Fixed Point Theorems for Generalized <i>α</i>-<i>β</i>-Weakly Contraction Mappings in Metric Spaces and Applications

Latif, Abdul; Mongkolkeha, Chirasak; Sintunavarat, Wutiphol

doi:https://doi.org/10.1155/2014/784207

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Volume 2014 | Article ID 784207 | https://doi.org/10.1155/2014/784207

Fixed Point Theorems for Generalized α-β-Weakly Contraction Mappings in Metric Spaces and Applications

Abdul Latif,¹Chirasak Mongkolkeha,²and Wutiphol Sintunavarat³

Academic Editor: S. A. Mohiuddine

Received17 Mar 2014

Accepted18 Apr 2014

Published07 May 2014

Abstract

We extend the notion of generalized weakly contraction mappings due to Choudhury et al. (2011) to generalized --weakly contraction mappings. We show with examples that our new class of mappings is a real generalization of several known classes of mappings. We also establish fixed point results for such mappings in metric spaces. Applying our new results, we obtain fixed point results on ordinary metric spaces, metric spaces endowed with an arbitrary binary relation, and metric spaces endowed with graph.

1. Introduction

The well-known Banach’s contraction principle has been generalized in many ways over the years [1–6]. One of the most interesting studies is the extension of Banach's contraction principle to a case of weakly contraction mappings which was first given by Alber and Guerre-Delabriere [7] in Hilbert spaces. In 2001, Rhoades [8] has shown that the result of Alber and Guerre-Delabriere [7] is also valid in complete metric spaces. Fixed point problems involving weak contractions and mappings satisfying weak contractive type inequalities have been considered in [9–13] and references therein.

On the other hand, the concept of the altering distance function was introduced by Khan et al. [14]. In 2011, Choudhury et al. [15] generalized weakly contraction mappings by using an altering distance control function and proved fixed point theorem for a pair of these mappings. Some generalizations of this function of fixed point problems in metric and probabilistic metric spaces have been studied [16–18].

Recently, Samet et al. [19] introduced the concepts of --contraction mappings and -admissible mappings and established various fixed point theorems for such mappings in complete metric spaces. Afterwards, many fixed point results via the concepts of -admissible mappings occupied a prominent place in many aspects (see [20–25] and references therein).

From the mentioned above, we introduce the concept of generalized weakly contraction mappings and give some examples to show the real generality of these mappings. We also obtain fixed point results for such mappings. Our result improves and complements several results in the literatures. As an application of our results, fixed point results on ordinary metric spaces, metric spaces endowed with an arbitrary binary relation, and metric spaces endowed with graph are also derived from our results.

2. Preliminaries

In this section, we give some notations and basic knowledge. Throughout this paper, denotes the set of positive integers.

Definition 1 (see [14]). A function is called an altering distance function if the following properties are satisfied:(i) is monotone increasing and continuous;(ii) if and only if .

Definition 2 (see [26]). Let be a metric space and let be a self-mapping on . A mapping is said to be contraction if, for each , one has where .

Definition 3 (see [8]). Let be a metric space and let be a self-mapping on . A mapping is said to be weak contraction if, for each , one has where is a continuous and nondecreasing function such that if and only if .

In fact, if we take for all , where , then the condition (2) becomes (1).

In 2011, Choudhury et al. [15] introduced the concept of a generalized weakly contractive condition as follows.

Definition 4 (see [15]). Let be a metric space and let be a self-mapping on . A mapping is said to be a generalized weakly contraction, if, for each , one has where is altering distance function, and is a continuous function with if and only if .

Remark 5. It is easy to see that a generalized weakly contractive condition (3) is more general than several generalized contractive conditions. The following conditions are an example of a special case of a generalized weakly contractive condition (3):(i) for all , where ;(ii) for all ;(iii) for all , where .Moreover, the contractive condition (1) is also a special case of condition (3).

Definition 6 (see [19]). Let be a nonempty set and let be a mapping. A self-mapping is said to be -admissible if the following condition holds:

Example 7 (see [19]). Let and define and by Then is -admissible.

3. Main Results

In this section, we introduce the concept of generalized weakly contraction mappings and prove the fixed point theorems for such mappings.

Definition 8. Let be a metric space, two given mappings, and a self-mapping on . A mapping is said to be a generalized weakly contraction type if, for each , one has where is altering distance function, and is a continuous function with if and only if .

Definition 9. Let be a metric space, two given mappings, and a self-mapping on . A mapping is said to be a generalized weakly contraction type if, for each , one has where is altering distance function, and is a continuous function with if and only if .

