Abstract
The purpose of this research article is to introduce new contractive conditions and to examine the existence and uniqueness of fixed points of self-mappings in the context of b-metric spaces by applying these different contractive conditions. Furthermore, some examples are given to illustrate its validity and superiority. Our results generalize and extend several well-known results in metric and b-metric spaces.
1. Introduction and Preliminaries
Many issues in engineering and science explained by nonlinear equations may be tackled by confining them to analogous fixed-point case. An operator sum Gx = 0 can be proved as fixed-point sum Fx = x, wherein F is a self-defining along with some relevant discipline. Fixed-point theory endows with some key modes to resolving problems ensuing from multiple offshoots of mathematical inspection such as split feasibility issues, supportive problems, equilibrium problems, and matching, as well as selective issues and such others. The theory of fixed points is the great vibrant and energetic zone of the study. This theory has previously been exposed as an excessive and major deployment for cramming nonlinear analysis. Specially, fixed-point procedures are being applied in a diversity of fields, for example, biology, chemistry, engineering, economics, physics, and game theory. Functional analysis is a very useful and important field of mathematics. Its results are supportive tools for other fields to solve many problems. Many researchers have put their efforts in obtaining these results; for further study, see [1–12] and literature. Because of Banach [4], the Banach-contraction theorem (1922) is indeed the most significant consequence in the theory of fixed points in metric spaces. This theorem promises the presence and distinctiveness of fixed points of self-mapping that satisfy the contraction condition on complete metric spaces as well as dispense a valuable approach for finding them. Since contractive condition deduces the uniform continuity of an operator f, so it was a natural question to raise the concern about existence of fixed point in the absence of continuity of f. In 1968, Kannan [12] answered this question by the introduction of Kannan contractive condition. One of the most famous generalizations of metric spaces was given by Bakhtin [3] in 1989. He presented the notion of b-metric space in which triangular inequality has been relaxed. In 1993, Czerwik [7] drew out the results in b-metric space. By accepting this idea, many researchers gave extensions of Banach’s principle in b-metric space. Boriceanu [5], Czerwik [7], Bota [13], and Pacurar [13] drew out the fixed-point theorems in b-metric space. Moreover, many authors examined and derived the existence of fixed point of a contraction function in the context of b-metric spaces; for detail, see [12, 14–17] In this paper, we have established fixed-point results in the context of b-metric spaces for two different contractive conditions. Some direct consequences from our main results are also presented. In the support of these results, examples are created.
Definition 1 (see [4]). Consider a metric space with metric d. A function is known as Banach contraction on if there exists a number such that , :
Definition 2 (see [9]). Consider a metric space and is a function if such that, for all we haveThen, is known as Kannan contraction.
Definition 3 (see [3]). Let be any nonempty set and be a real number. A function is called b-metric if axioms given below are fulfilled for all : Then, is called b-metric space.
If we take , then b-metric space becomes ordinary metric space. Hence, set of all metric spaces is a subset of set of all b-metric spaces.
Example 1. Let ; then, is a b-metric space with .(1)Obviously, is real, finite, and nonnegative.(2)Consider and So, .(3)To solve the triangular property in b-metric space, we will use the convexity of function, i.e., “If , then convexity of function implies that , i.e.,holds,” and we haveUsing (3),Hence, is a b-metric with .
Example 2 (see [5]). The set with , where together with the function , is where is b-metric space with . Notice that the abovementioned result holds with .
Definition 4 (see [5]). Let be b-metric space and be a sequence in . Then,(1) is called a convergent sequence if and only if there exists , such that for any such that, for all , we get . In this case, we write .(2) is said to be a Cauchy sequence if and only if for any such that, for each , we get .(3) is called complete if every Cauchy sequence in is convergent in .Let Ω be a nonempty set and be a self-map. We say that is a fixed point of if .
Let Ω be any set and be a self-map. For any given , we define inductively by and , and we recall , the iterative of under . For any , the sequence is given bywhich is known as the sequence of successive approximations, where is the initial value. This is also known as the Picard iteration starting at .
2. Fixed Points for Contractive Mappings
This section consists of our main results. We investigate the existence and uniqueness of fixed points of some new contractive conditions in the context of b-metric spaces. Moreover, results are supported with example.
Theorem 1. Let be a complete b-metric space with coefficient . Let be a function such thatwhere such that and Then, there is a unique fixed point of R.
Proof. Let and be a sequence in Ω defined by the recursion:By (8) and (9),So,where .
Case 1. If , then Let ; then, and .
