Several fixed point theorem results are obtained in the case of a metric space as shown in figures [13]- [25]. Khan, Berzig and Chandok have obtained in [10] some fixed point results for self-map pings in the case of two metric space endowed with binary relation. Generalization of the cont raction princip le to the case of non-self-mappings T : P ?? Q is presented and is natural to look for in an element p * such that d (p * , T p * ) is minimum. Such element a * is called best proximity point of the non-self-mapping T .

The best approximation t heorem was introduced by Fan [2] in 1669. Sadi q Basha [8] revisited the the-orem and proposed necessary and sufficient conditions for existence for proximal contractions of first and second kind of suc h points. Furthermore, several variants of the non-self-contractions for the existence of a best proximity point were studied in [3]- [7].

The paper discusses the existence and the uniqueness of best proximity points for a class of continuous non-self-mapping in bimetric space. More precisely, let (X , d , ?) a bimetric space. Under additional hypotheses, we obtain a common best proximity point result of both metrics d and ? by introducing the class of proximal ?-quasi-contraction with respect to one metric ?.

The paper is divided into six sections; section two introduces the notation used herein, it presents some definitions and recalls some useful results. The best proximity p oint theorem is stated in section 3 with the proof. While a conformation of the theorem is described in section 4 . Several consequences are derived in Section 5 such as the Maia's Theorem of existence and uniqueness of fixed points in bimetric spaces [17] and the theorem of Kh an, and Berzik and Chandok [1] for continuous generalized quasi-contractive mappings in bimetric spaces. 


# Introduction


# Preliminaries and Definitions

Let (X , d , ?) be a bimetric space such that d (x, y) ? ?(x, y) for all x, y ? X . Let (P,Q) a pair of nonempty subsets of X. We consider the following notations d (P, K ) := inf{d (p, k) : p ? P, q ? Q}; d (x,Q) := inf{d (x, q) : q ? Q}; P d 0 := {p ? P : there exists q ? Q such that d (p, q) = d (P,Q)};

Q d 0 := {n ? N : there exists p ? P such that d (p, q) = d (P,Q)}.

P ? 0 := {p ? P : there exists q ? Q such that ?(p, q) = ?(P,Q)};

Q ? 0 := {q ? Q : there exists p ? P such that ?(p, q) = ?(P,Q)}.

[ 
? n=1 ? n ? (s) < ?.
The set of all ??comparison functions ? satisfying (P1)-(P3) will be denoted by
? ? . Let ?, ? ? (0, +?). If ? < ?, then ? ? ? ? ? .
We recall some useful lemmas concerning the comparison functions ?.

[11] Let ? ? (0,+?) and ? ? ? ? . Then (i) ? ? is nondecreasing;

(ii) ? ? (t ) < t for all t > 0;

(iii)
? n=1
? n ? (t ) < ? for all t > 0.

[9] Let (X ,?) a metric space and (P,Q) a pair of nonempty subsets of X. Let T : P ?? Q be a non-self-mapping. Suppose that the following conditions hold:
(1) P 0 = (2) T (P 0 ? Q 0 .
Then, for all p ? P ? 0 , there exists a sequence {x n } ? P ? 0 such that
(1) x 0 = p ?(x n+1 , T x n ) = ?(P,Q) ? n ? N.
To begin, we have the following lemma: LEMMA 3.1. Let (X , d , ?) be a bimetric space such that d (x, y) ? ?(x, y) for all x, y ? X . Let (P,Q) a pair of nonempty subsets of X . Suppose that d
(P,Q) = ?(P,Q), then P ? 0 ? P d 0 and Q ? 0 ? Q d 0 .
III Let p ? P ? 0 . Then ?(p, q) = ?(P,Q) for some q ? Q. Therefore,
d (P,Q) = ?(P,Q) ? d (p, q) ? ?(p, q) = ?(P,Q) = d (P,Q).
This means p ? P d 0 . The same idea for Q ? 0 ? Q d 0 . Let recall the following concept introduced in [12] .

Let (X , ?) be a metric space . Let ? ? (0, +?). A non-self-mapping T : P ? Q is said to be a proximal ?-quasi-contractive iff there exist ? ? ? ? and positive numbers ? 0 , . . . , ? 4 such that:
?(u, v) ? ?(max ? 0 ?(x, y), ? 1 ?(x, u), ? 2 ?(y, v), ? 3 ?(x, v), ? 4 ?(y, u) ). For all x, y, u, v ? P satisfying, ?(u, T x) = ?(P,Q) and ?(v, T y) = ?(P,Q).
Our main result is giving by the following best proximity point theorem.

