#### Abstract

We introduce the notion of quasi--metric space. After defining the basic topological properties of quasi--metric space, we investigate fixed point of certain mapping in the frame of complete quasi--metric space. Our results unify and cover several existing fixed point theorems in distinct structures (such as standard quasi-metric spaces, quasi--metric spaces, dislocated quasi-metric spaces, and quasi-modular spaces) in the literature.

#### 1. Introduction and Preliminaries

Jleli and Samet  combined a number of existing fixed point results, by introducing a new distance (that includes, as particular cases, standard metric spaces, -metric spaces, dislocated metric spaces, and modular spaces). In this paper, our aim is to refine the new distance by omitting a symmetry condition. Hence, our new approaches cover and combine several more interesting existing fixed point results (that includes, as particular cases, standard quasi-metric spaces, quasi--metric spaces, dislocated quasi-metric spaces, and quasi-modular spaces) including the results of Jleli and Samet .

For the sake of completeness, we collect some basic concepts and results from the literature. Let denote the set where represent the set of all positive integers. Let be a nonempty set and let be a given mapping. For every , define the sets

Definition 1. We say that is a quasi--metric space on a nonempty set if it fulfils the following conditions:(D1), for every .(D2)There exists such that In this case, the pair is called a quasi--metric space.

Remark 2. If, in addition to the conditions in Definition 1, the equality, (D3),is satisfied for each , then, is called -metric space .

In what follows we shall define some basic topological notions for quasi--metric space.

Definition 3. Let be a quasi--metric space. Let be a sequence in and Then,(i) is said to be left -convergent to if ; in this case is said to be a left -limit of .(ii) is said to be right -convergent to if ; in this case is said to be a right -limit of .(iii) is said to be -convergent to if is both left and right -convergent to ; in this case is said to be a -limit of (see ).

Proposition 4. The -limit of any sequence in a quasi--metric space is unique.

Proof. Let be a quasi--metric space. Let be a sequence in . Assume that are both -limits of . On account of (D2) and regarding the definition of -convergence, we find that and also Thus, we have . By (D1), we find .

Definition 5. Let be a quasi--metric space. Let be a sequence in .(i) is said to be left -Cauchy sequence if (ii) is said to be right -Cauchy sequence if(iii) is said to be -Cauchy sequence if it is both left and right -Cauchy sequence (see ).

Definition 6. Let be a quasi--metric space. (i) is said to be left -complete if every left -Cauchy sequence in is left -convergent to some element in (ii) is said to be right -complete if every right -Cauchy sequence in is right -convergent to some element in (iii) is said to be -complete if and only if it is left and right -complete, so that every -Cauchy sequence in is -convergent to some element in (see ).

Example 7. Let and define byThen, clearly, satisfies (D1). Let ; we have two cases:Case 1: if , then Case 2: if , let ; then and so except possibly for finite number of terms. Let be the smallest natural number such that for all . Now, let ; if , thenOn the other hand, if , then we get Thus, we find that Similarly, if , then for any we have Consequently, satisfies condition (D2) with . Therefore, is a quasi--metric space. Now, let be -Cauchy sequence. Then there exists a smallest such that for all . As the restriction of to is just the usual metric on , is -convergent to some Thus, is a complete quasi--metric space.

Definition 8. Let be a nonempty set. A mapping is called quasi-metric on , if the following conditions are fulfilled:(Q1)for every , we have ;(Q2)for every , we have .Here, the pair is called quasi-metric space.

Proposition 9. Any quasi-metric space is a quasi--metric space with .

Proof is straightforward, so we omit it.

Example 10. Let be a set and let be an arbitrary one to one function. SetThen is a quasi-metric space .

In 2012, Shah and Hussain  introduced the concept of quasi--metric spaces and verified some fixed point theorems in quasi--metric spaces.

Definition 11. Let be a nonempty set, let be a given real number, and let be a mapping satisfying the following conditions:(QBM1) for every , we have ;(QBM2) for every , we have .Then, is said to be a quasi--metric space, and is called quasi--metric space.

The following proposition followed immediately from the previous definition.

Proposition 12. Any quasi--metric space is a quasi--metric space with .

Proof. Let be a quasi--metric space. Since the first condition is straightforward, it is sufficient to show that fulfils the property (D2) of Definition 1. Let and . For every , by the property (QBM2), we have for each . Thus we have Analogously, for the case , we derive that

In 2005, Zeyada et al.  introduced the concept of complete dislocated quasi-metric space and obtain some fixed point results on it.

