A\times (0,\mathrm{\infty})\to [0,+\mathrm{\infty})$ as in Theorem 5. Also, assume $\alpha ({x}_{n},{x}_{n+1},t)\ge t$ for all $n\in \mathbb{N}$ such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$. Then ${x}_{n}\u2aaf{x}_{n+1}$ for all $n\in \mathbb{N}$. Hence, by (v) we get ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$ and so $\alpha ({x}_{n},x,t)\ge t$ for all $n\in \mathbb{N}$ and all $t>0$. All other conditions can be proved as in the proof of Theorem 5. Thus all conditions of Theorem 2 hold and T has a best proximity point. □
Similarly from Theorems 3 and 4 we can deduce the following results.
 (i)
T is proximally orderpreserving and $T({A}_{0}(t))\subseteq {B}_{0}(t)$ for all $t>0$;
 (ii)for $x,y,u,v\in A$,$\begin{array}{c}\begin{array}{l}x\u2aafy,\\ M(u,Tx,t)=M(A,B,t),\\ M(v,Ty,t)=M(A,B,t)\end{array}\}\hfill \\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\frac{1}{M(u,v,t)}1\le \left(\frac{\frac{1}{M(x,v,t)}1+\frac{1}{M(y,u,t)}1}{\frac{1}{M(x,v,t)}1+\frac{1}{M(y,u,t)}1+\frac{1}{t}}\right)(\frac{1}{M(x,y,t)}1)\hfill \end{array}$(4.2)
 (iii)
for any sequence $\{{y}_{n}\}$ in ${B}_{0}(t)$ and $x\in A$ satisfying $M(x,{y}_{n},t)\to M(A,B,t)$ as $n\to +\mathrm{\infty}$, then $x\in {A}_{0}(t)$ for all $t>0$;
 (iv)there exist elements ${x}_{0}$ and ${x}_{1}$ in ${A}_{0}(t)$ such that$M({x}_{1},T{x}_{0},t)=M(A,B,t)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t0\mathit{\text{and}}{x}_{0}\u2aaf{x}_{1}.$
Then there exists ${x}^{\ast}\in A$ such that $M({x}^{\ast},T{x}^{\ast},t)=M(A,B,t)$, for all $t>0$, that is, T has a best proximity point ${x}^{\ast}\in A$.
Theorem 8 If in the above theorem, in place of tuniform continuity of T, we assume that for any increasing sequence $\{{x}_{n}\}$ in X and ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, we have ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$. Then there exists ${x}^{\ast}\in A$ such that $M({x}^{\ast},T{x}^{\ast},t)=M(A,B,t)$, for all $t>0$, that is, T has a best proximity point ${x}^{\ast}\in A$.
5 Application to fixed point theory
In this section we deduce new fixed point results in intuitionistic fuzzy metric space and ordered intuitionistic fuzzy metric space. Moreover, we derive certain recent fixed point results as corollaries to our best proximity results.
First we introduce the following concepts.
for all $t>0$.
holds for all $t>0$, where $\psi \in \mathrm{\Psi}$.
 (i)
T is an αadmissible mapping;
 (ii)
T is αψcontractive mapping;
 (iii)
there exists ${x}_{0}$ in X such that $\alpha ({x}_{0},T{x}_{0},t)\ge t$.
 (iv)
Moreover, if $x,y\in Fix(T)$ implies $\alpha (x,y,t)\ge t$, then T has a unique fixed point.
 (i)
T is an αadmissible mapping;
 (ii)
T is αψcontractive mapping;
 (iii)
there exists ${x}_{0}$ in X such that $\alpha ({x}_{0},T{x}_{0},t)\ge t$;
 (iv)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1},t)\ge t$ for all n and all $t>0$ with ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\alpha ({x}_{n},x,t)\ge t$ for all $n\in \mathbb{N}$ and all $t>0$.
