#### Abstract

We consider the existence of a fixed point of -contractive mappings in the context of generalized quasimetric spaces without Hausdorff assumption. The obtained results extend several results on the topic in the literature.

#### 1. Introduction and Preliminaries

In the last decade, quasimetric spaces have been one of the interesting topics for the researchers in the field of fixed point theory due to two reasons. The first reason is that the assumptions of quasimetric are weaker than the more general metric. Consequently, the obtained fixed point results in this space are more general and hence the corresponding results in metric space are covered. The second reason is the fact that fixed point problems in -metric space (introduced by Mustafa and Sims ) can be reduced to related fixed point problems in the context of quasimetric space (see, e.g., [2, 3]). Very recently, Lin et al.  introduced the notion of generalized quasimetric spaces and investigated the existence of a certain operator on such spaces. In this paper , the authors assumed that the generalized quasimetric space is Hausdorff to get a fixed point.

In this paper, we examine the existence of -contractive mappings in the context of generalized quasimetric space without the Hausdorffness assumption. Consequently, our results extend, improve, and generalize several results in the literature.

In what follows we recall the basic definitions and results on the topics for the sake of completeness. Throughout the paper, the symbols , , and denote the real numbers, the natural numbers, and the positive integers, respectively.

Let be a nonempty set and let . Then is called a distance function if, for every , it satisfies; ;;. Notice that if satisfies the conditions , , and , then is called a dislocated metric on . If satisfies the conditions , , and , then is called a quasimetric on . On the other hand, if satisfies the conditions , then is called a metric on .

One of the very natural generalizations of the notion of a metric was introduced by Branciari  in 2000 by replacing the triangle inequality assumption of a metric with a weaker condition, quadrilateral inequality.

Definition 1 (see ). Let be a nonempty set and let be a mapping such that, for all and for all distinct points each of them different from and , one has (i) if and only if ;(ii);(iii) (quadrilateral inequality). Then is called a generalized metric space (or shortly ).

We present an example to show that not every generalized metric on a set is a metric on .

Example 2 (see, e.g., ). Let with be a constant, and we define by (1), for all ;(2), for all ;(3);(4);(5);(6), where is a constant. Then is a generalized metric space, but it is not a metric space, because

Despite the analogy between the definitions of metric and generalized metric their topological properties differ from each other. For example, for a generalized metric space , we have the following. (P1),a generalized metric, need not be continuous; (P2),a convergent sequence in generalized metric space, need not be Cauchy; (P3),an open ball (), need not be open set; (P4),a generalized metric space, need not be Hausdorff, and hence the uniqueness of limits cannot be guaranteed.

Example 3 (see , Example 1.1). Let , where and . Define in the following way: Notice that whenever and . Furthermore, is a complete generalized metric space. Clearly, we have . Indeed, the sequence converges to both and . There is no such that and hence it is not Hausdorff. It is clear that the ball since there is no such that ; that is, open balls may not be an open set. The function is not continuous since although . For more details, see, e.g., [6, 7].
Regarding the weakness of the topology of generalized metric space, mentioned above, the authors add some additional conditions to get the analog of existing fixed point results in the literature; see, e.g., . Very recently, Suzuki  underlined the importance of generalized metric space by emphasizing that generalized metric space and metric space have no compatible topology.

The following is the definition of the notion of generalized quasimetric space defined by Lin et al. 

Definition 4. Let be a nonempty set and let be a mapping such that, for all and for all distinct points each of them different from and , one has (i) if and only if ;(ii). Then is called a generalized quasimetric space (or shortly ).

It is evident that any generalized metric space is a generalized quasimetric space, but the converse is not true in general. We give an example to show that not every generalized quasimetric on a set is a generalized metric on .

Example 5 (see ). Let with be a constant, and we define by (1), for all ;(2);(3);(4);(5);(6), where is a constant. Then is a generalized quasi-metric space, but it is not a generalized metric space, because

We next give the definitions of convergence and completeness on generalized quasimetric spaces.

Definition 6 (see ). Let be a ; let be a sequence in and . We say that is convergent to if and only if

Definition 7 (see ). Let be a and let be a sequence in . We say that is left-Cauchy if and only if for every there exists such that for all . We say that is right-Cauchy if and only if for every there exists such that for all . We say that is Cauchy if and only if for every there exists such that for all .

Remark 8. A sequence in a is Cauchy if and only if it is left-Cauchy and right-Cauchy.

Definition 9 (see ). Let be a . We say that (1) is left-complete if and only if each left-Cauchy sequence in is convergent;(2) is right-complete if and only if each right-Cauchy sequence in is convergent;(3) is complete if and only if each Cauchy sequence in is convergent.

Notice that, in the literature in several reports for fixed point results in generalized metric space, an additional but superfluous condition, “Hausdorffness,” was assumed. Recently, Jleli and Samet , Kirk and Shahzad , Karapınar , Kadeburg, and Radenović , and Aydi et al.  reported new some fixed point results by removing the assumption of Hausdorffness in the context of generalized metric spaces. The following crucial lemma is inspired from [7, 17].

Lemma 10. Let be a generalized quasimetric space and let be a Cauchy sequence in such that whenever . Then the sequence can converge to at most one point.

Proof. Given , since is a Cauchy sequence, there exists such that We use the method of Reductio ad absurdum. Suppose, on the contrary, that there exist two distinct points and in such that the sequence converges to and ; that is, By assumption for any , , and since , there exists such that and for any . Due to quadrilateral inequality, we have Letting , we can obtain that by regarding (5) and (6). Hence, we get which is a contradiction.

