Abstract
Some common fixed point theorems satisfying contractive conditions involving rational expressions and product for four mappings that satisfy property (E.A) along with weak compatibility of pairs are proved and further some results using (CLR)-property are obtained in complex valued metric spaces which generalize various results of ordinary metric spaces.
1. Introduction
Fixed point theory is one of the fundamental theories in nonlinear analysis which has various applications in different branches of mathematics. In this theory, to prove the existence and the uniqueness of a fixed point of operators or mappings has been a valuable research area by using the Banach contraction principle. There are many generalizations of the Banach contraction principle particularly in metric spaces. So, many researches attempted various generalizations of the concept of metric spaces such as 2-metric spaces, D-metric spaces, G-metric spaces, K-metric spaces, cone metric spaces, and probabilistic metric space.
Recently, Huang and Zhang [1] generalized the concept of metric spaces, replacing the set of real numbers by an ordered Banach space; hence they have defined the cone metric spaces. They also described the convergence of sequences and introduced the notion of completeness in cone metric spaces. They have proved some fixed point theorems of contractive mappings on complete cone metric space with the assumption of normality condition of a cone. Subsequently, various authors have generalized the results of Huang and Zhang and have studied fixed point theorems in cone metric spaces over normal and nonnormal cones.
Many results of analysis cannot be generalized to cone metric since the definition of these spaces is based on a Banach space which is not a division ring. So, in a recent time, Azam et al. [2] introduced and studied the notion of complex valued metric space and established some common fixed point theorems for mappings involving rational expressions which are not meaningful in cone metric spaces. Later, several authors have studied the problem of existence of uniqueness of a fixed point for mappings satisfying different type contractive conditions in the framework of complex valued metric spaces.
In 2002, Aamri and Moutawakil [3] introduced the property (E.A) and pointed out that this property buys containment of ranges without any continuity requirements besides minimizing the commutativity conditions of the maps to the commutativity at their points of coincidence. Further, property (E.A) allows replacing the completeness condition of the space with a natural condition of closeness of the range. Subsequently, there are numerous papers which contain fixed point results related to property (E.A) in various metric spaces. Most recently, Sintunavarat and Kumam [4] defined the notion of the (CLR)-property (or common limit in the range property) which does not impose either completeness of the whole space or any of the range spaces or continuity of maps. The importance of this property ensures that one does not require the closeness of the range of subspaces. Various fixed point theorems have been proved by using the notion of (CLR)-property (see [4–15]).
The aim of this paper is to establish common fixed point theorems for two pairs of weakly compatible self-mappings of a complex valued metric space satisfying contractive condition involving product and rational expressions using (E.A) property. Moreover, we give some results using the property common limit in the range of one of the mappings.
2. Basic Facts and Definitions
We recall some notations and definitions which will be utilized in our subsequent discussion.
Let be a set of complex numbers and . Define a partial order on as follows: It follows that if one of the following conditions is satisfied: (i),(ii),(iii),(iv).
In (i), (ii), and (iii), we have . In (iv), we have . So . In particular, we will write if and one of (i), (ii), and (iii) is satisfied. In this case . We will write if and only if (iii) is satisfied.
Take into account some fundamental properties of the partial order on as follows.(i)If , then .(ii)If , then .(iii)If and is a real number, then .
Definition 1 (see [12]). The “max” function for the partial order relation “” is defined by the following. (i) if and only if .(ii)If , then , or .(iii) if and only if or .
Using Definition 1 one can have the following lemma.
Lemma 2 (see [12]). Let and the partial order relation is defined on . Then the following statements are easy to prove. (i)If , then if .(ii)If , then if .(iii)If , then if .
Now we give the definition of complex valued metric space which has been introduced by Azam et al. [2].
Definition 3. Let be a nonempty set. If a mapping satisfies (C1) for all and , (C2) for all , (C3) , for all ,
then is called a complex valued metric on and the pair is called a complex valued metric space.
Let be a sequence in a complex valued metric space and . If for every with there is such that, for all , , then is called the limit of and is written as as . If for every with there is such that, for all , , then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete complex valued metric space.
The following lemma has been given in [2] that we utilize to prove the theorems.
Lemma 4. Let be a complex valued metric space and a sequence in . Then (i) converges to if and only if as ;(ii) is a Cauchy sequence if and only if as .
Definition 5 (see [14]). A pair of self-mappings is called weakly compatible if they commute at their coincidence point; that is, if there is a point such that , then , for each .
The definition of property (E.A) has been introduced by Aamri and Moutawakil in [3] and redefined by Verma and Pathak [12] in complex valued metric spaces.
Definition 6. Let be two self-mappings of a complex valued metric space . The pair is said to satisfy property (E.A), if there exists a sequence in such that for some .
Example 7. Let be endowed with the complex valued metric as where and . Then is a complete complex valued metric space. Define the mappings as , for all and consider the sequence . Thus we obtain where is the limit of sequence . Hence the pair satisfies property (E.A).
Definition 8 (see [15]). Let and be two self-mappings of complex valued metric space . and are said to satisfy the common limit in the range of property if for some .
Example 9. Let and be given as in Example 7. Define by and for all . Consider the sequence . Then for , with an easy calculation, we see that
Hence satisfy the common limit in the range of property (-property).
