Journal of Mathematics
Volume 2019 (2019), Article ID 2846315, 8 pages
https://doi.org/10.1155/2019/2846315

Research Article

Common Fixed Points in Generalized Metric Spaces with a Graph

Karim Chaira,¹ Mustapha Kabil,² and Abdessamad Kamouss²

¹Laboratory of Algebra, Analysis and Applications, Faculty of Sciences Ben M'sik, University of Hassan II Casablanca, Casablanca, Morocco
²Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies Mohammedia, University Hassan II Casablanca, Morocco

Correspondence should be addressed to Abdessamad Kamouss; akamouss@gmail.com

Received 21 November 2018; Accepted 16 January 2019; Published 3 February 2019

Academic Editor: Nan-Jing Huang

Copyright © 2019 Karim Chaira et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to prove the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings defined on generalized metric spaces with a graph. Our results improve and extend several recent results of metric fixed point theory.

1. Introduction

The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous research activity. In 1976, Jungck [1] proved a common fixed point theorem for commuting maps such that one of them is continuous. In 1982, Sessa [2] generalized the concept of commuting maps to weakly commutating pair of self-mappings. In 1986, Jungck generalized this idea, first to compatible mappings [3] and then in 1996 to weakly compatible mappings [4]. Using the weakly compatibility, several authors established coincidence points results for various classes of mappings on metric spaces with Fatou property (see [5, 6]). In 2011, Haghi et al. showed that some coincidence point and common fixed point generalizations in fixed point theory are not real generalizations as they could easily be obtained from the corresponding fixed point theorems.

Recently, Jleli and Samet introduced a new concept of generalized metric spaces (also known as JS-metric spaces) recovering various topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces (see [7]).

Motivated by the ideas given in some recent works on metric space with a graph [8–21], we extend some common fixed point theorems of Banach, Chatterjea, and Kannan contractions in standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces (see, for example, [20, 22–24]) to common fixed point theorems in generalized metric spaces with a graph. As corollaries, we obtain some results in generalized metric spaces for these contractions. Finally, some examples are given to illustrate our results.

2. Some Basic Concepts

In this section, we give some basic notations, definitions, and useful results in generalized metric spaces endowed with a graph.

Let be a nonempty set and be a given mapping.

For every , let us define the set

Definition 1 (see [7]). We say that is a generalized metric on if it satisfies the following conditions:
for every , we have ;
for every , we have ;
there exists such that if , , then

In this case, we say that the pair is a generalized metric space.
Obviously, if the set is empty for every , then is a generalized metric space if and only if and are satisfied. A sequence in a generalized metric space is said to be -convergent to if . Note that if the set is not empty for some , then .
A sequence in a generalized metric space is said to be a -Cauchy sequence if . Note that, in generalized metric spaces, a sequence has at most one limit and a -convergent sequence may not be -Cauchy sequence. Moreover, is said to be -complete if every -Cauchy sequence in is -convergent to some element in .
The next example shows that a -convergent sequence may not be a -Cauchy sequence.

Example 2. Let , and let be defined as follows:

Now we check the axioms of a generalized metric space:
either or . Thus, and .
It is clear that, for all , .
For all , we have .
If , then we can always find a sequence such that . Taking , we have
Now, in this structure, let us consider the sequence where for all in . Then,

But,

This shows that is a -convergent sequence but not a -Cauchy sequence. Note that for any such that , .

Now, we recall some preliminaries from graph theory which are needed for the sequel. The basic concepts related to a graph may be found in any textbook on graph theory, see, for example, [25, 26]. A directed graph or digraph is determined by a nonempty set of its vertices and the set of its directed edges. Let denote the diagonal of the Cartesian product . A digraph is said to be reflexive if the set ) of its edges contains all loops; i.e., . By we denote the graph obtained from by reversing the direction of edges and by we denote the undirected graph obtained from by ignoring direction of the edges.

Throughout this paper, let be a generalized metric space. Consider a directed graph G such that the set of its vertices coincides with , and the set of its edges contains all loops. We assume that has no parallel edges. Moreover, we will use the concept of increasing or decreasing sequences in the sense of a digraph. Therefore, the following definitions are needed.

Definition 3 (see [27]). Let be a digraph. A sequence is said to be

(1)G-increasing, if for all ;(2)G-decreasing, if for all ;(3)G-monotone, if it is either G-increasing or G-decreasing.