Let (X , d , ?) be a bimetric space and (P,Q) be a pair of non empty subsets of the complete metric space (X , d ). Let T : P ?? Q be a non-self-mapping satisfying the following conditions :
(A1) The set P ? 0 is nonempty closed in (X , d ); (A2) d (x, y) ? ?(x, y) for all x, y ? P ; (A3) d (P,Q) = ?(P,Q); (A4) T is continuous with respect to d ; (A5) there exists ? ? max 0?k?4 {? k , 2? 4 } such that T

# is a proximal ?-quasi contractive with respect to ?; (A6) A best proximity for T with respect to d is a best proximity point of T with respect to ?.

Moreover, assume that ? > max{? 0 , ? 3 , ? 4 }.


# Then T has a unique common best proximity point for both metrics p
* ? P such that d (p * , T p * ) = ?(p * , T p * ) = d (P,Q). Let x 0 ? P 0 ? . Using Lemma (2.
2), we can find x n+1 ? P 0 ? such that
?(x n+1 , T x n ) = ?(P,Q),
for every positive integer k.

If x n = x n+1 are equals for some non negative integer n, then nothing to prove since
d (P,Q) ? d (x n , T x n ) ? ?(x n , T x n ) = ?(P,Q) = d (P,Q).
Then, we assume that x n = x n+1 for every negative integer n. Let us prove that the sequence
{x n } ? P ? 0 ? P d 0 is a Cauchy sequence in (X , ?). Since ?(x n+1 , T x n ) = ?(P,Q) and ?(x n , T x n?1 ) = ?(P,Q) and T is a proximal ?-quasi contractive with
respect to the metric ? , we get using triangular inequalities
?(x n+1 , x n ) ? ?(max{? 0 ?(x n , x n?1 ), ? 1 ?(x n , x n+1 ), ? 2 ?(x n?1 , x n ), ? 4 ?(x n?1 , x n+1 )}) ? ?(max{? 0 ?(x n , x n?1 ),? 1 ?(x n , x n+1 ), ? 2 ?(x n?1 , x n ), ? 4 ?(x n?1 , x n ) + ? 4 ?(x n , x n+1 )}) ? ?(? max{?(x n , x n?1 ), ?(x n , x n+1 )}).
Where ? ? max{? 0 , ? 1 , ? 2 , ? 3 , 2? 4 }.

If ?(x n , x n?1 ) ? ?(x n , x n+1 ). Then by Lemma (2.1) it follows that
?(x n+1 , x n ) ? ?(??(x n+1 , x n )) = ? ? (?(x n+1 , x n )) < ?(x n+1 , x n ).
Definition 2.1.

Theorem 3.1.

