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
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
References
 Fan K: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 1969,112(3):234–240. 10.1007/BF01110225MathSciNetView ArticleMATHGoogle Scholar
 AminiHarandi A: Best proximity points theorems for cyclic strongly quasicontraction mappings. J. Glob. Optim. 2013, 56: 1667–1674. 10.1007/s1089801299539MathSciNetView ArticleMATHGoogle Scholar
 AminiHarandi A, Hussain N, Akbar F: Best proximity point results for generalized contractions in metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 164Google Scholar
 Di Bari C, Suzuki T, Vetro C: Best proximity points for cyclic MeirKeeler contractions. Nonlinear Anal. 2008,69(11):3790–3794. 10.1016/j.na.2007.10.014MathSciNetView ArticleMATHGoogle Scholar
 Hussain N, Kutbi MA, Salimi P: Best proximity point results for modified α  ψ proximal rational contractions. Abstr. Appl. Anal. 2013., 2013: Article ID 927457Google Scholar
 Suzuki T, Kikkawa M, Vetro C: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 2009, 71: 2918–2926. 10.1016/j.na.2009.01.173MathSciNetView ArticleMATHGoogle Scholar
 Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S00199958(65)90241XMathSciNetView ArticleMATHGoogle Scholar
 Kramosil I, Michálek J: Fuzzy metric and statistical metric spaces. Kybernetika 1975, 11: 336–344.MathSciNetMATHGoogle Scholar
 Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/01650114(88)900644MathSciNetView ArticleMATHGoogle Scholar
 George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64: 395–399. 10.1016/01650114(94)901627MathSciNetView ArticleMATHGoogle Scholar
 Chauhan S, Radenović S, Imdad M, Vetro C: Some integral type fixed point theorems in nonArchimedean Menger PMspaces with common propertry (E.A) and application of functional equations in dynamic programming. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2014. 10.1007/s1339801301426Google Scholar
 Di Bari C, Vetro C: Fixed points, attractors and weak fuzzy contractive mappings in a fuzzy metric space. J. Fuzzy Math. 2005, 13: 973–982.MathSciNetMATHGoogle Scholar
 Gopal D, Imdad M, Vetro C, Hasan M: Fixed point theory for cyclic weak ϕ contraction in fuzzy metric spaces. J. Nonlinear Anal. Appl. 2012., 2012: Article ID jnaa00110Google Scholar
 Kadelburg Z, Radenović S: A note on some recent best proximity point results for nonself mappings. Gulf J. Math. 2013, 1: 36–41.Google Scholar
 Long W, Khaleghizadeh S, Selimi P, Radenović S, Shukla S: Some new fixed point results in partial ordered metric spaces via admissible mappings. Fixed Point Theory Appl. 2014., 2014: Article ID 117Google Scholar
 Saadati R, Kumam P, Jang SY: On the tripled fixed point and tripled coincidence point theorems in fuzzy normed spaces. Fixed Point Theory Appl. 2014., 2014: Article ID 136Google Scholar
 Salimi P, Vetro C, Vetro P: Some new fixed point results in nonArchimedean fuzzy metric spaces. Nonlinear Anal., Model. Control 2013,18(3):344–358.MathSciNetMATHGoogle Scholar
 Chauhan S, Bhatnagar S, Radenović S: Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces. Matematiche 2013,LXVIII(I):87–98. 10.4418/2013.68.1.8MathSciNetMATHGoogle Scholar
 Shen Y, Qiu D, Chenc W: Fixed point theorems in fuzzy metric spaces. Appl. Math. Lett. 2012, 25: 138–141. 10.1016/j.aml.2011.08.002MathSciNetView ArticleMATHGoogle Scholar
 Vetro C: Fixed points in weak nonArchimedean fuzzy metric spaces. Fuzzy Sets Syst. 2011, 162: 84–90. 10.1016/j.fss.2010.09.018MathSciNetView ArticleMATHGoogle Scholar
 Vetro C, Gopal D, Imdad M:Common fixed point theorem for $(\varphi ,\psi )$weak contractions in fuzzy metric spaces. Indian J. Math. 2010, 52: 573–590.MathSciNetMATHGoogle Scholar
 Vetro C, Vetro P: Common fixed points for discontinuous mappings in fuzzy metric spaces. Rend. Circ. Mat. Palermo 2008, 57: 295–303. 10.1007/s1221500800227MathSciNetView ArticleMATHGoogle Scholar
 Atanassov K: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20: 87–96. 10.1016/S01650114(86)800343MathSciNetView ArticleMATHGoogle Scholar
 Park JH: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22: 1039–1046. 10.1016/j.chaos.2004.02.051MathSciNetView ArticleMATHGoogle Scholar
 Alaca C, Turkoghlu D, Yildiz C: Fixed points in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2006, 29: 1073–1078. 10.1016/j.chaos.2005.08.066MathSciNetView ArticleMATHGoogle Scholar
 Coker D: An introduction to intuitionistic fuzzy metric spaces. Fuzzy Sets Syst. 1997, 88: 81–89. 10.1016/S01650114(96)000760MathSciNetView ArticleMATHGoogle Scholar
 Ionescu C, Rezapour S, Samei ME: Fixed points of some new contractions on intuitionistic fuzzy metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 168Google Scholar
 Mohamad A: Fixedpoint theorems in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2007, 34: 1689–1695. 10.1016/j.chaos.2006.05.024MathSciNetView ArticleMATHGoogle Scholar
 Park JS, Kwun YC, Park JH: A fixed point theorem in the intuitionistic fuzzy metric spaces. Far East J. Math. Sci. 2005, 16: 137–149.MathSciNetMATHGoogle Scholar
 Rafi M, Noorani MSM: Fixed point theorem on intuitionistic fuzzy metric spaces. Iran. J. Fuzzy Syst. 2006,3(1):23–29.MathSciNetMATHGoogle Scholar
 Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 314–334.MATHGoogle Scholar
 Samanta TK, Mohinta S: On fixedpoint theorems in intuitionistic fuzzy metric space I. Gen. Math. Notes 2011,3(2):1–12.MathSciNetMATHGoogle Scholar
 Di Bari C, Vetro C: A fixed point theorem for a family of mappings in a fuzzy metric space. Rend. Circ. Mat. Palermo 2003, 52: 315–321. 10.1007/BF02872238MathSciNetView ArticleMATHGoogle Scholar
 Samet B, Vetro C, Vetro P: Fixed point theorems for α  ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014MathSciNetView ArticleMATHGoogle Scholar
 Salimi P, Latif A, Hussain N: Modified α  ψ contractive mappings with applications. Fixed Point Theory Appl. 2013., 2013: Article ID 151Google Scholar
 Agarwal RP, Hussain N, Taoudi MA: Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations. Abstr. Appl. Anal. 2012., 2012: Article ID 245872Google Scholar
 Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–229. 10.1007/s1108300590185MathSciNetView ArticleMATHGoogle Scholar
 Hussain N, Khan AR, Agarwal RP: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces. J. Nonlinear Convex Anal. 2010,11(3):475–489.MathSciNetMATHGoogle Scholar
 Hussain N, Taoudi MA: Krasnosel’skiitype fixed point theorems with applications to Volterra integral equations. Fixed Point Theory Appl. 2013., 2013: Article ID 196Google Scholar
 Mohiuddine S, Alotaibi A: Coupled coincidence point theorems for compatible mappings in partially ordered intuitionistic generalized fuzzy metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 265Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.