Abstract
The aim of this paper is to introduce and study certain new concepts of α-ψ-proximal 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.
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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 . If T is not a self-mapping, the equation 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 is a continuous mapping, then there exists an element x satisfying the condition , 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 non-self-mapping has not a fixed point, it is quite natural to find an element such that is minimum. Best proximity point theorems provide the existence of an element such that ; this element is called a best proximity point of T. Moreover, if the mapping under consideration is a self-mapping, 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 G-complete fuzzy metric space) and extended the Banach contraction theorem to G-complete 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 , the topology generated by the intuitionistic fuzzy metric 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 is a continuous t-norm if ∗ satisfies the following conditions:
-
(1)
∗ is commutative and associative;
-
(2)
∗ is continuous;
-
(3)
for all
-
(4)
whenever and for all .
Examples of t-norm are and .
Definition 2 A binary operation is a continuous t-conorm if ⋄ satisfies the following conditions:
-
(a)
⋄ is commutative and associative;
-
(b)
⋄ is continuous;
-
(c)
for all ;
-
(d)
whenever and for all .
Examples of a t-conorm are and .
Definition 3 A 5-tuple is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm, ⋄ is a continuous t-conorm and M, N are fuzzy sets on satisfying the following conditions, for all and :
-
(i)
;
-
(ii)
;
-
(iii)
for all if and only if ;
-
(iv)
;
-
(v)
;
-
(vi)
is left continuous;
-
(vii)
;
-
(viii)
;
-
(ix)
if and only if ;
-
(x)
;
-
(xi)
;
-
(xii)
is right continuous;
-
(xiii)
.
Then is called an intuitionistic fuzzy metric on X. The functions and denote the degree of nearness and the degree of non-nearness between x and y with respect to t, respectively.
Remark 1 Note that, if is an intuitionistic fuzzy metric on X and be a sequence in X such that , then . Indeed, from (i) of Definition 3 we know that for all and all .
Definition 4 Let be an intuitionistic fuzzy metric space. Then
-
a sequence is said to be Cauchy sequence whenever and for all ;
-
a sequence is said to converge , if and for all ;
-
is called complete whenever every Cauchy sequence is convergent in X.
Definition 5 [28]
Let be an intuitionistic fuzzy metric space. We say the mapping is t-uniformly continuous if for each , there exists , such that and implies and for all and for all .
Lemma 1 [32]
Let be an intuitionistic fuzzy metric space and T be a t-uniformly continuous mapping on X. If as , then as .
Lemma 2 [32]
Let be an intuitionistic fuzzy metric space. If and as , then and , , for all .
Let be an intuitionistic fuzzy metric space. The fuzzy metric is called triangular whenever,
and
for all and all .
On the other hand, Samet et al. [34] defined the notion of α-admissible mappings as follows.
Definition 7 Let T be a self-mapping on X and be a function. We say that T is an α-admissible mapping if
Salimi et al. [35] generalized the notion of α-admissible mappings in the following ways.
Definition 8 [35]
Let T be a self-mapping on X and be two functions. We say that T is an α-admissible mapping with respect to η if
Note that if we take then this definition reduces to Definition 7. Also, if we take, then we say that T is an η-subadmissible mapping.
Definition 9 [5]
A non-self-mapping is called α-η-proximal admissible if
for all , where . Also, if we take for all then we say T is an α-proximal admissible mapping.
Clearly, if , T is α-proximal admissible implies that T is α-admissible.
3 Main results
In [34] the authors consider the family Ψ of non-decreasing functions such that for each , where is the n th iterate of ψ.
Let A and B be nonempty subsets of an intuitionistic fuzzy metric space . We denote by and the following sets:
where .
Definition 10 Let A and B be two nonempty subsets of intuitionistic fuzzy metric spaces . Let, , . We say that T is α-proximal admissible if for with
for all .
Let A and B be nonempty subsets of an intuitionistic fuzzy metric space and be a non-self-mapping. We define and as follows:
and
Definition 11 Let A and B be nonempty subsets of an intuitionistic fuzzy metric spaces . Let be a non-self-mapping and be a function. We say T is a α-ψ-proximal contractive mapping if for ,
holds for all , where .
Theorem 1 Let A and B be nonempty subsets of a complete triangular intuitionistic fuzzy metric space such that is nonempty for all . Let be a t-uniformly continuous non-self-mapping satisfying the following assertions:
-
(i)
T is an α-proximal admissible mapping and for all ;
-
(ii)
T is a α-ψ-proximal contractive mapping;
-
(iii)
for any sequence in and satisfying as , then for all ;
-
(iv)
there exist elements and in such that
Then there exists such that , for all , that is, T has a best proximity point .
-
(v)
Moreover, if , implies for all , then T has a unique best proximity point.
Proof By condition (iv) there exist elements and in such that
On the other hand , so there exists such that
Now, since T is α-proximal admissible mapping, so we have . That is,
Again, since , there exists such that
Thus we have
Again since T is α-proximal admissible mapping, so . Hence,
Continuing this process, we get
for all and all .
