#### Abstract

In this paper, a new type of non-self-mapping, called Berinde MT-cyclic contractions, is introduced and studied. Best proximity point theorems for this type of mappings in a metric space are presented. Some examples illustrating our main results are also given. Our results generalize and improve some known results in the literature.

#### 1. Introduction and Preliminaries

Several problems in a real world can be modeled in the form of operator equations. An equation , which is called the fixed point equation, is one of the important means for solving some problems among them. Fixed point theory is then considered as an important tool for solving such problems. The well-known fixed point theorem for contraction mappings was given by Banach [1]. It is known as the Banach contraction principle. This principle guarantees that each contraction mapping from a complete metric space into itself always has a unique fixed point.

In 2004, Berinde [2] introduced and studied the concept of weak contraction mappings in the context of a complete metric space. Let be a metric space. A mapping is called a weak contraction if there exist and such thatA fixed point theorem of this type of mapping was proved in [2]. It extended and generalized that of the Banach contraction principle and others; see [2] and references therein.

On the other hand, if the fixed point equation does not have a solution, then for all . In this situation, it is natural to ask whether we can find an approximate solution such that the error is In order to have a concrete lower bound, let us consider two nonempty subsets and of a metric space and a mapping . It is observed that for all , where . So we are interested to find a point such thatSuch point is called a best proximity point of the mapping , and is called the global minimum value of .

The best proximity point theorem was first studied by Fan [3], in 1969. He proved that if is a nonempty compact convex subset in a normed space and is a continuous mapping, then there exists such that where . Especially, if , then we get that is a fixed point of .

Several years later, the above result has been studied and generalized by many researchers, such as Reich [4], Sehgal and Singh [5], Vetrivel et al. [6], Anuradha and Veeramani [7], Basha [8, 9], Kirk et al. [10], Raj [11], Gabeleh [12], Abkar and Gabeleh [13], Eldred and Veeramani [14], and Du and Lakzian [15] and references therein. Some recent research papers worth mentioning are [1619].

Throughout this paper, we denote by and nonempty subsets of a metric space We also require the following notions:

A mapping is called a cyclic mapping if and . And a point is said to be a best proximity point of if . In 2006, Eldred and Veeramani [14] introduced the concept of cyclic contraction and proved the existence of a best proximity point for this type of mapping on a complete metric space.

Definition 1 (see [14]). A mapping is called a cyclic contraction if the following conditions hold: (i) is a cyclic mapping;(ii)there exists such that for all .

The concept of -function was used by Reich [20] and Mizoguchi and Takahashi [21] to define a class of multivalued mappings which is more general than that of contraction mappings. After that Du [22, 23] studied the class of multivalued mappings generated by Mizoguchi and Takahashi functions (or MT-functions) and gave characterizations of MT-functions.

Definition 2 (see [22]). A function is said to be an MT-function (or R-function) if for all .

Theorem 3 (see [23]). Let be a function. Then the following statements are equivalent. (a) is an MT-function; i.e., for all .(b)For each , there exist and such that   for all (c)For any nonincreasing sequence in ,

It is clear that if is a nondecreasing function or a nonincreasing function, then is an MT-function. For more examples and details, see [15, 22, 23].

Consequently, Du and Lakzian [15] introduced MT-cyclic contractions with respect to and proved the existence and convergence theorems for this type of non-self-mapping in metric spaces.

Definition 4 (see [15]). If a map satisfies the following: then is called an MT-cyclic contraction with respect to on (i) is a cyclic mapping;(ii)there exists an MT-function such that + for all and .

It is obvious that if with , then is a cyclic contraction, and, hence, an MT-cyclic contraction with respect to which is more general than that of cyclic contraction. For example of an MT-cyclic contraction with respect to , but is not a cyclic contraction, see [15].

In 2009, Suzuki et al. [24] introduced the concept of the property UC of two nonempty subsets of a metric space as follows.

Definition 5 (see [24]). Let be a pair of nonempty subsets of a metric space . The pair is said to satisfy the property UC if and are sequences in and is a sequence in such thatand then .

Later, in 2011, Kosuru and Veeramani [25] introduced the concept of semisharp proximal pair of two nonempty subsets of a metric space. This concept is again more general than that of the property UC.

Definition 6 (see [25]). Let be a pair of nonempty subsets of a metric space . The pair is said to be a semisharp proximal pair if for each and there exist at most one and such that .

