Best Proximity Point Results in Complex Valued Metric Spaces

We introduce the concept of proximity points for nonself-mappings between two subsets of a complex valued metric space which is arecentlyintroducedextensionofmetricspacesobtainedbyallowingthemetricfunctiontoassumevaluesfromthefieldofcomplexnumbers.Weapplythisconcepttoobtaintheminimumdistancebetweentwosubsetsofthecomplexvaluedmetricspaces.Wetreattheproblemasthatoffindingtheglobaloptimalsolutionofafixedpointequationalthoughtheexactsolutiondoesnotingeneralexist.WealsodefineandusetheconceptofP-propertyinsuchspaces.Ourresultsareillustratedwithexamples.


Introduction and Preliminaries
In this paper we prove certain proximity point results to obtain the minimum distance between two subsets of a complex valued metric space. Essentially it is a global optimization problem which we treat here as the problem of finding the global optimal solution of a fixed point iteration. It is a part of the more general category of problems of finding minimum distances between two objects. In geometry it has led to the concept of geodesics, a curve along which the optimal distance between two given points of the space is realized [1]. Examples abound in physical theories, especially in the general theory of relativity, where finding the physically possible shortest path is sometimes the main task [2].
In proximity point problems our objects are sets. Here our aim is to find the distance between two sets and with the help of a function defined from to . Precisely we want to find a solution to the problem of minimizing the distance between and where is varied over the set . Equivalently we want to find the optimal solution of the equation = although the exact solution does not in general exist as in the case where and are disjoint. It is at this point the best approximation theorems and the best proximity point theorems have their roles to play. The best approximation theorems provide the best approximate solutions which need not be globally optimal. For instance, let us consider the following Ky Fan's best approximation theorem.
The element in the above theorem need not give the optimum value of ‖ − ‖.
Proximity point result was first proved by [4]. After that several results on proximity have followed. Particularly, in the general setting of metric spaces there are a good number of results, and [5][6][7][8][9][10][11][12][13][14] are instances of these results. As we have already stated in this paper we introduce the concept of proximity points in complex valued metric spaces.
First we describe the complex valued metric spaces. It is a generalization of metric space introduced by Azam et al. [15] where the metric function assumes values from the field of complex numbers. Following this work several works on complex valued metric spaces, especially on fixed point and related topics, have been done, some of which are noted in [16][17][18]. It opens the scope of incorporating concepts from complex analysis in the domain of metric spaces. In fact, there are large efforts for generalizing metric spaces by changing the form and interpretation of the metric function. International Journal of Analysis Gähler [19] introduced 2-metric spaces where a real number is assigned to any three points of the space. Probabilistic metric spaces were introduced by Schweizer and Sklar [20,21] in which any pair of points is assigned to a suitable distribution function making possible a probabilistic sense of distance. Fuzzy metric spaces were introduced in more than one way by various means of fuzzification as, for example, in [22] by assigning any pair of points to a suitable fuzzy set and spelling out the triangular inequality by using a t-norm. Another example is in the work of Kaleva and Seikkala [23] where any pair of points is assigned to a fuzzy number. Gmetric space [24] is another generalization in which every triplet of points is assigned to a nonnegative real number but in a different way than in 2-metric spaces. There are also other extensions of the metric which are not mentioned above. It can be seen that in recent times efforts of extending the concept of metric space have continued in a rapid manner.
Below we describe the essential features of complex valued metric spaces which we require here.
Let C be the set of complex numbers and 1 , 2 ∈ C. Define a partial order ≾ on C as follows: It follows that 1 ≾ 2 if one of the following conditions is satisfied: In particular, we will write 1 ⋨ 2 if 1 ̸ = 2 and one of (i), (ii), and (iii) is satisfied and we will write 1 ≺ 2 if only (iii) is satisfied.
Note that Definition 2. Let be a subset of C. If there exists ∈ C such that ≾ , for all ∈ , then is bounded above and is an upper bound. Similarly, if there exists ∈ C such that ≾ , for all ∈ , then is bounded below and is a lower bound.
Definition 3. For a subset ⊆ C which is bounded above if there exists an upper bound of such that, for every upper bound of , ⪯ , then the upper bound is called the least upper bound (lub) of or sup . Similarly, for a subset ⊆ C which is bounded below if there exists a lower bound of such that, for every lower bound of , ⪯ , then the lower bound is called the greatest lower bound (glb) of or inf .
Suppose that ⊆ C is bounded above. Then there exists = + V ∈ C such that = + ≾ = + V, for all ∈ . It follows that ≾ and ≾ V, for all = + ∈ ; that is, = { : = + ∈ } and = { : = + ∈ } are two sets of real numbers which are bounded above. Hence both sup and sup exist. Let * = sup and * = sup . Then clearly, * = * + * is the least upper bound (lub) of or sup .
Any subset ⊆ C which is bounded above has the least upper bound (lub) or supremum. Equivalently, any subset ⊆ C which is bounded below has the greatest lower bound (glb) or infimum.
Definition 4 (see [15]). Let be a nonempty set. Suppose that the mapping : × → C satisfies (i) If for every ∈ C with 0 ≺ there is 0 ∈ N such that, for all > 0 , ( , ) ≺ , then { } is said to be convergent, { } converges to , and is the limit point of { }. We denote this by lim → ∞ = or → as → ∞.
(ii) If for every ∈ C with 0 ≺ there is 0 ∈ N such that, for all , > 0 , ( , ) ≺ , then { } is said to be a Cauchy sequence.
(iii) If every Cauchy sequence in is convergent, then ( , ) is a complete complex valued metric space.
Definition 11. Let and be two nonempty bounded subsets of a complex valued metric space ( , ) and : → a non-self-mapping. A point ∈ is called a best proximity point of if ( , ) = dist( , ).
The definition of -property and weak -property was introduced in [13] and [14], respectively. Now we define them in complex valued metric space.

Main Results
Theorem 16. Let ( , ) be a pair of nonempty closed and bounded subsets of a complete complex valued metric space ( , ) such that 0 is nonempty and the pair ( , ) satisfies the weak -property. Let : → be a mapping with ( 0 ) ⊆ 0 . If there exists a real number with 0 < < 1 such that, for all , ∈ , then has a unique best proximity point in .
Since is closed and { } is a sequence in converging to , we have ∈ . By the continuity of , we have Then, But according to (14), the sequence { ( +1 , )} is a constant sequence with the constant value dist( , ). Therefore, ( , ) = dist( , ); that is, ∈ is a best proximity point of .
Then has a unique fixed point in . Then has a unique fixed point in .
Example 19. Consider = . Let : × → C be given as It is verified that the pair ( , ) satisfies the weak -property (precisely, -property).
Let : → be defined as follows: Then satisfies the properties mentioned in Theorem 16. It can be verified that inequality (12) is satisfied. Hence the conditions of Theorem 16 are satisfied and it is seen that 0 is the unique best proximity point of .
Then satisfies the properties mentioned in Theorem 16. It can be verified that inequality (12) is satisfied. Hence the conditions of Theorem 16 are satisfied and it is seen that 1 is the unique best proximity point of .
Note 3. As explained in Example 14, the pair ( , ) in the above example satisfies the weak -property but does not satisfy the -property.