Case 2. If , thenLet ; then,
Case 3. If , thenLet ; then, Cases 1–3 show that is a contractive-type mapping.
Let ; then,Continue in the following way:By Cases 1–3,Since is geometric series with common ratio less than 1, soAs ,
Hence,Hence, is a Cauchy sequence in Ω. Due to completeness of Ω, there exists such thatNow,where .
Case 4. If , thenAs , soHence, is a fixed point of R.
Case 5. If , thenAs , soHence, . is a fixed point of R.
Case 6. If , thenAs , soHence, .
Hence, is a fixed point of R.
Case 7. If , thenAs , soHence, . is a fixed point of R.
Now, assume that is another fixed point of R, i.e., . Consider. That is not possible; therefore, . Hence, is unique fixed point of R.
Example 3. Suppose ; we define as below: (1, 1) = (2, 2) = (3, 3) = 0 (1, 2) = (2, 1) = (2, 3) = (3, 2) = 1 (1, 3) = (3, 1) = Then, (X, ) is a b-metric space with coefficient , where
Define by
Then, all conditions of the above theorems are satisfied for Hence, is the unique fixed point of R.
In the following, some direct consequences of Theorem 1 are as follows.
Corollary 1 (Banach-contraction theorem in b-metric space). Let be a complete b-metric space with coefficient . Let be a self-function on such thatwhere Then, there exists a unique fixed point of R.
Proof. In Theorem 1, by putting b = 0, the required result will be obtained.
Corollary 2. Let be a complete metric space and a function such thatwhere such that . Then, there exists a unique fixed point of R.
Proof. Since every b-metric space is a metric space by taking s = 1, so the proof is obvious.
Corollary 3 (Banach-contraction theorem in metric space). Let be a complete metric space. Let be a function such thatwhere Then, there exists a unique fixed point of R.
Proof. In Corollary 2, by putting b = 0, the required will be obtained.
Theorem 2. Let be a complete b-metric space. Let be a mapping such thatwhere such that . Then, there exists a unique fixed point of R.
Proof. Let and be a sequence in Ω defined asBy (44) and (45), we obtain thatwhere.
Case 8. If , then (44) becomes.
Continuing in this way,
Case 9. If , then (47) becomes
Continuing in this way,Cases 8 and 9 show that is a contractive-type mapping.
Now, let m and n be any two natural numbers and m > n; then,Continuing in this way,By Cases 1 and 2,Since is a geometric series with common ratio less than 1, so .
As Hence, is a Cauchy sequence in Ω. Due to completeness of Ω, there exists such that . Now,By (44),where .
Case 10. If , then (61) becomesAs Hence, is a fixed point of .
Case 11. If , then (61) becomesAs Hence, is a fixed point of .
Case 12. If , then (61) becomesAs Hence,where is a fixed point of .
For uniqueness, assume that is another fixed point of , i.e., . ConsiderBy (44),which is a contradiction. Hence,where is unique fixed point of .
Example 4. Suppose , and we define as below: (1, 3) = (3, 1) =
Then, (X, ) is a b-metric space with coefficient , where
Define by
Then, all conditions of the above theorems are satisfied for Hence, is the unique fixed point of T.
Corollary 4. (Kannan-contraction theorem in b-metric space). Let be a complete b-metric space. Let be a mapping such thatwhere . Then, R has a unique fixed point.
Proof. By putting a = 0 in Theorem 2, we get the required result.
Corollary 5. Let be a complete metric space and be a mapping such thatwhere such that . Then, there exists a unique fixed point of R.
Corollary 6. (Kannan-contraction theorem in metric space). Let be a complete metric space and a self-mapping such thatwhere and . Then, there exists a unique fixed point of R.
Proof. By putting a = 0 in Corollary 5, we get the required result.
3. Conclusion
We have established and proved fixed-point results for different contractive conditions in b-metric space. To furnish, these results supportive examples are created. This generalization will be helpful for further investigation and applications. We conclude this paper by indicating, in the form of open questions, some directions for further investigation and work:(1)Can the conditions a + 2sb in Theorem 1 and 2as + 2b in Theorem 2 be relaxed?(2)If the answer to 1 is yes, then what hypotheses on a and b are needed to guarantee the existence of fixed points?(3)Can the concept of fixed point for these contractions be extended to more than one mapping?
Data Availability
The data used to support the finding of the study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publications.
Acknowledgments
This work was funded by Jahangirnagar University, Savar, Dhaka, Bangladesh. The authors would like to thank for technical and financial support.