#### Abstract

First, we define cyclic ()-contractions of different types in a uniform space. Then, we apply these concepts of cyclic ()-contractions to establish certain fixed and common point theorems on a Hausdorff uniform space. Some more general results are obtained as corollaries. Moreover, some examples are provided to demonstrate the usability of the proved theorems.

#### 1. Introduction

Let be a nonempty set. A nonempty family, , of subsets of is called the uniform structure of if it satisfies the following properties:(i)if is in , then contains the diagonal ;(ii)if is in and is a subset of which contains , then is in ;(iii)if and are in , then is in ;(iv)if is in , then there exists in , such that, whenever and are in , then is in ;(v)if is in , then is also in .The pair () is called a uniform space and the element of is called entourage or neighbourhood or surrounding. The pair () is called a quasi-uniform space (see, e.g., [1, 2]) if property (v) is omitted.

Existence and uniqueness of fixed points for various contractive mappings in the setting of uniform spaces have been investigated by several authors; see, for example, [3–12] and the references therein.

Recently, an interesting and remarkable notion of cyclic mapping was introduced and studied by Kirk et al. [13]. Following this paper, a number of authors introduced contractive mapping via the cyclic mappings and reported certain fixed point results in the setting of different type of spaces; see, for example, [13–17].

In this paper, we will give the characterization of cyclic mapping in the context of uniform spaces and, further, prove the existence and uniqueness of fixed and common fixed points of such mappings via -distance and -distance, introduced by Aamri and El Moutawakil [18].

For the sake of completeness, we recollect some basic definitions and fundamental results. Let be the diagonal of a nonempty set . For , we will use the following setting in the sequel: For subset , a pair of points and are said to be -close if and . Moreover, a sequence in is called a Cauchy sequence for , if for any there exists such that and are -close for . For (), there is a unique topology on generated by , where .

A sequence in is convergent to for , denoted by , if for any there exists such that for every . A uniform space is called Hausdorff if the intersection of all the is equal to of , that is, if for all implies . If , then we say that a subset is symmetrical. Throughout the paper, we assume that each is symmetrical. For more details, see, for example, [1, 18–21].

Now, we recall the notions of -distance and -distance.

*Definition 1 (see, e.g., [18, 19]). *Let be a uniform space. A function is said to be an -distance if for any there exists such that if and for some , then .

*Definition 2 (see, e.g., [18, 19]). *Let be a uniform space. A function is said to be an -distance if is an -distance,.

*Example 3 (see, e.g., [18, 19]). *Let be a uniform space and let be a metric on . It is evident that is a uniform space, where is the set of all subsets of containing a “band” for some . Moreover, if , then is an -distance on .

Lemma 4 (see, e.g., [18, 19]). *Let be a Hausdorff uniform space and let be an -distance on . Let and be sequences in and and let be sequences in converging to 0. Then, for , the following hold. **(a)**If and for all , then . In particular, if and , then .**(b)**If and for all , then converges to .**(c)**If for all with , then is a Cauchy sequence in .*

Let be an -distance. A sequence in a uniform space with an -distance is said to be a -Cauchy if for every there exists such that for all .

*Definition 5 (see, e.g., [18, 19]). *Let be a uniform space and let be an -distance on . (1) is -complete if for every -Cauchy sequence there exists in with .(2) is -Cauchy complete if for every -Cauchy sequence there exists in with with respect to .

*Remark 6. *Let be a Hausdorff uniform space which is -complete. If a sequence is a -Cauchy sequence, then we have . Regarding Lemma 4(b), we derive that with respect to the topology and hence -completeness implies -Cauchy completeness.

*Definition 7. *Let be a Hausdorff uniform space and let be an -distance on . Two self-mappings and of are said to be weak compatible if they commute at their coincidence points; that is, implies that .

We denote by the class of functions nondecreasing and continuous satisfying for and .

*Definition 8 (see [17]). *A function is called a comparison function if it satisfies the following: (i) is increasing; that is, implies , for ;(ii) converges to 0 as , for all .

*Definition 9 (see [22]). *A function is called a ()-comparison function if (i) is increasing,(ii)there exist and a convergent series of nonnegative terms such that
for and any .

Let be the collection of all ()-comparison functions defined in Definition 9.

