International Scholarly Research Notices
Volume 2014 (2014), Article ID 608725, 10 pages
http://dx.doi.org/10.1155/2014/608725

Research Article

Common Coupled Fixed Point Theorems for Two Hybrid Pairs of Mappings under Contraction

Bhavana Deshpande and Amrish Handa

Department of Mathematics, Govt. PG Arts & Science College, Ratlam, Madhya Pradesh 457001, India

Received 27 March 2014; Accepted 1 October 2014; Published 30 November 2014

Academic Editor: George L. Karakostas

Copyright © 2014 Bhavana Deshpande and Amrish Handa. 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 concept of (EA) property and occasional w-compatibility for hybrid pair and . We also introduce common (EA) property for two hybrid pairs and . We establish some common coupled …fixed point theorems for two hybrid pairs of mappings under - contraction on noncomplete metric spaces. An example is also given to validate our results. We improve, extend and generalize several known results. The results of this paper generalize the common …fixed point theorems for hybrid pairs of mappings and essentially contain …fixed point theorems for hybrid pair of mappings.

1. Introduction and Preliminaries

Let be a metric space and let be the set of all nonempty closed bounded subsets of . Let denote the distance from to and let denote the Hausdorff metric induced by ; that is,

The study of fixed points for multivalued contractions and nonexpansive mappings using the Hausdorff metric was initiated by Markin [1]. The existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. The theory of multivalued mappings has application in control theory, convex optimization, differential inclusions, and economics. In 1969, Nadler [2] extended the famous Banach contraction principle [3] from single-valued mapping to multivalued mapping and proved the fixed point theorem for the multivalued contraction. Many authors proved fixed point theorems for hybrid pair of mappings without assuming the continuity of any mapping involved including [4–7].

In [8], Gnana Bhaskar and Lakshmikantham established some coupled fixed point theorems and applied these results to study the existence and uniqueness of solution for periodic boundary value problems. Luong and Thuan [9] generalized the results of Gnana Bhaskar and Lakshmikantham [8]. Berinde [10] extended the results of Gnana Bhaskar and Lakshmikantham [8] and Luong and Thuan [9]. Lakshmikantham and Ćirić [11] proved coupled coincidence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces and extended the results of Gnana Bhaskar and Lakshmikantham [8]. Jain et al. [12] extended and generalized the results of Berinde [10], Gnana Bhaskar and Lakshmikantham [8], Lakshmikantham and Ćirić [11], and Luong and Thuan [9].

Deshpande and Handa [13] generalized and intuitionistically fuzzified the results of Gnana Bhaskar and Lakshmikantham [8], Lakshmikantham and Ćirić [11], and Luong and Thuan [9], while Deshpande et al. [14] generalized and intuitionistically fuzzified the results of Berinde [10], Gnana Bhaskar and Lakshmikantham [8], Lakshmikantham and Ćirić [11], and Luong and Thuan [9]. In [15], Deshpande et al. proved a common coupled fixed point theorem for mappings under -contractive conditions on intuitionistic fuzzy metric spaces. As an application, the existence and uniqueness of solution to a nonlinear Fredholm integral equation have been studied.

Recently Samet et al. [16] claimed that most of the coupled fixed point theorems in the setting of single valued mappings on ordered metric spaces are consequences of well-known fixed point theorems.

These concepts were extended by Abbas et al. [17] to multivalued mappings and who obtained coupled coincidence point and common coupled fixed point theorems involving hybrid pair of mappings satisfying generalized contractive conditions in complete metric spaces. Very few authors studied coupled fixed point theorems for hybrid pair of mappings including [17–20].

In [17], Abbas et al. introduced the following concept.

Definition 1. Let be a nonempty set, (a collection of all nonempty subsets of ), and let be a self-mapping on . An element is called

(1)a coupled coincidence point of hybrid pair if and ,(2)a common coupled fixed point of hybrid pair if and .

We denote the set of coupled coincidence points of mappings and by ,. Note that if , then is also in .

Definition 2. Let be a multivalued mapping and let be a self-mapping on . The hybrid pair is called -compatible if whenever .

Definition 3. Let be a multivalued mapping and let be a self-mapping on . The mapping is called -weakly commuting at some point if and .

Aamri and El Moutawakil [21] defined (EA) property for self-mappings which contained the class of noncompatible mappings. Kamran [22] extended the (EA) property for hybrid pair and . Liu et al. [23] introduced common (EA) property for hybrid pairs of single and multivalued mappings and gave some new common fixed point theorems under hybrid contractive conditions. Abbas and Rhoades [24] extended the concept of occasionally weakly compatible mappings for hybrid pair and .

In this paper, we introduce the concept of (EA) property and occasional -compatibility for hybrid pair and . We also introduce common (EA) property for two hybrid pairs and . We establish some common coupled fixed point theorems for two hybrid pairs of mappings under - contraction on noncomplete metric spaces. The - contraction is weaker contraction than the contraction defined in Gnana Bhaskar and Lakshmikantham [8] and Luong and Thuan [9]. We improve, extend, and generalize the results of Berinde [10], Gnana Bhaskar and Lakshmikantham [8], Jain et al. [12], Lakshmikantham and Ćirić [11], Liu et al. [23], and Luong and Thuan [9]. The results of this paper generalize the common fixed point theorems for hybrid pairs of mappings and essentially contain fixed point theorems for hybrid pair of mappings.

