Abstract
The aim of this paper is to define generalized rational contractions in the setting of graphical -metric spaces and obtain some fixed-point theorems. Our results are significant generalizations and extensions of some well-known results in the existing theory. We also supply a nontrivial example to show the validity of the obtained theorems. As applications, we obtain some results on rational expressions in the background of graphical metric spaces.
1. Introduction
In the context of fixed-point theory, the underlying space and contractive mapping play an important and crucial role. One of the pillars in this theory is the concept of metric space which was introduced by Frechet [1]. In this theory, the pioneer result is the well-known Banach contraction principle [2] which states that if is a complete metric space and is a mapping satisfying the conditionwhere , then has a unique fixed point.
Due to the simplicity and significance of the notion of metric space, it has been improved, extended, and generalized in various directions. The famous extension of the concept of metric spaces has been done by Bakhtin [3] which was formally defined by Czerwik [4] in 1993. Jachymski [5] replaced the order structure on a metric space with a graph structure and gave the graphic version of the Banach contraction principle. In 2017, Shukla et al. [6] introduced the concept of graphical metric spaces by proposing the graphical structure on metric spaces. Subsequently, Chuensupantharat et al. [7] combined the notions of -metric spaces and graphical metric spaces and gave the concept of graphical -metric spaces.
On the other hand, Fisher [8] and Dass and Gupta [9] introduced rational expression in the contractive condition and generalized the famous Banach contraction principle. Isik et al. [10] proved some fixed-point theorems for rational contractions endowed with a graph and investigated the solution of a system of integral equations as applications.
In this article, we introduce Fisher’s graph contraction and Dass–Gupta’s graph contraction in the setting of graphical -metric spaces and obtain some fixed-point results. Based on this structure, we show that every Dass–Gupta’s contraction is Dass–Gupta’s graph contraction, but the converse is not generally true. Some nontrivial and significant examples are also provided, equipped with some worthy graphs to show the authenticity of established outcomes.
2. Preliminaries
Frechet [1] initiated the theory of metric space in the following manner.for all , there exists a unique point such that .
Definition 1 (see [1]). Let (nonempty set) and be a function satisfying if and only if for all .
Then, is called a metric space.
Banach [2] obtained a fixed-point result in 1922 in the following way.
Theorem 2. Let be a complete metric space and . If there exists a nonnegative constant such that
In 1980, Fisher [8] presented the following result.for all , there exists a unique point such that .
Theorem 3. Let be a complete metric space and . If there exist nonnegative constants with such that
Dass and Gupta [9] established a result in the following way.for all , there exists a unique point such that .
Theorem 4. Let be a complete metric space and . If there exist nonnegative constants with such that
Czerwik [4] presented the idea of -metric space as follows.
Definition 5 (see [4]). Let , be a constant and be a function satisfying if and only if for all .
Then, the pair is considered as a -metric space.
Some concepts from graph theory given by Jachymski [5] will be presented here. Let denotes the diagonal of , where is any nonempty set. Let be a directed graph such that is the set of its vertices, that is, corresponding to and is the set of its edges, that is, which contains all loops, i.e., . If we alter the direction of the edges of , the resultant graph is represented by . Moreover, the letter represents a directed graph with symmetric edges. Specifically, we defineIf and are vertices in a graph , then a path from to in of length is a sequence of vertices such that , , , and . A graph is connected if any two vertices of have a path between them. Moreover, a graph is weakly connected if there is a path between each two vertices in undirected graph . We say is a subgraph of if and .
Motivated by Shukla et al. [6], we representMoreover, a relation on is such thatIf there is a direct path in from to , and if is in the path . If in with , , then is said to be a -termwise connected (shortly -TWC) sequence.
Shukla et al. [6] introduced the notion of graphical metric space in the following way.
Definition 6 (see [7]). Let and is a function satisfying if and only if and implies for all . Then, the pair is said to be a graphical metric space.
Example 7. Every metric space is a graphical metric space with graph , where and .
In 2019, Chuensupantharat et al. [7] gave the notion of graphical -metric space as a generalization of -metric space as follows:
Definition 8 (see [7]). Let and be a constant and be a function satisfying if and only if and implies for all . Then, the pair is said to be a graphical -metric space with coefficient on .
