#### Abstract

In this article, we introduce a new type of generalized multivalued Hardy and Roger’s type proximal contractive and proximal cyclic contractive mappings of -metric spaces and develop some results for the existence of best proximity point(s). Moreover, we obtain some results for the existence and uniqueness of best proximity points for single-valued mappings. Examples are given to explain the main results.

#### 1. Introduction

The metric fixed point theory plays a very fundamental role in many fields of mathematics especially in nonlinear analysis and some related disciplines. The fundamental tool of this theory is the Banach contraction principle (shortly BCP) [1] which states that if a self-mapping of a complete metric space with metric satisfiesfor all , and for some , has a unique fixed point, that is, there exists a point , such that

A mapping that satisfies (1) is known as Banach contraction. After this remarkable result, many mathematicians contributed for the development of fixed-point theory by producing many results with different generalized contractive mappings in complete metric spaces, for details one can see [28] and the references therein. One of the important generalizations of BCP was presented by Edelstein [9] in 1962. Later on, many mathematicians generalized Edelstein’s result, for instance Meir and Keeler [10] in 1969 and Reich [11] in 1971. Reich’s result has been further generalized by Hardy and Roger [12] in 1973 as follows.

Theorem 1. Let be a metric space and a self-mapping satisfying the following conditions for all :(1)where are nonnegative reals.Set . Then,(a)If is complete and , then has a unique fixed point(b)If (1) is modified as() for all impliesand is compact, is continuous, and ; then, has a unique fixed point.

Nadler [13] in 1969 generalized the BCP in the context of multivalued mappings of complete metric spaces. Later on, Nadler’s result has been generalized by Prolla [14] in 1983.

Meanwhile, the metric space has been generalized to -metric space; by then, the fixed point theory has been further generalized for single-valued and multivalued mappings in the context of -metric space, for instance, Bakhtin [15] in 1989 and Czerwik [16] in 1993.

For nonself mapping, ( and are two nonempty sets), such that (empty set); then, it is not possible to find the fixed point of . The best way to deal with such situation is to explore a point in , such thatwhereand if such a point in exists, it is called the best proximity point of . If , then the best proximity point becomes a fixed point. So, best proximity point theory is the proper generalization of fixed-point theory. Fan’s result [17] in 1969 was probably the first attempt in this direction.

Later on, many mathematicians extended Fan’s result and developed some best proximity point results. For more details, one can see [18]. Best proximity point theory has been further developed by using different proximal contractions, for more details, one can see references [1923].

Kirk [24] in 2003 introduced cyclic contraction and developed some fixed points results. Later on in 2006, Eldered and Veeramani [25] developed some best proximity point results for cyclic contractions.

Basha in 2019 [21] introduced proximal contractive and proximal cyclic contractive mappings and developed some results for the existence and uniqueness of best proximity point.

Recently, in 2021, Hiranmoy et al. [26] introduced proximal Kannan-type and proximal cyclic Kannan-type contractive mappings in metric spaces (compare with [21]) and developed some best proximity point results.

Motivated by the contractive mappings of Hiranmoy, we introduce the notion of multivalued Hardy and Roger’s type proximal and cyclic proximal contractive mappings and develop some results for the existence of best proximity points in -metric space. Furthermore, we give some examples to explain the results.

#### 2. Preliminaries

Throughout this article, , , , and denote the set of reals, nonnegative reals, positive integers, nonnegative integers, and collection of nonempty subsets of , respectively.

Definition 2. Let be a nonempty set and a real number. The mapping is a -metric and is called -metric space if satisfies the following axioms:() if and only if ()(), for all .Throughout this paper, and denote metric and -metric, respectively. Now, suppose that and are two nonempty subsets of . Define

Definition 3. A -metric space is boundedly compact if every bounded sequence in has a convergent subsequence (compare with [27]).

Definition 4 (see [26]). Let and be two nonempty subsets of . A mapping is said to be a proximal Kannan-type contractive mapping ifhold for all .

Definition 5. Let and be two nonempty subsets of . Then, a mapping is said to be cyclic if and (compare with [26]).