Lemma 10 (see [22]). *If is a ()-comparison function, then the following hold: **(i)** is comparison function,**(ii)**, for any ,**(iii)** is continuous at 0,**(iv)**the series converges for any .*

In 1922, Banach proved that every contraction in a complete metric space has a unique fixed point. This celebrated result has been generalized and improved by many authors in the context of different abstract spaces for various operators (see [1–28] and the references therein). Recently, fixed point theorems for operators defined on a complete metric space with a cyclic representation of with respect to have appeared in the literature (see, e.g., [13–17]). Now, we present a modification of the main result of [16]. For this, we need the following definitions.

*Definition 11 (see [13]). *Let be a nonempty set, a positive integer, and a mapping. is said to be a cyclic representation of with respect to if (i), are nonempty sets;(ii).

*Definition 12. *Let be a metric space, a positive integer, nonempty subsets of , and . An operator is a cyclic -contraction if (i) is a cyclic representation of with respect to ,(ii), for any , , , where and .

The main result of [14] is the following.

Theorem 13 (Theorem of [14]). *Let be a complete metric space, m a positive integer, nonempty subsets of , and . Let be a cyclic -contraction with . Then, has a unique fixed point .*

The main aim of this paper is to prove results similar to the abovementioned theorems in uniform spaces and to present modifications of Theorem 2.1 [16], Theorems 3.1-3.2 in [18], and other related results.

#### 2. Main Result

First, we present the following definition.

*Definition 14. *Let be a uniform space, a positive integer, nonempty subsets of , and . An operator is a cyclic -contraction if (i) is a cyclic representation of with respect to ,(ii)for any , , ,
where and .

Our main result is the following.

Theorem 15. *Let be an -complete Hausdorff uniform space such that is an -distance on , a positive integer, and nonempty closed subsets of with respect to the topological space , and . Let be a cyclic -contraction. Then, has a unique fixed point .*

*Proof. *We first show that the fixed point of is unique (if it exists). Suppose, on the contrary, that with are fixed points of . The cyclic character of and the fact that are fixed points of imply that . Using the contractive condition, we obtain
and from the last inequality
Similarly, we can show that and, consequently, .

Now, we prove the existence of a fixed point. Note that is not symmetric. To show that the sequence is Cauchy, we will show that both and , for any .

For this aim, take and consider the sequence given by
If there exists such that , then the proof is completed. In this case, is the required fixed point of . Throughout the proof, we assume that
Notice that for any there exists such that and , since . Due to the fact that is a cyclic -contraction, we have
by taking and in (3). From (8) and taking the monotonicity of into account, we derive by induction that
As is an -distance, we obtain that
so for we have that
In the sequel, we will prove that is a -Cauchy sequence. Denoting
implies that
As is a ()-comparison function, supposing , by Lemma 10, (iv), it follows that
so there is such that
Then, by (13) we obtain that

By repeating the same arguments in the proof of (16), we conclude that

Consequently, we get that the sequence is a -Cauchy in the -complete space . Thus, there exists such that . In what follows we prove that is a fixed point of . In fact, since , as is a cyclic representation of with respect to , the sequence has infinite terms in each for .

Since is closed for every , it follows that ; thus we take a subsequence of with . Using the contractive condition, we can obtain
and since and belong to , letting in the last inequality, we have . Analogously, we can derive that and, therefore, is a fixed point of . This finishes the proof.

Corollary 16. *Let be an -complete Hausdorff uniform space such that is an -distance on , a positive integer, nonempty closed subsets of with respect to the topological space , and . Let operator satisfy **(i)**, for any , , , where and .**Then, has a unique fixed point .*

*Proof. *By Theorem 15, it is enough to set .

Corollary 17 (cf. [16]). *Let be a complete metric space, a positive integer, nonempty closed subsets of , and . Let be a cyclic -contraction. Then, has a unique fixed point .*

*Proof. *By Theorem 15, it is enough to set .

Corollary 18 (cf. [13]). *Let be a complete metric space, a positive integer, nonempty closed subsets of , and a cyclic representation of with respect to . Let satisfy
**
for any , , , where and . Then, has a unique fixed point .*

*Definition 19. *Let be a uniform space, a positive integer, nonempty subsets of , and self-mappings. An operator is a cyclic --contraction if (i) is a cyclic representation of with respect to ,(ii), for any , , , where and .