If we take for all , then generalized weakly contraction mappings type and type become generalized weakly contraction mappings due to Choudhury et al. [15]. Therefore, classes of generalized weakly contraction mappings type and type are larger than the class of generalized weakly contraction mappings. Next, we give some examples to show the real generality of classes of generalized weakly contraction mappings.

Example 10. Let . From [27], is a complete metric space with metric defined by Let a mapping be defined by First, we show that is generalized weakly contractive type with the functions and defined by Next we show that is a generalized weakly contraction mapping type . For , we distinguish the following cases.

Case 1 (). Without loss of generality, we may assume that . Now we obtain that
In case of , we have and then
In case of , we have .
If , then we have If , then we have Therefore, for , we get that satisfies condition (7).

Case 2 ( and ). In this case, we obtain that From (18), we obtain that Therefore, we conclude that satisfies condition (7) in this case.

Case 3 ( and ). This case is similar to Case 2.

Case 4 ( and ). Now we obtain that If , we have and so If , we have , and hence If , we have , and hence
Now we conclude that satisfies condition (7) in this case.

Case 5 (one of ). If , we obtain that
If , we get and so
Now we conclude that satisfies condition (7) in this case.
From all cases, we get that is generalized weakly contraction mapping type .

Remark 11. From Example 10, we can see that is not a generalized weakly contraction mapping. Indeed, putting and , we get
Before presenting the main results in this paper, we introduce the following concept, which will be used in our results.

Definition 12. Let be a nonempty set and . A self-mapping is said to be -subadmissible if the following condition holds:

Definition 13. Let be a nonempty set and a mapping.(i) is said to be forward transitive if for each for which and one has ;(ii) is said to be 0-backward transitive if for each for which and one has .

3.1. Generalized Weakly Contraction Mappings Type

In this subsection, we give the fixed point results for generalized weakly contraction mappings type .

Theorem 14. Let be a complete metric space, two given mappings, and a generalized weakly contraction type ; then the following conditions hold: () is continuous;() is -admissible and -subadmissible;() is forward transitive and is -backward transitive;()there exists such that .Then has a fixed point in .

Proof. Starting from in assumption and letting for all , if there exists such that , then is a fixed point of . This finishes the proof. Therefore, we may assume that for all . Since , we get It follows from is -admissible and -subadmissible that and then By repeating this process, we get that is a sequence in such that and for all . By using the generalized weakly contractive condition type of , we have for all . Now we obtain that for all . From (34) and (35), we get for all .
Suppose that for some . Then we have which is a contradiction. Hence for all . This means that is a monotone decreasing sequence. Since is bounded below, there exists such that Using (36), we get for all . Taking in the above inequality, we have This implies that ; that is,
Next, we will prove that is a Cauchy sequence. Suppose that is not a Cauchy sequence. Then there exists such that for all , where . Further, corresponding to , we can choose in such a way that it is the smallest integer with satisfying (42). Then we have By using (43) and triangular inequality, we get From (41) and (44), we have From the triangular inequality, we get Using (45) and (46), we get Again, by the triangular inequality, we get Using (48), we obtain that Similarly, we can prove Since is forward transitive, is -backward transitive, and , we can conclude that In view of the fact that is generalized weakly contractive type mapping and (51), we have Letting , by using (45), (47), (49), and (50), we obtain that which is a contradiction. Then, we deduce that is a Cauchy sequence. Since is a complete metric space, then there exists such that as . From the continuity of , it follows that Using the uniqueness of limit of the sequence, we conclude that and the proof is complete.

In the next theorem, the continuity of a generalized weakly contraction mapping type in Theorem 14 is replaced by the following condition:if is a sequence in such that for all and as , then for all .

Theorem 15. Let be a complete metric space, two given mappings, and a generalized weakly contraction type ; then the following conditions hold:()condition holds;() is -admissible and -subadmissible;() is forward transitive and is -backward transitive;()there exists such that .Then has a fixed point in .

Proof. As in the proof of Theorem 14, we can find a sequence in such that for all and is Cauchy sequence which converges to some point in . Moreover, we have for all . By using condition , we obtain that for all . Now, let us claim that . Supposing the contrary, from the fact that is a generalized weakly contractive type and (55), we get On the other hand, we obtain that for all . Letting in (57), by using (58) and the continuity of , we get which is a contradiction. Therefore and the proof is complete.

We obtain that Theorems 14 and 15 cannot claim the uniqueness of fixed point. To assure the uniqueness of the fixed point, we will add the following condition:for all there exists such that

Definition 16. Let be a nonempty set, a mapping, and . The orbit of at is denoted by and defined by where and .

Theorem 17. By adding condition to the hypotheses of Theorem 14 (or Theorem 15) and the limit of orbit exists, where is an element in satisfying (60). Then has a unique fixed point.

