Common Fixed Point and Coupled Coincidence Point Theorems for Geraghty’s Type Contraction Mapping with Two Metrics Endowed with a Directed Graph

Abstract

The purpose of this paper is to present some existence and uniqueness results for common fixed point theorems for - contraction mappings with two metrics endowed with a directed graph. In addition, by using our main results, we obtain some results about coupled coincidence points endowed with a directed graph. Our results generalize those presented in previous papers.

1. Introduction and Preliminaries

Geraghty [1] introduced an interesting class of functions satisfying that and they have shown some results which are a generalization of Banach’s contraction principle in 1973.

Recently, Martínez-Moreno et al. [2] gave some new common fixed point theorems for Geraghty’s type contraction mappings employing the monotone property with two metrics by using -compatibility and -uniform continuity where the concept of -compatibility is defined as follows.

Definition 1 (see [3]). Let be a metric space, and let be two mappings. The mappings and are said to be -compatible ifwhenever is a sequence in such that .

In [2], there are many results on multidimensional common fixed point theorems which generalize the results in paper [4].

The fixed point theorem employing the concept of metric spaces endowed with a graph was initiated by Jachymski [5], which generalizes the Banach contraction principle to mappings on metric spaces equipped with a graph. Also, the definitions of -continuous and property were given in [5].

Definition 2 (see [5]). A function is said to be -continuous if and only if, for any which has a sequence in , , and for , we have that

Definition 3 (see [5]). Suppose that is a metric space and is a directed graph. The triple is said to have property if and only if for any sequence in such that and , where , we have that .

After the discovery above, many researchers have studied the topic of existence of a fixed point for single-valued mappings and multivalued mappings in various spaces equipped with a graph; see [6–11].

In the case of coupled fixed point with a graph, which will be our topic of interest, we now give a collection of notions and definitions from [12].

Definition 4 (see [12]). The given functions and are said to be -edge preserving if and only if the following condition is satisfied:

Definition 5 (see [12]). Suppose that is a metric space and is the set of all edges of the graph . is said to have the transitivity property if and only if, for all , the following is true:

Definition 6 (see [12]). A function is said to be -continuous if and only if the following are true. For any and any sequence of positive integers such that , , and as , we have that as ,

In 2015, Suantai et al. [12] used the above definitions to investigate some results on existence and uniqueness for coupled coincidence points and common fixed points of - contraction mappings in complete metric spaces endowed with a directed graph. Their results also generalize others in partially ordered metric spaces.

The aim of this paper is to present some existence and uniqueness results for common fixed point theorems for - contraction mappings with two metrics endowed with a directed graph. Furthermore, by using our main results, we are able to prove some results about coupled coincidence point endowed with a directed graph. Our results generalize the results obtained in [2, 12].

2. Main Results

Throughout this paper we shall say that will satisfy the standard conditions as mentioned in [5].

In addition, we will introduce the concept of -Cauchy and edge preserving which are an effective tool as follows.

Definition 7. Let and be two metric spaces, and let and be two mappings. The mapping is said to be -Cauchy on if, for any sequence in such that is Cauchy sequence in , then is Cauchy sequence in .

It is easy to see that if is -uniformly continuous on , then is -Cauchy on .

Definition 8. Let be a directed graph, and let be two mappings. We say that is -edge preserving with respect to if

Let denote the class of all functions which satisfy the following conditions: is nondecreasing. is continuous..

We now introduce a new class of the Geraghty type contractions in the following definition.

Definition 9. Let be a metric space endowed with a directed graph , and let be given mappings. The pair is called a -contraction with respect to if(1) is -edge preserving with respect to ;(2)there exist two functions and such that, for all such that ,  where   =  .

Let be a metric space endowed with a directed graph satisfying the standard conditions, and let two mappings be given.

We define important subsets of as follows:that is, the set of all coincidence points of mappings and , andthat is, the set of all common fixed points of mappings and .

Now we are ready to present and prove the main results.

