Abstract
This paper is aimed at establishing some unique common fixed point theorems in complex-valued -metric space under the rational type contraction conditions for three self-mappings in which the one self-map is continuous. A continuous self-map is commutable with the other two self-mappings. Our results are verified by some suitable examples. Ultimately, our results have been utilized to prove the existing solution to the two Urysohn integral type equations. This application illustrates how complex-valued -metric space can be used in other types of integral operators.
1. Introduction
In 1922, Banach [1] proved a fixed point theorem (FP-theorem), which is stated as the following: “a single-valued contractive type mapping on a complete metric space has a unique fixed point.” After the publication of the Banach FP-theorem, many researchers have contributed their ideas to the theory of FP. Chandok [2, 3], Jungck and Rhoades [4], and Al-Shami and Abo-Tabl [5, 6] proved different contractive types of FPs and a-fixed soft point results in the context of metric spaces.
Bakhtin [7] introduced the idea of -metric space, while Czerwik [8] proved some fixed point results for nonlinear set-valued contractive type mappings in -metric spaces. Suzuki in [9] proved basic inequality and some FP-theorems. Jain and Kaur [10] presented a new class of functions to define new contractive maps and established FP-results for these maps. They also extended some results in the framework of -metric-like spaces. They presented examples and established the application of their main results. They also presented some open problems. Petrusel et al. [11] considered coupled FP-problems for single-valued operators satisfying contraction in said space. They discussed uniqueness, data dependence, and shadowing-property of coupled FP-problem and also established an application for main results. Ameer et al. [12], Boriceanu [13, 14], Bota et al. [15], Czerwik et al. [16, 17], Hussain and Shah [18], Karapinar et al. [19], and Samreen et al. [20] established different contractive type FP and common FP (CFP) results in the context of -metric spaces.
In 2011, the concept of complex-valued metric space was given by Azam et al. [21], and they proved some CFP-theorems for self-mappings. The notion of said space was proposed by Rouzkard and Imdad [22] which generalizes the results of Azam et al. [21] and established some CFP-results. Abbas et al. [23] presented some generalized CFP-results by using cocyclic mappings in complex-valued metric space. They provided examples to indicate the authenticity of his expressions. Sarwar and Zada [24] used the ideas of and properties and proved FP-results for six self-mappings. They showed the existence of their results by establishing some examples. Abbas et al. [25], Nashine et al. [26], Mohanta and Maitra [27], Sintunavarat and Kumam [28], and Verma and Pathak [29] proved some results in the context of complex-valued metric space.
In 2013, Rao et al. [30] introduced the notion of complex-valued -metric space. Mukheimer et al. [31] established CFP-results on said space by extending and generalizing the results of [30, 31]. In [32], Chantakun et al. extend the work of Dubey et al. [33] by introducing sufficient conditions to prove some CFP-results in complex-valued -metric space. Yadav et al. [34] used compatible and weakly compatible maps to find CFP-results. They proved the validity of the results by providing some examples and establishing an application. Berrah et al. [35], Hasana [36], Mehmood et al. [37], and Mukheimer [38] established some FP and CFP theorems in complex-valued -metric spaces.
In this paper, we provide some extended and effective CFP-results for commuting three self-maps on complex-valued -metric spaces. To verify the validity of our work, we present some illustrative examples in the main section. Further, our results have been utilized to prove the existing solution to the two Urysohn integral type equations. This application is also illustrative of how complex-valued -metric space can be used in other integral type operators. This paper is organized as follows: In Section 2, we present the preliminary concepts. In Section 3, we establish some extended and modified CFP-results for commuting self-maps in complex-valued -metric space under the generalized rational type conditions. We also provide authentic examples to indicate the effectiveness of these results. In Section 4, we present an application of the two UITEs to support our main work. Finally, in Section 5, we discuss the conclusion.
2. Preliminaries
Let be the set of all complex numbers and . Define as , iff and , where denotes the real part and denotes the imaginary part of a complex number. Accordingly, , if any one of the following conditions holds:
and
and
and
and
Know that if and one of (), (), and () is satisfied.
Remark 1 (see [31]). We can easily check the following: (i)If and , (ii)(iii) and
Definition 2 (see [8]). Let be a nonempty set and a given real number. A mapping is called a -metric on if the following conditions are satisfied:
if and only if
,
for all . Then, is called a -metric space.
Definition 3 (see [30]). Let be a nonempty set and a given real number. A mapping is called a complex-valued -metric on if the following conditions are satisfied:
and if and only if
,
for all . Then, is called a complex-valued -metric space.
Example 4. Let . Define the mapping by, for all .
Then, is a complex-valued -metric space with .
