Abstract
In this manuscript, a class of generalized -weak contraction is introduced and some fixed point theorems in the framework of -metric space are proved. The result presented in this paper generalizes some of the earlier results in the existing literature. Further, some examples and an application are provided to illustrate our main result.
1. Introduction and Preliminaries
Fixed point theory plays a vital role in the development of nonlinear functional analysis. It has been used in various branches of engineering and sciences. Banach contraction principle is one of the most important results in fixed point theory introduced by great Polish mathematician Stefan Banach [1]. The concept of -metric space or metric-type space was first introduced by Czerwik [2]. He provided a property which is weaker than the triangular inequality. The basic idea of -metric was commenced by Bourbaki [3] and Bakhtin [4]. Later on, Khamsi and Hussain [5] reintroduced such spaces under the name of metric-type spaces for some results of fixed and common fixed points in the setting of -metric spaces. Since then, several authors proved fixed point results of single valued and multivalued operators in -metric space and its different type generalizations, we refer [6–22]. Every one of these applications captivated us to present the idea of -metric space.
Definition 1. (see [23]). Let be a nonempty set. A function is said to be a -metric if it satisfies the following conditions:for all . The pair is called a -metric with coefficient .
Here, we observe that every metric space is a -metric with . Conditions (1) and (2) of Definition 1 are similar to metric space but it is important how to use (3) effectively. An example is given to expound the concept of a third condition.
Example 1. Let . We define a mapping such that
The first two conditions of Definition 1 are clearly shown. The solution of third condition is as follows:
Sincewe have
Then, we obtain
So the value of coefficient is .
In this section, the concept of generalized -weak contraction for metric space is provided with some basic notions and results. In 1997, Alber and Guerre-Delabriere [24] suggested a generalization of Banach contraction mapping by introducing the concept of -weak contraction in Hilbert space. In 2008, Dutta and Choudhury [25] gave a generalization of weakly contractive mapping by defining -weak contraction in complete metric spaces.
Definition 2. (see [26]). Let denote the class of function which satisfies the following conditions:(1) is continuous and nondecreasing
Definition 3. (see [26]). A self-map is said to be a weakly contractive map if there exists a function such that is continuous, nondecreasing, and and satisfying
Theorem 4. (see [26]). Let be a complete metric space and be a weakly contractive self-map on . Then, has a unique fixed point in .
Definition 5. (see [25]). A self-map is said to be -weak contraction, if for each ,where are both continuous and monotone nondecreasing functions with .
Theorem 6. (see [25]). Let be a complete metric space and a self-map be a -weak contraction. Then, has a unique fixed point.
Definition 7. (see [27]) Two self-maps and are said to be generalized -weakly contractive map if there exists a function such that is continuous, nondecreasing, and and satisfyingwhere
Theorem 8. (see [27]). Let be a complete metric space and and are generalized -weakly contractive self-maps on . Then, and have a unique common fixed point in .
Definition 9. (see [28]). Two self-maps and are said to be generalized -weakly contractive maps if they satisfy, where such that is continuous, nondecreasing, and , such that is a lower semicontinuous function, and
Theorem 10. (see [28]). Let be a complete metric space and and are generalized -weakly contractive self-maps on . Then, and have a unique common fixed point in .
Definition 11. (see [29]). Let be a partially ordered set. Two mappings are said to be weakly increasing if and for all .
Remark 12. (see [29]). Note that two weakly increasing mappings need not be nondecreasing.
Definition 13. (see [30]). Let be a metric space and are given two self-mappings on . The pair is said to be compatible if , whenever is a sequence in such that
Definition 14. (see [1]). Let be a partially ordered set and let be a -metric space with coefficient . Three maps , , and are said to be a generalized -weak contraction if for each ,where and is a continuous function with condition
Theorem 15. (see [1]). Let be a partially ordered set and assume that there exists a metric function in such that is a complete metric space. Let are generalized -weak contraction mappings satisfying the following properties:(1), , and are continuous(2)The pairs and are compatible(3) and weakly increasing with respect to (4) and are comparable(5), there exists such that and Then, , , and have a unique common fixed point .
The next section includes the concept of generalized -weak contraction for -metric space and theorem related to it.
2. Generalized -Weak Contractions
Definition 16. Let be a partially ordered set and let be a -metric space with coefficient . Three maps , , and are said to be a generalized -weak contraction if for each and .where and is a continuous function with the condition
Theorem 17. Let be a partially ordered set and assume that there exist a -metric function in such that is complete b-metric space. Let are generalized -weak contraction mappings satisfying the following properties:(1), , and are continuous(2)The pair and are compatible(3) and weakly increasing with respect to (4) and are comparable(5), there exist such that and Then, , , and have a unique common fixed point .
