#### Abstract

Recently, fixed point theory on graphs has been considered by many authors. In this paper, by combining some ideas in some published papers and introducing -type quasi-contractions, we give some fixed point results for -type quasi-contractions on graphs. The results improve some old results in the literature.

#### 1. Introduction

In 2009, Ilić and Rakočević proved that quasi-contraction maps on normal cone metric spaces have a unique fixed point [1]. Then, Kadelburg et al. generalized their results by considering an additional assumption [2]. Also, they proved that quasi-contraction maps on cone metric spaces have the property () whenever . Later, the authors proved same results without the additional assumption and for by providing a new technical proof [3]. Also, there are some works on quasi-contractive multifunctions (see, e.g., [4, 5]).

In 2008, Suzuki introduced a new type of mappings and a generalization of the Banach contraction principle [6]. Later, his method extended for mappings and multifunctions (see, e.g., [7] and the references therein and [8]). On the other hand, Echenique gave a short constructive proof for Tarski's fixed point theorem in 2005 by using graphs [9]. In 2006, Espínola and Kirk started combining fixed point theory and graph theory [10]. In 2008, Jachymski provided some fixed point results for Banach contractions on a graph [11]. Recently, fixed point theory on graphs has been considered by many authors (see, e.g., [12–16]).

Let be a metric space, , a directed graph such that , and the set of its edges contains all loops. We denote the conversion of a graph by ; that is, the graph obtained from by reversing the direction of the edges. Moreover, denotes the undirected graph obtained from by ignoring the direction of the edges. In this paper, we consider undirected graphs. We say that a self-map on preserves the edges of whenever which implies that for all . A finite path of length in from to is a sequence of distinct vertices such that , , and for (see, e.g., [12]). A graph is connected if there is a path between any two vertices. is weakly connected if is connected. We denote by the set of all vertices in that there is a path between and those.

In 2008, Jachymski used the notion of -graphs for obtaining the main results of [11]. We say that is a -graph whenever for each sequence in with and for all , there is a subsequence such that for all [11]. This notion has been used by many authors in the literature, specially on ordered metric spaces and obtaining solutions of some differential equations (see, e.g., [17]).

The condition that the graph is a -graph looks quite strong and in this reason, Aleomraninejad et al. defined the notion of -graphs and showed that these notions are independent on infinite graphs (see [12]). We say that is a -graph whenever is a convergent sequence to a point and for all , we have [12]. Here, is the sum of edges distance between and ; that is, . They proved the same results for -graphs and -graphs (see the results of [12]). We will use only -graphs in this paper.

In this paper, by combining all of these ideas and introducing -type quasi-contractions, we give some results about fixed points of -type quasi-contractions on graphs. The results improve some old results in the literature.

#### 2. Main Results

Now, we are ready to state and prove our main results. In 2008, Suzuki obtained the following interesting fixed point result [6].

Theorem 1. *Let be a complete metric space and let be a self-map on . Define the nonincreasing function from onto by
**
Assume that there exists , such that
**
for all . Then, there exists a unique fixed point of . Moreover, for all . *

Throughout this paper, suppose that and is a -graph.

*Definition 2. *Let be a metric space, a self map on , and a graph with . We say that is a -type quasi-contraction whenever preserves the edges of and there exists , such that
for all , where

Theorem 3. *Let be a complete metric space, a -type quasi-contraction map with such that , for all . Then, has a unique fixed point. *

*Proof. *Take . Since , we have
Since , we obtain . Hence,
for all natural number and so is a Cauchy sequence. Since is complete, converges to some . Since is a -Graph, there is a subsequence such that for all . Hence, for all . We claim that for some natural number . Arguing by contradiction, we assume that for all . Fix a natural number and put for all . Choose a natural number such that for all . If , then
It follows that
and so , . Since ≤ , we obtain
for all . Now, we assume that ; then by (6), we have
This is a contradiction, since . So, we have
and by (3), we obtain
By considering the above inequality and (9), we deduce that
that is a contradiction. Therefore, there exists such that . Since is a Cauchy sequence, we obtain . In fact, if , from for all , it follows that is not a Cauchy sequence. Thus, is a fixed point of . The uniqueness of the fixed point follows easily.

*Question 1. *Does Theorem 3 hold for each ?

Theorem 4. *Let be a complete metric space. Then, the following statements are equivalent*(i)*is weakly connected,*(ii)*for each -type quasi-contraction map and , the sequences and are Cauchy equivalent, where ,*(iii)*for each -type quasi-contraction map , . *

*Proof. * Let be a -type quasi-contraction map and . Since , there is a path in from to . Since , for all and . Let . Put and . If one of the following inequalities holds
Then, we have
If , then
If , then
Without loss of generality, suppose that . Then,
and so . Hence,
Now, suppose that both of the inequalities (14) do not hold. If
then
and so
If , then we can continue in a similar process for . In the general case, we get and so . Thus,
Therefore, and are Cauchy equivalent.

Let . By using and the above process, we obtain easily that .

If is not weakly connected, then there exists such that is not empty. Take and define
Clearly, . Now, we show that is a -type quasi-contraction. For this reason, let . Since , either or . In both cases, we get . Thus, is a -type quasi-contraction which has two fixed points. This contradiction completes the proof.

Theorem 5. *Let be a complete metric space and let be a -type quasi-contraction and orbitally -continuous self-map on . Then,*(i)*for each , is a Picard operator,*(ii)*. *

*Proof. *Let . Then, . It is easy to check that is a Cauchy sequence. Let . Since is a -Graph, there exists a subsequence such that for all . Thus, for all . Since , . Since is orbitally -continuous, which yields . To prove , define the mapping by for all . It is sufficient to show that is a bijection from onto . Since , we get which yields . On the other hand, if , then which implies that . Thus, is a surjection from onto . Now, if with , then and so by using we obtain
which implies that . Therefore, is an injective and this completes the proof.

We need the following results for our last result.

Lemma 6 (see [18]). *Let be a nonempty set and let be a mapping. Then, there exists a subset such that and is one-to-one. *

Lemma 7 (see [8]). *Let be a nonempty set and that the mappings have a unique point of coincidence in . If and are weakly compatible, then and have a unique common fixed point. *

Theorem 8. *Let be a metric space, and let and be two self-maps on such that and is complete. Suppose that and satisfy the following conditions:*(i)* implies that ,*(ii)*if and for some , then ,*(iii)*there exists such that and implies that
**Then, and have a unique coincidence point. Moreover, if and are weakly compatible, then and have a unique fixed point. *

*Proof. *By using Lemma 6, there exists such that is one-to-one and . Define the self-map by . Clearly, is well defined and preserves the edges of . In fact, implies that . Note that implies that
Also, and lie in for all . To see this, take . Then, for some and so . By using , . Since is complete, by using Theorem 3, has a unique fixed point in , namely, . Thus, is a coincidence point of and . Note that the assumption shows the uniqueness of the coincidence point of and . Now, by using Lemma 7, it is easy to see that if and are weakly compatible, then and have a unique fixed point.

#### Acknowledgment

The authors would like to thank the anonymous referee for his helpful comments on an earlier version. This article was funded by the Deanship of Scientic Research (DSR), King Abdulaziz University, Jeddah. N. Shahzad acknowledges with thanks DSR for financial support.