Theorem 10. Let be a complete metric space endowed with a directed graph , and let be another metric on Suppose that and is a -contraction with respect to . Suppose that(1) is continuous and is -closed;(2);(3) satisfies the transitivity property;(4)if , one assumes that is -Cauchy on ;(5) is -continuous, and and are -compatible. Then, under these conditions,

Proof. (⇐) Suppose that . Let . We have Then . Hence which means that and thus .
(⇒) Suppose now . Let such that . By the assumption that and , it easy to construct sequences in for which for all . If for some , then is a coincidence point of the mappings and . Therefore, we assume that, for each holds.
Since and is edge preserving with respect to , we have . Continuing inductively, we obtain that for each . Hence it follows from the contractive condition thatOn the other hand, we get If , then, by (12), we obtain that which is a contradiction. So, for all , we have Notice that, in view of (12), we get for all that . From of property of , we have Hence, we deduce that the sequence is nonnegative and nonincreasing. Consequently, there exists such that . We claim that . Suppose, on the contrary, that . Then, due to (12), we have It follows that . Owing to the fact that , we get which implies that , a contradiction. So, we conclude that We assert that is a Cauchy sequence. Suppose, on the contrary, that is not a Cauchy sequence. Thus, there exists such that, for all , there exists such that with the smallest number satisfying the condition below: Then, we have Letting in the above inequality, by (18), we obtain that By the transitivity property of , we get for all . Thus, we have where Hence, we conclude that Keeping (18) and (21) in mind and letting , we derive that By inequality (24), we get and hence , a contradiction. So, we conclude that is a Cauchy sequence in .
Next, we claim that is a Cauchy sequence with respect to .
If , it is trivial. Thus, suppose . Let . Since is a Cauchy sequence in and is -Cauchy on , we have that is Cauchy sequence in . Then there exists with whenever . So is a Cauchy sequence with respect to .
Since is a -closed subset of the complete metric space , there exists such that Now, since is -continuous, and and are -compatible, we have Using the triangle inequality, we have Letting , from (29), the continuity of , and the -continuity of , it follows that which implies that . So is a coincidence point of and .

If , we have the following theorem.

Theorem 11. Let be a complete metric space endowed with a directed graph . Suppose that and is a -contraction with respect to . Moreover, suppose that(1) is continuous and is closed;(2);(3) satisfies the transitivity property;(4)(a) is -continuous and and are -compatible or (b) has property . Then, under these conditions,

Proof. In order to avoid the repetition, following from the same proof in Theorem 10, we can only consider (b) of condition . Since is a Cauchy sequence in and is closed in , there exists such that Now, we show that is a coincidence point of and . Suppose, on the contrary, that . Then . Since has property , we have for each . By using the triangle inequality, we derive which implies that By the property of , we obtain that whereFrom (32), we obtain that Letting in inequality (35), we obtain that so , a contradiction. Therefore . Consequently, we conclude that and have a coincidence point.

Theorem 12. In addition to the hypotheses of Theorem 10, assume that(K)for any such that , we have . If and are -compatible and , then

Proof. Theorem 10 implies that there exists a coincidence point ; that is, . Suppose that there exists another coincidence point such that . Assume that . By assumption (K), , and we have which is a contradiction. Therefore, . Starting from , choose the sequences satisfying for each . Taking into account the properties of coincidence points, it is easy to see that it can be done so that ; that is, for all . Next, set so we get The definition of the sequence implies that for all and hence with respect to . Because and are -compatible, we have that is, . Therefore, we have which we can conclude that is another coincidence point of the functions and . By the above results, we obtain and so is a common fixed point of and . In other words,

If , we get the following.

Definition 13. Let be a metric space endowed with a directed graph , and let be given mappings. The pair is called a -contraction with respect to if(1) is -edge preserving with respect to ;(2)there exist two functions and such that for all such that ,

Applying the similar argument as in the proof of Theorems 10 and 11, we have the following.

Corollary 14. Let be a complete metric space endowed with a directed graph , and let be another metric on Suppose that and is a -contraction with respect to . Moreover, suppose that(1) is continuous and is -closed;(2);(3) satisfies the transitivity property;(4)if , one assumes that is -Cauchy on ;(5)(a)   is -continuous, and and are -compatible or (b) if , assume that has property . Then, under these conditions,

Remark 15. Put and in Corollary 14. In this case, we obtain the results of [2].

