 Research
 Open Access
 Published:
Best proximity point theorems for αψproximal contractions in intuitionistic fuzzy metric spaces
Journal of Inequalities and Applications volume 2014, Article number: 352 (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.
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
Definition 1 A binary operation \ast :[0,1]\times [0,1]\to [0,1] is a continuous tnorm if ∗ satisfies the following conditions:

(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.
Definition 2 A binary operation \diamond :[0,1]\times [0,1]\to [0,1] is a continuous tconorm if ⋄ satisfies the following conditions:

(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\}.
Definition 3 A 5tuple (X,M,N,\ast ,\diamond ) is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, ∗ is a continuous tnorm, ⋄ is a continuous tconorm and M, N are fuzzy sets on {X}^{2}\times (0,\mathrm{\infty}) satisfying the following conditions, for all x,y,z\in X and t,s>0:

(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.
Definition 4 Let (X,M,N,\ast ,\diamond ) be an intuitionistic fuzzy metric space. Then

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.
Let (X,M,N,\ast ,\diamond ) be an intuitionistic fuzzy metric space. The fuzzy metric (M,N) is called triangular whenever,
and
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.
Definition 7 Let T be a selfmapping on X and \alpha :X\times X\to [0,+\mathrm{\infty}) be a function. We say that T is an αadmissible mapping if
Salimi et al. [35] generalized the notion of αadmissible mappings in the following ways.
Definition 8 [35]
Let T be a selfmapping on X and \alpha ,\eta :X\times X\to [0,+\mathrm{\infty}) be two functions. We say that T is an αadmissible mapping with respect to η if
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]
A nonselfmapping T:A\to B is called αηproximal admissible if
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.
Let A and B be nonempty subsets of an intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ). We denote by {A}_{0}(t) and {B}_{0}(t) the following sets:
where M(A,B,t)=sup\{M(x,y,t):x\in A,y\in B\}.
Definition 10 Let A and B be two nonempty subsets of intuitionistic fuzzy metric spaces (X,M,N,\ast ,\diamond ). Let, T:A\to B, \alpha :A\times A\times (0,\mathrm{\infty})\to [0,\mathrm{\infty}). We say that T is αproximal admissible if for {x}_{1},{x}_{2},{u}_{1},{u}_{2}\in A with
for all t>0.
Let A and B be nonempty subsets of an intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ) and T:A\to B be a nonselfmapping. We define {\mathcal{M}}^{T}(x,y,u,v,t) and {\mathcal{N}}^{T}(x,y,u,v,t) as follows:
and
Definition 11 Let A and B be nonempty subsets of an intuitionistic fuzzy metric spaces (X,M,N,\ast ,\diamond ). Let T:A\to B be a nonselfmapping and \alpha :A\times A\times (0,\mathrm{\infty})\to [0,\mathrm{\infty}) be a function. We say T is a αψproximal contractive mapping if for x,y,u,v\in A,
holds for all t>0, where \psi \in \mathrm{\Psi}.
Theorem 1 Let A and B be nonempty subsets of a complete triangular intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ) such that {A}_{0}(t) is nonempty for all t>0. Let T:A\to B be a tuniformly continuous nonselfmapping satisfying the following assertions:

(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.
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.

(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.
Proof By condition (iv) there exist elements {x}_{0} and {x}_{1} in {A}_{0}(t) such that
On the other hand T({A}_{0}(t))\subseteq {B}_{0}(t), so there exists {x}_{2}\in {A}_{0}(t) such that
Now, since T is αproximal admissible mapping, so we have \alpha ({x}_{1},{x}_{2},t)\ge t. That is,
Again, since T({A}_{0}(t))\subseteq {B}_{0}(t), there exists {x}_{3}\in {A}_{0}(t) such that
Thus we have
Again since T is αproximal admissible mapping, so \alpha ({x}_{2},{x}_{3},t)\ge t. Hence,
Continuing this process, we get
for all n\in \mathbb{N}\cup \{0\} and all t>0.
Now from (3.2) with u=y={x}_{n}, v={x}_{n+1} and x={x}_{n1}, we get
for all t>0 and all n\in \mathbb{N} where
This implies
Also we have
Thus, from (3.4), (3.5), and (3.6) we have
Now if max\{\frac{1}{M({x}_{n1},{x}_{n},t)},\frac{1}{M({x}_{n},{x}_{n+1},t)}\}=\frac{1}{M({x}_{n},{x}_{n+1},t)}, then we get
which is a contradiction. Hence,
for all n\in \mathbb{N} and t>0. So we deduce
for all n\in \mathbb{N} and t>0. Fix \u03f5>0. Then there exists N\in \mathbb{N} such that
Let m,n\in \mathbb{N} with m>n\ge N. Then by triangular inequality we get
Consequently, {lim}_{m,n\to \mathrm{\infty}}[\frac{1}{M({x}_{n},{x}_{m},t)}1]=0, i.e., {lim}_{m,n\to \mathrm{\infty}}M({x}_{n},{x}_{m},t)=1. Hence \{{x}_{n}\} is a Cauchy sequence. Now, since (X,M,N,\ast ,\diamond ) is a complete intuitionistic fuzzy metric space, so there exists {x}^{\ast}\in X such that {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}. Since T is tuniformly continuous, so by Lemmas 1 and 2, we have
That is, {x}^{\ast} is a best proximity of T. We show that {x}^{\ast} is unique best proximity point of T. Assume, to the contrary, that there exists {t}_{0}>0 such that 0<M({x}^{\ast},w,{t}_{0})<1 and w\ne {x}^{\ast} is another best proximity point of T, that is, M({x}^{\ast},T{x}^{\ast},t)=M(A,B,t) and M(w,Tw,t)=M(A,B,t) for all t>0. Now if condition (v) holds, then, from (3.2), we have
where
and
Therefore,
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. □
Theorem 2 Let A and B be nonempty subsets of a complete triangular intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ) such that {A}_{0}(t) is nonempty for all t>0. Let T:A\to B be a nonselfmapping satisfying the following assertions:

