Research Article | Open Access

# A Best Proximity Point Theorem for Generalized Non-Self-Kannan-Type and Chatterjea-Type Mappings and Lipschitzian Mappings in Complete Metric Spaces

**Academic Editor:**Tomonari Suzuki

#### Abstract

The purpose of this paper is to provide and study a best proximity point theorem for generalized non-self-Kannan-type and Chatterjea-type mappings and Lipschitzian mappings in complete metric spaces. The significant mapping in a unified form which related to contractive mappings, Kannan-type mappings, and Chatterjea-type mappings is established. We also provide some examples to illustrate the situation corresponding to the main theorem. The main result of this paper can be viewed as a general and unified form of several previously existing results.

#### 1. Introduction

Fixed point theory can be looked upon as an important model that can be used in several real world problems and it is in close relationship with other branches of mathematics. It furnishes unified treatment and is a vital tool for solving equations of form , where is a self-mapping defined on a subset of some suitable spaces such as a metric space, a normed linear space, or a topological vector space. However, in case the mapping is not a self-mapping, the fixed point theorems are not specified to provide the existence of a solution for the equation . On the other hand, the best approximation theorems and the best proximity point theorems play an important rule to solve an approximate solution to the equation when is a non-self-mapping, in which case a solution does not necessarily exist. For some interesting best approximation theorems, let us refer to [1â€“6]. For a non-self-mapping , a best proximity point theorem investigates the situations which lead to the existence of an element nearest to . It can be said on the other hand that a best proximity point theorem explores an element for which the value is minimum in the setting of metric spaces. This means that it is to study the global minimization of the real valued function . A best proximity point theorem succeeds in finding the global minimum of by constraining an approximate solution of the equation to satisfy the condition that . The solutions of the equation are called best proximity points of the mapping . Furthermore, if is a self-mapping, then all best proximity points turn into the fixed points of . With all these reasons, the study on best proximity point theorems is interesting and will cover the fixed point theorems implicitly.

For the previous research involving best proximity point theorems for several types of some mappings and contractions, one can refer to [7â€“17]. Best proximity point theorems for set valued mappings have been found in [18â€“28]. Moreover, in case of common best proximity point theorems, there are some interesting results in [29â€“32].

In 2013, Sadiq Basha et al. [33] established some best proximity point theorems for non-self-nonexpansive mappings, non-self-Kannan-type mappings, and non-self-Chatterjea-type mappings. Their results are quite interesting and there are examples to illustrate the main results.

Motivated and inspired by the research mentioned above, we are interested in studying a best proximity point theorem for generalized non-self-Kannan-type and Chatterjea-type mappings and Lipschitzian mappings in complete metric spaces with the way of an optimal approximate solution for and , where is a generalized non-self-Kannan-type and Chatterjea-type mapping and is a Lipschitzian mapping, respectively. The results obtained in this paper can be viewed as some extensions of the corresponding ones announced by Sadiq Basha et al. [33], Kannan [34], Chatterjea [35], and many others.

#### 2. Non-Self-Lipschitzian Mappings and Generalized Non-Self-Kannan-Type and Chatterjea-Type Mappings

In this section, some definitions related to Lipschitzian mappings and their special forms are given and will be used in the sequel. The new mappings called* generalized non-self-Kannan *and* Chatterjea mappings* are established which are wider than non-self-Kannan mappings and non-self-Chatterjea mappings. Some other notions are provided and will be used in the next section.

*Definition 1. *Let and be nonempty subsets of a metric space . An element in is said to be a best proximity point of a mapping if .

It is noticed that best proximity becomes a fixed point if the underlying mapping is a self-mapping. Moreover, in light of the fact that for all in , the function attains its global minimum at a best proximity point.

*Definition 2. *Let and be two metric spaces. A mapping is said to be a* non-self-Lipschitzian mapping* if there exists a constant such thatfor all .

The smallest number for which (1) holds is called the* Lipschitz constant* of .

*Definition 3. *Let and be two metric spaces. A Lipschitzian mapping with the Lipschitz constant is said to be a* non-self-nonexpansive mapping*.

*Definition 4. *Let and be two metric spaces. A Lipschitzian mapping with the Lipschitz constant is said to be a* non-self-contractive mapping*.

*Definition 5. *Let and be nonempty subsets of a metric space . A mapping is said to be (1)a* non-self-Kannan mapping* (see [34] for the self-mapping case) if there exists a constant such thatâ€‰for all ;(2)a* non-self-Chatterjea mapping* (see [35] for the self-mapping case) if there exists a constant such thatâ€‰for all .

*Definition 6. *Let and be nonempty subsets of a metric space . A mapping is said to be a* generalized non-self-Kannan and Chatterjea mapping* if there exist nonnegative constants such that andfor all .

It is obvious that (4) is in a generalized form of (2) and (3).