Proof. Which is a contradiction. So ?n ? 1,
?(x n?1 , x n ) > ?(x n+1 , x n ), then, ?(x n+1 , x n ) ? ? ? (?(x n , x n?1 )) ?n.
Thus by Mathematical induction we obtain that
?(x n+1 , x n ) ? ? n ? (?(x 0 , x 1 )) ?n.
In addition, for k < j and using triangle inequalities we obtain,
?(x k , x j ) ? m?1 l =k ?(x l , x l +1 ), ? j ?1 l =k ? l ? (?(x 0 , x 1 )). Since ? l =1
? l ? (t ) < ?, for every > 0 there exists K > 0 such that
j ?1 l =k ? l ? (t ) < for all j > k > K . Thus, d (x k , x j ) ? ?(x k , x j ) < . This imply that the sequence {x k } is ?-Cauchy, so by (A2), {x k } is d -Cauchy too.
As the space (X , d ) is complete, and the sequence {x n } ? P ? 0 who closed in (X , d ) by (A1), then there exists p * such that lim For the uniqueness, suppose that p * and s * two distinct best proximity points of T with respect to d on M 0 . By hypothesis (A6), we deduce that they are also best proximity points of T with respect to ?. Let
n??+? d (x n , p * ) = 0. So d (P,Q) ? d (x n+1 , T x n ) ? ?(x n+1 , T x n ) = ?(P,Q) = d (P,Q) From (A4),? = ?(p * , s * ) ? d (p * , s * ) > 0.
Since T is a proximal ?-quasi contractive with respect to the metric ?, we obtain the following inequality  Let P = {(?, 0) : ? ? [0, 1]} and Q = {(?, 1) : ? ? [0, 1]}. Also, let T : P ?? Q be defined by T (?, 0) = ( ? 7 , 1). Then, it is easy to see that ?(P, q) = d (P,Q) = 1 and P ? 0 = P d 0 = P which is closed in (X , d ). Thus, the hypotheses (A1) and (A2) of Theorem 3.1 are satisfied as well. As mentioned in Remark 3.1, the conditions (A3) and (A6) are satisfied since X = R 2 is a finite dimensional normed space. The non-self-mapping T is continuous with respect to the metric d which confirm that the a ssertion (A4) holds. Now, we shall show that T is ?-quasi-contractive mapping with respect to the metric ? with ?(t ) = 3 7 t , ? = 3 7 and ? i = 1 3 i +1 for i = 0, 1, 2, 3, 4. Let x, y, u, v ? P where x = (? 1 , 0), y = (? 2 , 0), u = (u 1 , 0) and v = (v 1 , 0) such that ?(u, T x) = ?(v, T y).
? ? ?(max{? 0 , ? 3 , ? 4 }?) ? ?(??) = ? ? (?) < ?,
This means that u 1 = ? 1 7 and u 2 = ? 2 7 . Therefore, we get
?(u, v) = ?(( ? 1 7 , 0), ( ? 2 7 , 0)) = 1 7 |? 1 ? ? 2 | = 1 7 ?(x, y) = 3 7 ( 1 3 ?(x, y)) ? 3 7 max{ 1 3 ?(x, y), 1 9 ?(x, u), 1 27 ?(y, v), 1 81 ?(x, v), 1 243 ?(y, u)}.
So, since ? = 3 7 ? max 0?k?3 {? k , 2? 4 } = max{ 1 3 , 1 9 , 1 27 , 2 81 }, then T is proximal ?-quasi-contractive mapping with respect to the metric ? with ?(t ) = 3  7 t , ? = 3 7 and ? i = 1 3 i +1 for i = 0, 1, 2, 3, 4 . Hence, all conditions of Theorems 3.1 are satisfied and so T has a unique common proximity point with respect to the metrics ? and d which is p * = (0,0) ? P, ?((0, 0), T (0, 0)) = d ((0, 0), (0, 1)
) = 1 = d (P,Q).
As a consequence of our main theorem is Maia's fixed point theorem: THEOREM 5.1. [17] Let (X ,d,?) be a bimetric space. Assume that for T : X ?? X , the following conditions are satisfied:

(1) d (x, y) ? ?(x, y) for all x, y ? X ;

(2) X is complete with respect to d ;

(3) T is continuous with respect to d;

(4) there exits ? ? [0, 1) such that ?(T x, T y) ? ??(x, y).


# Then T has a unique fixed point in X

All hypotheses of our main Theorem are satisfied by taking P = Q = X . In addition, the function ?(t ) = ?t belongs to ? 1 . Then there exists a unique fixed point in X .

Also we reproduce the results of Khan, Berzik and Chandok [ ]. We recall first the definition of generalized contractive mapping.

V.


# Consequences

Proof. © 2019 Global Journals [1] Assume T : X ?? X , there exists ? ? ? such that ?(T x, T y) ? ?(M ? (x, y)).

A mapping T is called a generalized contractive with respect to ?, if
M ? (x, y) = max{?(x, y), 1 2 [?(x, T x) + ?(y, T y)], 1 2 [?(x, T y) + ?(y, T x)]}.
A mapping T is called a generalized quasi-contractive with respect to ?, if M ? (x, y) = max{?(x, y), ?(x, T x), ?(y, T y), ?(x, T y), ?(y, T x)}.

Let recall the following concept introduced in [11].

[11] Let X a non empty set. A mapping T : X ?? X is called ?-quasi-contractive, if there exists ? > 0 and ? ? ? ? such that
?(T x, T y) ? ?(M T (?(x, y))
where M T (x, y) = max{? 0 ?(x, y), ? 1 ?(x, T x), ? 2 ?(y, T y), ? 3 ?(x, T y), ? 4 ?(y, T x)}, with ? k ? 0 for 0 ? k ? 4.