(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.
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.

(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.
Proof Following the same lines in the proof of Theorem 1, we can construct a sequence \{{x}_{n}\} in {A}_{0}(t) satisfying
and {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}, that is, {lim}_{n\to +\mathrm{\infty}}M({x}_{n},{x}^{\ast},t)=1, for all t>0. Moreover,
This implies
Passing to the limit as n\to +\mathrm{\infty} in the above inequality, we get
that is,
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\}.
Suppose there exists {t}_{0}>0 such that M({x}^{\ast},z,{t}_{0})<1. Then from (3.2) with x={x}_{n}, y={x}^{\ast}, u={x}_{n+1}, and v=z we get
On the other hand we know that
and
Now by taking the limit as n\to \mathrm{\infty} in (3.8) we get
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. □
Example 1 Let X=\mathbb{R} be endowed with the usual metric d(x,y)=xy. Consider M(x,y,t)=\frac{t}{t+d(x,y)} and N(x,y,t)=\frac{d(x,y)}{t+d(x,y)} for all x,y\in X and all t>0. Moreover, consider A=(\mathrm{\infty},1], B=[1,+\mathrm{\infty}) and define T:A\to B by
Also, define \alpha :X\times X\times (0,\mathrm{\infty})\to [0,+\mathrm{\infty}) by
and \psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) by
Clearly, M(A,B,t)=sup\{M(x,y,t)\phantom{\rule{0.25em}{0ex}}x\in A,y\in B\}=\frac{t}{t+2}. Hence,
It is immediate to show that T({A}_{0}(t))\subseteq {B}_{0}(t) for all t>0, M(1,T(1),t)=M(A,B,t) and \alpha (1,1,t)\ge t. Suppose
then
Hence, u=v=1, that is, \alpha (u,v,t)\ge t. Therefore T is an αproximal admissible mapping. Further,
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.
Theorem 3 Let A and B be nonempty subsets of a complete triangular intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ) such that {A}_{0}(t) is nonempty for all t>0. Let T:A\to B be a tuniformly continuous nonselfmapping. Assume that following assertions hold true:

(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)
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{and}}\phantom{\rule{1em}{0ex}}\alpha ({x}_{0},{x}_{1},t)\ge t\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}t0.
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.

(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.
Proof Following the same lines in the proof of Theorem 1, we can construct a sequence \{{x}_{n}\} in {A}_{0}(t) satisfying
From (ii) with u=y={x}_{n}, v={x}_{n+1} and x={x}_{n1}, we get
As in the proof of Theorem 2.2 of [27], we deduce that \{{x}_{n}\} is a Cauchy sequence. The completeness of (X,M,N,\ast ,\diamond ) ensures that the sequence \{{x}_{n}\} converges to some {x}^{\ast}\in X, that is {lim}_{n\to +\mathrm{\infty}}M({x}_{n},{x}^{\ast},t)=1. Since T is tuniformly continuous, so by Lemmas 1 and 2, we have
That is, {x}^{\ast} is a best proximity of T. Now we show that {x}^{\ast} is unique best proximity point of T. Suppose, to the contrary, that there exists {t}_{0}>0 such that 0<M({x}^{\ast},w,{t}_{0})<1 and w\ne {x}^{\ast} is another best proximity point of T, that is, M({x}^{\ast},T{x}^{\ast},t)=M(A,B,t) and M(w,Tw,t)=M(A,B,t) for all t>0. Now if condition (v) holds, then, from (ii), we have
which is a contradiction. Hence, w={x}^{\ast}. That is, T has a unique best proximity point. □
Theorem 4 Let A and B be nonempty subsets of a complete triangular intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ) such that {A}_{0}(t) is nonempty for all t>0. Let T:A\to B be a nonselfmapping. Assume that the following assertions hold true:

(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.
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.