Proof. Suppose that and are two fixed points of . By condition there exists such that It follows from is -admissible and -subadmissible that for all . Since the limit of exists, we get that converges to some element in . Let us claim that as . Suppose the contrary; that is By the generalized weakly contractive condition type of , we have for all . On the other hand, we have for all . From (64), (65), (66), and the property of and , we obtain that which is a contradiction, and hence as . Similarly, we can show that as . Using the uniqueness of limit of the sequence, we conclude that and the proof is complete.

3.2. Generalized Weakly Contraction Mappings Type

In this subsection, we obtain the existence and uniqueness of fixed point theorems for generalized weakly contraction mappings type .

Theorem 18. Let be a complete metric space, two given mappings, and a generalized weakly contraction type ; then the following conditions hold:() is continuous;() is -admissible and -subadmissible;() is forward transitive and is -backward transitive;()there exists such that .Then has a fixed point in .

Proof. As in the proof of Theorem 14, we can find a sequence in such that , , and for all . Moreover, for each , we have Since is a generalized weakly contraction mapping type , for each , we get Suppose that for some . From (70), we get which is a contradiction. Therefore for all . This means that is a monotone decreasing sequence. It follows from a sequence bounded below that there exists such that From (70), we get for all . Taking in the above inequality, we have This implies that ; that is,
Next, we will prove that is a Cauchy sequence. Suppose that is not a Cauchy sequence. Then there exists such that for all , where . Further, corresponding to , we can choose in such a way that it is the smallest integer with satisfying (76). Then we have As the same argument in Theorem 14, we have Moreover, we have In view of the fact that is generalized weakly contraction mapping type , we have for all . Letting in the above relation, we obtain that which is a contradiction. Therefore, we deduce that is a Cauchy sequence and so it converges to some element . By the continuity of , we get and hence . Therefore the proof is complete.

Theorem 19. Let be a complete metric space, two given mappings, and a generalized weakly contraction type ; then the following conditions hold:()condition holds;() is -admissible and -subadmissible;() is forward transitive and is -backward transitive;()there exists such that .Then has a fixed point in .

Proof. As in the proof of Theorem 18, we can find a sequence in such that for all and is a Cauchy sequence converging to some point in .
Also, as in the proof of Theorem 15, for each , we get Now, let us claim that . On contrary, assume that that is . By using (83) and a generalized weakly contractive condition type , we get Letting in (85), we get which is a contradiction. Therefore and the proof is complete.

Theorem 20. By adding condition to the hypotheses of Theorem 18 (or Theorem 19) and the limit of orbit exists, where is an element in satisfying (60). Then has a unique fixed point.

Proof. Apply the proof of Theorem 18 (or Theorem 19) and Theorem 17.

4. Applications

In this section, we give the several fixed point results which are obtained by our results in Section 3.

4.1. Fixed Point Results on an Ordinary Metric Space

Setting for all in Theorem 14 (or Theorem 18), we get the following result.

Corollary 21. Let be a complete metric space and a continuous generalized weakly contraction mapping. Then has a fixed point in .

By using Remark 5, we obtain the following results.

Corollary 22. Let be a complete metric space and a continuous mapping and for all , where and Then has a fixed point in .

Corollary 23. Let be a complete metric space and a continuous mapping and for all , where Then has a fixed point in .

Corollary 24. Let be a complete metric space and let be a continuous mapping and for all , where Then has a fixed point in .

4.2. Fixed Point Results on Metric Spaces Endowed with an Arbitrary Binary Relation

In this section, we give the existence of fixed point theorems on a metric space endowed with an arbitrary binary relation. Before presenting our results, we give the following notions and definitions.

Definition 25. Let be a nonempty set and a binary relation over . One says that is a comparative mapping with respect to if

Definition 26. Let be a nonempty set and a binary relation over . One says that has a transitive property with respect to if

Definition 27. Let be a metric space and a binary relation over . A mapping is said to be a generalized weakly contraction with respect to if for each for which one has where is altering distance function, and is a continuous function with if and only if .

Theorem 28. Let be a metric space and a binary relation over and a generalized weakly contraction with respect to ; then the following conditions hold: () is continuous;() has a transitive property with respect to ;() is comparative mapping with respect to ;()there exists such that .Then has a fixed point in .

Proof. Consider two mappings defined by From condition , we get . Since is comparative mapping with respect to , we get is -admissible and -subadmissible. Also, is forward transitive and is -backward transitive since has a transitive property with respect to . Since is a generalized weakly contraction with respect to , we have, for all , This implies that is generalized weakly contraction mapping types and . Now all the hypotheses of Theorem 14 (or Theorem 18) are satisfied and thus the existence of the fixed point of follows from Theorem 14 (or Theorem 18).