Example 7. Consider the space of all real valued continuous functions on with the supremum norm; i.e., . Setwhereand It is easy to show that and for all . Hence and is a semisharp proximal pair.

Note that the property UC implies semisharp proximality. In 2015, R. Espinola, et al. [26] introduced the concept of a proximally complete pair of subsets of a metric space. They proved existences and convergence theorems of best proximity points for cyclic contraction mappings. They obtained a useful theorem presented as follows.

Theorem 8 (Espinola et al. [26, Theorem 3.3]). Let be a pair of complete subsets of a metric space satisfying the UC property. If is a sequence in with and , for all , then the sequences and have convergent subsequences in and , respectively.

By those works mentioned above, we aim to introduce a new type of single-valued, non-self-mapping which is more general than that of contractions, cyclic contractions, and -cyclic contractions. The best proximity point theorems for this type of mappings in metric spaces will be investigated. Our main results extend and generalize those of Du and Lakzian [15], Eldred and Veeramani [14], and others.

#### 2. Main Results

By using ideas of cyclic contractions, MT-functions, and weak contractions, we shall first introduce Berinde MT-cyclic contractions with respect to and prove the existence and convergence theorems for this type of non-self-mapping in metric spaces.

Definition 9. Let be a cyclic mapping. The mapping is said to be a Berinde MT-cyclic contraction with respect to if there exists an MT-function and such thatfor all and .

It is easy to see that a Berinde MT-cyclic contraction with respect to can be reduced to an MT-cyclic contraction with respect to .

Remark 10. If is a Berinde MT-cyclic contraction with respect to , then satisfies the following condition:for all and . To see this, we can write (9) in the form , for all and . Because of , it follows thatand, for all and , hence (10) is satisfied.

Example 11. Let be the metric space consisting of all bounded real sequences with supremum metric and let be the canonical basis of , where is the space of all null sequences. Let be a sequence of positive real numbers satisfying and for and for some positive real number . Thus is convergent. Put for and let . Then is a bounded and complete subset of , and hence is a complete metric space with if .
Let , and let be defined by and define by Then and . Since for all , we have that is an -function. Next, we show that is a Berinde -cyclic contraction with respect to . Obviously, for satisfy (9) with . We will consider three cases as follows.
Case 1. For and , we have Case 2. For and , we get Case 3. For and , , we have From all of the above cases, we can now conclude that is a Berinde -cyclic contraction with respect to and . We note from Case 1 that is not an -cyclic contraction with respect to .

Proposition 12. Let and be nonempty subsets of a metric space and be a Berinde MT-cyclic contraction with respect to . Then starting with any given , define a sequence in by for all ; we have as .

Proof. Let be given. Define a sequence in by Suppose that (when is similar); then , and so, . Since is a Berinde MT-cyclic contraction with respect to , we have Again, since and is a cyclic mapping, we get . By the Berinde MT-cyclic contraction with respect to of , we have By induction, we can show that, for each ,By Remark 10, we have for all . It means that is a nonincreasing sequence. By Theorem 3, we get .
Put . Thus , for all . It follows from (20) that Hence for each , we have Since and as , by taking in the above inequality, we obtain that The proof is now completed.

Example 13. Let for all . Then . Put for and let be a bounded and complete subset of . Then be a complete metric space with if . Set and . So . Let be defined byIt is easy to see that and and so is a cyclic mapping. Define as Then is an MT-function. Now, we will show that is a Berinde MT-cyclic contraction with respect to . For with , Hence is a Berinde MT-cyclic contraction with respect to . Therefore, all the assumptions of Proposition 12 hold.

The following result is obtained immediately from Proposition 12 because every nondecreasing function or nonincreasing function is an MT-function.

Corollary 14. Let and be nonempty subsets of a metric space and be a cyclic mapping. Let be given and define a sequence in by , for all . Suppose that there exists a nondecreasing (or nonincreasing) function such that for all and Then .

By Proposition 12, if is a cyclic contraction or an MT-cyclic contraction with respect to , then we obtain directly the following results which were proved by Eldred and Veeramani [14] and Du and Lakzian [15], respectively.

Corollary 15 (Eldred and Veeramani [14, Proposition 3.1]). Let and be nonempty subsets of a metric space and be a cyclic contraction. Then starting with any and defining a sequence in by , for all , then .

Corollary 16 (Du and Lakzian [15, Theorem 2.1]). Let and be nonempty subsets of a metric space and be an MT-cyclic contraction with respect to . Then starting with any , define a sequence in by for all ; we have as .