(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.
Proof Following the same lines in the proof of Theorem 3, we can construct a sequence \{{x}_{n}\} in {A}_{0}(t) satisfying
{x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}, and there exists z\in {A}_{0}(t) such that M(z,T{x}^{\ast},t)=M(A,B,t). Also, \alpha ({x}_{n},x,t)\ge t for all n and all t>0. Then from (ii) with x={x}_{n}, y={x}^{\ast}, u={x}_{n+1} and v=z we get
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.
Definition 12 Let A, B be two nonempty closed subsets of a partially ordered intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ,\u2aaf). Then T:A\to B is said to be a proximally orderpreserving, if for all x,y,u,v\in A,
holds for all t>0.
Theorem 5 Let A and B be nonempty subsets of a partially ordered complete triangular intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ,\u2aaf) such that {A}_{0}(t) is nonempty for all t>0. Let T:A\to B be a tuniformly continuous nonselfmapping satisfying the following assertions:

(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)
holds for all t>0, where \psi \in \mathrm{\Psi};

(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.
Proof Define \alpha :A\times A\times (0,\mathrm{\infty})\to [0,+\mathrm{\infty}) by
At first we prove that T is an αproximal admissible mapping. For this assume that
So
Now, since T is proximally orderpreserving so, u\u2aafv. That is, \alpha (u,v,t)\ge t which implies that T is αproximal admissible. Condition (ii) implies that T is αψproximal contractive mapping. Further by (iv) we have
Therefore all conditions of Theorem 1 hold and T has a best proximity point. □
Theorem 6 Let A and B be nonempty subsets of a partially ordered complete triangular intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ,\u2aaf) such that {A}_{0}(t) is nonempty for all t>0. Let T:A\to B be a nonselfmapping satisfying the following assertions:

(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.
Theorem 7 Let A and B be nonempty subsets of a partially ordered complete triangular intuitionistic fuzzy metric space (X,M,N,\ast ,\diamond ,\u2aaf) such that {A}_{0}(t) is nonempty for all t>0. Let T:A\to B be a tuniformly continuous nonselfmapping. Also suppose that the following assertions hold true:

(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)
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}.
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.
Definition 13 Let (X,M,N,\ast ,\diamond ) be an intuitionistic fuzzy metric space, T:X\to X and \alpha :X\times X\times (0,\mathrm{\infty})\to [0,\mathrm{\infty}). We say, T is an αadmissible mapping if
for all t>0.
Let (X,M,N,\ast ,\diamond ) be an intuitionistic fuzzy metric space, T:X\to X be a selfmapping. We define {\mathcal{M}}^{T}(x,y,t) and {\mathcal{N}}^{T}(x,y,t) as follows:
and
Definition 14 Let (X,M,N,\ast ,\diamond ) be an intuitionistic fuzzy metric space. Let T:X\to X be a selfmapping and \alpha :X\times X\times (0,\mathrm{\infty})\to [0,\mathrm{\infty}) be a function. We say T is an αψcontractive mapping if
holds for all t>0, where \psi \in \mathrm{\Psi}.
Theorem 9 Let (X,M,N,\ast ,\diamond ) be a complete triangular intuitionistic fuzzy metric space. Let T:X\to X be a tuniformly continuous selfmapping. Also suppose that the following assertions hold:

(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.
Then T has a fixed point.

(iv)
Moreover, if x,y\in Fix(T) implies \alpha (x,y,t)\ge t, then T has a unique fixed point.
Theorem 10 Let (X,M,N,\ast ,\diamond ) be a complete triangular intuitionistic fuzzy metric space. Let T:X\to X be a selfmapping. Also suppose that the following assertions hold:

(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.
Then T has a fixed point.

(v)
Moreover, if x,y\in Fix(T) implies \alpha (x,y,t)\ge t, then T has a unique fixed point.
Theorem 11 Let (X,M,N,\ast ,\diamond ) be a complete triangular intuitionistic fuzzy metric space. Let T:X\to X be a tuniformly continuous selfmapping. Also suppose that the following assertions hold:

(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}
for all t>0;

(iii)
there exists {x}_{0} in X such that \alpha ({x}_{0},T{x}_{0},t)\ge t.
Then T has a fixed point.

(iv)
Moreover, if x,y\in Fix(T) implies \alpha (x,y,t)\ge t, then T has a unique fixed point.
Theorem 12 Let (X,M,N,\ast ,\diamond ) be a complete triangular intuitionistic fuzzy metric space. Let T:X\to X be a selfmapping. Also suppose that the following assertions hold:

(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}
for all t>0;

(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.
Then T has a fixed point.