Remark 9. The notion of graphical -metric space is a real generalization of graphical metric space because if we take in the above definition, then we can get the notion of a graphical metric space.
Example 10 (see [7]). Let and is defined bywhere is constant. Then, it is very simple to prove that is a graphical -metric space with coefficient , where and and as shown in Figure 1.
Notice that it is not a graphical metric space because
Definition 11. Let be a graphical -metric space, and be a sequence in , then(i) is said to be a convergent sequence if there exists a point such that (ii) is said to be a Cauchy sequence if For more characteristics in the direction of graphic contractions, graphical metric spaces, and graphical -metric spaces, we refer the readers to [10–21].
3. Results and Discussion
Let be a subgraph of graph such that and moreover suppose that is a weighted graph. Let be a sequence with initial point in . Then, is said to be an -Picard sequence (-PS) for if , for all .
Now, we state a property (P) given by Shukla et al. [6] in the following way.
(P) A graph satisfies the property (P) if a -termwise connected -PS converging in guarantees that there is a limit of and such that or for all .
Now, we define Fisher-type graph contraction in a graphical -metric space.
Definition 12. Let be a graphical -metric space and Then, is said to be an Fisher-type graph contraction for on if(i) is graph preserving, i.e., for each if () implies (ii)There exist nonnegative constants with such that for every with () ∈, we have
Remark 13. Any Fisher contraction is Fisher graph contraction along with defined by and .(i)If be a Fisher graph contraction with parameters and , then is a Banach graph contraction for graphs in .(ii)Since every graphical -metric space is graphical metric space and also every is , thus our theorems in this paper are precise generalizations of some of the results regarding Fisher contraction (see e.g., [8]).
Theorem 14. Let be an -complete graphical -metric space such that is continuous functional and let be a Fisher graph contraction. Suppose that these assertions hold:(i) satisfies the property (ii)There exists with for some Then, there exists such that the -PS with initial point is -TWC and as .
Proof. Let be such that , for some . As is a -PS originating from , there is a path such that and and () ∈ for . Now, by assumption (i), we have () ∈ for . It yields that is a path from to having length and therefore . Continuing in this way, we get is a path from to of length , and hence, , for all . Hence, we have which is a -TWC sequence. Thus, () ∈ for and . Now, by (ii), we havewhich yields thatSince , so taking , it follows from the above inequality thatwhere . Repeating in this way, we haveAs the sequence is a -TWC sequence and is a subgraph of , so by using inequality (14) and the triangular inequality, we haveSetting in inequality (15), we haveAgain is a -TWC sequence, so for all , (), we haveSince , so we obtainThus, is a Cauchy sequence in . Also, since is -complete, so converges in and by assumption, there exists and such that () or () for every andwhich shows that converges to .
Theorem 15. If assumptions given in Theorem 14 are satisfied and, in addition, we assume that the graph is weakly connected, then the fixed point of is unique.
Proof. By Theorem 14, we have is a -PS with initial point converges to . By assumption, or and thus, we obtainBy (ii), we haveLetting and using the fact that , we have . Thus, and therefore is a fixed point of . Now, we suppose that is another fixed point of . Assume that , there exists in such a way that and with () ∈ for . Now, since is Fisher graph contraction, so by using assumption (i) repeatedly, we have () ∈ for all . Now, using (ii) as we did in Theorem 14, we havewhere . Now, by using the triangle inequality, we haveSince , so this implies that and . Now, letting the limit as , we obtain . Thus, is the unique fixed point.
Corollary 16. Let be an -complete graphical -metric space and let Suppose that there exists nonnegative constant such that for every with () ∈ we have(i) satisfies the property (ii)There exists with for some Then, there exists such that the -PS with initial point is -TWC and as .
Proof. Take and in Theorem 14.
Remark 17. If in Definition 8, then a graphical -metric space reduced to a graphical metric space, so the following result is a direct consequence of Theorem 14.
Corollary 18. Let be an -complete graphical metric space and let Suppose that there exist nonnegative constants with such that for every with () ∈, we have(i) satisfies the property (ii)There exists with for some Then, there exists such that the -PS with initial point is -TWC and as .
Corollary 19 (see [6]). Let be an -complete graphical metric space and let Suppose that there exists nonnegative constant such that for every with () , we have(i) satisfies the property (ii)There exists with for some Then, there exists such that the -PS {} with initial point is -TWC and as .
