Abstract
The object of this paper is to emphasize the role of a suitable implicit relation involving altering distance function which covers a multitude of contraction conditions in one go. By using this implicit relation, we prove a new coincidence and common fixed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings in a metric space employing the common limit range property. Our main result improves and generalizes a host of previously known results. We also utilize suitable illustrative examples to substantiate the realized improvements in our results.
1. Introduction and Preliminaries
Fixed point theory is one of the most rapidly growing research areas in nonlinear functional analysis. Apart from numerous extensions of Banach Contraction Principle for single valued mappings, it was also naturally extended to multivalued mappings by Nadler Jr. [1] in 1969 which is also sometimes referred to as Nadler Contraction Principle. Since then, there has been continuous and intense research activity in multimap fixed point theory (including hybrid fixed point results) and by now there exists an extensive literature on this specific theme (see, e.g., [2–7] and the references therein). The study of common fixed points of mappings satisfying hybrid contraction conditions has been at the center of vigorous research activity. Here, it can be pointed out that hybrid fixed theorems have numerous applications in science and engineering.
In the following lines, we present some definitions and their implications which will be utilized throughout this paper.
Let be a metric space. Then, on the lines of Nadler Jr. [1], we adopt that(1) is a nonempty closed subset of ,(2) is a nonempty closed and bounded subset of ,(3)for nonempty closed and bounded subsets of and ,
It is well known that is a metric space with the distance which is known as the Hausdorff-Pompeiu metric on . The following terminology is also standard.
Let be a metric space with and . Then(1)a point is a fixed point of (resp., ) if (resp., ). The set of all fixed points of (resp., ) is denoted by (resp., );(2)a point is a coincidence point of and if . The set of all coincidence points of and is denoted by ;(3)a point is a common fixed point of and if . The set of all common fixed points of and is denoted by .
In 1984, Khan et al. [8] utilized the idea of altering distance function in metric fixed point theory which is indeed a control function that alters distance between two points in a metric space. Thereafter, this idea has further been utilized by several mathematicians (see, e.g., [9–13]).
Definition 1 (see [8]). An altering distance function is a mapping which satisfies that is increasing and continuous and if and only if .
Certain ideas on commutativity and weak commutativity for a pair of hybrid mappings on metric spaces were utilized by Kaneko [14, 15]. In 1989, Singh et al. [16] extended the notion of compatible mappings to hybrid pair of mappings and proved some common fixed point theorems for nonlinear hybrid contractions. Such ways of proving new results continue to attract the attention of many researchers of this domain where it can be observed that under compatibility the fixed point results often require continuity of one of the underlying mappings.
Kamran [17] extended the property (E.A) (due to Aamri and El Moutawakil [18]) to a hybrid pair of mappings. Most recently, Imdad et al. [19] established common limit range property (essentially motivated by Sintunavarat and Kumam [20]) for a hybrid pair of mappings and proved some fixed point theorems in symmetric (or semimetric) spaces.
The notions of coincidentally idempotent and occasionally coincidentally idempotent hybrid pairs of mappings were, respectively, introduced and used by Imdad et al. [21] and Pathak and Rodríguez-López [22]. An easy and natural example is available in Kadelburg et al. [23, Example 1] exhibiting the importance of the occasionally coincidentally idempotent property over coincidentally idempotent property.
The technical definitions of the earlier mentioned notions are described in the following lines.
Definition 2. Let be a metric space with and . A hybrid pair of mappings is said to be(1)commuting on [14] if for all ,(2)weakly commuting on [15] if for all ,(3)compatible [16] if for all and , whenever is a sequence in such that and , as ,(4)noncompatible [24] if there exists at least one sequence in such that and , as , but is either nonzero or nonexistent,(5)with the property (E.A) [17] if there exists a sequence in such that
for some and ,(6)with common limit range property with respect to the mapping [19] if there exists a sequence in such that
for some and ,(7)coincidentally idempotent [21] if for every ; that is, is idempotent at the coincidence points of and, , and(8)occasionally coincidentally idempotent [22] if for some .
Some relations between the introduced notions can be seen in [25, 26].
Remark 3. 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 (see Theorem 7). However, common limit range property may be satisfied without the closedness of (e.g., Example 6).
In this paper, an attempt has been made to derive common fixed point theorems for a hybrid pair of mappings using the notion of common limit range property with occasionally coincidentally idempotent property involving implicit relations and altering distance. The presented theorems extend and unify various known fixed point results [21, 23, 24, 27–37]. Some examples are also furnished to exhibit that our results are proper extensions of the known ones.
2. Implicit Relations
In the recent past, implicit relations have been utilized to prove unified common fixed points results covering various kinds of contraction mappings in one go. In fact, this idea was initiated by Popa [38, 39], where he introduced an implicit function which covers a variety of contraction classes. In [33], Popa and Patriciu introduced the following implicit function and utilized the same to prove some coincidence and common fixed point results for hybrid pair of mappings covering several contraction conditions.
In what follows, will be the set of all continuous functions satisfying the following conditions: is nondecreasing in its first variable; implies .
