BEST PROXIMITY POINT THEORY ON VECTOR METRIC SPACES

In this paper, we rst give a new denition of -Dedekind complete Riesz space (E; ) in the frame of vector metric space ( ; ; E) and we investigate the relation between Dedekind complete Riesz space and our new concept. Moreover, we introduce a new contraction so called -vector proximal contraction mapping. Then, we prove certain best proximity point theorems for such mappings on vector metric spaces ( ; ; E) where (E; ) is -Dedekind complete Riesz space. Thus, for the rst time, we acquire best proximity point results on vector metric spaces. As a result, we generalize some xed point results proved on both vector metric spaces and partially ordered vector metric spaces. Further, we provide nontrivial and comparative examples to show the e¤ectiveness of our main results. 1. Introduction and Preliminaries Cevik et al. [11] brought to the literature a notion of vector metric and proved Banach xed point theorem [7] which is considered starting of metric xed point theory in these spaces. Then, many authors have studied to obtain various xed point results in context of vector metric spaces [2,12,16,17,18,21]. However, when consider the topological structure of vector metric spaces, it may not be easy to prove a result existing in the real valued metric spaces. This is an important and interesting point for the authors. Now, we state denition of Riesz space and related properties: Let E be nonempty set and be a relation on . Then, the relation is called partial order, if it satises (o1) & &; 2020 Mathematics Subject Classication. Primary 54H25; Secondary, 47H10, 35B15.


Introduction and Preliminaries
Cevik et al. [11] brought to the literature a notion of vector metric and proved Banach …xed point theorem [7] which is considered starting of metric …xed point theory in these spaces. Then, many authors have studied to obtain various …xed point results in context of vector metric spaces [2,12,16,17,18,21]. However, when consider the topological structure of vector metric spaces, it may not be easy to prove a result existing in the real valued metric spaces. This is an important and interesting point for the authors. Now, we state de…nition of Riesz space and related properties: Let E be nonempty set and be a relation on . Then, the relation is called partial order, if it satis…es (o1) & &; 2020 Mathematics Subject Classi…cation. Primary 54H25; Secondary, 47H10, 35B15. Keywords and phrases. Best proximity point, -admissible, proximal contraction, vector metric spaces.
hakan.sahin@amasya.edu.tr 0000-0002-4671-7950. for all &; ; z 2 E. Moreover, the pair (E; ) is said to be partially ordered set. An ordered vector space is real vector space equipped with an partial order which is compatible operations of vector space, that is, implies & for all positive real number and all &; 2 E.
Then, ordered vector space (E; ) is called Riesz space if it is satis…ed the supremum & _ and in…mum &^ in E for all &; 2 E. Let (E; ) be a Riesz space. We denote positive cone of E by E + .The notation a n # a means that the sequence fa n g in E is nonicreasing such that inffa n : n 2 Ng = a. If there exists a sequence fa n g in E satisfying a n # 0 such that jb n bj a n for all n 2 N, then the sequence fb n g in E is said to be order converges to b 2 E where the modul of any point a in E is de…ned by jaj = a _ ( a). Moreover, if there exists a sequence fa n g in E satisfying a n # 0 such that jb n b n+p j a n for all n; p 2 N, then the sequence fb n g in E is called order-Cauchy sequence. A Riesz space E is said to be order-Cauchy complete if every order-Cauhy sequence in E is order converges to a point in E. A Riesz space (E; ) is called an Archimedean if 1 n a # 0 for all a 2 E + . Further, if every subset of E with upper bound (lower bound) has a supremum (in…mum) in E, then (E; ) is said to be Dedekind complete. Note that every Dedekind complete Riesz space is an Archimedean Riesz space. For more details about Riesz spaces see [3,11].
We continue this section with de…nition of a vector metric space and its related properties. ! &, if there exists a sequence fa n g satisfying a n # 0 such that (& n ; &) a n for all n 2 N: (ii) The sequence f& n g is said to be E-Cauchy sequence if there exists a sequence fa n g satisfying a n # 0 such that (& n ; & n+p ) a n for all n; p 2 N: (iii) The vector metric space ( ; ; E) is said to be E-complete if every E-Cauchy sequence in E-converges to a point in . Because of these reasons, best proximity point theory is one of the area attracted the most attention lately [1,4,5,6,9,13,15,19,20].
In the present paper, we aim to extend some …xed point results obtained on both vector metric spaces and partially ordered vector metric spaces. Introducing two new concepts so called -Dedekind complete Riesz space and -vector proximal contraction mapping, we prove some best proximity point theorems for such mappings on vector metric spaces ( ; ; E) where (E; ) is -Dedekind complete Riesz space. Finally, we give nontrivial examples to support our main results.

