Fixed Point Theorems for Hybrid Mappings

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.


Introduction and Preliminaries
Throughout this paper is a metric space with metric . For ∈ and ⊆ , ( , ) = inf{ ( , ) : ∈ }. We denote by CL( ) the class of all nonempty closed subsets of and by CB( ) the class of all nonempty bounded closed subsets of . For every , ∈ CL( ), let if the maximum exists ∞, otherwise. (1) Such a map is called generalized Hausdorff metric induced by . Notice that is a metric on CB( ). A point ∈ is said to be a fixed point of : → CL( ) if ∈ . The point is called a coincidence point of : → and : → CL( ) 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 : → CL( ). ( . ). 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 [3][4][5][6] 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 2.3].

Compatibility and Property
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 [12][13][14][15][16]).
For the sake of completeness, we recall some basic definitions and results.
Definition 2. Let , be self-maps on and let , be multivalued maps from to CL( ).
(i) The maps and are said to be compatible [17] if ∈ CL( ) for all ∈ and ( , ) → 0 whenever { } is a sequence in such that → ∈ CL( ) and → ∈ .
(ii) The maps and are noncompatible if ∈ CL( ) for all ∈ and there exists at least one sequence { } in such that → ∈ CL( ) and → ∈ but lim → ∞ ( , ) ̸ = 0 or is nonexistent.

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 4.
Let be a self-map on and from to CL( ).
(i) The maps and are weakly compatible [19] if they commute at their coincidence points, that is, = whenever ∈ .
(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.

Main Results
We begin with the following proposition. Proposition 5. Let ( , ) be a metric space, let , be selfmaps on , and let , be mappings from to CL( ) such that for all , ∈ , where 1 , 2 , 3 ≥ 0 and 1 < 1. Suppose that either Proof. Suppose that (I) holds; then there exists a sequence { } in and ∈ CL( ) such that The Scientific World Journal 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 lim → ∞ = . Consider the following: Since is closed, there exists ∈ such that We claim that lim → ∞ = . From (25) we get Using (3) and (7) we get Since 1 < 1, it follows that lim → ∞ ( , ) = 0 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 1 < 1, using closedness of , it follows that ∈ .
Now we show that ∈ ( , ); from (25), (14), and (15) we have Since 1 < 1, closedness of implies ∈ . Similarly, the assertion of proposition holds if we assume (II). 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. 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. (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.
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]. 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 [ ( , )] = 0; 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 16. Let ( , ) be a metric space, let be a self-map on , and let be a mapping from to CL( ) such that for all , ∈ , where 1 , 2 , 3 ≥ 0 and 1 < 1.