2. Main Results

We first define the following.

Definition 4. Mappings and are said to satisfy the (EA) property if there exist sequences in , some in , and ,   in such that

Definition 5. Let and . The pairs and are said to satisfy the common (EA) property if there exist sequences , and in , some in , and in such that

Example 6. Let with the usual metric. Define and by

Consider the sequences

Clearly,

Therefore, the pairs and are said to satisfy the common (EA) property.

Definition 7. Mappings and are said to be occasionally -compatible if and only if there exists some point such that , and .

Example 8. Let with usual metric. Define by

It can be easily verified that and are coupled coincidence points of and , but and . So and are not -compatible. However, the pair is occasionally -compatible.

Let denote the set of all functions satisfying the following:

    is continuous and strictly increasing,    for all ,    for all .

And let denote the set of all functions which satisfies

    for all and ,    for all and .

Note that, by and , we have that if and only if . For example, functions where , and are in ,   where ,  , and are in .

Now, we prove our main results.

Theorem 9. Let be a metric space. Assume and to be mappings satisfying the following.

(1) and satisfy the common (EA) property.(2)For all , there exist some and some such that

(3) and are closed subsets of . Then

(a) and have a coupled coincidence point,(b) and have a coupled coincidence point,(c) and have a common coupled fixed point, if is -weakly commuting at and and for ,(d) and have a common coupled fixed point, if is -weakly commuting at and and for ,(e),  and have common coupled fixed point provided that both (c) and (d) are true.

Proof. Since and satisfy the common (EA) property, there exist sequences , and in , some in , and in such that

Since and are closed subsets of , then there exist ,

Now, by using condition (2) of Theorem 9, we get

Letting in the above inequality, by using (9), (10), ,  , and , we obtain

which, by and , implies

Since and , it follows that

That is, is a coupled coincidence point of and . This proves (a). Again, by using condition (2) of Theorem 9, we get

Letting in the above inequality, by using (9), (10), ,  , and , we obtain

which, by and , implies

Since and , it follows that

That is, is a coupled coincidence point of and . This proves (b).
Furthermore, from condition (c), we have which is -weakly commuting at ; that is, and . Thus, and ; that is, and . This proves (c). A similar argument proves (d). Then (e) holds immediately.

Put in Theorem 9, and we get the following result.

Corollary 10. Let be a metric space. Assume and to be mappings such that

(1) and satisfy the common (EA) property,(2)for all ,,,, there exist some and some such that

(3) is a closed subset of . Then

(a) and have a coupled coincidence point,(b) and have a coupled coincidence point,(c) and have a common coupled fixed point, if is -weakly commuting at and and for ,(d) and have a common coupled fixed point, if is -weakly commuting at and and for ,(e),  and have common coupled fixed point provided that both (c) and (d) are true.

Put and in Theorem 9, and we get the following result.

Corollary 11. Let be a metric space. Assume and to be mappings such that

(1) satisfies the (EA) property,(2)for all , there exist some and some such that

If (3) of Corollary 10 holds. Then

(a) and have a coupled coincidence point,(b) and have a common coupled fixed point, if is -weakly commuting at and and for .

Corollary 12. Let be a metric space. Assume and to be mappings satisfying (1) of Theorem 9 and

(1)for all ,,,, there exists some such that

If (3) of Theorem 9 holds, then

(a) and have a coupled coincidence point,(b) and have a coupled coincidence point,(c) and have a common coupled fixed point, if is -weakly commuting at and and for ,(d) and have a common coupled fixed point, if is -weakly commuting at and and for ,(e), and   have common coupled fixed point provided that both (c) and (d) are true.

Proof. If , then for all . Now divide condition (1) of Corollary 12 by 4 and take , and then the above condition reduces to condition (2) of Theorem 9 with and hence by Theorem 9 we get Corollary 12.

Put in Corollary 12, and we get the following result.

Corollary 13. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 10 and

(1)for all , there exists some such that

If (3) of Corollary 10 holds, then

(a) and have a coupled coincidence point,(b) and have a coupled coincidence point,(c) and have a common coupled fixed point, if is -weakly commuting at and and for ,(d) and have a common coupled fixed point, if is -weakly commuting at and and for ,(e), and   have common coupled fixed point provided that both (c) and (d) are true.

Put and in Corollary 12, we get the following result.

Corollary 14. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 11 and

(1)for all , there exists some such that

If (3) of Corollary 10 holds, then

(a) and have a coupled coincidence point,(b) and have a common coupled fixed point, if is -weakly commuting at and and for .

Theorem 15. Let be a metric space. Assume and to be mappings satisfying (1) of Theorem 9 and (2) of Theorem 9 and

(1) and are -compatible.(2)Suppose that either

(a) is a closed subset of and or(b) is a closed subset of and .

