#### Abstract

We collect, improve, and generalize very recent results due to Mongkolkeha et al. (2014) in three directions: firstly, we study *g*-best proximity points; secondly, we employ more general test functions than can be found in that paper, which lets us prove best proximity results using different kinds of control functions; thirdly, we introduce and handle a weak version of the *P*-property. Our results can also be applied to the study of coincidence points between two mappings as a particular case. As a consequence, the contractive condition we introduce is more general than was used in the mentioned paper.

#### 1. Introduction

*Fixed point theory* is a branch of nonlinear analysis which has attracted much attention in recent times due to its possible applications. After the appearance of the pioneering* Banach contractive mapping principle* in 1922, many mathematicians have intensively investigated sufficient conditions to ensure that certain contractive mappings have a fixed point. Some of the most well-known generalizations are due to Zabreĭko and Krasnoselĭ [1], Edelstein [2], Browder [3], and Caristi [4].

When a mapping from a metric space into itself has no fixed points, it could be interesting to study the existence and uniqueness of some points that minimize the distance between an origin and its corresponding image. These points are known as* best proximity points* and they were introduced by Fan [5] and modified by Sadiq Basha in [6]. The study of this kind of points and their properties has become one of the newest branches of fixed point theory, and many interesting results, generalizing the notion of fixed point, have been presented. In fact, many theorems in fixed point theory have been very useful so as to introduce their corresponding extensions to this new field of study (see also [7–13] and references therein).

On the other hand, in the past years, fixed point theorems in partially ordered metric spaces have also attracted much attention, especially after the works of Ran and Reurings [14], Nieto and Rodríguez-López [15], Gnana Bhaskar and Lakshmikantham [16], Berinde and Borcut [17, 18], Karapınar and Berinde [19, 20], Berzig and Samet [21], and Roldán et al. [22–24], among others. Their results were extended to more general contractivity conditions in which* altering distance functions* play a key role. Very recently, Alghamdi and Karapınar [25] used a similar notion in -metric spaces, and Berzig and Karapınar [26] also considered a more general kind of contractivity conditions using a pair of generalized altering distance functions.

In order to consider a contractive condition on the whole metric space that can be particularized to partially ordered metric spaces, some advances have been done in recent times (see, for instance, [25–27] and references therein). This subject has been extended by Mongkolkeha et al. [28] to the field of determining best proximity points, describing a wide class of contractive mappings and using very general control functions. The main aim of this paper is to collect, generalize, and improve their results using contractive conditions and control functions that can be particularized in a wide kind of different results applicable to several frameworks.

#### 2. Preliminaries

Let denote the set of all nonnegative integers. Throughout this paper, let be a metric space, let and two nonempty subsets of , and let , , and be three mappings. Define Notice that, if and verify , then and . Therefore, is nonempty if, and only if, is nonempty. Thus, if is nonempty, then , , and are nonempty subsets of . It is clear that, if , then is nonempty. In [29], the authors discussed sufficient conditions in order to guarantee the nonemptiness of . In general, if and are closed subsets of a normed linear space such that , then is contained in the boundary of (see [6]).

The main aim of this paper is to study sufficient conditions to ensure the existence and, in some cases, the unicity of the following kind of points.

*Definition 1. *One will say a point is a -*best proximity point of * if and is a best* proximity point of * if .

If , a -best proximity point of is called a* coincidence point of* * and* (i.e., ), and if is the identity mapping on , then is a* fixed point of* (i.e., ).

We describe the families of functions that we will use henceforth.

*Definition 2. *(i) One will denote by the family of all functions such that, for all , the series converges (functions in are called* (c)-comparison functions*).

(ii) One will denote by the family of all functions such that and for all .

(iii) One will denote by the family of all continuous mappings such that if one or more arguments take the value zero (i.e., if ).

(iv) One will denote by the family of all mappings such that if one or more arguments take the value zero (i.e., if ).

(v) One will denote by the family of all mappings such that .

(vi) One will denote by the family of all mappings such that whatever the sequences such that, at least one of them, is convergent to zero (i.e., there exists verifying ).

*Remark 3. * It is easy to see that, if , then for all .

We point out that we do not impose any monotone condition on the control function we will use.

Clearly and . Notice that functions in , and have not to be continuous.

*Example 4. *Examples of functions in are the following ones (where ):

The mappings of have been very useful in the framework of fixed point theory (see [30–32]). The following lemma can be found in the literature but we recall it here for the sake of completeness.

Lemma 5. *Let be a mapping and let be a sequence. If and for all , then .*

In the following result, denotes the family of all permutations .

Lemma 6. *Given and , define by
**
Then and is symmetric. Furthermore, if , then .*

*Definition 7. *If is a binary relation on , one will consider the mapping given, for all , by

*Definition 8. *A* preorder* (or a* quasiorder*) on is a binary relation on that is* reflexive* (i.e., for all ) and* transitive* (if verify and , then ). In such a case, we say that , *≼* is a* preordered space* (or a* preordered set*). If a preorder is also* antisymmetric* ( and implies ), then is called a* partial order*.