Proof. The proof is done by using the concept of Banach contraction principle in which a Cauchy sequence is taken in complete -metric space. Every Cauchy sequence is convergent in a complete metric space, and converging point of that sequence is proved to be a fixed point of contraction. ☐
Let us assume that be an arbitrary point in . By property (1), there exist such that and . Continuing this process, sequences and can be constructed in , defined as
By using property (4), we obtain
Similarly,
Continuing this process, we get
Thus,
According to our first supposition, if there exists such that , then from (18),which implies that . Consequently, for any . Hence, for every , we have which implies that is a -Cauchy sequence.
According to our second supposition, for any integer . Let . Now, we have to show that as . Since and are comparable, then from (18), we have
By property (2) of and the fact that , we get
Similarly, we have
By combining (28) and (29), we obtain
This shows that the sequence is monotonically decreasing. So there exists such that
Suppose . Then,
Taking the limit as , we get . Since , by using (19), we have but this is a contradiction. Then, . Hence,
Next, we have to show that is a -Cauchy sequence. We prove this by contradiction. Now, we suppose that is not a -Cauchy sequence. Then, for any , there exist two subsequences of positive integers and such that for all positive integers ,
From (34) and (35) and by using triangle inequality, we get
Let in the above inequality and by using (33), we obtain
Again by using triangle inequality, we have
By taking limit as in above inequality and using (33)–(35), we get
Moreover, we obtain
Using inequalities (33)–(37) and letting , we have
However, , therefore
From (18), we have
Taking limit as in the above inequality and using that fact that , we have
From (43) and (44), and using (19), we obtain
This is a contradiction. Therefore, is a -Cauchy sequence, and hence, is a -Cauchy sequence for all . Hence, there exists such that
Next, claim that is a coincidence point of , , and . From (46) and the continuity of , we get
From triangle inequality, we have
Since pair are compatible, then
Using the continuity of and (58), we getletting in (48) and using (47)–(50) together with (51), we findwhich means that . Similarly, from triangle inequality, we have
In a similar way, we obtain which means that . Thus, we find that , that is, is a coincidence point of , , and .
Now, we use the property (6) to show that is a common fixed point of , , and . For this, we prove that , , and have a common fixed point. To prove this, we show that if and are coincidence points of , , and , i.e.,
Then,
From our assumption mentioned in property (6), there exists such that
Now, we can define a sequence as follows:
Again, we have
Now, putting and in (18), we get
Since , we have the next inequality:
Similarly, again writing and in (18), we find
By combining (60) and (61), we obtain for all ,
Consequently, by using property of and , we get
Therefore, there exists such that
Let , then from (18), we obtainon taking limit as and using (19), we get
This is a contradiction. Thus, and from (64), we obtain
In the same pattern, we can show that
Now, using the fact that the limit is unique and by using (55)–(68), we can write
Since the pair and are compatible, we have
Let us take . Consider
Letting and using the continuity of , we get the above inequality as
That is, and is coincidence point of and . Similarly, . That is, and is the coincidence point of and . Hence, from (55), we have
This proves that is a common fixed point of , , and . Now, we show that is unique common fixed point. We will show this by contradiction. Assume that is not unique, therefore, there exists another fixed point as
By using (55), we have . Hence, we getbut this is a contradiction to our assumption, and hence, a common fixed point is unique.
An example is given to support the main result because there is limited examples in the literature.
Example 2. Let is the set of real numbers and a -metric is defined aswith coefficient and . Consider three self-maps defined asDefine maps and asAfter substituting values in (18),we haveNow, for , we getFor , we haveThen, clearly, three maps , , and are generalized -weak contraction for all values of .
3. An Application to Fredholm Integral Equations
In this section, applying Theorem 17, we give an existence theorem for common solutions of Fredholm integral equations where the upper limits of equations are taken to be the coefficient of -metric space . Here, we consider the following integral equations:for all where . Let us consider the space is a set of continuous functions defined on . Obviously, the space with the metric for all is a complete metric space.
Here, -metric space is defined on the partially ordered set. So, can be prepared with partial order given by
Theorem 18. Suppose the following hypotheses hold:(1) and are continuous(2)The following inequalities hold:(3)There exists a continuous function such thatfor all and such that Then, the integral equations (83) and (84) have the unique solution .
Proof. Let us define bywhere and . Here, and are considered to be weakly increasing function according to the requirement of our result.
Now, for all such that , we havewhere .
Using the Cauchy-Schwarz inequality in the R.H.S., we get
By using hypotheses (4), we have
This implies the following:
Suppose we choose the values of as , , and , respectively. Therefore, from inequality (93),
Since all the hypotheses of Theorem 17 are satisfied so there exists a unique common fixed point of and , that is, the solution of equations (83) and (84).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The authors equally conceived the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Acknowledgments
The authors thank the Basque Government for its support of this work through Grant IT1207-19. This study is supported by Ege University Scientific Research Projects Coordination Unit (Project Number FGA-2020-22080). The second author is thankful to Higher Education Commission of Pakistan (HEC).