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`Journal of Function SpacesVolume 2015, Article ID 270971, 9 pageshttp://dx.doi.org/10.1155/2015/270971`
Research Article

## Fixed Points of Modified -Contractive Mappings in Complete Metric-Like Spaces

1Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Atılım University, Incek, 06836 Ankara, Turkey
3Department of Mathematics, Faculty of Basic Science, University of Bonab, P.O. Box 5551-761167, Bonab, Iran

Received 29 August 2014; Revised 21 October 2014; Accepted 22 October 2014

Copyright © 2015 Hamed H. Alsulami et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce the notion of modified -contractive mappings in the setting of complete metric-like spaces and we investigate the existence and uniqueness of fixed point of such mappings. The presented results unify, extend, and improve several results in the related literature.

#### 1. Introduction and Preliminaries

Throughout the paper, and denote the set of positive integers and the set of nonnegative integers, respectively. Similarly, , , and represent the set of real, positive real, and nonnegative real numbers, respectively. In what follows, we recall the notion of partial metric which is an interesting generalization of the notion of metric.

Definition 1 (see [1]). Let be a nonempty set. A mapping is said to be a partial metric on if for all the following conditions are satisfied: () if and only if ;();();().In this case, the pair is called a partial metric space (PMS).

Notice that the function defined by satisfies the conditions of a metric on . Each partial metric on generates a topology on , whose base is a family of open -balls where for all and . Consequently, several topological concepts can be easily defined as follows.

A sequence in the PMS converges to the limit if and is said to be a Cauchy sequence if exists and is finite. A PMS is called complete if every Cauchy sequence in converges with respect to , to a point such that . For more details, see, for example, [112] and the related references therein.

Lemma 2 (see, e.g., [3, 4]). Let be a complete PMS. Then consider the follwing. (A)If , then .(B)If , then .(C)A sequence is a Cauchy sequence in the PMS if and only if it is a Cauchy sequence in the metric space .(D)A PMS is complete if and only if the metric space is complete. Moreover (E)Assume as in a PMS such that . Then for every .

Now, we state the definition of metric-like (dislocated) function that was first introduced by Hitzler [13] and reintroduced later by Amini-Harandi [14].

Definition 3 (see [13]). Let be a nonempty set. A mapping is said to be metric-like (dislocated) on , if for all the following conditions are satisfied: ()if then ;();().The pair is called dislocated (metric-like) space.

Remark 4 (see [14]). Every partial metric space is a metric-like space.

A sequence , in a metric-like space ,(a)converges to if ,(b)is called Cauchy in , if exists and is finite.A metric-like space is said to be complete if and only if every Cauchy sequence in converges to so that

We recall next some basic definitions and crucial results on the topic. In this paper, we follow the notations of Amini-Harandi [14].

Definition 5 (see [14]). Let be a metric-like space and a subset of . One says that is -open subset of , if for all there exists such that . Also, is called a -closed subset of if is -open subset of .

Lemma 6 (see [15]). Let be a metric-like space. Then, (A)if then ;(B)if is a sequence such that , then one has (C)if then ;(D) holds for all where ;(E)if is a sequence in a -closed subset of with as , then ;(F)if is a sequence in such that as and , then for all .

Definition 7. Let and be metric-like spaces and a sequence in such that . A mapping is said to be continuous at a point if .

In this paper, we modify the notion of -contraction that was introduced by Wardowski [16] and investigate the existence of a fixed point of such modified -contractive mapping in the context of complete metric-like spaces. We consider also an example to illustrate the main result.

#### 2. Main Result

In this section we present our main theorems. We start with the following definition.

Definition 8. Let be a metric-like space. A self-mapping is said to be modified -contraction of type I if there exists such that for all with where and are real numbers such that and is a mapping satisfying the following conditions: () is strictly increasing; that is, for all such that , ,()for any sequence of positive real numbers if and only if .

Theorem 9. Let be a complete metric-like space and a modified -contraction of type I. Then, has a fixed point ; that is, .

Proof. For an arbitrary , we construct a sequence in the following way: If there exists such that , then is the desired fixed point of which completes the proof. Consequently, we suppose that for every . Thus, we have By the hypothesis of the theorem, we have and hence Since , we get So from , we conclude that Therefore, is a decreasing sequence of real numbers which is bounded below. This implies that converges and We will show that . Suppose, on the contrary, that . For every there exists , such that Hence from , we get On the other hand from (7), we have Due to assumption of the theorem, we obtain which is equivalent to Consequently, we derive that since . On account of (7), we have ; thus by assumption of the theorem, we have which yields Owing to the fact that , we obtain that Now by using (14) and continuing in the same way as in the derivation of (18) and (21), we deduce This implies that . Regarding , we have , and thus, there exists such that , . Therefore, from (6), we get
This contradicts the definition of given in (12). Then we get and from (12) we conclude In the next step, we claim that Suppose, on the contrary, that there exist and sequences and of natural numbers such that By triangular inequality, we have It follows from (24), (27), and Squeezing Theorem that From (24), (26), and (28), there exists such that Hence from (29), , and the hypothesis of the theorem, we have From (24) and it follows that and hence we get However, this contradicts the relation (26). Hence . Therefore is a Cauchy sequence in . By the completeness of there exists such that Next, we will prove that, for every , Arguing by contradiction, we assume that there exists such that From (18) and , we have It follows from (35) and (36) that Obviously, this is a contradiction. Hence, inequality (34) is satisfied. Regarding the assumption of the theorem, (34) implies that either or for every . In the first case, because of , the limits in (24) and (33) imply Thus, letting in (38), we conclude that Again by , we observe that On the other hand, from (6), we have It follows from (33) and (42) that ; therefore .
In the second case from (6), we have Then employing (24), (33), and , we conclude that . Equivalently, from we get Using (6), we obtain Finally, from (33) and (45) it follows that ; therefore . Hence, is a fixed point of .

