Abstract and Applied Analysis

Volume 2014 (2014), Article ID 803729, 11 pages

http://dx.doi.org/10.1155/2014/803729

Research Article

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

Marwan Amin Kutbi,¹ Jamshaid Ahmad,² Mujahid Abbas,³ and Muhammad Arshad⁴

¹Department of Mathematics, King Abdul Aziz University, Jeddah, Saudi Arabia

²Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Islamabad 44000, Pakistan

³Department of Mathematics & Applied Mathematics, University of Pretoria, Pretoria 002, South Africa

⁴Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan

Received 26 August 2013; Accepted 24 December 2013; Published 30 January 2014

Academic Editor: Patricia J. Y. Wong

Copyright © 2014 Marwan Amin Kutbi 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.

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.

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