Definition 6 (see [26]). Let and be two nonempty subsets of . A cyclic mapping is said to be a proximal cyclic Kannan-type contractive mapping ifthat hold for all .

In the following, we introduce a compact weak proximal pair in -metric space.

Definition 7. Let and be two nonempty subsets of . The pair is said to be a compact weak proximal pair if for bounded sequences in and in with as , the sequences and have convergent subsequences in and , respectively (compare with [26]).

Remark 8. Note that if in above definition, then is a compact weak proximal pair if and only if is boundedly compact.

Now, we present a lemma in the context of -metric space (analogous to [[26], Lemma 2.2]) that will be used in the sequel to prove our main results.

Lemma 9. Let and be two nonempty subsets of , such that at least one of and is bounded, and is a compact weak proximal pair. Then, , and hence, so is .

Proof. Asso for each , there exists and , such thatTherefore, the sequence converges to . Now, we assume that is bounded. So, there exists a positive real number , such that for all , so we havewhich impliesTherefore, and are bounded sequences. So by compact weak proximality of the pair , there exist subsequences of and of , such that converges to and converges to . Therefore,Thus, we haveSo, and . Hence, and . Similarly, if is bounded, then and .

Theorem 10 (see [26]). Let and be two nonempty subsets of , such that at least one of and is bounded, and is a compact weak proximal pair. Let be a proximal Kannan-type contractive mapping and assume that(i)(ii)If and are two bounded sequences in and , respectively, such that converges to , then .Then, has a unique best proximity point in .

Theorem 11 (see [26]). Let and be two nonempty subsets of , such that at least one of and is bounded, and is a compact weak proximal pair. Let be a proximal cyclic Kannan-type contractive mapping and assume that the following conditions hold:(i) and (ii)If and are two bounded sequences in and , respectively, such that converges to , then .Then, the following conditions hold:(a)There exist and , such that and (b)If and , such that and , then .

Now, we introduce the notions of a new type of generalized multivalued Hardy and Roger’s proximal contractive and proximal cyclic contractive mappings.

Definition 12. Let and be two nonempty subsets of . A mapping is said to be a new type of generalized multivalued Hardy and Roger’s proximal contractive mapping ifwhich hold for all , where,.

Remark 13. If in the Definition 12, we replace by , then is said to be a new type of generalized Hardy and Roger’s proximal contractive mapping.

Definition 14. Let and be two nonempty subsets of . A multivalued cyclic mapping is said to be a new type of generalized multivalued Hardy and Roger’s proximal cyclic contractive mapping ifwhich hold for all , where

Remark 15. If in the Definition 14, we replace by , then is said to be a new type of generalized Hardy and Roger’s proximal cyclic contractive mapping.

#### 3. Best Proximity Points Results for a New Type of Multivalued Hardy and Roger’s Proximal Contractive Mappings in -metric Space

The following is our main result of this section.

Theorem 16. Let and be two nonempty subsets of , such that at least one of and is bounded and is a compact weak proximal pair. Let be a new type of generalized multivalued Hardy and Roger’s proximal contractive mapping. Further assume that(i)For each , (ii)If and are two bounded sequences in and , respectively, such that converges to , then .Then, has a best proximity point.

Proof. Lemma 9 implies . Let ; then, . We can pick an element , so that there exists , such thatContinuing this way, we can construct sequences in and in , such thatfor all . Therefore,that is,If for some , then is the best proximity point of , and the proof is completed. So, we may assume that for all . Now, we show that and are bounded sequences. As we haveso by the given condition, we obtainwhich impliesIf , then , so (17) implies , and so , a contradiction. Therefore, and . Hence, we getthat is,Asso, if , then we haveIf , thenwhich impliesand by (2), . Therefore,Hence, is a bounded sequence. Furthermore, for all , we haveTherefore, is also a bounded sequence. From (69), it is clear that is a decreasing sequence of nonnegative real numbers and hence convergent. Using hypothesis (ii), converges to 0. Now, by compact weak proximality of the pair , there exist two subsequences of and of , such that converges to some and converges to some . Consequently,Consequently,Thus, , which implies . For each , there exists , such that . Now,which impliesMoreover, we haveLetting , we getthen using the factswe getLetting , we getIt implies . Thus, we have , that is, is a best proximity point of . This completes the proof.
Now, we give an example to explain our claim.

