#### Abstract

We investigate the existence and uniqueness of a fixed point of an operator via simultaneous functions in the setting of complete quasi-metric spaces. Our results generalize and improve several recent results in literature.

#### 1. Introduction and Preliminaries

One of the attractive research subjects in the fixed point theory is the investigation of the existence and uniqueness of (common) fixed point of various operators in the setting of quasi-metric space. Very recently, Jleli and Samet [1] and Samet et al. [2] reported that -metrics, introduced by Mustafa and Sims [3], can be deduced from quasi-metrics by taking . Consequently, the authors in [1, 2] proved that several fixed point results in the setting of -metric spaces can be deduced from the corresponding theorems in the context of quasi-metric spaces. The importance of these results follows from the simplicity of construction of quasi-metric despite the notion of -metric.

In this paper, we investigate the existence and uniqueness of a fixed point of operators via simultaneous functions, defined by Khojasteh et al. [4], in the setting of complete quasi-metric spaces. We also observed that several existing results can be concluded from our main results. We also show that some result in the context of -metric spaces can be deduced from the corresponding theorems in the framework of quasi-metric spaces.

For the sake of completeness, we recollect basic notions, definitions, and fundamental results. Let be two nonempty subsets of a set and let be a mapping. A point is called a* fixed point* of the mapping if .

*Definition 1. *Let be a nonempty set and let be a given function which satisfies(1) if and only if ;(2) for any points . Then is called a quasi-metric and the pair is called a quasi-metric space.

It is evident that any metric space is a quasi-metric space, but the converse is not true in general. Now, we recall convergence and completeness on quasi-metric spaces.

*Definition 2. *Let be a quasi-metric space and let be a sequence in and . The sequence converges to if

*Remark 3. *A convergent sequence in a quasi-metric space has a unique limit.

*Remark 4. *If converges to in a quasi-metric space , then
In other words, is a continuous mapping on its first argument. This property follows from and . Therefore,

*Definition 5 (see, e.g., [1, 2]). *Let be a quasi-metric space and let be a sequence in . We say that is left-Cauchy if, for every , there exists a positive integer such that for all .

*Definition 6 (see, e.g., [1, 2]). *Let be a quasi-metric space and let be a sequence in . We say that is right-Cauchy if, for every , there exists a positive integer such that for all .

*Definition 7 (see, e.g., [1, 2]). *Let be a quasi-metric space and let be a sequence in . We say that is Cauchy if, for every , there exists a positive integer such that for all .

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

*Definition 9 (see, e.g., [1, 2]). *Let be a quasi-metric space. We say that(1) is left-complete if each left-Cauchy sequence in is convergent;(2) is right-complete if each right-Cauchy sequence in is convergent;(3) is complete if each Cauchy sequence in is convergent.

#### 2. Simulation Functions

The notion of* simulation function* was introduced by Khojasteh et al. in [4].

*Definition 10 (see [4]). *A simulation function is a mapping satisfying the following conditions:();() for all ;()if , are sequences in such that then

Let be the family of all simulation functions .

Before presenting our main fixed point results using simulation functions, we show a wide range of examples to highlight their potential applicability to the field of fixed point theory. In the following results, the mapping is defined from into .

*Definition 11 (Khan et al. [5]). *An altering distance function is a continuous, nondecreasing mapping such that .

*Example 12. *Let and be two altering distance functions such that for all . Then the mapping
is a simulation function.

If, in the previous example, and for all , where , then we obtain the following particular case of simulation function:

*Example 13. *If is a lower semicontinuous function such that and we define by
then is a simulation function.

If, in the previous example, is continuous, we deduce the following case.

*Example 14. *If is a continuous function such that and we define
then is a simulation function.

*Example 15. *Let be two continuous functions with respect to each variable such that for all and define
Then is a simulation function.

*Example 16. *If is a function such that for all and we define
then is a simulation function.

*Example 17. *If is an upper semicontinuous mapping such that for all and and we define
then is a simulation function.

*Example 18. *If is a function such that exists and , for each , and we define
then is a simulation function.

*Example 19. *Let be a function such that for all and provided that and are two sequences such that , and we define
and then is a simulation function.

The following results are more theoretical.

Proposition 20. *Let be a function such that and there exists verifying that for all . Then .*

*Proof. *For all , . If and are sequences in such that , then .

Proposition 21. *Let . Then the following statements hold.*(a)*For each , the function defined by
* *is a simulation function (i.e., for any ).*(b)*For each , the function defined by
* *is a simulation function (i.e., for any ).*

*Proof. *Since for all , the conclusion (a) is a direct consequence of Proposition 20. Next, we prove the conclusion (b). Let be given. It is obvious that for all because
Let be two sequences such that . For any , we have

#### 3. Main Results

In this section we use simulation functions to present a very general kind of contractions on quasi-metric spaces, and we prove related existence and uniqueness fixed point theorems.

