Some Extensions of Fixed Point Results over Quasi--Spaces

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 [1] 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 [1].

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 [1].

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 [1]).

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 [1]).

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 [1]).

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 [2].

In 2012, Shah and Hussain [3] 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. [4] 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 [5]; 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 [1]. 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.