#### Abstract

In this paper, we discuss some (coincidence) best proximity point results for generalized proximal contractions and -proximal Geraghty contractions in controlled metric type spaces. To clarify our study, various examples are given and some conclusions are drawn.

#### 1. Introduction and Preliminaries

To solve the equation ( is a mapping defined on a subset of a metric space, a simplified linear space, or a topological vector space), fixed point theory is an important tool. A nonself-mapping may not have a fixed point. From this perspective, the best approximation theorem and the best proximity point are relevant. A classical best approximation theorem was due to Fan [1], i.e., if is a nonempty compact convex subset of a Hausdorff locally convex topological vector space with a seminorm and is a continuous mapping, then there is an element in satisfying the condition that . Many subsequent extensions and variations of Fan’s theorem have occurred, including references [2, 3].

However, even though the best approximation theorems provide an approximate solution to the equation , they do not provide an ideal approximate solution. Moreover, the theorem of the best proximity point specifies adequate criteria for the presence of an element to reduce the error . For a nonself-mapping , is at least for all in , then the best proximity point theorem establishes a globally optimal solution of error by constraining an approximate solution of the equation to the condition that . Such an ideal approximate solution is the best proximity point of the nonself-mapping . For sure, the best proximity point hypotheses are a logical augmentation of fixed point hypotheses, on the grounds that the best proximity point is a fixed point in the light of self-mappings.

The best proximity point hypotheses have been demonstrated in [4]. Anuradha and Veeramani have tested the proximal pointwise contractions for the presence of a best proximity point [2]. Generally, several best proximity point theorems were analyzed for multiple variants of contractions in [5–14]. A best proximity point theorem for contraction mappings was presented in [15]. Some interesting common best proximity theorems have been discussed in [7, 15].

Nadler [16] was the first who generalized the Banach contraction principle for multivaluated mappings. Later, several works appeared in this direction. For more details, see [17–20]. The best proximity point hypotheses for different sorts of multivalued mappings have likewise been obtained in [21, 22].

Recently, the authors in [23] introduced a controlled metric type space in which the function of extended -metric spaces was substituted by a function depending on the parameters of the left-hand side of the triangular inequality. The primary goal of this article is to include the best proximity point theorems for generalized and modified proximal contractions in the context of complete controlled type metric spaces, thus providing an optimal approximate solution to the equation . It is acknowledged that the previous best proximity point theorems include the well-known Banach contraction principle and some of its generalizations.

First, we state the following useful definitions in the sequel.

*Definition 1. *(see [6]). Let be a metric space having a pair of nonempty subsets such that is nonempty. The pair has the -property if and only ifwhere and .

*Definition 2. *(see [24]). Let be a metric space having a pair of nonempty subsets and . Let and . The mapping is said to be -proximal admissible iffor all .

*Definition 3. *(see [25]). Let represent the closed and bounded subsets of . Let be the Pompeiu–Hausdroff metric induced by metric defined byfor , where

*Definition 4. *(see [23]). Let be a nonempty set, and consider as a function. Let satisfying(1) if and only if (2)(3), for all , then is called a controlled metric type spaceFrom now on, is a controlled metric type space.

*Definition 5. *(see [23]). A sequence in a controlled metric type space converges to some in if for each positive , there is some positive such that for each . It can be written as

*Definition 6. *(see [23]). The sequence in a controlled metric type space is said to be a Cauchy sequence, if for every , for all , where .

*Definition 7. *(see [23]). A controlled metric type space is said to be complete if every Cauchy sequence is convergent in .

*Definition 8. *(see [23]). Let and .(1)The open ball is defined as follows:(2)The mapping is said continuous at if for all , there exists such thatClearly, if is continuous at in the controlled metric type space , then implies that as .

*Definition 9. *(see [26]). Define the function byfor (it represents the set of closed subsets of ), whereLet and be two nonempty subsets of . Definewhereand we will denote

Theorem 1 (see [26]). *The function is a generalized Pompeiu–Hausdroff controlled metric space on .*

*Remark 1. *(see [26]). Let be a generalized Pompeiu–Hausdroff-controlled metric type space. Then, the following assertions hold (for all bounded and closed subsets of ):(1) is equivalent to (2)(3)

Theorem 2 (see [26]). *If is a complete controlled metric space with , for all , where , then is complete.*

#### 2. Coincidence Best Proximity Points for Generalized Proximal Contractions

In this section, we will discuss some best proximity point theorems using the multivalued concept on a controlled metric space .

From now and onward, and are nonempty subsets of a controlled metric type space (until otherwise stated). Define by and , where and and are nonempty subsets of .

*Definition 10. *(see [26]). A mapping is continuous in a controlled metric type space at if for all , there exists such thatwhere is given asClearly, if is continuous at , then implies that as

We introduce the following.

*Definition 11. *Let be a controlled metric type space having two nonempty subsets and . Let be a mapping. A point is said to be a best proximity point of the mapping if

*Definition 12. *Let be a controlled metric type space having two nonempty subsets and . A nonempty set is said to be approximately compact with respect to if every sequence in satisfying the condition that for some in has a convergent subsequence.