In order to remove the continuity of , we need the following condition:if is the sequence in such that for all and it converges to the point , then for all .

Theorem 29. Let be a metric space and a binary relation over and a generalized weakly contraction with respect to ; then the following conditions hold: ()the condition holds on ;() has a transitive property with respect to ;() is comparative mapping with respect to ;()there exists such that .Then has a fixed point in .

Proof. The result follows from Theorem 15 (or Theorem 19) by considering the mappings and given by (97) and by observing that condition implies property .

To assure the uniqueness of the fixed point, we will add the following condition:for all there exists such that

Theorem 30. By adding condition to the hypotheses of Theorem 28 (or Theorem 29) and the limit of orbit exists, where is an element in satisfying (99). Then has a unique fixed point.

Proof. The result follows from Theorem 17 (or Theorem 20) by considering the mappings and given by (97) and by observing that condition implies property .

4.3. Fixed Point Results on Metric Spaces Endowed with Graph

Throughout this section, let be a metric space. A set is called a diagonal of the Cartesian product and is denoted by . Consider a graph such that the set of its vertices coincides with and the set of its edges contains all loops; that is, . We assume has no parallel edges, so we can identify with the pair . Moreover, we may treat as a weighted graph by assigning to each edge the distance between its vertices. A graph is connected if there is a path between any two vertices.

In this section, we give the existence of fixed point theorems on a metric space endowed with graph. Before presenting our results, we give the following notions and definitions.

Definition 31. Let be a metric space endowed with a graph and mapping. One says that preserves edges of if

Definition 32. Let be a metric space endowed with a graph and mapping. One says that has a transitive property with respect to graph if

Remark 33. It is easy to see that if is a connected graph, then has a transitive property with respect to graph .

Definition 34. Let be a metric space endowed with a graph . A mapping is said to be a generalized weakly contraction with respect to graph if for each for which one has where is altering distance function, and is a continuous function with if and only if .

Example 35. Let be a metric space, a given mapping, an arbitrary altering distance function, and an arbitrary continuous function with if and only if . If for all , then is trivially generalized weakly contraction with respect to graph , where .

Theorem 36. Let be a metric space endowed with a graph and a generalized weakly contraction with respect to graph ; then the following conditions hold: () is continuous;() has a transitive property with respect to graph ;() preserves edges of ;()there exists such that .Then has a fixed point in .

Proof. Consider two mappings defined by From condition , we get . Since preserves edges of , we get is -admissible and -subadmissible. Also, is forward transitive property and is -backward transitive since has transitive property with respect to graph . Since is a generalized weakly contraction with respect to graph , we get for all . This implies that is generalized weakly contraction mapping types and . Therefore, all the hypotheses of Theorem 14 (or Theorem 18) are satisfied. Now the existence of the fixed point of follows from Theorem 14 (or Theorem 18).

In order to remove the continuity of , we need the following condition.

Definition 37. Let be a metric space endowed with a graph . One says that has -regular property if is the sequence in such that for all and it converges to the point ; then for all .

Theorem 38. Let be a metric space endowed with a graph and a generalized weakly contraction with respect to graph ; then the following conditions hold: () has -regular property;() has a transitive property with respect to graph ;() preserves edges of ;()there exists such that .Then has a fixed point in .

Proof. The result follows from Theorem 15 (or Theorem 19) by considering the mappings and given by (104) and by observing that -regular property implies property .

To assure the uniqueness of the fixed point, we will add the following condition:for all there exists such that

Theorem 39. By adding condition to the hypotheses of Theorem 36 (or Theorem 38) and the limit of orbit exists, where is an element in satisfying (106). Then has a unique fixed point.

Proof. The result follows from Theorem 17 (or Theorem 20) by considering the mappings and given by (97) and by observing that condition implies property .

By using Remark 33, we get the following results.

Corollary 40. Let be a metric space endowed with a graph and a generalized weakly contraction with respect to graph ; then the following conditions hold:() is continuous;() is connected graph;() preserves edges of ;()there exists such that .Then has a fixed point in .

Corollary 41. Let be a metric space endowed with a graph and a generalized weakly contraction with respect to graph ; then the following conditions hold:() has -regular property;() is connected graph;() preserves edges of ;()there exists such that .Then has a fixed point in .

Corollary 42. By adding condition to the hypotheses of Corollary 40 (or Corollary 41), the limit of orbit exists, where is an element in satisfying (106). Then has a unique fixed point.

Conflict of Interests

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

Authors’ Contribution

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

Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR for the technical and financial support.

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Copyright © 2014 Abdul Latif 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.

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