 (v)
Moreover, if $x,y\in Fix(T)$ implies $\alpha (x,y,t)\ge t$, then T has a unique fixed point.
 (i)
T is an αadmissible mapping;
 (ii)$\begin{array}{c}x,y\in X,\phantom{\rule{1em}{0ex}}\alpha (x,y,t)\ge t\hfill \\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\frac{1}{M(Tx,Ty,t)}1\le \left(\frac{\frac{1}{M(x,Ty,t)}1+\frac{1}{M(y,Tx,t)}1}{\frac{1}{M(x,Ty,t)}1+\frac{1}{M(y,Tx,t)}1+\frac{1}{t}}\right)(\frac{1}{M(x,y,t)}1)\hfill \end{array}$
 (iii)
there exists ${x}_{0}$ in X such that $\alpha ({x}_{0},T{x}_{0},t)\ge t$.
 (iv)
Moreover, if $x,y\in Fix(T)$ implies $\alpha (x,y,t)\ge t$, then T has a unique fixed point.
 (i)
T is an αadmissible mapping;
 (ii)$\begin{array}{c}x,y\in X,\phantom{\rule{1em}{0ex}}\alpha (x,y,t)\ge t\hfill \\ \phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\frac{1}{M(Tx,Ty,t)}1\le \left(\frac{\frac{1}{M(x,Ty,t)}1+\frac{1}{M(y,Tx,t)}1}{\frac{1}{M(x,Ty,t)}1+\frac{1}{M(y,Tx,t)}1+\frac{1}{t}}\right)(\frac{1}{M(x,y,t)}1)\hfill \end{array}$
 (iii)
there exist elements ${x}_{0}$ in X such that $\alpha ({x}_{0},T{x}_{0},t)\ge t$;
 (iv)
if $\{{x}_{n}\}$ is a sequence in X such that $\alpha ({x}_{n},{x}_{n+1},t)\ge t$ for all n and all $t>0$ with ${x}_{n}\to x$ as $n\to +\mathrm{\infty}$, then $\alpha ({x}_{n},x,t)\ge t$ for all n and all $t>0$.
 (v)
Moreover, if $x,y\in Fix(T)$ implies $\alpha (x,y,t)\ge t$, then T has a unique fixed point.
By taking $\alpha (x,y,t)=t$ for all $x,y\in X$ and all $t>0$, we obtain the following corrected version of Theorem 2.2 in [27].
Corollary 1 (Theorem 2.2 of [27])
holds for all $x,y\in X$ and all $t>0$. Then T has a fixed point.
 (i)
T is an increasing mapping;
 (ii)assume$\frac{1}{M(Tx,Ty,t)}1\le \left(\frac{\frac{1}{M(x,Ty,t)}1+\frac{1}{M(y,Tx,t)}1}{\frac{1}{M(x,Ty,t)}1+\frac{1}{M(y,Tx,t)}1+\frac{1}{t}}\right)(\frac{1}{M(x,y,t)}1)$
 (iii)
there exists ${x}_{0}$ in X such that ${x}_{0}\u2aafT{x}_{0}$.
Then T has a fixed point.
 (i)
T is an increasing mapping;
 (ii)assume$\frac{1}{M(Tx,Ty,t)}1\le \left(\frac{\frac{1}{M(x,Ty,t)}1+\frac{1}{M(y,Tx,t)}1}{\frac{1}{M(x,Ty,t)}1+\frac{1}{M(y,Tx,t)}1+\frac{1}{t}}\right)(\frac{1}{M(x,y,t)}1)$
 (iii)
there exist elements ${x}_{0}$ in X such that ${x}_{0}\u2aafT{x}_{0}$;
 (iv)
if $\{{x}_{n}\}$ be an increasing sequence in X such that ${x}_{n}\to x$ as $n\to \mathrm{\infty}$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$.
Then T has a fixed point.
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first and fourth authors acknowledge with thanks DSR, KAU for financial support.
Authors’ Affiliations
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