We will need to assume a property introduced in [28, 29] in partially ordered sets and in [19] in metric spaces with a graph. The digraph is said to satisfy the (JNRL) property, if for any G-increasing sequence (resp., G-decreasing sequence) which -converges to some , we have (resp., ) for any . Let and be two self-mappings on . The following definitions and proposition will be needed in the sequel.

Definition 4 (see [4, 30]). If there exists such that , then is called a coincidence point of and , while is called a point of coincidence (or coincidence value) of and . If , then is called a common fixed point of and .
The pair of mappings and is said to be weakly compatible if they commute at their coincidence points.
The digraph is said to satisfy the property (P) for and , if are points of coincidence of and in , then and .

Proposition 5 (see [30]). Let and be weakly compatible self-mappings on a nonempty set . If and have a unique coincidence point , then is the unique common fixed point of and .

Now, we introduce G-Banach, G-Chatterjea, and G-Kannan S-contractions mappings in generalized metric spaces.

Definition 6. Let , be two self-mappings. We say that is

(1)G-Banach S-contraction if there exists such that for every

(2)G-Chatterjea S-contraction if there exists such that for every

(3)G-Kannan S-contraction if there exists such that for every

The number is called the constant of .

3. Main Results

In this section we establish common fixed point theorems for a pair of weakly compatible self-mappings and such that is a G-Banach, G-Chatterjea, or G-Kannan -contraction in the framework of generalized metric spaces with a reflexive digraph .

Throughout this section we assume that is a generalized metric space with , and is a reflexive directed graph such that , and the graph G has no parallel edges.

Let and be two self-mappings on such that .

If is arbitrary, we can choose a point in such that . Continuing in this way, for a value in , we can find such that

By , we denote the set of all elements of such that for . The following notation is useful in the sequel:

Theorem 7. Let be a generalized metric space endowed with a reflexive digraph such that and has no parallel edges and satisfies the (JNRL) property. Let and be two self-mappings on such that is a G-Banach -contraction, is a -complete subspace of , and .
Suppose that there exists such that ; then and have a point of coincidence . Moreover, and have a unique point of coincidence in if the digraph has the property (P) for and .
Furthermore, if and are weakly compatible, then and have a unique common fixed point in .

Proof. Suppose that there exists in such that .
Let and be in such that . Since and is a G-Banach S-contraction, then

and then, by induction on m, we can get

Then when , which implies that is a -Cauchy sequence in .
Since is a -complete subspace of X, then the sequence -converges to some . Thus there exists such that .
Let . Since satisfies the (JNRL) property, then and since is a G-Banach S-contraction, then

Since , then .
With the uniqueness of the limit we get . Thus is a point of coincidence of and ( is a coincidence point of and ).
Suppose that there exists another point of coincidence and such that . With the property (P), we have and .
Since is a G-Banach S-contraction, then we have ; thus ; then , proving that . Since and are weakly compatible, then by Proposition 5, is the unique common fixed point of and T.

In the next theorem, we establish a common fixed point result for G-Chatterjea S-contraction.

Theorem 8. Let be a generalized metric space endowed with a reflexive digraph such that and has no parallel edges and satisfies the (JNRL) property. Let and be two self-mappings on such that is a G-Chatterjea -contraction and is a -complete subspace of and .

(1)Suppose that there exists such that ; then the sequence defined by (9) -converges to with . Moreover, if , then is a point of coincidence of and in .(2)Moreover, and have a unique point of coincidence in if the digraph has the property (P) for and . Furthermore, if and are weakly compatible, then and have a unique common fixed point in .

Proof. (i) Suppose that there exists in such that . Let and be in such that . Let us prove that

with .
We prove statement (14) by two-dimensional induction on for every .
Observe that

Thus, inequality (14) holds for with .
Now, assume that inequality (14) holds for any such that and , and let such that and .
Since is a G-Chatterjea S-contraction and , then

Thus,

Since and , the inductive hypothesis gives

From inequality (17), we get

proving that inequality (14) holds for such that .
Since , , and , then, for any such that , we have

Then when and , which implies that is a -Cauchy sequence. Since is a -complete subspace of X, then the sequence -converges to some ; that is, there exists such that . Since satisfies the (JNRL) property, then . Since is a G-Chatterjea S-contraction, then for any we have

Now we prove that for any integer we have

For ,

Let . Suppose that inequality (22) holds for and prove that it holds for .

