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The Scientific World Journal
Volume 2015 (2015), Article ID 938165, 7 pages
http://dx.doi.org/10.1155/2015/938165
Research Article

Fixed Point Theorems for Hybrid Mappings

1School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan
2Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
3Department of Mathematics, Incek, 06586 Ankara, Turkey
4Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia

Received 3 July 2014; Accepted 23 July 2014

Academic Editor: Kishin Sadarangani

Copyright © 2015 Maria Samreen 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 obtain some fixed point theorems for two pairs of hybrid mappings using hybrid tangential property and quadratic type contractive condition. Our results generalize some results by Babu and Alemayehu and those contained therein. In the sequel, we introduce a new notion to generalize occasionally weak compatibility. Moreover, two concrete examples are established to illuminate the generality of our results.

1. Introduction and Preliminaries

Throughout this paper is a metric space with metric . For and , . We denote by the class of all nonempty closed subsets of and by the class of all nonempty bounded closed subsets of . For every , let Such a map is called generalized Hausdorff metric induced by . Notice that is a metric on . A point is said to be a fixed point of if . The point is called a coincidence point of and if . The set of coincidence points of and is denoted by . If and are both self-maps on . The point is called a coincidence point of and if . A pair is known as hybrid pair where and .

1.1. Compatibility and Property

Sessa [1] introduced the concept of weakly commuting maps. Jungck [2] defined the notion of compatible maps in order to generalize the concept of weak commutativity and showed that weakly commuting maps are compatible but the converse is not true [2]. Pant [36] initiated the study of noncompatible maps. Sastry and Krishna Murthy [7] defined the notion of tangential single-valued maps. Aamri and El Moutawakil [8] rediscovered the notion of tangential maps and named it as property . The class of maps satisfying property has remarkable property that it contains the class of compatible maps as well as the class of noncompatible maps [8]. Kamran [9] extended the notion of property to a hybrid pair. Liu et al. [10] defined common property for two hybrid pairs. Kamran and Cakic [13] introduced the hybrid tangential property and showed that it properly generalizes the notion of common property   [22, Example ]. In [11], the authors discussed fixed point theory problems in the context of -metric space. Furthermore, in [11] the authors investigated the existence of a fixed point for multivalued mappings of integral type employing strongly tangential property (see also [1216]).

For the sake of completeness, we recall some basic definitions and results.

Definition 1. Let and be self-maps on . The pair is said to (i)be compatible [2] if , whenever is a sequence in such that , for some ;(ii)be noncompatible if there is at least one sequence in such that , for some , but is either nonzero or nonexistent;(iii)satisfy property [8] if there exists a sequence in such that , for some .

Definition 2. Let , be self-maps on and let , be multivalued maps from to . (i)The maps and are said to be compatible [17] if for all and whenever is a sequence in such that and .(ii)The maps and are noncompatible if for all and there exists at least one sequence in such that and but or is nonexistent.(iii)The maps and are said to satisfy property [9] if there exists a sequence in , some , and such that .(iv)The hybrid pairs and are said to satisfy common property [10] if there exist two sequences , in , some , and such that , , and .(v)The hybrid pair is said to be -tangential at [13] if there exist two sequences , in , such that and .

1.2. Weak Compatibility and Weak Commutativity

Jungck [18] introduced the notion of weak compatibility and in [19] Jungck and Rhoades further extended weak compatibility to a hybrid pair of single-valued and multivalued maps. Singh and Mishra [20] introduced the notion of -commutativity for a hybrid pair to generalize the notion of weak compatibility. Kamran [9] introduced the notion of -weak commutativity and showed that -commutativity implies -weak commutativity but the converse is not true in general [9, Example 3.8]. Al-Thagafi and Shahzad [21] introduced the class of occasionally weakly compatible single-valued maps and showed that the weakly compatible maps form a proper subclass of the occasionally weakly compatible maps [21, Example]. Abbas and Rhoades [23] generalized the notion of occasionally weak compatibility to a hybrid pair.

Definition 3. Let and be self-maps on . The pair is said to(iv)be weakly compatible [18] if whenever , ;(v)be occasionally weakly compatible (owc) [21] if for some .

Definition 4. Let be a self-map on and from to . (i)The maps and are weakly compatible [19] if they commute at their coincidence points, that is, whenever .(ii)The maps and are said to be -commuting [20, 22] at if .(iii)The map is said to be -weakly [9] commuting at if .(iv)The maps and are said to be occasionally weakly compatible [23] if and only if there exists some point such that and .
Recently, Babu and Alemayehu [24] obtained some fixed point theorems for single-valued mappings using property , common property , and occasionally weak compatibility. The purpose of this paper is to extend the main results of [24] to hybrid pairs. We also introduce a new notion for a hybrid pair that generalizes occasionally weak compatibility.

2. Main Results

We begin with the following proposition.

Proposition 5. Let be a metric space, let , be self-maps on , and let , be mappings from to such that for all , where and . Suppose that either (I), the pair satisfies property and is closed subspace of , or(II), the pair satisfies property and is closed subspace of .
Then and .

Proof. Suppose that (I) holds; then there exists a sequence in and such that Since then for all . Now for we have Now by using the definition of Hausdorff metric, we have Applying limit throughout we have which infers that . Therefore, there exists a sequence in such that . Consider the following: Since is closed, there exists such that We claim that . From (25) we get Using (3) and (7) we get Since , it follows that and hence Now we show that . Using (25) we have Letting and using (3), (7), (8), (11), and definition of Hausdorff metric the above inequality yields Since , using closedness of , it follows that Since , there exists such that Now we show that ; from (25), (14), and (15) we have Since , closedness of implies . Similarly, the assertion of proposition holds if we assume (II).

