Abstract
In this study, we establish the existence and uniqueness of common fixed point on complex partial b-metric space. An example and application to support our result is presented.
The fourth author Y. U. Gaba would like to dedicate this publication to his wife, Clémence A. Epse G., in celebration of her 33rd birthday
1. Introduction
In 1989, Backhtin [1] and Czerwik [2] introduced the concept of b-metric spaces and provided a framework to extend the results in the classical setting of metric spaces which are known already. Azam et al. [3]introduced complex-valued metric spaces in 2011 and proved some common fixed-point theorems under the contraction condition. Then, in 2013, Rao et al. [4]introduced the definition of complex valued b-metric space and provided a method to extend the results. Later, in 2017, the concept of complex partial metric space was introduced by Dhivya and Marudai [5], and they proved common fixed-point theorems. Recently, Gunaseelan [6]introduced the concept of complex partial b-metric space in 2019. Many authors have discussed significant results and application on complex metric spaces [7–23]. In this study, we establish common fixed-point theorems on complex partial b-metric space using continuity property.
2. Preliminaries
Let be the set of complex numbers and . Define a partial order on as follows:
if and only if and .
Then, if one of the following properties is fulfilled:(i), (ii), (iii), (iv),
In particular, we write if and one of , and is fulfilled, and we write if only is fulfilled.
Definition 1. (see [4]). Let be a nonvoid set and let be a given real number. A function is called a complex valued b-metric on if, for all , the following conditions are fulfilled:(i) and if and only if (ii)(iii)
The pair is called a complex valued b-metric space.
Definition 2. (see [5]). A complex partial metric on a nonvoid set is a function such that, for all ,(i)(ii)(iii)if and only if(iv)A complex partial metric space is a pair such that is a nonvoid set and is the complex partial metric on .
Definition 3. (see [6]). A complex partial b-metric on a nonvoid set is a function such that, for all ,(i)(ii)(iii)(iv)such thatA complex partial b-metric space (b-CPMS) is a pair such that is a nonvoid set and is the complex partial b-metric on . The number is called the coefficient of .
Definition 4. (see [6]). Let be a complex partial b-metric space with coefficient . Let be any sequence in and . Then,(i)The sequence is said to be convergent with respect to and converges to if (ii)The sequence is said to be Cauchy sequence in if exists and is finite(iii) is said to be a complete complex partial b-metric space if, for every Cauchy sequence in , there exists such that
In 2019, Gunaseelan [6] proved some fixed-point theorems on complex partial b-metric space as follows.
Theorem 1. Let be any complete complex partial b-metric space with coefficient and be a mapping satisfyingfor all , where . Then, has a unique fixed point and .
We prove the existence and uniqueness of common fixed point on complex partial b-metric space, inspired by his work.
3. Main Results
Theorem 2. Let be a complete b-CPMS with the coefficient and be two continuous mappings such thatfor all , where . Then, and have a unique common fixed point and .
Proof. Let . DefineThen, by (1) and (2), we obtain
Case 1. If , then we haveThis implies , which is a reductio ad absurdum.
Case 2. If , then we haveFrom the next step, we haveWe consider three cases.
Case 3. which implies , is a reductio ad absurdum.
Case 4. From (6) and (9), , we obtainFor , with , we haveMoreover, by using (9), we obtainTherefore,Then, we haveHence, is a Cauchy sequence in .
Case 5. This implies thatSince , we get . Therefore, is a Cauchy sequence in .
Case 6. If , then we haveHence,For the next step, we haveThen, we consider three cases.
Case 7. which implies and is a reductio ad absurdum.
Case 8. Then, by (18) and (21), we get , where . Hence, is a Cauchy sequence in .
Case 9. Hence, we obtainBy using (18) and (21) yieldswhere .