Main Results
We start to this section with the following our new de…nitions which are useful in the sequel. Note that every Dedekind complete Riesz space (E; ) is an -Dedekind complete Riesz space for all ( ; ; E) vector metric space. However, the converse may not be true. The following example shows this fact: be the family of all continuous function from [0; 1] to real numbers and = [0; 1). Then, E is Riesz space with the pointwise ordering but it is not Dedekind complete. Indeed, consider a sequence of functions ff n g de…ned by: Then, it can be seen that ff n g n 3 E and the set of functions ff n : n 2 Ng is bounded from above with the function 1 : and hence f = 2 E. Now, consider the mapping : ! E de…ned by Considering structure of vector metric spaces, we restate the de…nitions introduced in the frame of metric space as follows.
for all &; ; u; v 2 M , then T is said to be vector proximal contraction.
Now, we introduce a new concept which is more general than proximal contraction mapping.
for all u; v; &; 2 M , then the mapping T is said to be -vector proximal contraction mapping.
Let ( ; ; E) be a vector metric space where (E; ) is an -Dedekind complete Riesz space. We will consider the following sets in the rest of paper. Since Since T is -vector proximal contraction mapping, from (1) and (2), there exists q 2 [0; 1) such that and by condition (ii), we have (& 1 ; & 2 ) 1: Continuing this process, we construct a sequence f& n g such that and (& n ; & n+1 ) 1 for all n 2 N. For arbitrary n; p 2 N which implies that f& n g is E-Cauchy sequence because of the fact that (E; ) is an Archimedean Riesz space. Since ( ; ; E) is E-complete vector metric space and M 0 is E-closed, there exist a sequence fa n g E satisfying a n # 0 and & 2 M 0 such that To show the uniqueness of best proximity point, assume that there exists another best proximity point 2 M which is distinct from & such that Since T is -vector proximal contraction mapping, from (7), (8)  The following example shows the e¤ectiveness of Theorem 10 because of the fact that any real valued metric can not be applied to this example. Therefore, all hypotheses of Theorem 10 are satis…ed. Hence, the mapping T has a unique best proximity point. On the other hand, it is clear that any real valued metric cannot applicable to this example.
Using Theorem 10, we can obtain the following result in partially ordered vector metric spaces. Proof. If we de…ne a mapping : M M ! [0; 1) as for all &; 2 , then from condition (ii) T is an -vector proximal admissible mapping. Further, using (iv) we have that T is an -vector proximal contraction mapping. Therefore, all hypotheses of Theorem 10 are satis…ed and so T has a best proximity point.
If we take (&; ) = 1 in Theorem 10, we deduce a best proximity point result as follows. Since Since T is -vector proximal contraction mapping, from (9) and (10) (& n ; & n+1 ) 1 (12) for all n 2 N. For arbitrary n; p 2 N, it can be seen that which implies that f& n g is E-Cauchy sequence because of the fact that (E; ) is an Archimedean Riesz space. Since ( ; ; E) is E-complete vector metric space and M 0 is E-closed, there exists a sequence fa n g satisfying a n # 0 and & 2 M 0 such that (& n ; & ) a n (13) for all n 2 N.
On the other hand, from (12), (13) and the condition (H) there exists a subsequence f& n k g of f& n g such that (& n k ; & ) 1 for all k 2 N. Because of (11) and (14), we get ;E ! z . Because of the fact that the limit of f& n k +1 g is unique, it is obtained that & = z . Therefore, from (14), & is a best proximity point of T . It can be shown the uniqueness of best proximity point as in Theorem 10.
The following example is important to show the di¤erence between Theorem 10 and Theorem 14. Therefore, all hypotheses of Theorem 14 are satis…ed. Hence, the mapping T has a best proximity point. On the other hand, since the mapping T is not vectorially continuous, Theorem 10 cannot be applied to this example.
Using Theorem 14, we can obtain the following result in partially ordered vector metric spaces. 3.

Conclusion
In this paper, we …rst give a new de…nition of -Dedekind complete Riesz space (E; ) on a vector metric space ( ; ; E), and so we obtain a new family of Riesz space which is a larger than the class of Dedekind complete Riesz space in the frame of vector metric space. We also introduce a new notion so called -vector proximal contraction mapping including the concept of -proximal contraction de…ned Samet et al. [14]. Then, for the …rst time, we obtain some best proximity point results on vector metric spaces ( ; ; E) where (E; ) is -Dedekind complete Riesz space and hence we extend some …xed point results proved on both vector metric spaces and partially ordered vector metric spaces such as [7,8,12].

Declaration of Competing Interests
The authors declare that they have no known competing …nancial interest or personal relationships that could have appeared to in ‡uence the work reported in this paper.