(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])
Let (X,M,N,\ast ,\diamond ) be a complete triangular intuitionistic fuzzy metric space. Let T:X\to X be a tuniformly continuous mapping satisfying
holds for all x,y\in X and all t>0. Then T has a fixed point.
Theorem 13 Let (X,M,N,\ast ,\diamond ,\u2aaf) be a partially ordered complete triangular intuitionistic fuzzy metric space. Let T:X\to X be a tuniformly continuous selfmapping. Also assume the following assertions hold true:

(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)
holds for all x,y\in X with x\u2aafy and t>0;

(iii)
there exists {x}_{0} in X such that {x}_{0}\u2aafT{x}_{0}.
Then T has a fixed point.
Theorem 14 Let (X,M,N,\ast ,\diamond ,\u2aaf) be a partially ordered complete triangular intuitionistic fuzzy metric space. Let T:X\to X be a selfmapping. Also assume the following assertions hold true:

(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)
holds for all x,y\in X with x\u2aafy and t>0;

(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.
References
Fan K: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 1969,112(3):234–240. 10.1007/BF01110225
AminiHarandi A: Best proximity points theorems for cyclic strongly quasicontraction mappings. J. Glob. Optim. 2013, 56: 1667–1674. 10.1007/s1089801299539
AminiHarandi A, Hussain N, Akbar F: Best proximity point results for generalized contractions in metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 164
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.2007.10.014
Hussain N, Kutbi MA, Salimi P: Best proximity point results for modified α  ψ proximal rational contractions. Abstr. Appl. Anal. 2013., 2013: Article ID 927457
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.2009.01.173
Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S00199958(65)90241X
Kramosil I, Michálek J: Fuzzy metric and statistical metric spaces. Kybernetika 1975, 11: 336–344.
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/01650114(88)900644
George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64: 395–399. 10.1016/01650114(94)901627
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/s1339801301426
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.
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 jnaa00110
Kadelburg Z, Radenović S: A note on some recent best proximity point results for nonself mappings. Gulf J. Math. 2013, 1: 36–41.
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 117
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 136
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.
Chauhan S, Bhatnagar S, Radenović S: Common fixed point theorems for weakly compatible mappings in fuzzy metric spaces. Matematiche 2013,LXVIII(I):87–98. 10.4418/2013.68.1.8
Shen Y, Qiu D, Chenc W: Fixed point theorems in fuzzy metric spaces. Appl. Math. Lett. 2012, 25: 138–141. 10.1016/j.aml.2011.08.002
Vetro C: Fixed points in weak nonArchimedean fuzzy metric spaces. Fuzzy Sets Syst. 2011, 162: 84–90. 10.1016/j.fss.2010.09.018
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.
Vetro C, Vetro P: Common fixed points for discontinuous mappings in fuzzy metric spaces. Rend. Circ. Mat. Palermo 2008, 57: 295–303. 10.1007/s1221500800227
Atanassov K: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20: 87–96. 10.1016/S01650114(86)800343
Park JH: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22: 1039–1046. 10.1016/j.chaos.2004.02.051
Alaca C, Turkoghlu D, Yildiz C: Fixed points in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2006, 29: 1073–1078. 10.1016/j.chaos.2005.08.066
Coker D: An introduction to intuitionistic fuzzy metric spaces. Fuzzy Sets Syst. 1997, 88: 81–89. 10.1016/S01650114(96)000760
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 168
Mohamad A: Fixedpoint theorems in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2007, 34: 1689–1695. 10.1016/j.chaos.2006.05.024
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.
Rafi M, Noorani MSM: Fixed point theorem on intuitionistic fuzzy metric spaces. Iran. J. Fuzzy Syst. 2006,3(1):23–29.
Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 314–334.
Samanta TK, Mohinta S: On fixedpoint theorems in intuitionistic fuzzy metric space I. Gen. Math. Notes 2011,3(2):1–12.
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/BF02872238
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.2011.10.014
Salimi P, Latif A, Hussain N: Modified α  ψ contractive mappings with applications. Fixed Point Theory Appl. 2013., 2013: Article ID 151
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 245872
Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–229. 10.1007/s1108300590185
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.
Hussain N, Taoudi MA: Krasnosel’skiitype fixed point theorems with applications to Volterra integral equations. Fixed Point Theory Appl. 2013., 2013: Article ID 196
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 265
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Latif, A., Hezarjaribi, M., Salimi, P. et al. Best proximity point theorems for αψproximal contractions in intuitionistic fuzzy metric spaces. J Inequal Appl 2014, 352 (2014). https://doi.org/10.1186/1029242X2014352
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029242X2014352
Keywords
 αproximal admissible mapping
 fuzzy αψproximal contractions
 best proximity point
 intuitionistic fuzzy ordered metric space