Observe that if and are nonempty subsets of a metric space and is a cyclic mapping with , define a sequence in by , for all ; then and are subsequences of in and , respectively. Similarly, if , then and are subsequences of in and , respectively. Moreover, , for all .

Theorem 17. Let and be nonempty subsets of a metric space and be a cyclic mapping. Let be given and define a sequence in by , for all . Suppose that the following conditions hold: (i) for all and with ;(ii) has a convergent subsequence in ;(iii). Then there exists such that .

Proof. Let and be a subsequence of such that In fact, for each ,It follows by that From , for each , we have Taking in the above inequality, we obtain thatThe proof is completed.

Using the same proof as Theorem 17, we obtain a similar result.

Theorem 18. Let and be nonempty subsets of a metric space and be a cyclic mapping. Let be given and define a sequence in by , for all . Suppose that the following conditions hold: (i) for all and with ;(ii) has a convergent subsequence in ;(iii). Then there exists such that .

Applying Proposition 12 and Theorems 17 and 18, we establish the following new best proximity point theorems for a Berinde MT-cyclic contraction with respect to .

Theorem 19. Let and be nonempty subsets of a metric space and be a Berinde MT-cyclic contraction with respect to . Let be given and define a sequence in by , for all . If has a convergent subsequence in , then there exists such that .

Theorem 20. Let and be nonempty subsets of a metric space and be a Berinde MT-cyclic contraction with respect to . Let be given and define a sequence in by , for all . If has a convergent subsequence in , then there exists such that .

By Theorems 19 and 20, we obtain the next result.

Corollary 21. Let and be nonempty subsets of a metric space and be a Berinde MT-cyclic contraction with respect to . Let be given and define a sequence in by , for all . Suppose that or is compact. Then there exists such that .

Theorem 22. Let and be nonempty subsets of a metric space such that is a semisharp proximal pair and is a Berinde MT-cyclic contraction with respect to . Let be given and define a sequence in by , for all . If has a convergent subsequence in , then the following hold: (i)there exists such that ;(ii) and are fixed point of in and , respectively.

Proof. By Theorem 19, there exists such that and it follows that Hence . In the semisharp proximality of , we have . Considerwhich implies that Therefore, and are fixed points of in and , respectively.

Using the proof of Theorem 22, we obtain the following result.

Theorem 23. Let and be nonempty subsets of a metric space such that is a semisharp proximal pair, and is a Berinde MT-cyclic contraction with respect to . Let be given and define a sequence in by , for all . If has a convergent subsequence in , then the following hold: (i)there exists such that ;(ii) and are fixed point of in and , respectively.

By using Theorem 8, we have the following corollary.

Corollary 24. Let and be nonempty complete subsets of a metric space such that satisfies the property UC, and is a Berinde MT-cyclic contraction with respect to . Let be given and define a sequence in by , for all . Then the following hold: (i)there exists such that ;(ii) and are fixed points of in and , respectively.

We have discussed that, under some specific conditions, a Berinde MT-cyclic contraction with respect to can be reduced to a cyclic contraction or an MT-cyclic contraction with respect to . Thus, Theorems 17 and 19 are generalizations of the results proved by Eldred and Veeramani [14] and Du and Lakzian [15], respectively. Hence, the three following corollaries are obtained directly from those theorems.

Corollary 25 (Eldred and Veeramani [14, Proposition 3.2]). Let and be nonempty closed subsets of a complete metric space Let be a cyclic contraction, , and define for all . Suppose that has a convergent subsequence in Then there exists such that

Corollary 26 (Du and Lakzian [15, Theorem 2.3]). Let and be nonempty subsets of a metric space and be a cyclic mapping. Let be given and define a sequence in by , for all . Suppose that the following conditions hold: (i) for all and ;(ii) has a convergent subsequence in ;(iii). Then there exists such that

Corollary 27 (Du and Lakzian [15, Theorem 2.4]). Let and be nonempty subsets of a metric space and be a MT-cyclic contraction with respect to . Let be given and define a sequence in by , for all . Suppose that has a convergent subsequence in . Then there exists such that .

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank Professor Dr. Suthep Suantai for his helpful advice. This research is supported by Center of Excellence in Mathematics and Applied Mathematics, Chiang Mai University, and Faculty of Science, Chiang Mai University, Chiang Mai, Thailand.