Abstract

We introduce the concept of proximity points for nonself-mappings between two subsets of a complex valued metric space which is a recently introduced extension of metric spaces obtained by allowing the metric function to assume values from the field of complex numbers. We apply this concept to obtain the minimum distance between two subsets of the complex valued metric spaces. We treat the problem as that of finding the global optimal solution of a fixed point equation although the exact solution does not in general exist. We also define and use the concept of P-property in such spaces. Our results are illustrated with examples.

1. 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.

Theorem 1 (see [3]). Let be a nonempty compact convex subset of a normed linear space and let be a continuous function. Then there exists such that .

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 [5–14] 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 [16–18]. 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. 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. G-metric 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 be the set of complex numbers and . Define a partial order on as follows: It follows that if one of the following conditions is satisfied: (i), ;(ii), ;(iii), ;(iv), .In particular, we will write if and one of (i), (ii), and (iii) is satisfied and we will write if only (iii) is satisfied.

Note that

Definition 2. Let be a subset of . If there exists such that , for all , then is bounded above and is an upper bound. Similarly, if there exists such that , for all , then is bounded below and is a lower bound.

Definition 3. For a subset 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 . Similarly, for a subset 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 .

Suppose that is bounded above. Then there exists such that , for all . It follows that and , for all ; that is, and are two sets of real numbers which are bounded above. Hence both and exist. Let and . Then clearly, is the least upper bound (lub) of or .

Similarly, if is bounded below, then is the greatest lower bound (glb) of or , where and .

Any subset which is bounded above has the least upper bound (lub) or supremum. Equivalently, any subset 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 satisfies (i), for all , if and only if ;(ii), for all ;(iii), for all .Then is called a complex valued metric on and is called a complex valued metric space.

Definition 5. Let be a complex valued metric space and a sequence in and .(i)If for every with there is such that, for all , , then is said to be convergent, converges to , and is the limit point of . We denote this by or as .(ii)If for every with there is such that, for all , , 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 6. Let be a complex valued metric space and . The set is said to be bounded if there exists such that , for all .
Clearly, for any bounded subset , the set is both bounded below and above and hence sup and inf exist.

Lemma 7 (see [15]). Let be a complex valued metric space and a sequence in . Then converges to if and only if as .

Note 1. We can also replace the limit in Lemma 7 by the equivalent limiting condition as .

Lemma 8. Let be a complex valued metric space and a sequence in . Then is a Cauchy sequence if and only if as .

Note 2. We can also replace the limit in Lemma 8 by the equivalent limiting condition as .

Definition 9. Let be a complex valued metric space, and . Then the function is continuous at if for any sequence in ,

Definition 10. Let be a complex valued metric space. A set is called closed if for any sequence in , implies .

Next we define the proximity point and some related concepts in complex valued metric space.

Let and be two nonempty bounded subsets of a complex valued metric space . Then is always bounded below by and hence inf exists. Here we define From the above definition, it is clear that for every there exists such that and conversely, for every there exists such that .

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 .

The definition of -property and weak -property was introduced in [13] and [14], respectively.

Now we define them in complex valued metric space.

Definition 12. Let and be two nonempty subsets of a metric space with . Then the pair is said to have the -property if, for any and ,

Definition 13. Let be a pair of nonempty subsets of a complex valued metric space with . Then the pair is said to have the weak -property if and only if where and .

Example 14. Let us consider the complex valued metric space where and let be given as Let and be two subsets of given by where is a fixed positive integer.
Notice that It can be verified that for and that is, the pair satisfies the weak -property.
Particularly, let , and , . Then, So, the pair does not satisfy the -property.

Remark 15. Let be a pair of nonempty subsets of a complex valued metric space with . If the pair satisfies the -property then it also satisfies the weak -property. But the converse is not true.

2. Main Results

Theorem 16. Let be a pair of nonempty closed and bounded subsets of a complete complex valued metric space such that is nonempty and the pair satisfies the weak -property. Let be a mapping with . If there exists a real number with such that, for all , where then has a unique best proximity point in .

Proof. Let . Since , there exists such that . Again, since , there exists such that . Continuing the process, we construct a sequence in such that Since the pair satisfies the weak -property, we conclude that If , then it follows from (14) that is a best proximity point of . So we assume for all ; that is, for all . Applying (12) in (15), we have where that is, So, we have Then it follows that For any , which implies that is a Cauchy sequence. From the completeness of , there exists such that 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 is a constant sequence with the constant value . Therefore, ; that is, is a best proximity point of .
Finally, we will prove that such a point is unique.
Suppose that is another best proximity point of ; that is, . Now, and imply that and . Since the pair satisfies the weak -property, we necessarily have Now, where So, we have which is a contradiction unless ; that is, . Hence the best proximity point of is unique.

Corollary 17. Let be a complete complex valued metric space. Let be a mapping such that, for all , where Then has a unique fixed point in .

Corollary 18. Let be a complete complex valued metric space. Let be a mapping such that, for all , Then has a unique fixed point in .

Example 19. Consider . Let be given as Then is a complex valued metric space with the required properties of Theorem 16.
Let Then is a pair of nonempty closed and bounded subsets of such that 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 is the unique best proximity point of .

Example 20. We take the complex valued metric space considered in Example 19. Let Then is a pair of nonempty closed and bounded subsets of such that It is verified that the pair satisfies the weak -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 is the unique best proximity point of .