#### 2. Main Results

In this section, we state and prove the main result of this paper. We start by introducing the following family of functions.

Let be the family of functions satisfying the following conditions: is nondecreasing; for all , where is the th iterate of . These functions are known in the literature as (c)-comparison functions. It is easily proved that if is a (c)-comparison function, then for any . For more details about such function, we refer the reader to [21, 22]. In this study, we discuss the notion of -admissible mappings; see, e.g., . The following definition was introduced in .

Definition 11. Let be a self-mapping of a set and . Then is called a -admissible if

In what follows we define the -contractive mapping in the setting of generalized quasimetric space.

Definition 12. Let be a . and let be a given mapping. We say that is an -contractive mapping if there exist two functions and such that

Now, we state the following fixed point theorem.

Theorem 13. Let be a complete , and let be an -contractive mapping. Suppose that (i) is admissible;(ii)there exists such that , , , and ;(iii) is continuous. Then has a periodic point.

Proof. Due to statement (ii) of theorem, there exists which is an arbitrary point such that and . We will construct a sequence in by for all . If we have for some , then is a fixed point of . Hence, for the rest of the proof, we presume that Since is -admissible, we have Utilizing the expression above, we obtain that By repeating the same steps with starting with the assumption , we conclude that In a similar way, we derive that Recursively, we get that Analogously, we can easily derive that
Step  1. We will show that and . Regarding (8) and (12), we deduce that for all .
Iteratively, we find that Similarly, By the properties of we can conclude that that is, Similarly, , that is; Step  2. We will prove that is a right-Cauchy sequence; that is, The cases and are proved, respectively, by (20) and (21). Now, take arbitrary. It is sufficient to examine two cases.
Case (I). Suppose that , where . Then, by using Step 1 and the quadrilateral inequality together with (18), we find Case (II). Suppose that , where . Again, by applying the quadrilateral inequality and Step 1 together with (18) and (19), we find By combining the expressions (23) and (24), we have We conclude that is a right-Cauchy sequence in .
In the same way is a left-Cauchy sequence in . So it is a Cauchy sequence. Since is a complete , there exists such that Also, we can easily see that for whenever . Indeed, if , for some with , then which is a contradiction. Analogously, we derive the same conclusion for the case . Therefore, we conclude that the sequence cannot have two limits due to Lemma 10.
Step  3. We claim that has a periodic point in . Suppose, on the contrary, that has no periodic point. Since is continuous, from Step 2, we have , () which contradicts the assumption that has no periodic point. Therefore, there exists such that for some . So has a periodic point in .

Now, we state the following fixed point theorem.

Theorem 14. Let be a complete , and let be an -contractive mapping. Suppose that (i) is admissible;(ii)there exists such that , , , and ;(iii)if is a sequence in such that for all and as , then for all . Then has a periodic point.

Proof. Following the proof of Theorem 13, we know that the sequence defined by for all converges for some . It is sufficient to show that admits a periodic point. Suppose, on the contrary, that has no periodic point. Notice that and for sufficiently large . By the quadrilateral inequality, for this , we have On account of the fact that , for all , and regarding the assumption (iii), we get that Letting in the above equality, from (20) we find that Again from (20), (26), and (30), we can obtain and hence is a periodic point of .

In what follows we give an example to illustrate Theorem 13.

Example 15. In Example 5 define the mapping as
First, we can see easily that the classic Branciari contraction  cannot be applied in this case since Now we define the mapping from by for all . For , where , we have Obviously is -admissible and also for we have Finally is continuous. Therefore satisfies in Theorem 13 and we can see that has two fixed points and .

##### 2.1. Existence Theorem

Let be a and denote the set of periodic points of .

Property (E). For all , we have and , for any .

Theorem 16. Adding Property (E) to the hypotheses of Theorem 13 (res. Theorem 14), one obtains existence of a fixed point of .

Proof. Suppose that is a periodic point of ; that is, . If , then is a fixed point of ; that is, . Assume that . We will show that is a fixed point of .
Suppose, on the contrary, that . Then , ,, and .
By Property (), we have and . Due to (9), we can obtain Due to property , we get that Again by (9), we have
Consequently, we get the following contradiction: . Hence, the assumption that is not a fixed point of is not true and thus is a fixed point of .

To assure the uniqueness of the fixed point, we will consider the following properties.

Property (U). For all , we have .

Theorem 17. Adding property (U) to the hypotheses of Theorem 16, one obtains uniqueness of the fixed point of .

Proof. Suppose that and are two distinct fixed points of . By property (), .
Thus by -admissibility of and the above relation, we can obtain which is a contradiction. Therefore .

#### 3. Consequences

Definition 18. Let be a , and let be a function satisfying for all . Then is said to be a -contractive mapping.

Theorem 19. Let be a complete , and let be a continuous -contractive mapping. Then is a unique point in .

Proof. It is sufficient to take for all .

Theorem 20. Let be a complete , and let be a continuous mapping. Suppose that there exists such that Then is a unique point in .

Proof. Take , where . It is clear that all conditions of Theorem 19 are satisfied.

#### 4. Conclusion

It is evident that almost all fixed point theorems, in the context of generalized metric spaces, can be represented in the setting of generalized quasimetric spaces. Thus, all fixed point results obtained in Section 2 infer the analog of the fixed point theorems in the context of generalized metric spaces. Consequently, several results in the literature (see, e.g., [5, 8, 9, 19, 28]) can be derived from our main results.

#### Conflict of Interests

The authors declare that they have no conflict of interests.

#### Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

#### Acknowledgment

The authors thank the anonymous referees for their remarkable comments, suggestion, and ideas that helped to improve this paper.