*Definition 22. *Let be a quasi-metric space. We will say that a self-mapping is a -contraction if there exists such that

For clarity, we will use the term *-contraction* when we want to highlight that is a *-contraction* on a quasi-metric space involving the quasi-metric . In such a case, we will say that is a *-contraction with respect to *.

Next, we observe some useful properties of -contractions in the context of quasi-metric spaces.

*Remark 23. *By axiom , it is clear that a simulation function must verify for all . Consequently, if is a -contraction with respect to , then
In other words, if is a -contraction, then it cannot be an isometry.

We will prove that if a -contraction has a fixed point, then it is unique.

Lemma 24. *If a -contraction in a quasi-metric space has a fixed point, then it is unique.*

*Proof. *Let be a quasi-metric space and let be a -contraction with respect to . We are reasoning by contradiction. Suppose that there are two distinct fixed points of the mapping . Then . By (18), we have
which is a contradiction due to Remark 23.

Inspired by Browder and Petryshyn’s paper [6], we will characterize the notions of asymptotically right-regularity and asymptotically left-regularity for a self-mapping in the context of quasi-metric space .

*Definition 25. *We will say that a self-mapping on a quasi-metric space is(i)asymptotically right-regular at a point if ;(ii)asymptotically left-regular at a point if ;(iii)asymptotically regular if it is both asymptotically right-regular and asymptotically left-regular.

Now, we show that a -contraction is asymptotically regular at every point of .

Lemma 26. *Every -contraction on a quasi-metric space is asymptotically regular.*

*Proof. *Let be an arbitrary point of a quasi-metric space and let be a -contraction with respect to . If there exists some such that , then is a fixed point of ; that is, . Consequently, we have that for all , so
for sufficient large . Thus, we conclude that
Similarly, , so is asymptotically regular at . On the contrary, suppose that for all ; that is,
On what follows, from (18) and , we have that, for all ,

In particular,
The above inequality yields that is a monotonically decreasing sequence of nonnegative real numbers. Thus, there exists such that . We will prove that . Suppose, on the contrary, that . Since is -contraction with respect to , by , we have
which is a contradiction. Thus, and this proves that . Hence, is an asymptotically right-regular mapping at . Similarly, it can be demonstrated that is asymptotically left-regular at .

Given a self-mapping , a sequence is called a* Picard sequence of * (or* generated by *) if for all .

*Remark 27. *In the proof of the previous result we have proved that if is a -contraction on a quasi-metric space and is a Picard sequence of , then either there exists such that is a fixed point of (i.e., ) or

Now, we show that every Picard sequence generated by a -contraction is always bounded.

Lemma 28. *Let be a quasi-metric space and let be a -contraction with respect to . If is a Picard sequence generated by , then is bounded.*

*Proof. *Let be arbitrary and let be defined iteratively by for all . If there exists some and such that , then the set is finite, so it is bounded. Hence, assume that for all and . In this case, by Remark 27, we have that
Notice that by Lemma 26,
In particular, there exists such that
We will prove that is bounded reasoning by contradiction. We distinguish between right and left boundedness. Suppose that the set
is not bounded. Then we can find such that . If is the smallest natural number, greater than , verifying this property, then we can suppose that
Again, as is not bounded, there exists such that
Repeating this process, there exists a partial subsequence of such that, for all ,
Therefore, by the triangular inequality, we have that, for all ,
Letting in (35) and using (29) we obtain
By (28), we have . Therefore using the triangular inequality we obtain
Letting and using (29) we obtain
Owing to the fact that is a -contraction with respect to , we deduce from that, for all ,
which is a contradiction. This proves that is bounded. Similarly, it can be proved that is also bounded. Therefore, the set is bounded.

In the next theorem we prove the existence of fixed point of a -contraction.

Theorem 29. *Every -contraction on a complete quasi-metric space has a unique fixed point. In fact, every Picard sequence converges to its unique fixed point.*

*Proof. *Let be a complete quasi-metric space and let be a -contraction with respect to . Take and consider the Picard sequence . If contains a fixed point of , the proof is finished. In other case, Lemma 26 and Remark 27 guarantee that
We are going to show that is a left Cauchy sequence. For this purpose, taking into account that Lemma 28 guarantees that is bounded, we can consider the sequence given by
It is clear that the sequence is a monotonically nonincreasing sequence of nonnegative real numbers. Therefore, it is convergent; that is, there exists such that . Let us show that reasoning by contradiction. If then, by definition of , for every there exists such that and
Hence,
By using (40) and the triangular inequality, we have, for all ,
Letting in the above inequality and using (41) and (44), we derive that
Due to fact that is a -contraction with respect to and by using , (18), (44), and (46), we have
which is a contradiction. This contradiction concludes that and, hence, is a left Cauchy sequence. Similarly, it can be proved that is a right Cauchy sequence. Therefore, is a Cauchy sequence. Since is a complete quasi-metric space, there exists such that .

