#### Abstract

By using the weaker Meir-Keeler function and the triangular -admissible mapping , we introduce the notion of -weaker Meir-Keeler contractive mappings and prove a theorem which assures the existence of a periodic point for these mappings on generalized quasimetric spaces.

#### 1. Introduction and Preliminaries

Let be a nonempty set and let . Then is called a distance function if for every , satisfies; ;;.If satisfies conditions , then is called a metric on . If satisfies conditions , , and , then is called a quasimetric on . If satisfies conditions , , and , then is called a dislocated metric on . If satisfies conditions and , then is called a dislocated quasimetric on .

In 2000, Branciari [1] introduced the notion of generalized metric as a natural extension of the concept of a metric, where the triangle inequality condition of a metric had been replaced by a weaker condition, namely, quadrilateral inequality. At the first glance, both metric and generalized metric seem to have almost the same topological properties. Despite the first impression, the generalized metric does possess some fundamental topological feature, such as(P1)generalized metric needs not to be continuous;(P2)a convergent sequence in generalized metric space needs not to be Cauchy;(P3)generalized metric space needs not to be Haussdorf and hence the uniqueness of limits cannot be guaranteed.The question whether the analog of existing fixed point results in the literature are still valid in generalized metric space without assuming an extra conditions, such as, continuity of generalized metric function, and/or Hausdorffness of the corresponding space, and so forth. Several authors worked on this interesting questions and this space (e.g., [118]).

Definition 1 (see [1]). 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 g.m.s).

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

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

We now introduce the new notion of generalized quasimetric space as follows.

Definition 3. 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 quasimetric space (or shortly g.q.m.s).

Remark 4. Any generalized metric space is a generalized quasimetric space, but the converse is not true in general.

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

Example 5. Let with as a constant, and we define by (1), for all ;(2);(3);(4);(5);(6),where is a constant. Then, let be a generalized quasimetric 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. Let be a g.q.m.s, let be a sequence in , and let . We say that is g.q.m.s convergent to if and only if

Definition 7. Let be a g.q.m.s 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. Let be a g.q.m.s 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. Let be a g.q.m.s 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 g.q.m.s is Cauchy if and only if it is left-Cauchy and right-Cauchy.

Definition 11. Let be a g.q.m.s. 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 the sequel, we let the function satisfy the following conditions: () is a nondecreasing weaker Meir-Keeler function;() for and ;()for all , is decreasing;()for , if , then , where .

Now, we recall the notion of -admissible mappings. The following definition was introduced in [3].

Definition 12. Let be a self-mapping of a set and . Then is called a -admissible if

In the sequel, we use the notion of triangular -admissible which was defined in [4] as follows.

Definition 13. Let and . The mapping is said to be a triangular -admissible if, for all , we have (1)for all , implies ;(2)for all , and imply .

#### 2. Main Results

In this section we state our main result. First we introduce the notion of -weaker Meir-Keeler contractive mappings via the weaker Meir-Keeler function and the triangular -admissible mapping .

Definition 14. Let be a g.q.m.s, let , and let be a function satisfying for all . Then is said to be a -weaker Meir-Keeler contractive mapping.

Now, we state our main periodic point theorems as follows.

Theorem 15. Let be a Hausdorff and complete g.q.m.s, and let be a -weaker Meir-Keeler contractive mapping. Suppose that (i) is triangular -admissible;(ii)there exists such that and ;(iii) is continuous.Then has a periodic point in ; that is, there exists a such that for some .

Proof. Regarding the assumption (ii) of the 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 ; that is, is a periodic point in . Hence, for the rest of the proof, we presume that Since is triangular admissible, we also have Utilizing the expression above, we obtain that By repeating the same steps with starting with the assumption , we conclude that Since, is triangular -admissible, 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 first assert that
Regarding (5) and (8), we deduce that for all . Since is nondecreasing, by iteration, we derive the following inequality: Since is decreasing, it must converge to some . We claim that . Suppose that, on the contrary, . 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 show that We repeat the same argument that was used in Step 1. On account of (5) and (11), we observe that for all . Since is nondecreasing, by iteration, we derive the following inequality: Since is decreasing, it must converge to some . We claim that . Suppose that, on the contrary, . 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 next 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 Let . Then, by using condition , we have
Case (II). Suppose that and is even. Let , . Then, by using the quadrilateral inequality, we also have Let . Then, by using 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 that, on the contrary, has no periodic point. Since is continuous, we obtain from (25) that From (25) and (26), we get immediately that . As is Hausdorff, 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 .

Following the proof of Theorem 15, we can easily get the following periodic point theorem.

Theorem 16. Let be a Hausdorff and complete g.q.m.s, and let be a -weaker Meir-Keeler contractive mapping. Suppose that (i) is triangular -admissible;(ii)there exists such that and ;(iii)if is a sequence in such that and for all and as , then and for all . Then has a periodic point in .

Proof. Following the proof of Theorem 15, we know that the sequence defined by , for all , converges for some . From (25) and condition (iii), there exists a subsequence of such that for all . Applying (5), for all , we get that Letting in the above equality, we find that Therefore, we have . As is Hausdorff, 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 .

#### 3. Consequences

Using the weaker Meir-Keeler function , we introduce the notion of -weaker Meir-Keeler contractive mappings and prove a theorem which assures the existence of a periodic point for these mappings on generalized quasimetric spaces.

Definition 17. Let be a g.q.m.s, and let be a function satisfying for all . Then is said to be a -weaker Meir-Keeler contractive mapping.

Theorem 18. Let be a Hausdorff and complete g.q.m.s, and let be a -weaker Meir-Keeler contractive mapping. Suppose that is continuous. Then has a periodic point in ; that is, there exists such that for some .

Proof. It is sufficient to take , for all , in Theorem 15.

Theorem 19. Let be a Hausdorff and complete g.q.m.s. Suppose that is continuous and there exists such that for all . Then has a periodic point in ; that is, there exists such that for some .

Proof. It is sufficient to take , for all , in Theorem 18, where .

Definition 20. Let be a partially ordered set and let be a given mapping. We say that is nondecreasing with respect to if

Definition 21. Let be a partially ordered set. A sequence is said to be nondecreasing with respect to if for all .

Definition 22. Let be a partially ordered set and let be a metric on . We say that is regular if, for every nondecreasing sequence such that as , we have for all .

We have the following result.

Corollary 23. Let be a partially ordered set and let be a metric on such that is complete. Let be a nondecreasing mapping with respect to . Suppose that there exists a function such that for all with . Suppose also that the following conditions hold: (i)there exists such that ;(ii) is continuous or is regular.Then has a periodic point.

Proof. Define the mapping by It is evident that is a contractive mapping; that is, for all . From condition (i), we have that and . Moreover, for all , from the monotone property of , we have Thus is -admissible. Now, if is continuous, the existence of a fixed point follows from Theorem 15. Suppose now that is regular. Let be a sequence in such that for all and as . From the regularity hypothesis, we have for all . This implies from the definition of that for all . In this case, the existence of a fixed point follows from Theorem 16.

Corollary 24. Let be a partially ordered set and let be a metric on such that is complete. Let be a nondecreasing mapping with respect to . Suppose that there exists a function such that for all with . Suppose also that the following conditions hold: (i)there exists such that ;(ii) is continuous or is regular.Then has a periodic point.

Proof. It is sufficient to take , for all , in the corollary above, where .

#### Conflict of Interests

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