Since , then . With the uniqueness of the limit we get . Thus is a point of coincidence of and ( is a coincidence point of and ).
(ii) Assume that there exists another point of coincidence and such that , , and .
Since is a G-Chatterjea S-contraction, then we have

and thus , and then and . Since and are weakly compatible, then, by Proposition 5, is the unique common fixed point of and T.

In the next theorem, we establish a common fixed point result for G-Kannan S-contraction.

Theorem 9. Let be a generalized metric space endowed with a reflexive digraph such that and has no parallel edges and satisfies the (JNRL) property. Let and be two self-mappings on such that is a G-Kannan -contraction with constant and is a -complete subspace of and .

(1)Suppose that there exists such that ; then the sequence defined by (9) -converges to with . Moreover, if then is a point of coincidence of and in .(2)Moreover, and have a unique point of coincidence in if the digraph has the property (P) for and . Furthermore, if and are weakly compatible, then and have a unique common fixed point in .

Proof. (i) Suppose that there exists in such that . Let and be in such that .
Since is a G-Kannan S-contraction and , then we have

which implies that

Then, by induction, we can get

where . Now by using (8) we get

Since , then when and , which implies that is a -Cauchy sequence. Since is a -complete subspace of X, then the sequence -converges to some . Thus there exists such that .
Since satisfies the (JNRL) property, then . Since is a G-Kannan S-contraction, then for any we have

Taking superior limit as , we get

Thus Since , then . Thus is a point of coincidence of and ( is a coincidence point of and ).
(ii) Suppose that there exists another point of coincidence and such that . With the property (P), we have and .
Since is a G-Kannan S-contraction, then we have

and then .
Since and are weakly compatible, then, by Proposition 5, is the unique common fixed point of and T.

Example 10. Let and define by for all . Then is a -complete generalized metric space with the coefficient . Consider the graph on defined by and

The digraph satisfies the (JNRL) property; in fact let be a G-monotone sequence in which -converges to some .

(1)Let be a G-increasing sequence; i.e., for any ; then we have the following:

(a)If such that and , then for all (b)If , then is a null sequence or is a stationary sequence.(c)If with , then is a stationary sequence. In the three cases, we have .

(2)Let be a G-decreasing sequence; i.e., for any ; then we have the following:

(a)If such that , then is a constant sequence.(b)If with , then is a stationary sequence or for any such that is a increasing integer sequence and -converges to 0. In both cases, we have .

Let and be two self-mappings on defined by

and for all . Obviously, .

Let such that .

(1)If , then and so . Thus .(2)If , then , and so .

(3)If , , then , and so .

In all cases we have

proving that is a G-Banach, G-Chatterjea, and G-Kannan S-contraction, respectively.

For we have and we can verify that . In fact, (9) gives that and so . Proceeding in this way, we get for and hence for . Then -converges to 0. Since and , then is a point of coincidence of and . Obviously satisfies the property (P) for and .

Furthermore, and are weakly compatible. Thus, we have all the conditions of Theorems 7, 8, and 9, proving that and have a unique common fixed point in which is .

The next remark shows that we cannot relax the weak compatibility condition in Theorem 7.

Remark 11. In Example 10, if we take for all , and , then we can easily verify that is a G-Banach S-contraction, and ; i.e., 2 is a coincidence point of and . Since , then and are not weakly compatible. However, we can check that all other conditions of Theorem 7 are satisfied. Despite the fact that and have a unique point of coincidence 1, they do not have any common fixed point.

Remark 12.

(1)By taking , where is the complete graph, i.e., , we get corollaries for Theorems 7, 8, and 9 in the framework of generalized metric space without using graph for Banach, Chatterjea, and Kannan S-contractions.(2)Since standard metric spaces and b-metric spaces are particular generalized metric spaces, then our results can be viewed as generalizations and extensions of corresponding results in [20, 22–24] and several other comparable results.(3)By taking in Theorems 7, 8, and 9, we obtain versions of Banach, Chatterjea, and Kannan fixed point theorems in generalized metric spaces with a graph.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors thank the organizers of the International Workshop of Mathematics and Applications WIMA’17 for accepting their oral presentation of the summary of this work.

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