*Definition 13. *Given and . A pair of mappings is said to be a -generalized proximal contraction if there exists a real number such thatfor all in .

*Definition 14. *A mapping is said to be a -generalized proximal contraction if there exists such thatfor all in .

Note that, if we take (the identity mapping on ), then every -generalized proximal contraction will reduce to a -generalized proximal contraction.

*Definition 15. *Let be a controlled metric type space having two nonempty subsets and . Let and be mappings. A point is said to be a coincidence best proximity point of the pair of mappings if

*Remark 2. *If we take (the identity mapping over ), then every coincidence best proximity point becomes a best proximity point of the mapping .

If or , then every best proximity point will reduce to a fixed point of the mapping .

Our first main result is stated as follows:

Theorem 3. *Let be a controlled metric type space having two nonempty subsets and . Let and be one-to-one and continuous mappings. Assume that is a closed subset and is approximately compact with respect to with and . Further, assume that the pair is a -generalized proximal contraction such thatand , where . Then, there exists a coincidence best proximity point of the pair .*

*Proof. *Let be an arbitrary element in . Since is contained in and is contained in , there exists an element in such thatAgain, since is an element of which is contained in and is contained in , it follows that there is an element in such thatThis process can be continued by selecting in satisfying the condition as follows:Having selected satisfying the condition, there exists an element in satisfyingfor every integer .

Since the pair is a -generalized proximal contraction, by using equations (22) and (23), we obtainWe deduce thatNow, we have to prove that is a Cauchy sequence, for all natural numbers with ,Assume thatThen, we obtainUsing the ratio test, we havewhere . Taking limit as , we obtainThat is, is a Cauchy sequence in the complete generalized Pompeiu–Hausdroff controlled metric type space ; hence, it converges to some in (as the set is closed). Therefore,Taking on both sides of the above inequality, we haveTherefore, . In view of the fact that is approximately compact with respect to , has a subsequence converging to some for some . Thus,Therefore, is a member of . Since is contained in and for some in , and is a one-to-one continuous mapping, so . Since is continuous, it can be concluded that . This implies thatThat is, is a coincidence best proximity point of the pair .

To prove the uniqueness of the coincidence best proximity point of the pair of mappings , suppose that there is another coincidence best proximity point of the pair . We haveAs the mapping is one-to-one on the set and , one has . Since the pair is a -generalized proximal contraction, one can writeIt is a contradiction.

Corollary 1. *Let and be mappings, where is a closed subset and is approximately compact with respect to with . Suppose that is a continuous and - generalized proximal contraction such thatthen there exists a unique best proximity point of .*

*Proof. *If we take identity mapping ( is identity on ), the remaining proof is same as in Theorem 3

*Definition 16. *Let and . A pair of mappings is said to be a -modified proximal contraction if there exists such thatfor all in .

*Definition 17. *A mapping is said to be a -modified proximal contraction if there exists such thatfor all in .

Note that if we take (the identity mapping on ), then every -modified proximal contraction is a -modified proximal contraction.

Theorem 4. *Let and be two continuous and one-to-one mappings, where is a closed subset and is approximately compact with respect to with and . If the pair is a - modified proximal contraction andthen there exists a unique coincidence best proximity point of the pair .*

*Proof. *Let be an arbitrary element in . Since is contained in and is contained in , there exists an element in such thatSince is an element of which is contained in and is contained in , it follows that there exists an element in such thatBy continuing this process, we can construct a sequence in , satisfying the condition as follows:Having chosen in , there exists an element in , such thatfor every positive integer . Since the pair is a -modified proximal contraction from equations (43) and (44), we obtainRecursively, we haveNow, we have to prove that is a Cauchy sequence. For all natural numbers with , we haveAssume thatIt follows thatUsing the ratio test, we haveBy applying limit in inequality (49), we getwhich shows that is a Cauchy sequence; hence, it is convergent to some in (as the set is closed). Therefore,Taking limit on both sides of the above inequality, we haveTherefore, . In view of the fact that is approximately compact with respect to , has a subsequence converging to some for some . It follows thatTherefore, is an element of . Since is contained in , we have for some in . As and is a one-to-one continuous mapping, . Since is continuous, it can be concluded that . Hence,To prove the uniqueness, suppose that is another coincidence best proximity point of the pair such that . Then,Since the pair is a -modified proximal contraction, we havewhich is a contradiction (as is one-to-one mapping on ). Hence, the pair has a unique coincidence best proximity point.

Corollary 2. *Let be a given continuous mapping, where is a closed subset and is approximately compact with respect to with . If is a - modified proximal contraction and suppose thatthen there exists a unique best proximity point of .*

*Proof. *If we take (the identity mapping over the set ), the remaining proof is same as Theorem 4.

*Example 1. *Let . Consider the function given as and , where

Take to be symmetric which is defined as . It is easy to see that is a *controlled metric type space*. Take and . Obviously, , , and . Now, consider as follows:Clearly, . Define byWe get . Now, we have to show that the pair satisfieswhere , and . Since the pair is a -*modified proximal contraction:*for every , the pair is a -*modified proximal contraction*. Hence, 0 is the unique *coincidence best proximity point* of and .

*Definition 18. *Let and