Definition 5 (see [30, 31]). Let be a complex-valued -metric space and a sequence in and . Then, (1) is said to converge to if for every there exists such that , . We denote this by or as (2)if for every there exists such that for all , , then is called a Cauchy sequence(3)if every Cauchy sequence is convergent, then is called a complete complex-valued -metric space
Lemma 6 (see [30, 31]). Let be a complex-valued -metric space and let be a sequence in . Then, converges to iff as .
Lemma 7 (see [30, 31]). Let be a complex-valued -metric space and let be a sequence in . Then, is a Cauchy sequence iff as .
Definition 8 (see [39]). Let be a complex-valued -metric space. The self-mappings and are said to be commuting if for all .
3. Main Result
In this section, we prove some CFP theorems in complex-valued -metric space under the generalized rational type contraction conditions for three self-mappings in which one is continuous. We present some examples for the validation of our work.
Theorem 9. Let be a complete complex-valued -metric space and be three self-maps satisfying for all , such that and . If is continuous and , are commutable pairs, then , and have a unique CFP in .
Proof. Fix , and define a sequence sequences in such that
Now, by using (1),
This implies that
After simplification, we get that
This implies that
After simplification, we get that
Now, from (8) and (5) and by induction, we have that
So, for with ,
Therefore, the sequence is Cauchy. Since is complete, there exists such that , as , or , and from (2), we have
As is continuous, so
Since, and are commutable pairs, therefore, from (12), we have that
Now, we have to show that , so by putting and , in (1),
This implies that
Taking and using (11), (12), and (13), we get that
After simplification, we get that
Since , hence, we get that
Next, we have to show that , by the view of (1),
This implies that
Now, again applying on both sides and by using (11) and (18), we have that
This implies that . Hence,
Now, we have to show that , by using (1),
This implies that
Taking and using (11) and (18), we get
This implies that . Hence,
Thus, from (18), (22), and (26), we find that is a CFP of , and , i.e.,
Uniqueness: suppose that is another CFP of , and such that
Then, from (1), we have that
This implies that . Since . Hence, prove that , and have a unique CFP in .
If we put in Theorem 9, we get the following corollary.
Corollary 10. Let be a complete complex-valued -metric space and be three self-maps satisfying for all , such that and . If is continuous and , are commutable pairs, then , and have a unique CFP in .
If we put in Theorem 9, we can get the following corollary.
Corollary 11. Let be a complete complex-valued -metric space and be three self-maps satisfying: for all , and . If is continuous and , are commutable pairs, then , and have a unique CFP in .
Theorem 12. Let be a complete complex-valued -metric space and be three self-maps satisfying for all , such that , and . If is continuous and , are commutable pairs, then , and have a unique CFP in .
Proof. Fix , and define a sequence sequences in such that
Now, by using (32),
This implies that,
After simplification, we get that
Now, there are two possibilities: (i)If is a maximum term in , then after simplification, (36) can be written as(ii)If is a maximum term in , then after simplification, (36) can be written as
Let , then from (37) and (38), for all , we have
Similarly,
Now, from (40) and (39) and by induction, we have that
Now, for with ,
Therefore, sequence is Cauchy. Since is complete, there exists such that , as , or , and from (33), we have
As is continuous, so
Since, and are commutable pairs, therefore, from (44), we have that
Now, we have to show that , so by putting and , in (32):
This implies that
Taking and using (43), (44), and (45), we get that
After simplification, we get that
Since ; hence, we get that
Next, we have to show that , by the view of (32),
This implies that
Now, again applying on both sides and by using (43) and (50), we have that
This implies that . Since . Hence,
Now, we have to show that , by using (32),
This implies that
Taking and using (43) and (50), we get
This implies that . Since . Hence,
Thus, from (50), (54), and (58), we find that is a CFP of , and , i.e.,
Uniqueness: suppose that is another CFP of , and such that
Then, from (32), we have that
This implies that . Since , hence proving that , and have a unique CFP in .
If we put in Theorem 12, we can get the following corollary.
Corollary 13. Let () be a complete complex-valued -metric space and , , be three self-maps satisfying for all , such that , and , where . If is continuous and , are commutable pairs, then , and have a unique common fixed point in .
Corollary 14. Let be a complete complex-valued -metric space and be three self-maps satisfying for all , such that , and , where . If is continuous and , are commutable pairs, then , and have a unique common fixed point in .
Example 15. Let be a complex-valued -metric space, where and with , for all . Now, we find ,
That is , where . Now, define as
Notice that
In all regards, it is enough to show that , for all and , such that and , where , we have
For , we discuss different cases with , and . Hence,
Case 1. Let , then from (67) and (68), directly, we get that . Hence, (32) is satisfied with , and .
Case 2. Let , then from (67) and (68), we find , satisfied with , i.e., Thus, (32) is true for , and .
Case 3. Let ; then, from (67) and (68), we find is satisfied with , i.e., Thus, (32) is true for , and .