Proof. Take and in Corollary 18.
Now, we state Dass–Gupta graph contraction in the background of a graphical -metric space.
Definition 20. Let be a graphical -metric space and Then, is said to be Dass–Gupta graph contraction for on if(i) is edge preserving, i.e., for each if () implies (ii)There exist nonnegative constants with such that for every with () ∈, we have
Example 21. Let be equipped with which is defined byThen, is a graphical -metric space with the coefficient . Define by Now, taking such that andThen, is a Dass and Gupta graph contraction for and . It is shown in Figure 2.
Observe that, is not a Dass–Gupta contraction, as
Remark 22. Any Dass–Gupta contraction is Dass–Gupta graph contraction along with and defined by and .(i)If is a Dass–Gupta graph contraction with parameters and , then is a Banach graph contraction in .(ii)Since every is a and also every is a , thus our theorems in this paper are precise generalizations of some of the results regarding Dass–Gupta contraction (see e.g., [9]).Now, we present our leading theorem regarding Dass–Gupta graph contraction in the following way.
Theorem 23. Let be an -complete graphical -metric space and let be a Dass and Gupta’s graph contraction. Suppose that these assertions hold:(i) satisfies the property (ii)There exists with for some Then, there exists such that the -PS with initial point is -TWC and as .
Proof. Let be such that , for some . As is a -PS originating from , there is a path such that and and () ∈ for . Now, by assumption (i), we have () for . It yields that is a path from to having length and therefore . Continuing in this way, we get is a path from to of length , and hence, , for all . Hence, is a -TWC sequence. Thus, () ∈ for and . Now, by (ii), we havewhich yields thatSince , so taking , it follows from the above inequality thatwhere . Repeating in this way, we haveAs the sequence is a -TWC sequence and is a subgraph of , so by using inequality (35) and the triangular inequality, we haveSetting in inequality (36), we haveAgain is a -TWC sequence, so for all , , we haveSince , so we obtainThus, is a Cauchy sequence in . Also, since is -complete, so converges in and by assumption, there exists and such that () ∈ or () ∈ for every andwhich shows that converges to .
Theorem 24. If assumptions given in Theorem 23 are satisfied and, in addition, we assume that the graph is weakly connected, then the fixed point of is unique.
Proof. By Theorem 23, we have is a -PS with initial point converges to . By assumption, or and thus, we obtainBy (ii), we haveTaking the limit as and using the fact that , we have . Thus, and therefore is a fixed point of . Now, we suppose that is another fixed point of . Assume that , there exists in such a way that and with () ∈ for . Now, since is Dass–Gupta graph contraction, so by using assumption (i) repeatedly, we have () for . Now, using (ii) as we did in Theorem 23, we havewhere . Now, by using the triangle inequality, we haveSince , so this implies that and . Now, letting the limit as , we obtain . Thus, is a unique fixed point.
Remark 25. If in Definition 8, then the graphical -metric space reduced to a graphical metric space, then following result is a direct consequence of Theorem 23.
Corollary 26. Let be an -complete graphical metric space and let Suppose that there exist nonnegative constants with such that for every with () ∈, we have(i) satisfies the property (ii)There exists with for some Then, there exists such that the -PS with initial point is -TWC and as .
Example 27. Let be equipped with such that andDefine the graphical -metric byClearly, is a graphical -metric space with the coefficient . Define byfor all . There exists such that , that is, and the condition (27) is satisfied for and Hence, is Dass–Gupta graph contraction on . Hence, all assumptions of Theorem 23 are satisfied and 0 is the required fixed point of mapping .
Figure 3 exemplifies the weighted graph for , where the value of is equal to the weight of edge and .
4. Conclusion
In this article, we have introduced the notion of Dass–Gupta graph contraction in the background of graphical -metric spaces and established some fixed-point theorems. We also supplied some nontrivial examples to show the validity of obtained results. We hope that the obtained theorems in this article will make new relations for those people who are employing in graphical -metric spaces.
The prospective work in this direction will focus on finding the common fixed points of self-mappings and set valued mappings in the context of graphical -metric spaces. Differential and integral inclusions can be investigated as applications of these results.
Data Availability
The required data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declared that there are no conflicts of interest.