Definition 13. Let be a nonempty set and . is said to be dislocated quasi-metric (or quasi-metric-like) if it satisfies the following conditions for every :(QML1) ;(QML2) .In this case is called dislocated quasi-metric space (or quasi-metric-like space). If in addition satisfies(QML3) for every ,then it is called dislocated metric.

Proposition 14. Any dislocated quasi-metric space is a quasi--metric space.

Example 15. Let and define byThen is a dislocated quasi-metric.

In 1988, Kozlowski introduced the notion of modular spaces ; before we generalize this notion to the quasi form we need the following definitions.

Definition 16. Let be a linear space over . A function is said to be quasi-modular if the following conditions hold:();()for every , we havewhenever and . If in addition satisfies then is called modular on .

Definition 17. Let be a linear space and let be a quasi-modular space on . The setis called a quasi-modular space.

The convergence in quasi-modular spaces is defined as follows.

Definition 18. Let be a quasi-modular space, let be a sequence in , and .(i) is said to be left -convergent to if .(ii) is said to be right -convergent to if .(iii) is said to be -convergent to if it is both left and right -convergent to .

Definition 19. Let be a quasi-modular space and let be a sequence in .(i) is said to be left -Cauchy if .(ii) is said to be right -Cauchy if .(iii) is said to be -Cauchy if it is both left and right -Cauchy.

Definition 20. (i) A quasi-modular space is said to be left (right) -complete if every left (right) -Cauchy sequence converges to some .
(ii) A quasi-modular space is said to be -complete if and only if it is both left and right -complete.

Definition 21. Let be a quasi-modular space.(i) is said to have left Fatou property if for every whenever is left -convergent to .(ii) is said to have right Fatou property if for every whenever is right -convergent to .(iii) is said to have Fatou property if it has left and right Fatou property.

Let be a quasi-modular space. Define the mapping by

We have the following example of a quasi--space.

Proposition 22. Let be a quasi-modular space such that has Fatou property. Then is quasi--metric on with .

Proof. Clearly, satisfies (D1). Let us prove that satisfies (D2). Let and let As has Fatou property for any we haveSimilarly, if , then for every we haveThus, is a quasi--metric space on .

Consequently, we have the following result.

Proposition 23. Let be a quasi-modular space where has Fatou property. Then(i)a sequence is left -convergent (right -convergent or -convergent) to some if and only if it is left -convergent (right -convergent or -convergent) to ;(ii)a sequence is left -Cauchy (right -Cauchy or -Cauchy) if and only if it is left -Cauchy (right -Cauchy or -Cauchy);(iii) is left -complete (right -complete or -complete) if and only if it is left -complete (right -complete or -complete).

#### 2. The Banach Contraction Principle in a Quasi--Metric Space

The Banach contraction principle was extended to a -metric space by Jleli and Samet in . We shall extend this principle to a quasi--metric space.

Definition 24. Let be a quasi--metric space and let be a function. We say that is -contraction if for every , where

Proposition 25. Let be a quasi--metric space. Suppose that the function is -contraction for some Then any fixed point of with satisfies

Proof. Let be a fixed point of with . Then, as is -contraction, we havewhich is possible only if

For each let us definewhere

The following theorem is an extension of Banach contraction principle in the context of a quasi--metric space.

Theorem 26. Let be a -complete quasi--metric space and let be a -contraction mapping for some Suppose that there exists such that Then, has a fixed point and is -convergent to . Moreover, if is another fixed point of such that and , then .

Proof. We shall prove that is a -Cauchy sequence. Let , as is -contraction, for each we havewhich implies thatSo, we obtain that Taking (30) into account and regarding the definition of , for every , we haveUsing the fact that and we havewhich implies that is right -Cauchy.
Analogously, we have which implies Thus, is left -Cauchy and, hence, it is -Cauchy sequence in By completeness of there exists such that . Since is -contraction, we have Hence So is another -limit for the sequence By the uniqueness of the limit in a quasi--metric space (Proposition 4) we have . Now, if is another fixed point of with , then, as is -contraction, we have which implies that . Similarly, using the fact that , we can prove that Therefore, .

Since any quasi-metric space and any quasi--metric space is a quasi--metric space, we derive the following results.

Corollary 27. Let be a complete quasi-metric space and let be -contraction mapping for some . Suppose that there exists such thatThen has a unique fixed point . Moreover, the sequence converges to .

Corollary 28. Let be a complete quasi--metric space and let be -contraction mapping for some . Suppose that there exists such thatThen has a unique fixed point . Moreover, the sequence converges to .

