#### Abstract

We introduce the notions of -weaker Meir-Keeler contractive mappings and -stronger Meir-Keeler contractive mappings. We discuss the existence of periodic points in the setting of generalized quasi-metric spaces. Our results improve, extend, and generalize several results in the literature.

#### 1. Introduction and Preliminaries

Very recently, Lin et al. [1] introduced the notion of generalized quasi-metric inspired from the notion of generalized metric, defined by Branciari [2]. It is a very well-known fact that the concept of generalized metric can be derived from the definition of metric by replacing the triangle inequality with a weaker condition, namely, quadrilateral inequality. In spite of the analogy between the definitions of metric and generalized metric, the topological structure of these spaces is completely different. It was proved that the topologies of these two spaces are incomparable [3].

In what follows that 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.

A quite natural generalization of the notion of a metric was introduced by Branciari [2] in 2000 by replacing the triangle inequality assumption of a metric with a weaker condition, quadrilateral inequality.

Definition 1 (see [2]). Let be a nonempty set and let be a mapping such that for all and for all distinct point each of them different from and , one has(d1) if and only if ;(d2);(d3) (quadrilateral inequality).Then is called a generalized metric space (or shortly ).

The following example illustrates that not every generalized metric on a set is a metric on .

Example 2 (see e.g. [1, 4]). Let with is 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 Now, we will mention that some standard properties can not be possesed by generalized metric: more precisely, (P1)open ball need not be open set,(P2)a convergent sequence in generalized metric space needs not to be Cauchy,(P3)generalized metric needs not to be continuous,(P4)generalized metric space needs not to be Hausdorff, and hence the uniqueness of limits can not be guaranteed.Several authors noticed these weak points of the generalized metric space and inserted some additional assumptions to get the analog of celebrated fixed point theorems in the context of generalized metric space. In particular, generalized metric space assumed Hausdorff. Later, several authors proved that this assumption is superfluous; see for example [59].

Example 3 (see [10], 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 (P1)–(P4). 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, for example, [4, 8, 10].

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, for example, [1117].

The following is the definition of the notion of generalized quasi-metric space defined by Lin et al. [1].

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

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

Example 5 (see [1]). Let with being 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 quasi-metric spaces.

Definition 6 (see [1]). Let be a , and let be a sequence in and . We say that is convergent to if and only if

Definition 7 (see [1]). Let be a and let be a sequence in . We say that is left-Cauchy if and only if for every there exits such that for all .

Definition 8 (see [1]). Let be a and let be a sequence in . We say that is right-Cauchy if and only if for every there exits such that for all .

Definition 9 (see [1]). Let be a and let be a sequence in . We say that is Cauchy if and only if for every there exits such that for all .

Remark 10. A sequence in a is Cauchy if and only if it is left-Cauchy and right-Cauchy.

Definition 11 (see [1]). 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.

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

#### 2. Periodic Points of Weaker Meir-Keeler Contractive Mappings

In this section, we recall the weaker Meir-Keeler function and a weaker Meir-Keeler function, as follows.

Definition 12 (see [18]). A function is said to be a Meir-Keeler type function, if, for each , there exists such that for with , we have .

Definition 13. We call a weaker Meir-Keeler function if the function satisfies the following condition:

In the sequel, we need the following classes of auxiliary functions. Let denote the set of the nondecreasing functions satisfying the following conditions:() is a weaker Meir-Keeler function;() for and ;()for all , is decreasing;()for , if , then , where .

Furthermore, let denote the set of functions satisfying the following conditions:() is continuous;() for and .

The following lemma plays a crucial role in the proof of the main result that were inspired from [5, 8], proved first in [4].

Lemma 14 (see [4]). Let be a generalized quasi-metric 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 (6) and (7). Hence, we get which is a contradiction.

In this study, we also recall the following notions of -admissible mappings.

Definition 15 (see [19]). Let be a self-mapping of a set and . Then is called a -admissible if

We now introduce the notion of -weaker Meir-Keeler contractive mappings in the following way.

Definition 16. Let be a , let , and let be a function satisfying for all . Then is said to be a -weaker Meir-Keeler contractive mapping.

We state two main periodic point theorems of -weaker Meir-Keeler contractive mapping, as follow.

Theorem 17. Let be a complete , and let . Suppose is a -weaker Meir-Keeler contractive mapping which satisfies (i) is -admissible;(ii)there exists such that , and , ;(iii) is continuous.Then has a periodic point in .