Now from (3.2) with , and , we get
for all and all where
This implies
Also we have
Thus, from (3.4), (3.5), and (3.6) we have
Now if , then we get
which is a contradiction. Hence,
for all and . So we deduce
for all and . Fix . Then there exists such that
Let with . Then by triangular inequality we get
Consequently, , i.e., . Hence is a Cauchy sequence. Now, since is a complete intuitionistic fuzzy metric space, so there exists such that as . Since T is t-uniformly continuous, so by Lemmas 1 and 2, we have
That is, is a best proximity of T. We show that is unique best proximity point of T. Assume, to the contrary, that there exists such that and is another best proximity point of T, that is, and for all . Now if condition (v) holds, then, from (3.2), we have
where
and
Therefore,
which is a contradiction. Hence, for all . i.e., . Thus T has unique best proximity point. □
Theorem 2 Let A and B be nonempty subsets of a complete triangular intuitionistic fuzzy metric space such that is nonempty for all . Let be a non-self-mapping satisfying the following assertions:
-
(i)
T is an α-proximal admissible mapping and for all ;
-
(ii)
T is a α-ψ-proximal contractive mapping such that ψ is continuous;
-
(iii)
for any sequence in and satisfying as , then for all ;
-
(iv)
there exist elements and in such that
-
(v)
if is a sequence in X such that for all and n with as , then for all and all n.
Then there exists such that , for all , that is, T has a best proximity point .
-
(vi)
Moreover, if , implies for all , then T has a unique best proximity point.
Proof Following the same lines in the proof of Theorem 1, we can construct a sequence in satisfying
and as , that is, , for all . Moreover,
This implies
Passing to the limit as in the above inequality, we get
that is,
and so, by condition (iii), . Since , then there exists such that . Also from (iv) we have for all .
Suppose there exists such that . Then from (3.2) with , , , and we get
On the other hand we know that
and
Now by taking the limit as in (3.8) we get
which implies , i.e., , which is a contradiction. Hence, for all . So, . Therefore, T has a best proximity point. □
Example 1 Let be endowed with the usual metric . Consider and for all and all . Moreover, consider , and define by
Also, define by
and by
Clearly, . Hence,
It is immediate to show that for all , and . Suppose
then
Hence, , that is, . Therefore T is an α-proximal admissible mapping. Further,
that is, T is an α-ψ-proximal contractive mapping. Moreover, if is a sequence such that for all and such that as , then and hence . Consequently, for all and all . Therefore all the conditions of Theorem 2 hold and T has a unique best proximity point. Here is the best proximity point of T.
Theorem 3 Let A and B be nonempty subsets of a complete triangular intuitionistic fuzzy metric space such that is nonempty for all . Let be a t-uniformly continuous non-self-mapping. Assume that following assertions hold true:
-
(i)
T is an α-proximal admissible mapping and for all ;
-
(ii)
for ,
(3.9)
holds for all ;
-
(iii)
for any sequence in and satisfying as , then for all ;
-
(iv)
there exist elements and in such that
Then there exists such that , for all , that is, T has a best proximity point .
-
(v)
Moreover, if , implies for all , then T has a unique best proximity point.
Proof Following the same lines in the proof of Theorem 1, we can construct a sequence in satisfying
From (ii) with , and , we get
As in the proof of Theorem 2.2 of [27], we deduce that is a Cauchy sequence. The completeness of ensures that the sequence converges to some , that is . Since T is t-uniformly continuous, so by Lemmas 1 and 2, we have
That is, is a best proximity of T. Now we show that is unique best proximity point of T. Suppose, to the contrary, that there exists such that and is another best proximity point of T, that is, and for all . Now if condition (v) holds, then, from (ii), we have
which is a contradiction. Hence, . 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 such that is nonempty for all . Let be a non-self-mapping. Assume that the following assertions hold true:
-
(i)
T is an α-proximal admissible mapping and for all ;
-
(ii)
(3.9) holds for all ;
-
(iii)
for any sequence in and satisfying as , then for all ;
-
(iv)
there exist elements and in such that
-
(v)
if is a sequence in X such that for all n and all such that as , then for all n and all .
Then there exists such that , for all , that is, T has a best proximity point .
-
(vi)
Moreover, if , imply for all , then T has a unique best proximity point.
Proof Following the same lines in the proof of Theorem 3, we can construct a sequence in satisfying
as , and there exists such that . Also, for all n and all . Then from (ii) with , , and we get
Taking the limit as in the above inequality we get , i.e., . Therefore 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 . Then is said to be a proximally order-preserving, if for all ,
holds for all .
Theorem 5 Let A and B be nonempty subsets of a partially ordered complete triangular intuitionistic fuzzy metric space such that is nonempty for all . Let be a t-uniformly continuous non-self-mapping satisfying the following assertions:
-
(i)
T is proximally order-preserving and for all ;
-
(ii)
for ,
(4.1)
holds for all , where ;
-
(iii)
for any sequence in and satisfying as , then for all ;
-
(iv)
there exist elements and in such that
Then there exists such that , for all , that is, T has a best proximity point .