[1] Let (X ,d,?) be a bimetric space. Assume that for T : X ?? X , the following conditions are satisfied:

(1) d (x, y) ? ?(x, y) for all x, y ? X ;

(2) X is complete with respect to d ;

(3) T is continuous with respect to d;

(4) T is a generalized contractive with respect to ? .


# Then T has a unique fixed point in X

In our main Theorem 3.1, taking P = Q = X . In addition, since T is a generalized contractive with respect to ?, M ? (x, y) ? max{?(x, y), ?(x, T x), ?(y, T y), ?(x, T y), ?(y, T x)}.

The function ?(t ) belongs to ? 2 . Then there exists a unique fixed point in X .

[1] Let (X ,d,?) be a bimetric space. Assume that for T : X ?? X , the following conditions are satisfied:

(1) d (x, y) ? ?(x, y) for all x, y ? X ;

(2) X is complete with respect to d ;

(3) T is continuous with respect to d;

(4) T is a generalized quasi-contractive with respect to ? .

Then T has a unique fixed point in X .

Proof. Also the same idea with ? ? ? 2 .

We suggest another corollary, which is an immediate consequence of our Theorem 3.1.  Let (X , d , ?) be a bimetric space. Assume that for T : X ?? X , the following conditions are satisfied:

(1) d (x, y) ? ?(x, y) for all x, y ? X ;

(2) X is complete with respect to d ;

(3) T is continuous with respect to d;

(4) T is a ?quasi-contractive with respect to ?, where ? ? max 0?k?3 {? k , 2? 4 }.

Moreover, assume ? > max{? 0 , ? 3 , ? 4 }.

Then T has a unique fixed point in X .

PROOF. All hypotheses of Theorem 3.1 are satisfied by taking P = Q = X .

A new class of non-self-mappings in bimetric space is given in this paper. This has been achieved by introducing the proximal ?-quasi-contraction mappings involving ?comparison functions. Under some additional conditions, we establish the existence and uniqueness of best proximity points for such mappings. As an application, we derive some fixed point results which are known in the literature. We believe that the approach used in the current contribution, may be extended for non-self-mappings which are not necessarily continuous.


# VI.


# Conclusion

Corollary 5.1. 
![Best Proximity Point Theorem on the Context of Bimetric Spaces M. Iadh. Ayari ? , M. Ali. Ayari ? & Sahbi. Ayari ?](image-2.png "Common")
2![Fan. K. Extentions of two fixed point theorems of F.E. Brower.Math. Z.., 112 (1669), 234-240.II.](image-3.png "Ref 2 .")
![we have T is continuous with respect to d , and, so it follows that lim n??+? d (x n+1 , T x n ) = d (p * , T p * ) = d (P,Q).](image-4.png "")
![which is a contradiction. Thus ? = 0 and therefore ?(p * , s * ) = 0.The conditions (A3) and (A6), which occur in Theorem 3.1, are always satisfied in finite dimensional normed spaces.Consider the complete Euclidian space X = R 2 with the metrics ?((x 1 , y 1 ), (x 2 , y2 )) = |x 1 ? x 2 | + |y 1 ? y 2 |and the Euclidian distance d = (x 1 ? x 2 ) 2 + (y 1 ? y 2 ) 2 . Using the inequality a + b ? a + b, we get d ((x 1 , y 1 ), (x 2 , y 2 )) ? ?((x 1 , y 1 ), (x 2 , y 2 )).So the assertion A2 of Theorem 3.1 is satisfied.](image-5.png "")
RefDefinition 2.1.Year 201961ersion I VIIIssueRemark 2.1.lemma 2.1.( F )lemma 2.2.8. Sadiq. Bacha. Extentions of Banach's contraction principle. J. Num. Func. Anal. Optim Theory Appl., 31 (2010), 569-576. © 2019 Global Journals
Proof.Ref12. M. Iadh. Ayari, M.M.M. Jaradat, ZeadMustfa. Generalization of Best 1 Global Journal of Science Frontier Research Volume XIX Issue ersion I V II Year 2019 62 Proximity Points Theorem for Non-Self Proximal Contractions of First Kind ( F )Submitted.© 2019 Global Journals
RefYear 201965ersion I VIIIssue( F )© 2019 Global Journals

		
		
			

				


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