A Hybrid Common Fixed Point Theorem under Certain Recent Properties

We prove a common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings via common limit range property. Our result improves some results from the existing literature, especially the ones contained in Sintunavarat and Kumam (2009). Some illustrative and interesting examples to highlight the realized improvements are also furnished.


Introduction and Preliminaries
Nadler's contraction principle [1] is the generalization of the classical Banach contraction principle to the case of multivalued mappings. Hybrid fixed point theory for singlevalued and multivalued mappings is a new development in nonlinear analysis (see, e.g., [2][3][4][5][6][7] and references therein). The concepts of commutativity and weak commutativity were extended to multivalued mappings on metric spaces by Kaneko [8,9]. In 1989, Singh et al. [10] extended the notion of compatible mappings and obtained some coincidence and common fixed point theorems for nonlinear hybrid contractions. It was observed that under compatibility the fixed point results always require continuity of one of the underlying mappings. Afterwards, Pathak [11] generalized the concept of compatibility by defining weak compatibility for hybrid pairs of mappings (including single-valued case) and utilized the same to prove common fixed point theorems. Naturally, compatible mappings are weakly compatible but not conversely. For an extensive collection of hybrid contraction conditions, we refer to [12][13][14][15][16][17][18][19][20][21].
The following definitions and results will be needed in the sequel.
Let ( , ) be a metric space. Then, on the lines of Nadler [1], we adopt that (1) ( ) = { : is a nonempty closed subset of }, It is well known that ( ) is a metric space with the distance which is known as the Hausdorff-Pompeiu metric on ( ).
Obviously, in this case ( , ) is also noncompatible and occasionally weakly compatible, but simple modification of this example shows that the occasionally coincidentally idempotent property is independent of these two notions. For example, if then the pair ( , ) is occasionally coincidentally idempotent but not occasionally weakly compatible.

Remark 4.
It was shown in [25] that, in the case of singlevalued mappings, the occasionally weakly compatible property does not produce new common fixed point results, since it reduces to weak compatibility in the presence of a unique point of coincidence. However, [25,Example 2.5] shows that an analogue conclusion does not hold in the case of hybrid pairs. Hence, in this case, the notion of occasionally weak compatibility might still produce new results. A similar example can be presented, showing the possibility of usage of the occasionally coincidentally idempotent notion.
Inspired by the work of Aamri and Moutawakil [26], Kamran [27] extended the notion of property (E.A) for a hybrid pair of mappings.
In 2011, Sintunavarat and Kumam [28] introduced the notion of common limit range property for single-valued mappings and showed its superiority over property (E.A). Motivated by this fact, Imdad et al. [29] established common limit range property for a hybrid pair of mappings and proved some fixed point results in symmetric (semimetric) spaces.
Definition 6 (see [29]). Let ( , ) be a metric space with : → and : → ( ). Then the hybrid pair of mappings ( , ) is said to satisfy the common limit range property with respect to the mapping if there exists a sequence { } in such that for some ∈ X and ∈ ( ).
One can verify that the pair ( , ) enjoys the property (E.A) as, considering the sequence { } = {1 − 1/ } ∈N , one gets that However, there exist no in for which = .
Example 8. In the setting of Example 7, replace the mapping by the following, besides retaining the rest: Then the pair ( , ) satisfies the common limit range property with respect to the mapping (for the sequence { } = {1 − 1/ } ∈N ) as Remark 9. Note that, if a pair ( , ) satisfies the property (E.A) along with the closedness of ( ), then the pair also satisfies the common limit range property with respect to the mapping .
The aim of this note is to prove a common fixed point theorem for a hybrid pair of mappings by using the notion of common limit range property (due to Imdad et al. [29]) along with occasionally coincidentally idempotent property (due to Pathak and Rodriguez-Lopez [24]).

Main Result
We first state the following theorem due to Sintunavarat and Kumam [30] proved for a pair of hybrid pair of mappings by using the property (E.A).

Theorem 10 (see [30, Theorem 3.1]). Let be a self mapping of a metric space ( , ) and let be a mapping from into
( ) such that the following conditions are satisfied: (1) and satisfy the property (E.A), If ( ) is a closed subset of , then and have a common fixed point. Now we utilize the occasionally coincidetally idempotent notion (which is weaker than coincidentally idempotent one in the case when the set of coincidence points is not empty; see Example 3) and common limit range property (instead of property (E.A)) and prove a respective result without any requirement of closedness of the range of . Proof. Suppose that the pair ( , ) satisfies the common limit range property with respect to the mapping ; then there exists a sequence { } in such that for some ∈ and ∈ ( ). We assert that ∈ . If not, then, using condition (10), we get Taking the limit as → ∞, we have 4 The Scientific World Journal Since ∈ , the above inequality implies that which is a contradiction. Hence ∈ which shows that the pair ( , ) has a coincidence point (i.e., ( , ) ̸ = 0). If the mappings and are occasionally coincidentally idempotent, two cases arise: and may be or may not be coincidentally idempotent at .
Case I. If and are coincidentally idempotent at , then we have = ∈ . Now we show that = .
If not, using condition (10), we get Since ∈ , the above inequality implies that which is a contradiction. Thus we have = ∈ = which shows that is a common fixed point of the mappings and .
Hence, all the conditions of Theorem 11 are fulfilled and the pair ( , ) has a common fixed point (which is 0). The same conclusion cannot be obtained using Theorem 10 since ( ) is not closed and ( , ) is not coincidentally idempotent.
In view of Remark 9, we have the following natural result which still improves the results of Sintunavarat and Kumam [30] as the notion of occasionally coincidentally idempotent is more general than coincidentally idempotent.

Corollary 13.
Let be a self mapping of a metric space ( , ) and let be a mapping from into ( ) satisfying condition (10) of Theorem 10. Suppose that the pair ( , ) satisfies the property (E.A) along with the closedness of ( ). Then the mappings and have a coincidence point (i.e., ( , ) ̸ = 0). Moreover, if the pair ( , ) enjoys occasionally coincidentally idempotent property (i.e., V = V for some V ∈ ( , )), then the pair ( , ) has a common fixed point.
Notice that a noncompatible hybrid pair always satisfies the property (E.A). Hence, we get the following corollary.

Corollary 14.
Let be a self mapping of a metric space ( , ) and let be a mapping from into ( ) satisfying condition (10) of Theorem 10. Suppose that the pair ( , ) is noncompatible and ( ) is a closed subset of . Then the mappings and have a coincidence point (i.e., ( , ) ̸ = 0). Moreover, if the pair ( , ) enjoys occasionally coincidentally idempotent property (i.e., V = V for some V ∈ ( , )), then the pair ( , ) has a common fixed point.