Then , and   have a common coupled fixed point.

Proof. Since and satisfy the common (EA) property, there exist sequences , and in , some in , and in satisfying (9). Suppose (a) holds; that is, is a closed subset of , and then there exist , and we have

As in Theorem 9, we can prove that

That is, is a coupled coincidence point of and . Hence, . From -compatibility of , we have ; hence, and ; that is, and . Now, we shall show that and . Suppose, not. Then, by condition (2) of Theorem 9, we get

Letting in the above inequality, by using (9) and , we obtain

Since , and , therefore, by , we get

which is a contradiction. Thus, and . Hence, we have

Since , then there exist such that and . Now, by condition (2) of Theorem 9, ,  , and , we get

which, by and , implies

Thus,

That is, is a coupled coincidence point of and . Hence, . From -compatibility of , we have ; hence and ; that is, and . Now, we shall show that and . Suppose, not. Then, by condition (2) of Theorem 9 and , we get

which is a contradiction. Thus, and . Hence, we have

Therefore, is a common coupled fixed point of the pairs and . The proof is similar when (b) holds.

If we put in Theorem 15, we get the following result.

Corollary 16. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 10 and (2) of Corollary 10 and

(1) and are -compatible;(2)suppose that either

(a) is a closed subset of and or(b) is a closed subset of and .

Then , and   have a common coupled fixed point.

If we put and in Theorem 15, we get the following result.

Corollary 17. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 11 and (2) of Corollary 11 and

(1) is -compatible;(2) is a closed subset of and .

Then and have a common coupled fixed point.

Corollary 18. Let be a metric space. Assume and to be mappings satisfying (1) of Theorem 9, (1) of Corollary 12, (1) of Theorem 15, and (2) of Theorem 15; then ,  and have a common coupled fixed point.

Proof. If , then for all . If we divide condition (1) of Corollary 12 by 4 and take , then it reduces to condition (2) of Theorem 9 with and hence by Theorem 15 we get Corollary 18.

If we put in Corollary 18, we get the following result.

Corollary 19. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 10, (1) of Corollary 13, (1) of Corollary 16, and (2) of Corollary 16; then , and have a common coupled fixed point.

If we put and in Corollary 18, we get the following result.

Corollary 20. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 11, (1) of Corollary 14, (1) of Corollary 17, and (2) of Corollary 17; then and have a common coupled fixed point.

Theorem 21. Let be a metric space. Assume and to be mappings satisfying (2) of Theorem 9 and

(1) and are occasionally -compatible.

Then ,  and have a common coupled fixed point.

Proof. Since the pairs and are occasionally -compatible, therefore there exist some points ,, such that

It follows that

Now, we shall show that and . Suppose, not. Then, by condition (2) of Theorem 9 and , we have

which is a contradiction. Thus, and . Hence,

Thus, by (36), we get

Now, we shall show that and . Suppose, not. Then, by condition (2) of Theorem 9 and , we have

which is a contradiction. Thus,

Similarly, we can show that

Thus, by (39), (41), and (42), we get

That is, is a common coupled fixed point of ,  and .

Put in Theorem 21, and we get the following result.

Corollary 22. Let be a metric space. Assume and to be mappings satisfying (2) of Corollary 10 and

(1) and are occasionally -compatible.

Then ,  and have a common coupled fixed point.

Put and in Theorem 21, and we get the following result.

Corollary 23. Let be a metric space. Assume and to be mappings satisfying (2) of Corollary 11 and

(1) is occasionally -compatible.

Then and have a common coupled fixed point.

Corollary 24. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 12 and (1) of Theorem 21; then , and   have a common coupled fixed point.

Proof. If , then for all . If we divide condition (1) of Corollary 12 by 4 and take , then it reduces to condition (2) of Theorem 9 with and hence by Theorem 21 we get Corollary 24.

Put in Corollary 24, and we get the following result.

Corollary 25. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 13 and (1) of Corollary 22; then , and have a common coupled fixed point.

Put and in Corollary 24, and we get the following result.

Corollary 26. Let be a metric space. Assume and to be mappings satisfying (1) of Corollary 14 and (1) of Corollary 23; then and have a common coupled fixed point.

Example 27. Suppose that , equipped with the metric defined as and for all . Let be defined as

Suppose be defined as

Define by

and by

Now, for all with , we have
Case ( a). If , then

Case (b). If with , then

Similarly, we obtain the same result for . Thus, the contractive condition (2) of Theorem 9 is satisfied for all with . Again, for all with and , we have

Thus, the contractive condition (2) of Theorem 9 is satisfied for all with and . Similarly, we can see that the contractive condition (2) of Theorem 9 is satisfied for all with . Hence, the hybrid pairs and satisfy condition (2) of Theorem 9, for all . In addition, all the other conditions of Theorem 9, Theorem 15, and Theorem 21 are satisfied and is a common coupled fixed point of , and  .

Conflict of Interests

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

Acknowledgment

The authors thank the referee for the extremely careful reading and very deep and useful comments and suggestions that contributed to the improvement of the paper.

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