We will show that the point is a fixed point of reasoning by contradiction. Suppose that ; that is, . By Remark 4,
Therefore, there is such that
In particular, . This also means that for all . As and , axiom and property (18) imply that, for all ,
In particular, for all , which means that
Similarly, it can be proved that . Therefore, converges, at the same time, to and to . By the unicity of the limit, , which contradicts . As a consequence, is a fixed point of . Notice that the uniqueness of the fixed point follows from Lemma 24.

Next, we show a variety of cases in which Theorem 29 can be applied. Firstly, we mention the analog of the celebrated Banach contraction principle [7] in quasi-metric spaces.

Corollary 30 (see, e.g., [1]). *Let be a complete quasi-metric space and let be a mapping such that
**
where . Then has a unique fixed point in .*

*Proof. *The result follows from Theorem 29 taking into account that is a -contraction with respect to , where is defined by for all (see (6)).

The following example shows that the above theorem is a proper generalization of the analog of Banach contraction principle.

*Example 31. *Let be such that . Let and be a function defined by
Then is a complete quasi-metric space (but it is not a metric space). Consider the mapping defined as for all . It is clear that it is a -contraction with respect to , where
Indeed, if , then . Hence, we get that
If then . Hence, we get that
Notice that all conditions in Theorem 29 are satisfied and has a unique fixed point, which is .

In the following corollaries we obtain some known and some new results in fixed point theory via simulation functions.

Corollary 32 (Rhoades type). *Let be a complete quasi-metric space and let be a mapping satisfying the following condition:
**
where is a lower semicontinuous function and . Then has a unique fixed point in .*

*Proof. *The result follows from Theorem 29 taking into account that is a -contraction with respect to , where is defined by for all (see Example 13).

*Remark 33. *Note that Rhoades assumed in [8] that the function was continuous and nondecreasing and it verified . In Corollary 32, we replace these conditions by the lower semicontinuity of , which is a weaker condition. Therefore, our result is stronger than Rhoades’ original version.

Corollary 34. *Let be a complete quasi-metric space and let be a mapping. Suppose that for every ,
**
for all , where is a function such that for all . Then has a unique fixed point.*

*Proof. *The result follows from Theorem 29 taking into account that is a -contraction with respect to , where is defined by for all (see Example 16).

Corollary 35. *Let be a complete quasi-metric space and let be a mapping. Suppose that, for every ,
**
for all , where is an upper semicontinuous mapping such that for all and . Then has a unique fixed point.*

*Proof. *The result follows from Theorem 29 taking into account that is a -contraction with respect to , where is defined by for all (see Example 17).

Corollary 36. *Let be a complete quasi-metric space and let be a mapping satisfying the following condition:
**
where is a function such that exists and , for each . Then has a unique fixed point in .*

*Proof. *The result follows from Theorem 29 taking into account that is a -contraction with respect to , where is defined by
(see Example 18).

Corollary 37. *Let be a complete quasi-metric space and let be a mapping satisfying the following condition:
**
where is a function such that and provided that and are two sequences such that . Then has a unique fixed point in .*

*Proof. *The result follows from Theorem 29 taking into account that is a -contraction with respect to , where is defined by for all (see Example 19).

*Example 38. *The following example is inspired by Remark 3 in Boyd and Wong [9]. Let and let us define
It is apparent that is a complete quasi-metric space, but it is not a metric space (for instance, ). Let us consider the mappings , , and defined by
Although is not an upper semicontinuous mapping, it is easy to show that is a simulation function (if and , then ). Furthermore, it can be proved that
Therefore, is a -contraction with respect to . Using Theorem 29, has a unique fixed point, which is .

As Boyd and Wong pointed out in [9], as
there can be no decreasing function with for and for which (58) holds. Furthermore, since
there is no increasing function with for and for which (58) holds.

*Example 39. *Let (it is also possible to consider , where ) and let us define
It is clear that is a complete quasi-metric space, but it is not a metric space since . Let us define and by
Then and is a -contraction with respect to . Therefore, has a unique fixed point, which is .

#### 4. Consequences: Fixed Point Results in the Context of -Metric Spaces

In this section, we show the applicability of our main results to the framework of -metric spaces, and we indicate that some existing fixed point results in that setting can be easily derived from our main theorems. First, we recall some basic definitions and fundamental results on this topic which can be found in the literature.

*Definition 40 (Mustafa and Sims [3]). *A generalized metric (or a -metric) on a nonempty set is a mapping