Example 4 (see [33]). The following functions are the examples of implicit function belonging to the set : (1)(2) where ,(3) where and ,(4) where and ,(5) where and ,(6) where and ,(7) where ,(8) where , and ,(9) where and ,(10) where ,(11) where and , and(12)
Certainly, apart from foregoing examples, there are many other functions that meet the requirements and .
3. Main Results
Now we prove our main result.
Theorem 5. Let be a self-mapping of a metric space and a mapping from into satisfying
for all , where and is an altering distance function. Suppose that the pair satisfies the common limit range property with respect to the mapping . Then the mappings and have a coincidence point (i.e., ).
Moreover, the mappings and have a common fixed point in provided that the pair is occasionally coincidentally idempotent.
Proof. Suppose that the pair enjoys the common limit range property with respect to the mapping . Then there exists a sequence in such that
for some and . First we show that . To accomplish this, using inequality (16) with and , we obtain
Taking the limit as , we have
Since , we have . Using in inequality (19), we have
or, equivalently,
From the condition , we have which implies . Hence which shows that the pair has a coincidence point (i.e., ) so that the set of coincidence points is nonempty.
In the case that the mappings and are occasionally coincidentally idempotent; then for some , which implies . Now we assert that . On using inequality (16) with and , we get
Since , we have . Using in inequality (22), we get
or, equivalently,
In view of , we have which implies ; that is, . Thus in all, we have which shows that is a common fixed point of the mappings and .
Example 6. Consider equipped with the standard metric. Define mappings and as follows:
Then(i) is not closed in ;(ii);(iii)for , it is ; hence, enjoys the common limit range property with respect to the mapping ;(iv); hence, is occasionally coincidentally idempotent;(v); hence, is not coincidentally idempotent.
Define (which is an altering distance function), while is given by
(see Example 4 (10) with ). Write
In order to check the contractive condition (16) of Theorem 5, without loss of generality, we can suppose that . Then . Consider the following possible cases.(1)For the case , we have
(2)In the case , one finds
(3)If , then one can show on the lines of case (1) that .(4)If , then , , and
(5)If , , then
(6)The case is similar to the case (2).
Thus, all the conditions of Theorem 5 are satisfied and the pair has common fixed points (which are and ).
The same conclusion cannot be reached using [35, Theorem 3.1] or [33, Theorem 4.1], as is not closed and is not coincidentally idempotent.
Theorem 7. Let be a self-mapping of a metric space and a mapping from into satisfying inequality (16) of Theorem 5. Suppose that the pair satisfies the property (E.A) and is a closed subset of ; then the mappings and have a coincidence point (i.e., ).
Moreover, the mappings and have a common fixed point in provided that the pair is occasionally coincidentally idempotent.
Proof. If the pair enjoys the property (E.A), then there exists a sequence in such that for some and . Since is a closed subset of , there exists some such that . Hence condition (32) implies for some and which shows that the pair also satisfies the common limit range property with respect to the mapping . Now, the conclusions follow from Theorem 5.
Example 8. Consider equipped with the standard metric. Define mappings and as
Then(i) is closed in ;(ii);(iii)for , it is ; hence, enjoys the property (E.A);(iv); hence, is occasionally coincidentally idempotent;(v); hence, is not coincidentally idempotent.
Take (which is an altering distance function) and given by
(see Example 4(3) with , ). Denote
In order to check the contractive condition (16) of Theorem 5, without loss of generality, we can suppose that . Consider the following possible cases.(1)Consider . Then
(2)Consider . Then .(3)Consider . Then , , , and
Hence, all the conditions of Theorem 7 are fulfilled and the pair has a common fixed point (which is ).
The same conclusion cannot be obtained using, for example, [35, Theorem 3.1] or [33, Theorem 4.1], since is not coincidentally idempotent.
Notice that a noncompatible hybrid pair always satisfies the property (E.A). Hence, we get the following corollary.
Corollary 9. Let be a self-mapping of a metric space and a mapping from into satisfying inequality (16) of Theorem 5. Suppose that the pair is noncompatible and is a closed subset of . Then the mappings and have a coincidence point (i.e., ).
Moreover, the mappings and have a common fixed point in provided that the pair is occasionally coincidentally idempotent.
Corollary 10. The conclusions of Theorems 5 and 7 and Corollary 9 remain true if inequality (16) is replaced by one of the following contraction conditions. For all and some ,
(1)
(2)
where ,
(3)
where and ,
(4)
where and ,
(5)
where and ,
(6)
where and ,
(7)
where ,
(8)
where , , and ,
(9)
where and ,
(10)
where ,
(11)
where and ,
(12)
Proof. The conclusion follows from Theorem 5 in view of Example 4, (1)–(12).
On setting in the earlier defined theorems involving altering distance, we can get some natural results which improve hybrid type fixed point results given in the literature. For the sake of simplicity, we only derive the following corollary by putting in Theorem 5.
Corollary 11. Let be a self-mapping of a metric space and a mapping from into satisfying
for all and some . Suppose that the pair satisfies the common limit range property with respect to the mapping . Then the mappings and have a coincidence point (i.e., ).
Moreover, the mappings and have a common fixed point in provided that the pair is occasionally coincidentally idempotent.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the referees for careful reading of the text and useful remarks that helped them to improve it. The third author (Professor Zoran Kadelburg) is grateful to the Ministry of Education, Science and Technological Development of Serbia.