*Definition 7. *Let and be nonempty subsets of a metric space and let be a mapping. A mapping is said to be (1)a* non-self-Kannan mapping with respect to the mapping * if there exists a constant such thatâ€‰for all ;(2)a* non-self-Chatterjea mapping with respect to the mapping * if there exists a constant such thatâ€‰for all .

*Definition 8. *Let and be nonempty subsets of a metric space and let be a mapping. A mapping is said to be a* generalized non-self-Kannan and Chatterjea mapping with respect to the mapping * if there exist nonnegative constants such that andfor all .

It is clear that (7) is in a generalized form of (5) and (6).

*Definition 9 (see [33]). *Let and be nonempty subsets of a metric space . Given mappings and , the pair is said to form a weak *-cyclic contraction* if there exists a nonnegative real number such that for all and .

*Definition 10 (see [33]). *Let and be nonempty subsets of a metric space. Given mappings and , the pair is said to form a -cyclic contraction if there exists a nonnegative real number such that for all and .

It is easy to observe that every -cyclic contraction is a weak -cyclic contraction.

#### 3. Main Results

In this section, we establish and prove a best proximity point theorem for generalized non-self-Kannan-type and Chatterjea-type mappings and Lipschitzian mappings in complete metric spaces. Before going into the main theorem, it is useful to know the following observation.

*Remark 11. *Let , and be nonnegative real numbers with and satisfyThen the following hold: (i).(ii).(iii).(iv) and .

*Proof. *Notice that (i) is directly obtained from (10). For (ii), we observe that (iii) and (iv) are not hard to verify from (ii).

Theorem 12. *Let and be nonempty closed subsets of a complete metric space. Let and satisfy the following conditions: *(a)* is a Lipschitzian mapping with Lipschitz constant .*(b)*There exist nonnegative constants such that *(c)* for all *(d)*The pair forms a weak -cyclic contraction.**Then, there exist elements and such thatIf is any fixed element in , , and , then the sequences and converge to some best proximity points of and , respectively. Further, if is another best proximity point of , then *

*Proof. *Let be any fixed element in . Then, it can generate the sequences and by for all and for all , respectively. It is observed thatIt follows from (14) and Remark 11(i) that By mathematical induction, we obtain By using (b), it is not hard to verify that is a Cauchy sequence in and hence converges to some element in . Similarly, we observe thatIt follows from (17) and Remark 11(i) that Therefore, mathematical induction yields It follows from (b) that is a Cauchy sequence in and hence converges to some element in . Furthermore, it can be observed thatLetting in (20), it yields Notice that ; it is not hard to verify that and then .

On the other hand, we also found thatLetting in (22), it yields This implies that and hence . Since the pair forms a weak -cyclic contraction, it follows that there exists such that Hence This shows that is a best proximity point of and is a best proximity point of . Similarly, if we suppose that is another best proximity point of , it can be proved that Consequently, it can be observed thatIt follows from (29) and Remark 11(iii) that This completes the proof of the theorem.

To achieve a better understanding of the situation of the main theorem even more, let us consider the following example to illustrate.

*Example 13. *Let with supremum norm. Define the following two sets:It is not hard to verify that and . Define the mappings and by for all as shown in Figure 1.

Then, (1)All the conditions are consistent with all of the assumptions in Theorem 12.(2)The mapping is a non-self-Lipschitzian mapping with constant which is not a non-self-nonexpansive mapping.(3)There exist such that . That is, the mapping is not a non-self-Kannan-type mapping with respect to the mapping .(4)There exist such that . That is, the mapping is not a non-self-Chatterjea-type mapping with respect to the mapping .

*Solution 1. *() It is found thatOn the other hand, let us considerFor convenience for writing, we will let and . From (29), (34), and (35), it is sufficient to show thatfor all . Now, we let (1).(2) + for all . The two surfaces and can be illustrated as in Figure 2.

The idea to prove (36) is to divide the unit square of -plane into six parts as shown in Figure 3.

() We will show that (36) is true on the area . It is clear that , , and for all . Since , then . It implies that , , and and then , , and . Thus, we haveThus it is sufficient to show that or equivalently to show thatDefine for all . It is not hard to compute that for all . Thus is an increasing function on . This implies thatfor all . On the other hand,for all . It is found that is the critical point of providing the absolute maximum valueon . It is not hard to see that is an increasing function on and is a decreasing function on . Notice that the starting point provides the value and then it implies that the end point provides the absolute minimum value on . Therefore, for all . Thus, we have for all .

() For the area , one can study from the proof of the area to verify that (36) holds for all .

() For the area , one can study from the proof of the area to conclude that (36) is true for all .

Notice that the graph of the two surfaces and has symmetry. Therefore, (36) is also true for the areasHence we can conclude that (36) holds for all . This shows that is a generalized non-self-Kannan and Chatterjea mapping with respect to the mapping with constants , .

Further, it can be verified thatWe observe that the pair is a weak -cyclic contraction. For this fact, let us consider and then Therefore, for any it can be seen that It can be observed that the mapping is a non-self-Lipschitzian mapping with constant but it is not a non-self-nonexpansive mapping as follows: