#### Abstract

We introduce the concept of the generalized -contraction mappings and establish the existence of fixed point theorem for such mappings by using the properties of -distance and -admissible mappings. We also apply our result to coincidence point and common fixed point theorems in metric spaces. Further, the fixed point theorems endowed with an arbitrary binary relation are also derived from our results. Our results generalize the result of Kutbi, 2013, and several results in the literature.

#### 1. Introduction

It is well known that many problems in many branches of mathematics can be transformed to a fixed point problem of the form for self-mapping defined on framework of metric space . In 1992, Banach [1] introduced the concept of contraction mapping and proved the fixed point theorem for such mapping, which is called the Banach contraction principle, which opened an avenue for further development of analysis in this field. Several mathematicians used different conditions on self-mappings and proved several fixed point theorems in metric spaces and other spaces.

In 1969, Nadler [2] established the fixed point theorem for multivalued contraction mapping by using the concept of Hausdorff metric which in turn is a generalization of the classical Banach contraction principle. Afterward, Kaneko [3] extended the corresponding results of Nadler [2] to single valued mapping and multivalued mapping which is also generalization of the result of Jungck [21]. Subsequently, there are a number of results that extend this result in many different directions (see in [4–13]).

On the other hand, Kada et al. [14] introduced the concept of -distance on a metric space. Using this concept, they improved Caristi’s fixed point theorem, Ekland’s variational principle, and Takahashi’s existence theorem. Afterward, Suzuki and Takahashi [15] established the fixed point result for multivalued mapping with respect to -distance. In fact, this result is an improvement of the Nadler’s fixed point theorem. Several fixed point theorems have been proved by many mathematicians in framework of metric spaces via -distance; for example, see [16–19]. Recently, Kutbi [20] established useful lemma for -distance which is an improved version of the lemma given in [21] and proved a key lemma on the existence of -orbit for generalized -contraction mappings. Also, he gave the existence of coincidence points and common fixed points for generalized -contraction mappings not involving the extended Hausdorff metric.

The purpose of this work is to introduce the generalized -contraction mapping and prove fixed point theorem for such mapping via the concept of -admissible mapping of Mohammadi et al. [22], which is multivalued mapping version of -admissible mapping of Samet et al. [23] and different from the notion of -admissible which has been provided in [24] (also seen in [25–28]). The applications for coincidence point and common fixed point theorems in metric spaces and fixed point theorems endowed with an arbitrary binary relation are also derived from our results. Our results improve and complement the main result of Kutbi [20] and many results in the literature.

#### 2. Preliminaries

In this section, we recall some definitions and lemmas of -distance that will be required in the sequel. For metric space , let , , and denote the collection of nonempty subsets of , nonempty closed subsets of , and nonempty closed bounded subsets of , respectively. For , we define the Hausdorff distance with respect to by for every , where . It is well known that is a metric space.

*Definition 1. *Let be a metric space, a single valued mapping, and a multivalued mapping.(1)A point is called a fixed point of if and the set of fixed points of is denoted by .(2)A point is called a coincidence point of and if . One denotes by the set of coincidence points of and .(3)A point is called a common fixed point of and if . One denotes by the set of common fixed points of and .

*Definition 2. *Let be a metric space, a single valued mapping, and a multivalued mapping. The sequence in is said to be an -orbit of at if for all . In particular case, the sequence in is said to be an orbit of at if is the identity mapping on ; that is, for all .

*Definition 3. *Let a metric space, a single valued mapping, and a multivalued mapping.(1) is said to be a contraction [2] if there exists a constant such that for each ,
(2) is said to be an -contraction [3] if there exists constant , and for each ,

*Definition 4 (see [14]). *Let be a metric space. A function is called -distance on if it satisfies the following for each :);
() a mapping is lower semicontinuous; () for any , there exists such that and imply .

Let us give some examples of -distance.

*Example 5. *Let be a metric space. Then, the metric is -distance on , but the converse is not true in general case. Therefore, the -distance is a generalization of the metric.

*Example 6. *Let be a metric space. Then, a function defined by for all is -distance on , where is a positive real number.

*Example 7. *Let be a normed linear space. Then, a function defined by for all is -distance on .

*Example 8. *Let be a normed linear space. Then, a function defined by for all is -distance on .

*Remark 9. *We obtain that in general for , and neither of the implications necessarily holds.

*Definition 10 (see [29]). *Let be a metric space. One says that the -distance on is a -distance if for all .

For more details of other examples and properties of the -distance, one can refer to [14, 15, 29]. The following lemmas are useful for the main results in this paper.

Lemma 11 (see [14]). *Let be a metric space and a -distance on . Suppose that and are sequences in and and are sequences in converging to . Then, the following hold for every :*(1)*if and for any , then ; in particular, if and , then ;*(2)*if and for any , then converges to ;*(3)*if for any with , then is a Cauchy sequence;*(4)*if for any , then is a Cauchy sequence.*

Next, we give the definition of some type of mapping. Before giving next definition, we give the following notation. Let be a metric space and a -distance on . For and , we denote .

*Definition 12. *Let be a metric space, a singlevalued mapping, and a multivalued mapping.(1) is a -contraction [15] if there exist a -distance on and such that, for any and , there is with
(2) is a generalized -contraction [20] if there exist a -distance on and such that, for any and , there is with

*Definition 13 (see [22, 24]). *Let be a nonempty set, , where is a collection of subset of and . One says that(1) is an -admissible if, for all , one has
where ;(2) is an -admissible whenever, for each and with , one has for all .

*Remark 14. *It is easy to see that is an -admissible mapping which implies as an -admissible mapping.

#### 3. Fixed Point Results

In this section, we introduce the new mapping, the so-called generalized -contraction mapping, and prove the fixed point results for this mapping by using -distance.

*Definition 15. *Let be a metric space and a given mapping. The multivalued mapping is said to be a generalized -contraction if there exist a -distance on and such that, for any and , there is with

Theorem 16. *Let be a complete metric space, , and a generalized -contraction mapping. Suppose that the following conditions hold:*(a)* is an -admissible mapping;*(b)*there exist and such that ;*(c)*if for every with , one has
**then, .*

*Proof. *For and in (b), by the definition of generalized -contraction of , there exists such that
Since , , and is -admissible mapping, we have
From (9) and (10), we have
Similarly, using the definition of generalized -contraction of , there exists such that
From , , and is -admissible mapping; we have
From (12) and (13), we have
Continuing this process, we can construct the sequence in such that ,
for all . Therefore, for each , we have
If there exists such that , then we have , and then, . By property of -distance, we get
Since and , using Lemma 11, we get , and thus, . This implies that is fixed point of . Therefore, we may assume that for all . From (17), we get
for all . By Lemma 11, we have that converges in .

By repeating (19), we obtain that
for all .

For for which , we get
Since , we get as . Using Lemma 11, we get as Cauchy sequence in . By completeness of , we get as for some . Since is lower semicontinuous, we have
Assuming that , then by hypothesis, we get
which is contradicting. Therefore, ; that is, is fixed point of . This completes the proof.

Corollary 17 (see Corollary 2.1 in [20]). *Let be a complete metric space, let be -distance on , and let be a multivalued map satisfying the following:*(a)* for each and , there exists such that
* *where ;*(b)* for every with , one has
**Then, .*

*Proof. *Setting for all in Theorem 16, we obtain the desired result.

Next, we give the notion of -contraction mapping and prove the existence of fixed point theorem for such mapping.

*Definition 18. *Let be a metric space and . The multivalued mapping is said to be -contraction if there exist a -distance on and such that for any and , there is with

Theorem 19. *Let be a complete metric space, , and a -contraction mapping. Suppose that the following conditions hold:*(a)* is -admissible mapping;*(b)*there exist and such that ;*(c)*for every with , one has
**Then, .*

*Proof. *We obtain that this result can be proven by using similar method in Theorem 16. Then, in order to avoid repetition, the details are omitted.

#### 4. Applications

##### 4.1. Application to Coincidence Point and Common Fixed Point Results

First of all, we introduce the following concept.

*Definition 20. *Let be a nonempty set, , such that , where
and . One says that is -admissible whenever, for each and with , one has for all .

*Remark 21. *If is -admissible and is the identity mapping, then is -admissible.