Proof Define by
At first we prove that T is an α-proximal admissible mapping. For this assume that
So
Now, since T is proximally order-preserving so, . That is, which implies that T is α-proximal admissible. Condition (ii) implies that T is α-ψ-proximal 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 such that is nonempty for all . Let be a non-self-mapping satisfying the following assertions:
-
(i)
T is proximally order-preserving and for all ;
-
(ii)
(4.1) holds for all ;
-
(iii)
for any sequence in and satisfying as , then for all ;
-
(iv)
there exist elements and in such that
-
(v)
if is an increasing sequence in X such that as , then for all n.
Then there exists such that , for all , that is, T has a best proximity point .
Proof Define as in Theorem 5. Also, assume for all such that as . Then for all . Hence, by (v) we get for all and so for all and all . 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 such that is nonempty for all . Let be a t-uniformly continuous non-self-mapping. Also suppose that the following assertions hold true:
-
(i)
T is proximally order-preserving and for all ;
-
(ii)
for ,
(4.2)
holds for all ;
-
(iii)
for any sequence in and satisfying as , then for all ;
-
(iv)
there exist elements and in such that
Then there exists such that , for all , that is, T has a best proximity point .
Theorem 8 If in the above theorem, in place of t-uniform continuity of T, we assume that for any increasing sequence in X and as , we have for all . Then there exists such that , for all , that is, T has a best proximity point .
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 be an intuitionistic fuzzy metric space, and . We say, T is an α-admissible mapping if
for all .
Let be an intuitionistic fuzzy metric space, be a self-mapping. We define and as follows:
and
Definition 14 Let be an intuitionistic fuzzy metric space. Let be a self-mapping and be a function. We say T is an α-ψ-contractive mapping if
holds for all , where .
Theorem 9 Let be a complete triangular intuitionistic fuzzy metric space. Let be a t-uniformly continuous self-mapping. Also suppose that the following assertions hold:
-
(i)
T is an α-admissible mapping;
-
(ii)
T is α-ψ-contractive mapping;
-
(iii)
there exists in X such that .
Then T has a fixed point.
-
(iv)
Moreover, if implies , then T has a unique fixed point.
Theorem 10 Let be a complete triangular intuitionistic fuzzy metric space. Let be a self-mapping. Also suppose that the following assertions hold:
-
(i)
T is an α-admissible mapping;
-
(ii)
T is α-ψ-contractive mapping;
-
(iii)
there exists in X such that ;
-
(iv)
if is a sequence in X such that for all n and all with as , then for all and all .
Then T has a fixed point.
-
(v)
Moreover, if implies , then T has a unique fixed point.
Theorem 11 Let be a complete triangular intuitionistic fuzzy metric space. Let be a t-uniformly continuous self-mapping. Also suppose that the following assertions hold:
-
(i)
T is an α-admissible mapping;
-
(ii)
for all ;
-
(iii)
there exists in X such that .
Then T has a fixed point.
-
(iv)
Moreover, if implies , then T has a unique fixed point.
Theorem 12 Let be a complete triangular intuitionistic fuzzy metric space. Let be a self-mapping. Also suppose that the following assertions hold:
-
(i)
T is an α-admissible mapping;
-
(ii)
for all ;
-
(iii)
there exist elements in X such that ;
-
(iv)
if is a sequence in X such that for all n and all with as , then for all n and all .
Then T has a fixed point.
-
(v)
Moreover, if implies , then T has a unique fixed point.
By taking for all and all , we obtain the following corrected version of Theorem 2.2 in [27].
Corollary 1 (Theorem 2.2 of [27])
Let be a complete triangular intuitionistic fuzzy metric space. Let be a t-uniformly continuous mapping satisfying
holds for all and all . Then T has a fixed point.
Theorem 13 Let be a partially ordered complete triangular intuitionistic fuzzy metric space. Let be a t-uniformly continuous self-mapping. Also assume the following assertions hold true:
-
(i)
T is an increasing mapping;
-
(ii)
assume
holds for all with and ;
-
(iii)
there exists in X such that .
Then T has a fixed point.
Theorem 14 Let be a partially ordered complete triangular intuitionistic fuzzy metric space. Let be a self-mapping. Also assume the following assertions hold true:
-
(i)
T is an increasing mapping;
-
(ii)
assume
holds for all with and ;
-
(iii)
there exist elements in X such that ;
-
(iv)
if be an increasing sequence in X such that as , then for all .
Then T has a fixed point.
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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.
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Latif, A., Hezarjaribi, M., Salimi, P. et al. Best proximity point theorems for α-ψ-proximal contractions in intuitionistic fuzzy metric spaces. J Inequal Appl 2014, 352 (2014). https://doi.org/10.1186/1029-242X-2014-352
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DOI: https://doi.org/10.1186/1029-242X-2014-352