3. Main Results
In this section, initially, some common fixed point results for the pairs, which are weakly compatible and satisfy property (E.A), have been proved, by reconstructing the contractive conditions given in [16].
Theorem 10. Let be a complex valued metric space and let be four self-mappings satisfying the following: (i), ;(ii)
for all , where ;(iii)the pairs and are weakly compatible;(iv)one of the pairs or satisfies property .
If the range of one of the mappings or is a complete subspace of , then the mappings , and have a unique common fixed point in .
Proof. Suppose that the pair satisfies property (E.A). Then there exists a sequence in such that
for some . Further, since , there exists a sequence in such that . Hence . Our claim is . Using condition (7), we have
By dividing two sides of the above inequality with we get
Thus and letting we have
Now, suppose that is complete subspace of ; then for some . Subsequently, we obtain
We claim that . To prove this, in (7)
and letting and using (12) we have
and consequently . Thus is a coincidence point of . Weak compatibility of the pair implies that
Conversely, since , there exists such that . Hence . Now we show that is a coincidence point of ; that is, . Putting , in (7), we get
thus . Hence and is a coincidence point of and . Weak compatibility of the pair implies that
Therefore, is a common coincidence point of , and .
In order to show that is a common fixed point of these mappings, we write in (7)
Thus,
A similar argument derives if we assume that is a complete subspace of and also using the property (E.A) of the pair gives us the same result.
Uniqueness. To prove that is a unique common fixed point, let us suppose that is another common fixed point of , and . In (7) take and ; then
is a contradiction. Thus . Consequently, and is the unique common fixed point of , and .
Putting in Theorem 10 we have the following corollary.
Corollary 11. Let , and be three self-mappings of a complex valued metric space satisfying the inequality
for all in , where . Suppose that the following conditions hold: (i),(ii)both the pairs and are weakly compatible,(iii)one of the pairs and satisfies the property .
If is complete subspace of , then , and have a unique common fixed point in .
In Theorem 10, if we put and , we have the following.
Corollary 12. Let be a complex valued metric space and let and be two self-mappings satisfying the following: (i);(ii)(iii) is a weakly compatible pair;(iv)the pair satisfies property .
If is complete subspace of , then and have the unique common fixed point in .
Theorem 13. Let and be four self-mappings of a complex valued metric space satisfying the following: (i) and ;(ii)
if where and is as in Definition 1, or
for all ;(iii)the pairs and are weakly compatible;(iv)one of the pairs or satisfies property .
If the range of one of the mappings or is a complete subspace of , then the mappings , and have a unique common fixed point in .
Proof. Let us suppose that so and the pair satisfies property (E.A). Then there exists a sequence in such that
for some . Since , there exists a sequence in such that . Hence . We show that . In inequality (23), putting and we get
Thus,
and letting we have
which is a contradiction since . Therefore,
Assuming is complete subspace of , then for some . Right after, we obtain
Our aim is to prove and for this putting and in (23) we get
Letting and using (30)
and hence , and since . Therefore, is a coincidence point of . Weak compatibility of the pair implies that .
Otherwise, since , there exists such that . Hence, . To show that is a coincidence point of pair , by using similar arguments in Theorem 10 and inequality (23) we have
and then because . With the same assertions as in Theorem 13 one gets that is a common coincidence point of , and .
Other details of Theorem 13, in which is a unique common fixed point of the mappings , and , can be obtained in view of the final part of the proof of Theorem 10 with suitable modifications.
In concluding, we note that the conclusions of Theorem 13 are still valid if we replace inequality (23) with the following inequality:
where and the mappings and are defined as in Theorem 13.
Finally, at the end of the section some common fixed point theorems for weakly compatible pairs which satisfy the -property have been proved.
Theorem 14. Let and be four self-mappings of a complex valued metric space satisfying the following: (i) and ;(ii)
for all where and is as in Definition 1;(iii) and are weakly compatible pairs.
If the pair satisfies -property, or the pair satisfies -property, then the mappings , and have a unique common fixed point in .
Proof. Let us suppose that the pair satisfies -property; then by Definition 8 there exists a sequence such that
for some . And also, since , we have for some . We claim that . Then putting and in inequality (35) we have
If tends to infinity and with equality (36), then
which is possible for since . Therefore, ; that is, is a coincidence point of the pair . Also weak compatibility of the mappings and implies the following equality:
Besides, since , there exist some such that . We claim that . Then from (35), we have
Thus,
which implies that , since . Hence,
and this shows that is a coincidence point of the pair . Weak compatibility of the pair yields that . In conclusion we show that is a common fixed point of , and . Using (35), we get
and hence which is the desired result. The uniqueness of common fixed point follows easily. The details of the proof of this theorem can be obtained by using the argument that the pair satisfies -property with suitable modifications. This completes the proof.
Theorem 14 is still true if we replace condition (35) with the following condition: where and the mappings and are defined as in Theorem 14.
Theorem 15. Let be a complex valued metric space and let be four self-mappings satisfying the following: (i), ;(ii), for all , where ;(iii)the pairs and are weakly compatible.
If the pair satisfies -property, or the pair satisfies -property, then the mappings , and have a unique common fixed point in .
Proof. This theorem can be obtained by using a similar technique as in the above theorem. So we omit it.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author is grateful to the reviewers for their careful reviews and useful comments.