Example 16. Let , and let the metrics be defined by for all , respectively, where is a constant real number such that . It is easy to see that .
Now, we consider given by Consider the mappings and defined by for all , respectively.
Next, we show that conditions - in Definition 13 hold as follows.
(1) Let , if then , if , then we obtain , and , and we have and . This implies that .
(2) Let , and let be defined by Let be arbitrary points in and . If , we have and hence the contractive condition (42) holds for this case. In another case, we have , and . Also, we have Therefore, condition holds for the functions and .
Next, the proof for conditions – in Corollary 14 is shown as follows.
(1) The function can be easily seen to be continuous. Moreover, we immediately obtain that is -closed.
(2) By the construction of and , we can conclude that .
(3) is clearly satisfying the transitivity property.
(4) Since , we need to prove that the function is -Cauchy on . Let , and let be a sequence in such that is a Cauchy sequence in . Then, there exist such that, for all . Then we have This implies that is -Cauchy on .
(5) It can be easily shown that is -continuous. As a result, we will only need to prove that and are -compatible. Let be a sequence in such that Then we have and so it follows that . Now, we have as .
It is easy to see that there exists a point such that , and this implies that
Consequently, all the conditions of Corollary 14 hold. Therefore, and have a coincidence point and, furthermore, a point is a coincidence point of the mappings and .

3. Coupled Coincidence Point Theorems

We begin with some useful background.

Definition 17 (see [4]). Let be a metric space, and let . The mappings and are said to be -compatible if hold whenever and are sequences in such that and .

Let be a nonempty set, and let be two mappings.

We define two mappings by for all

The following lemma guarantees that 2-dimensional notion of common fixed coincidence points can be interpreted in terms of two mappings and .

Lemma 18 (see [2]). Suppose that is a nonempty set and and are functions. We say that the point is a coupled coincidence point of and , that is, and (see [13, 14]), if and only if it is a coincidence point of the functions and .

Definition 19. Suppose that is a directed graph and and are two functions. is said to be -edge preserving with respect to if and only if the following holds:

The concept of the cross product of the graphs , is defined by where

Now, the following lemma shows that being edge preserving and the transitivity property for 2-dimensional case can be interpreted in terms of two mappings and

Lemma 20. Let and be two mappings, and let be a metric space endowed with a directed graph . Then, is -edge preserving with respect to if and only if is edge preserving with respect to .

Proof. Let be -edge preserving with respect to , and let be such that Let be edge preserving with respect to , and let be such that

Lemma 21. Let be two mappings, and let be a metric space endowed with directed graphs and . Then, and satisfy the transitivity property if and only if satisfies the transitivity property.

Proof. Suppose that and satisfy the transitivity property. Let be such that Suppose that satisfies the transitivity property. Let be such that

Let be a metric space endowed with a directed graph satisfying the standard conditions, and let and be two functions. We will define the following sets:

Suantai et al. gave the notion of a --contraction as in [12]. We are ready to present an application of Corollary 14 in order to deduce results about coupled fixed points.

Theorem 22. Let be a complete metric space endowed with a directed graph , let be another metric on , and let and be two functions so that the pair of and is a -contraction. Suppose that the following are true:(1) is a continuous function such that is -closed.(2) is a subset of .(3) satisfies the transitivity property.(4)if , then assume further that is -uniformly continuous on .(5)(a) is -continuous, and and are -compatible or (b) if , assume that has property . Under these conditions if and only if .

Proof. We only need to apply Corollary 14 to the functions and on the complete metric space and the metric space , where the metrics and are defined as follows: If we put , we havefor all .
By Lemmas 20 and 21, we have that is -edge preserving with respect to and satisfies the transitivity property. Since the pair of and is a --contraction, we get that is a --contraction with respect to . Since , for any given , we can construct sequences for whichFrom this, it easy to see that if is -continuous, then is -continuous. Observe that This completes the proof.

Taking in Theorem 22, we have the following Corollary which implies the result in [12].

Corollary 23. Let be a complete metric space endowed with a directed graph , and let and be two mappings such that the pair of and is a -contraction. Suppose that the following hold:(1) is continuous and is closed.(2).(3) satisfies the transitivity property.(4)(a) is -continuous and and are compatible or (b) has property . Under these conditions if and only if .

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by Chiang Mai University, Thailand.