Example 17. Let , , and . Considerfor all . Then, is a -metric on with . It implies , , and ; now, define a mapping as follows:It implies for each , . Now, we check is a new type of generalized multivalued Hardy and Roger’s proximal contractive mapping. LetThen, we discuss two possible cases.Case 1: ifthenCase 2: ifthenIt implieswhere. So, all axioms of Theorem 16 are satisfied. Hence, has the best proximity points set .

Theorem 18. Let and be two nonempty subsets of , such that at least one of and is bounded, and is a compact weak proximal pair. Let be a new type of generalized Hardy and Roger’s proximal contractive mapping and assume that(i)For each , (ii)If and are two bounded sequences in and , respectively, such that converges to , then Then, has a unique best proximity point.

Proof. Existence of best proximity point follows from Theorem 16. Now, to prove the uniqueness, consider and be two distinct best proximity points of . Then, we haveIt impliessoa contradiction as . Hence, has a unique best proximity point.

Corollary 19. If we take in Theorem 18 and , then we get Theorem 10.

#### 4. Best Proximity Points Results for a New Type of Multivalued Hardy and Roger’s Proximal Cyclic Contractive Mappings in -metric Space

In this section, we consider new type of multivalued Hardy and Roger’s proximal cyclic contractive mapping for the existence of best proximity points.

Theorem 20. Let and be two nonempty subsets of , such that at least one of and is a bounded subset of and is a compact weak proximal pair. Let be a new type of generalized multivalued Hardy and Roger’s proximal cyclic contractive mapping and assume that(i)For each , , and for each , and(ii)If and are two bounded sequences in and , respectively, such that converges to , then .Then, there exist and , such that and , and furthermore, .

Proof. Since is a compact weakly proximal pair and at least one of and is bounded, so by Lemma 9, it follows that and . Let and imply and , so there exists and , such that. Continuing this way, we construct sequences in , in , in , and in , such thatandIt impliesandfor all . Thus, we haveFirst, we assume that is bounded. Then, there exists a positive real number , such that for all . Therefore, for all , we haveimpliesTherefore, is bounded. Also, is cyclic, so for each , , and so , for all . Therefore, there exists a positive real number , such that . It implies is bounded, soIt implies is bounded. Thus, , , , and all are bounded sequences. On a similar line, we can prove , , , and are bounded whenever is bounded. Since is a compact weak pair, therefore, there exist subsequences , , , and of , , , and , respectively, such that , , , and , as . First, we show that . As we haveandso if , then we haveSo,If , then , so (19) implies , and is a contradiction. Therefore,IfthenTherefore,for all , and hence, the sequence is a convergent sequence of real numbers. By hypothesis (ii), it follows that converges to 0. Now,.
Therefore, . Again, we have as , and hence, we getSo, implies . Thus, there exists , such thatwhere . It impliesIfonly for finitely many , then we can exclude those from and then assumefor all . Iffor infinitely many , then we can extract a subsequence from , such thatfor all . This givesFrom the relationsandwe getfor all . Taking limit as , we getso . Therefore, we haveNext, we assume thatfor all ; then, from relationswe getTaking limit as in above, we getIt impliesIf , then (19) implies which implies , a contradiction, soHence,Similarly, we can proveNow, let , such thatandIf , thensoIt implieswhich further impliesa contradiction. So, . This completes the proof.

Theorem 21. Let and be two nonempty subsets of , such that at least one of and is a bounded subset of and is a compact weak proximal pair. Let be a new type of generalized Hardy and Roger’s proximal cyclic contractive mapping and assume that(i)For each , , and for each , ,(ii)If and are two bounded sequences in and , respectively, such that