Proof. Regarding the assumption (ii) of theorem, we let be 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 In the sequel, we prove that the sequence is Cauchy; that is, is both right-Cauchy and left-Cauchy.
Step 1. We will prove that Since is a -weaker Meir-Keeler contractive mapping, we have that, for each , Since is nondecreasing, by iteration, we derive the following inequality: Due to fact that is weak Meir-Keeler function, we find that Since is decreasing, it must converge to some . We claim that . Suppose, on the contrary, that . Then by the definition of weaker Meir-Keeler function , corresponding to the given , there exists such that for with , and such that . Since , there exists such that , for all . Thus, we conclude that , which is a contradiction. Therefore , that is,
Step 2. We will prove that Since is a -weaker Meir-Keeler contractive mapping, we have that, for each , Inductively, we find that by using the fact that is nondecreasing. Since is decreasing, it must converge to some . We claim that . Suppose, on the contrary, that . Then by the definition of weaker Meir-Keeler function , corresponding to the given , there exists such that for with , and such that . Since , there exists such that , for all . Thus, we conclude that , which is a contradiction. Therefore ; that is,
Step 3. We will prove that the sequence is right-Cauchy by standard technique. For this purpose, it is sufficient to examine two cases.
Case (I). Suppose that and is odd. Let , . Then, by using the quadrilateral inequality, we have Letting , then, by using the condition , we have
Case (II). Suppose that and is even. Let , . Then, by using the quadrilateral inequality, we also have Letting . Then, by using the condition , we have By above argument, we get that is a right-Cauchy sequence.
Analogously, we derive that the sequence is left-Cauchy. Consequently, the sequence is Cauchy. Since is a complete , there exists such that
Step 4. We claim that has a periodic point in . Suppose, on the contrary, that has no periodic point. Since is continuous, we obtain from (31) that From (31) and (32), we get immediately that . Due to Lemma 14, we conclude that which contradicts the assumption that has no periodic point. Therefore, there exists such that for some . So has a periodic point in .

Theorem 18. Let be a complete , and let . Suppose is a -weaker Meir-Keeler contractive mapping which satisfies (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 in .

Proof. Following the proof of Theorem 17, we know that the sequence defined by for all , converges for some . From (31) and condition (iii), there exists a subsequence of such that for all . Applying (10), for all , we get that Letting in the above equality, we find that Therefoe, we have . Owing to Lemma 14, we conclude that which contradicts the assumption that has no periodic point. Thus, there exists such that for some . So has a periodic point in .

#### 3. Periodic Points of Stronger Meir-Keeler Contractive Mappings

In this section, we recall the notion of stronger Meir-Keeler function, as follows.

Definition 19. We call a stronger Meir-Keeler function if the function satisfies the following condition:

And, we let the function satisfy the following conditions:() is a stronger Meir-Keeler function;() for and .

Next, we introduce the notion of -stronger Meir-Keeler contractive mappings via the stronger Meir-Keeler function and the -admissible mapping .

Definition 20. Let be a , let , and let be a function satisfying for all . Then is said to be a -stronger Meir-Keeler contractive mapping.

We state two main periodic point theorms of -stronger Meir-Keeler contractive mapping, as follows.

Theorem 21. Let be a complete , and let . Suppose is a -stronger Meir-Keeler contractive mapping which satisfies (i) is -admissible;(ii)there exists such that , and , ;(iii) is continuous.Then has a periodic point in .

Proof. Following the proof of Theorem 17, we obtained that Next, we prove that the sequence is Cauchy; that is, is both right-Cauchy and left-Cauchy.
Step 1. First, we will prove that Taking into account (36) and the definition of stronger Meir-Keeler function , we have that, for each , Thus the sequence is decreasing and bounded below and hence it is convergent. Let . Then there exists and such that for all with Taking into account (40) and the definition of stronger Meir-Keeler function , corresponding to use, there exists such that Thus, we can deduce that for each with and so Since , we get
Step 2. We will prove that Taking into account (36) and the definition of stronger Meir-Keeler function , we have that for each Thus the sequence is decreasing and bounded below and hence it is convergent. By the same above proof process of Step  1, we also conclude that
Step 3. We will prove that the sequence is right-Cauchy by standard technique. For this purpose, it is sufficient to examine two cases.
Case (I). Suppose that and is odd. Let , . Then, by using the quadrilateral inequality, we have Letting , then, we have
Case (II). Suppose that and is even. Let , . Then, by using the quadrilateral inequality, we also have Letting , then, by using the condition , we have By above argument, we get that is a right-Cauchy sequence.
Analogously, we derive that the sequence is left-Cauchy. Consequently, the sequence is Cauchy.
Since is a complete , there exists such that
Step 4. We claim that has a periodic point in . Suppose, on the contrary, that has no periodic point. Since is continuous, we obtain from (52) that From (52) and (53), we get immediately that . Regarding Lemma 14, we deduce that which contradicts the assumption that has no periodic point. So, there exists such that for some . So has a periodic point in .

Apply Theorems 18 and 21, and we can easily deduce the following theorem.

Theorem 22. Let be a complete , and let . Suppose is a -stronger Meir-Keeler contractive mapping which satisfies (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 in .

#### Conflict of Interests

The authors declare that they have no competing interests.

#### Authors’ Contribution

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

#### Acknowledgment

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