Definition 10. Let be a metric-like space. A mapping is said to be a modified -contraction of type II if there exists such that for all with where is a mapping satisfying the conditions and stated in Definition 8.

Theorem 11. Let be a complete metric-like space and a modified -contraction of type II. Then, has a fixed point ; that is, .

Proof. It is sufficient to take and in Theorem 9.

Definition 12. Let be a partial metric space. A self-mapping is said to be a modified -contraction of type I if there exists such that for all with where and are real numbers such that and is a mapping satisfying the following conditions: () is strictly increasing; that is, for all such that , ,()for any sequence of positive real numbers if and only if .

Theorem 13. Let be a complete partial metric space and a modified -contraction of type I. Then, has a unique fixed point ; that is, .

Proof. Since every partial metric space is metric-like space (see, e.g., Remark 4), the existence of a fixed point of the mapping is guaranteed by Theorem 9. Thus, it is sufficient to show that is the unique fixed point of . Suppose, on the contrary, that is another fixed point of such that . Then, we have . If , we have If , then the inequality follows from the condition in Definition 1. In any case, the left-hand side of (48) is fulfilled. Hence, we have On the other hand, from , we have and . Regarding , we get that Combining (51) and (52), we conclude that since and . This is a contradiction and hence .

Analogously, we conclude a result similar to Theorem 11 by introducing the next definition.

Definition 14. Let be a partial space. A mapping is said to be a modified -contraction of type II if there exists such that for all with where is a mapping satisfying the conditions and stated in Definition 12.

Theorem 15. Let be a complete partial space and a modified -contraction of type II. Then, has a unique fixed point ; that is, .

Proof. It is sufficient to take and in Theorem 13.

Definition 16. Let be a metric-like space. A self-mapping is said to be modified -contraction of type III if there exists such that for all with where and are real numbers such that and is a mapping satisfying the conditions and introduced in Definition 8.

Theorem 17. Let be a complete metric-like space and a continuous modified -contraction of type III. If for all , then has a fixed point ; that is, .

Proof. As in the proof of Theorem 9, we construct an iterative sequence in the following way. Take and arbitrary and set and Notice that if for some , the proof is completed. Suppose that for all . Thus, (55) yields that which can be written as Regarding the assumption , we get From , we conclude that Therefore, is a decreasing sequence of real numbers which is bounded from below. Hence, it converges and We will show that by method of Reductio ad absurdum. Suppose that . Thus, for every there exists , such that . Because of , we have On the other hand, it follows from (57) that , which implies due to (55). Consequently, we have Since , we obtain that Again from (57), we have ; thus, by the hypothesis of the theorem which results in . Taking into account that , we derive
Now we employ (63) and applying a procedure similar to that used in derivation of (66) and (68), we obtain
This implies that Then, from () we have , so that there exists such that
However, this contradicts the definition of given in (62). Thus, and from (62) we conclude
In the sequel, we will show that . Assume the contrary; that is, let there exist and sequences and of natural numbers such that Observe that by the triangle inequality we have It follows from (72), (74), and the Squeeze Theorem that Therefore, there exists such that Hence from (76), , and assumption of theorem, we have Now, using (72), (75), and , we obtain However, due to it follows that
This is a contradiction with the relation (73). Hence . By the completeness of there exists such that Since and is continuous, we deduce by Definition 7. Consequently, we have According to the assumption of the theorem . Then, from (80) and (81) it follows that Since , we obtain and thus the condition gives , which completes the proof.

In the following result, we proved that Theorem 17 is valid in the context of partial metric space.

Theorem 18. Let be complete partial metric space and let be a continuous self-mapping on such that, for all , . Let and be real numbers such that . Assume that there exists such that, for all , where satisfies the conditions and . Then, has a unique fixed point ; that is, .

Proof. Since every partial metric space is a metric-like space, Theorem 17 provides the existence of a fixed point; that is, has a fixed point . Therefore, it is sufficient to show the uniqueness of the fixed point of . Indeed, if there is another fixed point of , such that , due to (), we have and equivalently . By the assumption of theorem, we have On the other hand, () implies Moreover, from , we get Due to the fact that and , from (84) and (86), we conclude However, this is a contradiction, and hence, .

Definition 19. Let be metric-like spaces. A self-mapping is said to be modified -contraction of type IV if there exists such that for all with where is a mapping satisfying the conditions and introduced in Definition 8.

Theorem 20. Let be a complete metric-like space and a continuous modified -contraction of type IV. If for all , then has a fixed point ; that is, .

Proof. The proof is obvious by taking and in Theorem 17.

Theorem 21. Let be complete PMS and let be a continuous self-mapping on such that, for all , . Assume that there exists such that, for all , where satisfies conditions and . Then, has a unique fixed point ; that is, .

Proof. The proof is trivial by taking and in Theorem 18.

Last, we provide an example which illustrates our results.

Example 22. Let be a real number such that , , and Let and be defined as Define a mapping as follows: Then satisfies in the conditions of Theorem 21.
Observe that is a complete partial metric space, but not a metric space. Define the function in Theorem 21 as . Then we get Note also that, for all , we have On the other hand, Hence, we get Therefore is an -contraction satisfying the conditions of Theorem 21, , and ; that is, is the unique fixed point of .

#### Conflict of Interests

The authors declare that they have no competing interests.

#### Authors’ Contribution

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

#### Acknowledgments

This research was supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors thank anonymous referees for their remarkable comments, suggestions, and ideas that helped to improve this paper.

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