Then, , we obtainFor , with ,Using (24), we obtainTherefore,Hence, we haveHence, is a Cauchy sequence in . In all cases, is a Cauchy sequence. Since is complete, there exists such that as andBy the continuity of , as :However,Next, we prove that is a fixed point of :As , we obtain . Thus, . Hence, and . In the same way, we have such that as andBy the continuity of as ,However,Next, we prove that is a fixed point of :As , we obtain . Thus, . Hence, and . Therefore, and have a common fixed point .
Let be another common fixed point for the mappings and . Then,This implies that .
Theorem 3. Let be a complete b-CPMS with the coefficient and be two continuous mappings such thatfor all , where . Then, and have a unique common fixed point and .
Proof. Following from Theorem 2, we can easily prove is a Cauchy sequence. Since is complete, there exists such that as .
Suppose that .
Then, we estimateThis yieldsHence, , which is a reductio ad absurdum. Then, . Similarly, we derive that . Therefore, and have a common fixed point . Following from Theorem 2, we can easily prove uniqueness part.
Theorem 4. Let be a complete b-CPMS with the coefficient and be two continuous mappings such thatfor all , where . Then, and have a unique common fixed point and .
Proof. Let . DefineThen, by (42) and (43), we obtainIf , thenThis shows that , which is a reductio ad absurdum. Therefore,Similarly, we obtainFrom (46) and (47), , we obtainFor , with , we haveBy using (48), we obtainTherefore,Hence, we haveHence, is a Cauchy sequence in . Since is complete, there exists such that as andSince is continuous, we obtainSimilarly, we derive that . Then, and have a common fixed point. Let be another common fixed point for the mappings and . Then,This implies that .
Theorem 5. Let be a complete b-CPMS with the coefficient and be two continuous mappings such thatfor all , where . Then, and have a unique common fixed point and .
Proof. Following from Theorem 5, we can easily prove is a Cauchy sequence. Since is complete, there exists such that as andSuppose that .
Then, we estimateThis yieldsHence, , which is a reductio ad absurdum. Then, . Similarly, we derive that . Therefore, and have a common fixed point . Following from Theorem 5, we can easily prove the uniqueness part.
Example 3.5. Let be endowed with the partial order iff . We define in Tables 1 and 2.
It is easy to verify that is a complete b-CPMS with the coefficient for . Define by :Clearly, and are continuous functions. Now, for , we consider the following cases:(A)If and , then . Hence, all the conditions of Theorem 2 are fulfilled.(B)If and , then and :(C)If and , then and :(D)If and , then and :(E)If and , then and :All the conditions of Theorem 1, with , are fulfilled. Therefore, and have a unique common fixed point 1.
Example 3.6. Let , where and be endowed with the partial order iff . Define by , for all or and or .
It is easy to verify that is a complete b-CPMS with the coefficient for . Define byClearly, and are not continuous functions. Now, we consider the following cases:(A)If, then and :(B)If and , then and :(C)If and , then and :(D)If , then and :All the conditions of Theorems 2 and 3, with , are fulfilled except continuous mapping. Therefore, and have no common fixed point.
Remark 1. In view of the fact in Theorems 2 and 3, we cannot drop the continuous mapping.
4. Application
Consider the following systems of integral equations:where(i) and are unknown variables for each , (ii) and are deterministic kernels defined for
Let be the set of continuous functions defined on . Define by. Then, is a complete b-CPMS. Define partial order given by
Theorem 6. Assume that(A) are continuous functions satisfyingwhere
Then, systems (71) and (72) have a unique common solution.
Proof. For and , define the continuous mappings byThen,Hence, all the conditions of Theorem 2 are fulfilled for with . Therefore, integrals (71) and (72) have a unique common solution.
5. Conclusion
In this paper, we proved common fixed-point theorems on complex partial b-metric space. An illustrative example and application on complex partial b-metric space is given. Recently, Khalehoghli et al. [24, 25] introduced -metric spaces and obtained a generalization of Banach fixed-point theorem. It is an interesting open problem to study the relation instead of complex partial b-metric space and obtain common fixed-point results on R-complete complex partial b-metric spaces.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.