*Definition 22. *Let be a metric space, , and . The multivalued mapping is said to be generalized -contraction if there exist a -distance on and such that, for any and , there is with

*Remark 23. *If is generalized -contraction and is the identity mapping, then is generalized -contraction.

Next, we give useful lemma of Haghi et al. [30].

Lemma 24 (see [30]). *Let be a nonempty set and a mapping. Then, there exists a subset such that and is one-to-one.*

Now, we apply our result in Section 3 to the coincidence point theorem by using Lemma 24.

Theorem 25. *Let be a complete metric space, , , and a generalized -contraction. Suppose that the following conditions hold:*(A)*;*(B)* is -admissible;*(C)* there exist such that and ;*(D)* for all with , one has
**Then, . *

*Proof. *Consider the mapping . Using Lemma 24, there exists such that and is one-to-one. Now, we can define a mapping by
for all . Follows from is one-to-one that is well-defined.

Since is a generalized -contraction, there exist a -distance on and such that, for any and , there is with
By the construction of , for any , and , there is such that
This implies that is a generalized -contraction. Since is -admissible, we have as -admissible. It is obtained that condition (C) implies condition (B) in Theorem 16. From (D), for all with , we have
Using Theorem 16 with mapping , we can find a fixed point of mapping .

Let be fixed point of ; that is, . Since , we can find such that . Now, we have
Therefore, is a coincident point of and ; that is, . This completes the proof.

Finally, we obtain a common fixed point result. Before giving our results, we need a few definitions.

*Definition 26. *Let be a metric space, , and . Mappings and are said to commute weakly if
for all .

Theorem 27. *Suppose that all the hypotheses of Theorem 25 hold. Further, if and commute weakly and satisfy the following condition for :
**
then, .*

*Proof. *From Theorem 25, and have a coincidence point ; that is, . By the hypothesis, we get . It follows from and which commute weakly that
This implies that is a common fixed point of and , and thus, . This completes the proof.

*Remark 28. *If we set for all in Theorems 25 and 27, then we get Theorems 2.1 and 2.2 of Kutbi [20].

##### 4.2. Application to Fixed Point on Metric Space Endowed with an Arbitrary Binary Relation

In this section, we give the existence of fixed point theorems on a metric space endowed with an arbitrary binary relation.

Before presenting our results, we need a few definitions. Let be a metric space and a binary relation over . Denote that this is the symmetric relation attached to . Clearly,

*Definition 29. *Let be a metric space and a binary relation over . One says that is a comparative mapping with respect to if, for each and , implies for all .

*Definition 30. *Let be a metric space and a binary relation over . The multivalued mapping is said to be a generalized -contraction with respect to if there exist a -distance on and such that, for any for and , there is with

Theorem 31. *Let be a complete metric space, a binary relation over , and a generalized -contraction with respect to . Suppose that the following conditions hold:*(A)* is a comparative mapping with respect to ;*(B)* there exist and such that ;*(C)* for all with , one has
**Then, .*

*Proof. *Consider the mapping defined by
From condition (B), we get . It follows from as comparative mapping with respect to that is -admissible mapping. Since is generalized -contraction with respect to , for any and , there is with
This implies that is generalized -contraction mapping. Now, all the hypotheses of Theorem 16 are satisfied, and so the existence of the fixed point of follows from Theorem 16. Therefore, .

Next, we deduce Theorem 31 to the special case in the context of partially ordered metric spaces. Before studying the next results, we give the following concepts.

*Definition 32. *Let be a nonempty set. Then, is called a partially ordered metric space if is a metric space and is a partially ordered set.

For partially ordered metric space and , we denote that

*Definition 33. *Let be a partially ordered metric space. One says that is a comparative mapping with respect to if, for each and , implies for all .

*Definition 34. *Let be a partially ordered metric space. The multivalued mapping is said to be a generalized -contraction with respect to if there exist a -distance on and such that, for any for which and , there is with

Corollary 35. *Let be a partially ordered metric space and a generalized -contraction with respect to . Suppose that the following conditions hold: *(A)* is a comparative mapping with respect to ;*(B)* there exist and such that ;*(C)* for all with ,
**Then, .*

*Proof. *Since is a binary operation on , this result follows from Theorem 31.

#### Acknowledgment

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.