Tripled Coincidence and Common Fixed Point Results for Two Pairs of Hybrid Mappings

Abstract

The tripled fixed point is a generalization of the well-known concept of “coupled fixed point.” In this paper, tripled coincidence and common fixed point results for two hybrid pairs consisting of multivalued and single valued mappings on a metric space are proved. We give examples to illustrate our results. In the process, several comparable coincidence and fixed point results in the existing literature are improved, unified, and generalized.

1. Introduction and Preliminaries

The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Nadler Jr. [1]. After this, fixed point theory has been developed further and applied to many disciplines to solve functional equations. Banach contraction principle has been extended in different directions. Some authors used generalized contractions for multivalued mappings and hybrid pairs of single and multi-valued mappings, while others used more general spaces. Dhage [2, 3] established hybrid fixed point theorems and obtained some applications of presented results. Gnana Bhaskar and Lakshmikantham [4] introduced the notion of a coupled fixed point and proved some coupled fixed point results under certain contractive conditions in a complete metric space endowed with a partial order. They applied their results to study the existence and uniqueness of solution for a periodic boundary value problem associated with a first-order ordinary differential equation. Later, Lakshmikantham and Ćirić [5] established the existence of coupled coincidence point results to generalize the results of Gnana Bhaskar and Lakshmikantham [4]; Karapınar [6] generalized these results on a complete cone metric space endowed with a partial order. Recently, Berinde and Borcut [7, 8] introduced the concept of a tripled fixed point for nonlinear contractive mappings in partially ordered complete metric spaces and obtained tripled coincidence and fixed point results for commuting maps. Hussain et al. [9, 10] obtained some coupled and tripled coincidence results without compatibility. Ilić et al. [11] obtained coupled coincidence and common fixed point theorems for a hybrid pair of mappings. For other related results in this direction, we refer to [12–16] and references mentioned therein. The purpose of this paper is to obtain tripled coincidence and common fixed point results for two hybrid pairs consisting of multivalued and single valued mappings.

Let us recall some definitions and well known results needed in the sequel.

Let be a metric space. For and , we denote . The set of all nonempty bounded and closed subsets of is denoted by . Let be the Hausdorff metric induced by the metric on ; that is, for every .

Lemma 1 (see [1]). Let and . Then, for every , there exists such that

Lemma 2 (see [1]). Let and . Then, for every , there exists such that

Lemma 3 (see [1]). Let . If , then .

Definition 4. Let be a nonempty set, (collection of all nonempty subsets of and . An element is called (i) a tripled fixed point of if , , and (ii) tripled coincidence point of a hybrid pair if , and (iii) tripled common fixed point of a hybrid pair if , , and .

We denote the set of tripled coincidence point of a hybrid pair by . Note that if , then and are also in .

Definition 5. Let and . Then the hybrid pair is called -compatible if whenever .

Definition 6. Let and . The mapping is called -idempotent at some point if , , and .

2. Main Result

Theorem 7. Let be a metric space, and let be mappings such that for all , where , , are nonnegative real such that If and is complete subset of , then and have tripled coincidence point. Moreover and have tripled common fixed point if one of the following conditions holds:(i) and are w-compatible, , and for some , and is continuous at , , ;(ii)if , , and and is -idempotent for ;(iii) is continuous at , , for some and for some ; ; and and .

Proof. Let be arbitrary. Choose such that , and . Choose such that , , and . This can be done because . If then Imply that As and are closed, Similarly Hence and are tripled coincidence points of pairs and , respectively. Now assume that , for some which gives that ; therefore, there exist such that Since , there exist , , , , , and in such that , , , , , and . Thus Continuing this process, we obtain three sequences , , and in such that with
By (4), which further gives Similarly it can be shown that Again which implies Similarly, it can be shown that Let From (17) and (18), we get From (20) and (21) we get Adding (23) and (24), we obtain Since by inequality (5), we get Hence Then from (27), we get As , By the similar process as above, we can show that Thus we have Continuing this process, we obtain Similarly Continuation of this process implies that By (32) and (34), we have That is, holds true for all , where Now for every with , we have Since , we conclude that , , and are Cauchy sequences in . By completeness of , there exists such that ,, and . Then from (4), we get On taking limits as , we get which implies that And hence . Similarly , . And ,  , . Thus is a tripled coincidence point of and . Suppose that (i) holds; then, for some and , we have ,  , , and ,   ,. Since and are -compatible, we have for and . Since , and . So , , and . Similarly , , and . Continuing in this way, we get which implies for all and Since and for and and is continuous at , and , so we have , , and . Now using (4), we get On taking limits as , we get which implies that and hence Similarly, Consequently, Similarly, Hence is a tripled fixed point of and . Now suppose that (ii) holds. Since is , -idempotent for some and , we have , , , and , , . Since we have , and . So Hence is a tripled common fixed point of and . Now suppose that (iii) holds. For some and , we get Since is continuous at and , we get , , and . Thus This implies that is a tripled common fixed point of and .

If in Theorem 7,   ,  (the identity mapping), then we have the following result.

Corollary 8. Let be a metric space and such that for all , where , , are nonnegative real satisfy (5). Then has a fixed point.

Corollary 9. Let be a metric space and   and   be mappings such that for all , where . If and is complete subset of . Then and have tripled coincidence point. Moreover and have tripled common fixed point if anyone of the conditions (i)–(iii) of Theorem 7 holds.

Theorem 10. Let be a metric space and and be mappings such that for all , where . If and is complete subset of . Then and have tripled coincidence point. Moreover and have tripled common fixed point if one of the conditions (i)–(iii) of Theorem 7 holds.

Proof. Let be arbitrary. Choose such that , and . Note that , and are well defined. Choose such that , , and . If , then following similar arguments to those given in Theorem 7, we obtain that and are tripled coincidence point and , respectively. Now assume that , set . Then , so there exists such that Since , there exist , , , , , and in such that , , , , , and . Thus Continuing this process, we obtain sequences , , and in as ,   , , and , , such that
From (58), we have Hence, if we suppose that , then Therefore, Similarly, we obtain Using (65) and (66), we obtain for all where .Thus for with . Hence we conclude that is a Cauchy sequence in . Similarly, we obtain that and are Cauchy sequences in . Since is complete, there exists such that , , and . Thus from (58), On taking limit as , we get which implies that As is closed, . Similarly, , . Therefore is a tripled coincidence point of and . Similarly, is a tripled coincidence point of and . Thus is a tripled coincidence point of and . Suppose that (i) holds; then, for some and , we have , , and , , . Since and are w-compatible, we have for and . Since , , and . So , , and   . Similarly , and , . Continuing in this way, we get for all and And ,   and for and . Since is continuous at , and , we have , and . Now using (58), we get Taking limit as , we get Hence and therefore Similarly, Consequently, Similarly, Hence is a tripled common fixed point of and . Now suppose that (ii) holds. Since is , -idempotent for some and , we have , , , and , , . Since we have , and . So, Hence is a tripled common fixed point of and . Now suppose that (iii) holds. For some and , we get Since is continuous at and , we get , , and . Thus This implies that is a tripled common fixed point of and .

Conclusion. In this paper, we established tripled coincidence and common fixed point results for a pair of hybrid mappings in the context of complete metric spaces. These results are extensions of results in [4, 5, 11] to the case tripled coincidence and common fixed points. Our results may be the motivation to other authors for extending and improving these results to be suitable tools for their applications.

Conflict of Interests

The authors declare that they have no competing interests.

Authors' Contribution

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

Acknowledgment

Marwan Amin Kutbi gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.