*Definition 9 (Raj [33]). *Let and be two subsets of a metric space such that is nonempty. We say that the pair has the* P-property* if

In [28], the authors introduced the following find of contractive mappings and succeed in proving the following result.

*Definition 10 (Mongkolkeha et al. [28], Definition 3.1). *Let and be nonempty subsets of a metric space . A mapping is said to be a* generalized almost* * contraction* if
for all , where , , , and

Theorem 11 (Mongkolkeha et al. [28], Theorem 3.2). *Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the -property. Let satisfy the following conditions:*(a)*is an -proximal admissible and generalized almost -contraction;*(b)* is continuous;*(c)*there exist elements such that and ;*(d)*. **Then there exists an element such that
**
Moreover, for any fixed , the sequence , defined by
**
converges to the element .*

#### 3. Existence of -Best Proximity Points under Different Conditions

The main aim of this paper is to study the following kind of mappings and to ensure that, under some conditions, they have a -best proximity point.

*Definition 12. *Let , , , , and be five mappings. One will say that is a *-contraction* if, for all such that and , we have that
where

In the previous definition, we have not supposed that or because the main aim of the present paper is to introduce sufficient conditions on the involved mappings (, , , and ) and on the ambient space to ensure the existence and, in some cases, the unicity of -best proximity points of .

*Remark 13. * Some other authors used to impose that their contractive condition must be verified for all . However, our condition (10) must only be satisfied for all . Later, we will discuss when it is necessary to assume that this property holds for all .

The mapping need not be symmetric. However, if and is a -contraction, then is also a -contraction, where is defined as in Lemma 6. In such a case, when , without loss of generality, we can consider that is symmetric; that is, in this case, the order of the arguments of in (10) is not important.

The following definitions are very useful in order to establish weaker conditions than the -property (see also [34]) or the notion of -proximal admissible mapping.

*Definition 14. *Let and be two subsets of a metric space such that is nonempty, and let and be two mappings. One will say that the quadruple has the following:(i)the* weak* *-property of the first kind *if
(ii)the* weak* *-property of the second kind* if
(iii)the* weak* *-property of the third kind* if

Lemma 15. *If the pair has the -property, then the quadruple has the weak -property of the first, the second, and the third kind, whatever the mappings and .*

*Remark 16. *Obviously, if is a metric space, then the pair has the -property. Therefore, the quadruple has the weak -property of the first, the second, and the third kinds whatever the mappings and .

*Definition 17. *Let and be two subsets of a metric space such that is nonempty, and let , , and be three mappings. One will say that is *-proximal admissible* if

Lemma 18. *If is -proximal admissible, then is -proximal admissible, whatever .*

*Definition 19. *Let and be two mappings and let and . We will say that is *-transitive on* if

Indeed, one will only use the notion of *-transitive mapping on* ; that is,

Next we prove our first main result.

Theorem 20. *Let and be two closed subsets of a complete metric space and let , , , , and be five mappings. Assume that the following conditions hold:*(a)* and ;*(b)*the quadruple has the weak -property of the first kind;*(c)* is a -proximal admissible -contraction;*(d)*if is a sequence such that is Cauchy, then also is Cauchy;*(e)*there exists such that and ;*(f)* is a continuous mapping;*(g)* is a continuous mapping;*(h)* and .**Then there exists a convergent sequence verifying
**
whose limit is a -best proximity point of .**Actually, every sequence verifying (18) and converges to a -best proximity point of .*

*Remark 21. * Although the previous result seems to have too many hypotheses, actually, this is its best advantage. As we will see in Section 5, there are a lot of different ways to particularize this theorem which generate many independent results. For instance, our control functions do not need any kind of monotone property.

This result improves the main theorem in [28] in several aspects: firstly, we introduce a mapping which is not necessarily the identity mapping on ; secondly, need not have the -property; thirdly, the contractive condition on is weaker; finally, we only suppose ; that is, is not necessarily continuous.

Taking into account the completeness of the ambient space , the condition (d) can be interpreted as the continuity of the inverse mapping of , if is invertible. A simple way to guarantee this condition is to suppose that there are such that for all . For instance, the condition for all can be found in [35].

Notice that the second part of the thesis does not clarify whether the -best proximity point of is unique or not.

*Proof. *Given , we know that . Then, there is such that . Therefore, . Since , there is such that , so . Repeating the same argument starting from , there is such that . By induction, we can consider a sequence such that
If there exists some such that , then , so is a -best proximity point of . In such a case, if we define for all , we have that is constant, so converges to a -best proximity point of . In this case, the proof is finished.