Case 4. Let ; then, from (67) and (68), we find is satisfied with , i.e.,
Hence, (32) is satisfied with , and . The pairs of self-mappings and are commutable; that is,
Thus, all the conditions of Theorem 12 are satisfied with noticing that the point , which remains fixed under mappings and , is indeed unique.
Theorem 16. Let be a complete complex-valued -metric space and be three self-maps satisfying for all , , such that , and . If is a continuous self-map and , are commutable pairs, then and have a unique CFP in .
Proof. Fix , and define a sequence sequences in such that
Now, by the view of (74) and (75),
This implies that
After simplification, we get that
Again, by the view of (74) and (75),
This implies that
After simplification, we get that
Now, from (81) and (78) and by induction, we have
So, for with ,
Therefore, sequence is Cauchy. Since is complete, there exists such that , as , and from (75), we have that
As is continuous, so
Since, and are commutable pairs, therefore, from (85), we have that
Now, we prove . So, for this, we put and in (74),
This implies that
Taking and using (84), (85), and (86), we get that
This implies that . Since, . Hence,
Next, we have to show that , by using (74),
This implies that
Taking and using (84) and (90), we get
Thus, we get that . Since , therefore, . Hence,
Now, we have to show that , by using (74),
This implies that
Taking and using (84) and (90), we get
So, we get that . Since , therefore, . Hence,
Thus, from (90), (94), and (98), we find that is a CFP of , and , i.e.,
Uniqueness: suppose that is another CFP of , and such that
Then, from (74), we have that
This implies that . Since , therefore, , hence proving that and have a unique CFP in .
Corollary 17. Let be a complete complex-valued -metric space and , , be three self-maps satisfying for all , , , , , such that and , with . If is a continuous self-mapping and (), () are commutable pairs, then , , and have a unique common fixed point in .
Example 18. Let and be defined as for all . Then, is a complex-valued -metric space. Now, we find :
That is , where . Now, we define by
Notice that
In all regards, it is enough to show that , for all and , such that and , where , we have
For , we discuss different cases with , where . Hence,
Case 1. Let . Then, from (106) and (107), directly, we get that . Hence, (74) is satisfied with , and .
Case 2. Let , then from (106) and (107), we find is satisfied with , as
By using and after simplifying, we get that
Thus, (74) is true for , and .
Case 3. Let , then from (106) and (107), we find is true for , as
By using and after simplifying, we get that
Thus, (74) is true for , and .
Case 4. Let , then from (106) and (107), we get that is true for , as
By using and after simplifying, we get that
Thus, (74) is true for , and .
So, all conditions of Theorem 16 are satisfied to get a unique CFP, that is “” of the mappings and .
4. Applications
Here, we provide an application to support our main result. To do this, we take a couple of UITEs to obtain the existing result of a common solution to check the effectiveness of our result. Let the set contain real-valued continuous functions defined on . In the following, we use Theorem 9 to obtain the existing result of a common solution. This enables us to establish a theorem based on UITEs to attain the existing result of a common solution.
Theorem 19 (see [28]). Let , where and is defined as for all and . Consider that the UITEs are where . Let be such that for every , we have that
If there exists such that, for all , where with where
Then, the two UITEs, i.e., (41), have a unique common solution.
Proof. Define as
Then, we have the following three cases: (1)If is the maximum term in , then from (118), (119), and (123), we have that
for all . Thus, , and satisfy all conditions of Theorem 9 with and in (1). Then, two UITEs, i.e., (116), have a unique common solution in . (2)If is the maximum term in , then from (118), (119), and (123), we have that
for all . Thus, , and satisfy all conditions of Theorem 9 with and in (1). Then, two UITEs, i.e., (116), have a unique common solution in . (3)If is the maximum term in , then from (119), we have that
Then, there are furthermore five subcases arising: (i)If is the minimum term in . Then from (118), (121), (123), and (126), we have that
for all (ii)If is the minimum term in . Then from (118), (121), (123), and (126), we have that
for all (iii)If is the minimum term in . Then from (118), (121), (123), and (126), we have that
for all (iv)If is the minimum term in . Then from (118), (121), (123), and (126), we have that
for all (v)If is the minimum term in . Then from (118), (121), (123), and (126), we have that
for all . Thus, the subcases of Case 3 (Case (i-v)) for the mappings , and satisfy all the conditions of Theorem 9 with and in (1). Then, two UITEs, i.e., (116), have a unique common solution in .
5. Conclusions
We have established some unique CFP-results in complex-valued -metric space by using rational contraction conditions for three self-mappings in which one self-map is continuous and commutable with the other two self-mappings. In our main work, we have generalized the results (e.g., see [28, 37, 38]). To show the authenticity of our results, we have given some useful examples in the main section. We have also provided an application for our main result to indicate its utility. In this direction, many results can be contributed to the said space by applying different contractions with different types of integral operators.
Data Availability
Data sharing is not applicable to this article as no data set was generated or analyzed during the current study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. FP-043-43.