We can obtain a similar result in the context of complete dislocated quasi-metric spaces.

#### 3. Ćirić Type Contraction in a Quasi--Metric Space

In this section, we consider the existence and uniqueness of fixed point for Ćirić type contraction in the setting of quasi--metric space.

Definition 29. Let be a function and We say that is a generalized -quasi-contraction mapping if it satisfies the following condition:for every .

Proposition 30. Suppose that is a generalized -quasi-contraction mapping for some . If has a fixed point with , then .

Theorem 31. Let be a -complete quasi--metric space with constant and let be a generalized -quasi-contraction mapping for some . Suppose that there exists such that Then converges to some . If , , and , then is a fixed point of . Moreover, if is another fixed point of with , , and , then .

Proof. Let , since is generalized -quasi-contraction mapping, for all ; we haveAswe haveTherefore, (40) yields Hence for any we have Then for every we find Since and , we derive This implies that is both left and right -Cauchy sequence and hence -Cauchy sequence. By completeness of there exists some such that converges to ; that is, .
Note that as in (44) for any we have Now, assume that and . Then, by using (46) and condition (D2) there exists such that for every .
On the other hand, as is generalized -quasi-contraction mapping we haveBy using (46) and (47) we have Hence, we derive Again by using the fact that is generalized -quasi-contraction mapping and the technique used above, we observe that By continuing in the same manner, we deduce for every . Now, as and , we haveRegarding the condition (D2) and the fact that and , we getThus, . By analogy, as in the above, we can conclude thatHence, we have . Therefore, we get ; that is, is a fixed point of .
Now, if is another fixed point of with , , and , then as is generalized -quasi contraction mapping we haveas and . So, by Proposition 30, we have which implies that . In a similar way, we derive that so that Since , , and , we deduce that which yields .

#### 4. Fixed Point Theorems in Quasi--Metric Space with Partial Order

Definition 32. Let be a quasi--metric space with partial order and let be a mapping. We say that is weakly continuous if the following condition holds: if is -convergent to , then there exists a subsequence of such that is -convergent to as .

Definition 33. Let be a nonempty set with partial order . A mapping is said to be nondecreasing if

Definition 34. The pair is said to be -left regular (resp., -right regular) if the following condition holds: for every sequence satisfies () for each , with being -convergent to ; then there exists a subsequence of such that () for every .
The pair is said to be -regular if and only if it is left and right -regular.

Definition 35. A function is said to be weakly -contraction if or implies That is, whenever are comparable condition (61) is satisfied.

Theorem 36. Let be a quasi--metric space with partial order and let be a function. Assume that the following conditions hold:(i) is -complete;(ii) is weakly continuous;(iii) is weakly -contraction for some ;(iv) is nondecreasing;(v)there exists such that and .Then has a fixed point and is -convergent to . Moreover, if , then .

Proof. Since is nondecreasing and , then for every we haveand by the transitivity of , for every , we haveTherefore, for each ,  , and are always comparable. As is weak -contraction for each ,  , we haveHenceThus, for every we haveUsing the above inequality we have for every Since and we havewhich means that is right and left -Cauchy and hence -Cauchy sequence. By the completeness of there exists such that is -convergent to . Since is weakly continuous, there exists a subsequence of such that is -convergent to as . By the uniqueness of the limit in a quasi--metric space we have and is a fixed point of .
Now, if then as and is weak -contraction we havewhich is possible only if .

The weak continuity condition of in the previous theorem can be replaced by the regularity of the pair as in the following result.

Theorem 37. Let be a quasi--metric space with partial order and let be a function. Assume that the following conditions hold:(i) is -complete;(ii) is regular;(iii) is weakly -contraction for some ;(iv) is nondecreasing;(v)there exists such that and .Then has a fixed point and is -convergent to . Moreover, if , then .

Proof. Following the steps of the previous proof we can prove that is -convergent to Moreover, we havefor every . Since is regular it is right regular and so there exists a subsequence of such that for each . As is weakly -contraction we haveUsing the inequality above, we getSimilarly, we can prove that which implies that is a -limit for the sequence . By Proposition 4,  ; and is a fixed point of . As in the previous proof

Example 38. Let be the complete quasi--metric space introduced in Example 7. Define the function by Then clearly is a -contraction mapping with . Let then . So by Theorem 26   has a fixed point.

#### Competing Interests

The authors declare that they have no competing interests.

#### Authors’ Contributions

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