On the contrary, suppose that
Notice that, in particular, and for all . We claim that
If , then by hypothesis. Suppose that for some . Hence, taking into account that is -proximal admissible, we have that
This proves that (21) holds. Moreover, using the weak -property of the first kind, for all ,
Next we use (21), (23), and the -contractive property of to see that, for all ,
(the last equality holds since the first argument of is zero), where
Joining (24) and (25), we have that
Using (20) and the fact that for all , if there exists some such that
then we have that , which is impossible. Then for all and (26) yields to
In particular, for all ,
Next we prove that is a Cauchy sequence. Fix arbitrary and consider . Since , the series converges. In particular, there exists such that
Therefore, if , we have that
This means that is a Cauchy sequence. Using the hypothesis (d), also is a Cauchy sequence. By the completeness of , there exists such that . From for all , we deduce that (because is closed). Since and are continuous mappings, and . Taking limit in (19) as , we conclude that is a -best proximity point of .

Next we change the conditions on the control functions.

Theorem 22. *Theorem 20 also holds if one replaces condition (h) by the following one:*(h^{'})*, , and is -transitive.*

*Proof. *Taking into account that for all and following the lines of the proof of Theorem 20, we deduce that
By Lemma 5, we have that
Next, we are going to prove that is a Cauchy sequence reasoning by contradiction. Assume that is not Cauchy. In this case (following, for instance, [27]), there exist and two subsequences and verifying that, for all ,
Notice that
Therefore
Similarly,
Furthermore,
where, for all ,
Notice that
Taking limit as and using (34),
Taking limit as in (39) and using (34), (36), (37), and (41), we deduce that
This means that is a sequence of real numbers converging to and whose terms are strictly greater than . In particular, since ,
From the fact that for all and using that is -transitive, we deduce that
Since has the weak -property of the first kind, for all ,
Therefore, from the -contractivity condition on , it follows that, for all ,
Using (36), the third and the fourth arguments of converge to zero as . Since , all the terms tend to zero as . Hence, letting in (46) and using (34) and (43), we conclude that
which is impossible. This contradiction proves that is a Cauchy sequence. Then, the rest of the proof is similar to the proof of Theorem 20.

In the following theorem, we replace the continuity of by another condition.

Theorem 23. *Theorem 20 also holds if one supposes that the contractive condition (10) is valid for all and all , and one replaces conditions (b) and (g) by the following ones:*(b^{'})*the quadruple has the weak P-property of the second kind;*(g^{'})*if is a sequence verifying and for all , then there exists a partial subsequence of such that for all .*

*Proof. *Following the lines of the proof of Theorem 20, we deduce that and are Cauchy sequences, contained in the closed subset , of the complete metric space . Then, there is such that and, using that is continuous, . We are going to prove that is a -best proximity point of .

Since has the weak -property of the second kind, for all ,
It follows that is also a Cauchy sequence in the closed subset . Hence, there is such that . This means that
Since for all , we deduce that
that is, and . Using condition (g^{'}), we deduce that there exists a partial subsequence of such that
Notice that
Therefore
The first and the second arguments of
tend to zero, and the last argument tends to
Therefore,
Next we are going to show that is a -best proximity point of reasoning by contradiction. Suppose that ; that is,
Since the first and the second terms in the maximum in (54) tend to zero, and the fourth term tends to , then there exists such that
Using the contractivity condition (notice that but ), for all ,
Since the third argument of in (59) tends to zero and , its limit as is zero. Therefore, letting in (59), we have that
As , item of Remark 13 guarantees that . Thus,
which is impossible. This contradiction shows that must verify ; that is, is a -best proximity point of .

*Remark 24. *When is -transitive, condition (g^{'}) is equivalent to the following one.(g^{”})If is a sequence verifying and for all , then for all .

*Remark 25. *Notice that, following the same sketch of proof with appropriate changes, Theorem 23 remains true under the hypothesis of Theorem 22.

#### 4. Uniqueness of -Best Proximity Points

In this section, we introduce a sufficient condition in order to demonstrate that the -best proximity point, whose existence is guaranteed by the previous results, is unique.

*Definition 26. *Let , , and be three mappings. One will say that is *-regular* if, for all such that , there exists such that and .

Theorem 27. *Under the hypothesis of Theorem 20, assume that and is -regular. Then for all -best proximity points and of in One has that .**In particular, if is injective on the set of all -best proximity points of in , then has a unique -best proximity point.*

*Proof. *Let be two -best proximity points of in . Since and is a -proximal admissible, we deduce that
We distinguish whether or . Firstly, assume that . In such a case, the contractivity condition yields to
where the last equality holds since and the last two arguments of are zero. Since
it follows that

Therefore
which is only possible when ; that is, .

Next, suppose that . In this case, by the -regularity of , there exists such that and . Based on , we are going to define a sequence such that will converge, at the same time, to and to . By the unicity of the limit, this will prove that . We only reason with , but the same argument is valid for .

Indeed, since , there is such that , and since , there is verifying . Therefore, . Repeating this argument, there exists a sequence such that for all . In particular, and .

Now we reason using . We claim that
If , by the choice of . Suppose that for some . In such a case, taking into account that is -proximal admissible, we have that
This concludes that (67) holds. Taking into account that, for all ,
it follows that, for all ,
Therefore, using the weak -property of the first kind,
and, hence, by the contractivity condition, for all ,