Remark 6. Note that if is a self-map on , Proposition 5 reduces to [24, Proposition 2.1].

Now we introduce the notion of occasionally weak commutativity.

Definition 7. Let be a hybrid pair. The mapping is said to be occasionally -weakly commuting if and only if there exists some such that and .

Note that if a hybrid pair is occasionally weakly compatible at then is occasionally -weakly commuting at . The following example shows that the converse of the above statement is not true.

Example 8. Let with the usual metric. Define , by and for all . Then for all , , , and . Therefore is occasionally weakly compatible at any .

Our next result extends [24, Theorem 2.2] to hybrid pairs. Note that in the hypothesis of our result we assumed that hybrid pairs satisfy occasionally weak commutativity rather than using the notion of occasionally weak compatibility.

Theorem 9. In addition to the hypothesis of Proposition 5 on , , , and , (i)if is occasionally -weakly commuting at and then and have a common fixed point;(ii)if is occasionally -weakly commuting at and then and have a common fixed point;(iii), , , and have a common fixed point if both (i) and (ii) hold.

Proof. By (i), we have and . Thus . This proves (i). (ii) can be proved on the same lines; then (iii) is immediately followed.

Example 10. Let with the usual metric. Define mappings and by We observe that , is closed, and is open; neither nor . There exists a sequence ; ,    in with , so that the hybrid pair satisfies property but it is not compatible. Inequality (25) is satisfied for , , and . Also note that is occasionally -weakly commuting at point and is occasionally -weakly commuting at each point in the interval . Furthermore (i), (ii), and (iii) of Theorem 9 are also satisfied at point . Hence , , , and have common fixed point .

In the next result we will use the notion of hybrid tangential property and occasionally weak commutativity to extend and improve [24, Proposition 2.5].

Theorem 11. Let be a metric space, let , be self-maps on , and let , be mappings from to satisfying inequality (25). Assume , are closed subspaces of X and further suppose that either (I) is -tangential or(II) is -tangential.
Then and . Furthermore,(i)if is occasionally -weakly commuting at and then and have a common fixed point;(ii)if is occasionally -weakly commuting at and then and have a common fixed point;(iii), , , and have a common fixed point if both (i) and (ii) hold.

Proof. Suppose that hybrid pair is -tangential; then there exist sequences and in such that Now we prove that ; from (25) we have On taking limit and using (18), we get which implies ; hence . Since and are closed there exists such that The rest of the proof runs on the same lines as that of Proposition 5 and Theorem 9.

Corollary 12. Let be a metric space, let , be self-maps on and , and let be mappings from to satisfying inequality (25) of Proposition 5. Suppose that pairs    satisfy common property and , are closed subsets of X; then and . Furthermore, (i)if is occasionally -weakly commuting at and then and have a common fixed point;(ii)if is occasionally -weakly commuting at and then and have a common fixed point;(iii), , , and have a common fixed point if both (i) and (ii) hold.

Remark 13. If and are self-maps on then Corollary 12 coincides with [24, Proposition 2.5].

Example 14. Let with the usual metric. Define mappings and by In this example and are closed subspaces of ; neither nor . There exists a sequence ; , in with , where . Hence and satisfy common property . It can be easily shown that the hybrid pairs and satisfy inequality (25) with , , and . Furthermore, is occasionally -weakly commuting at point while is occasionally -weakly commuting at each point in the interval . Conditions (i), (ii), and (iii) of Corollary 12 hold true for ; so , , , and have common fixed point .

In the following we include some of the consequences of Theorem 9.

Corollary 15. Let be a metric space, let , be self-maps on , and let be mappings from to such that for all , where and . Suppose that either (I), the pair satisfies property and is closed subspace of , or(II), the pair satisfies property and is closed subspace of .
Then and . Furthermore (i)if is occasionally -weakly commuting at and then and have a common fixed point;(ii)if is occasionally -weakly commuting at and then and have a common fixed point;(iii), , and have a common fixed point if both (i) and (ii) hold.

Proof. Take in Theorem 9.

Corollary 16. Let be a metric space, let be a self-map on , and let be a mapping from to such that for all , where and . Suppose that : the pair satisfies property and is closed subspace of . Then . Furthermore if is occasionally -weakly commuting at and then and have a common fixed point.

Proof. Take and in Theorem 9.

Corollary 17. Let be a metric space, let be a self-map on , and let be a mapping from to such that for all , where and . Suppose is closed and the pair satisfies property , where is an identity map on . Then has a fixed point.

Proof. Take and in Theorem 9.

Corollary 18 (see [24, Theorem 2.2]). Let , , , and be four self-maps on a complete metric space satisfying the inequality for all , where and . Suppose that either (i), the pair satisfies property and is closed subspace of , or(ii), the pair satisfies property and is closed subspace of , holds.
Then and . Furthermore if both the pairs and are occasionally weakly compatible on , then the maps , , , and have a unique common fixed point in .

Proof. Take and in Theorem 9. Moreover, uniqueness of fixed point is followed from inequality (26) as .

Corollary 19. Let be a metric space and let , be self-maps on such that for all , where and . Suppose that : the pair satisfies property and is closed subspace of . Then . Furthermore if the pair is occasionally weakly compatible, then and have a unique common fixed point.

Proof. Take and in Theorem 9. Moreover, uniqueness of fixed point is followed from inequality (27) as .

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.

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