Inspired by [28], we now prove a common fixed point theorem as an application of our Theorem 15.

Theorem 20. *Let be a uniform space, self-maps such that is cyclic --contraction, and -complete Hausdorff uniform space together with being an -distance on . Suppose that are nonempty closed subsets of with respect to the uniform topology and . Then, and have a unique coincidence point. Moreover, if and are weakly compatible, then they have a unique common fixed point .*

*Proof. *As , so there exists such that and is one-to-one. Now, since , we define mappings by . Since is one-to-one on , so is well defined. As is cyclic --contraction, so
for any , , . Thus,
for any , , , which implies that is cyclic -contraction on . Hence, all the conditions of Theorem 15 are satisfied by , so has a unique fixed point in . That is, , so and have a unique coincidence point as required. Moreover, if and are weakly compatible, then they have a unique common fixed point.

Corollary 21 (cf. Theorem 3.2 [18]). *Let be a uniform space, self-maps such that is --contraction, and -complete Hausdorff uniform space together with being an -distance on . Suppose that and and are commuting. Then, and have a unique common fixed point .*

*Proof. *Take for all in Theorem 20.

*Example 22. *Let be a metric space, where and . Set , , and . Define . It is easy to see that is a uniform space. If we define by for and by and , then for every we have
Also, for the above inequality obviously holds. This shows that the contractive condition of Corollary 16 is satisfied and 0 is fixed point .

*Definition 23. *Let be a uniform space, let be two mappings, and let and be nonempty closed subsets of . The is said to be a cyclic representation of with respect to the pair if and .

*Definition 24. *Let be a uniform space, nonempty subsets of , and . Two self-maps are called cyclic -contraction pair if (i) is a cyclic representation of with respect to the pair ,(ii), for any , , where .

Theorem 25. *Let be an -complete Hausdorff uniform space such that is an -distance on and let be nonempty closed subsets of with respect to the topological space and . Suppose that are cyclic -contraction pair. Then, and have a unique common fixed point .*

*Proof. *Take and consider the sequence given by
Since , for any , , and , and are cyclic -contraction pair, we have
Hence,
Similarly, we have
Hence,

From inequalities (25) and (27) and taking into account the monotonicity of , we get by induction that
Since is an -distance, we find that
so for we have that
In the sequel, we will prove that is a -Cauchy sequence. Denote
By relation (31), we have
Regarding together with Lemma 10(iv), we get that
since . Thus, there is such that
Then, by (32) we obtain that
In an analogous way, we derive that
Hence, we get that is a -Cauchy sequence in the -complete space . So there exists such that . In what follows, we prove that is a fixed point of . In fact, since and as is a cyclic representation of with respect to , the sequence has infinite terms in each .

Since are closed, it follows that ; thus we take subsequences of with and . Using the contractive condition, we can obtain
and since and belong to , letting in the last inequality, we have ; hence. Similarly, we can show that and, therefore, is a fixed point of . Similarly, we can show that is a fixed point of . Finally, in order to prove the uniqueness of the fixed point, we have with and fixed points of . The cyclic character of and the fact that are fixed points of imply that . Using the contractive condition, we obtain
and from the last inequality we get
Using the same arguments above, we can show that and, consequently, . This finishes the proof.

Corollary 26. *Let be a complete metric space and nonempty closed subsets of and . Let be cyclic -contraction pair. Then, and have a unique common fixed point .*

*Proof. *By Theorem 25, it is enough to set .

Corollary 27. *Let be an -complete Hausdorff uniform space such that is an -distance on and let be nonempty closed subsets of with respect to the topological space and . Suppose that the maps satisfy the following inequality: **(i)**, for any , , where .**Then, have a unique common fixed point .*

*Proof. *By Theorem 25, it is enough to set .

*Example 28. *Let be a partial metric space, where and . Set and . Define , where . It is easy to see that is a uniform space. If we define by for and by , , and , . Then, for every we have
for any , . Also, for the above inequality obviously holds. This shows that the contractive condition of Corollary 27 is satisfied and 0 is a common fixed point of and .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU, for the financial support.