Fixed Points of 𝐺-Type Quasi-Contractions on Graphs

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 𝐺-type quasi-contractions, we give some fixed point results for 𝐺-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 (𝑃) whenever 𝜆∈(0,1/2). Later, the authors proved same results without the additional assumption and for 𝜆∈(0,1) 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 (𝑋,𝑑) be a metric space, Δ={(𝑥,𝑥)∶𝑥∈𝑋}, 𝐺 a directed graph 𝐺 such that 𝑉(𝐺)=𝑋, and the set 𝐸(𝐺) of its edges contains all loops. We denote the conversion of a graph 𝐺 by 𝐺⁻¹; that is, the graph obtained from 𝐺 by reversing the direction of the edges. Moreover, 𝐺 denotes the undirected graph obtained from 𝐺 by ignoring the direction of the edges. In this paper, we consider undirected graphs. We say that a self-map 𝑇 on 𝑋 preserves the edges of 𝐺 whenever (𝑥,𝑦)∈𝐸(𝐺) which implies that (𝑇𝑥,𝑇𝑦)∈𝐸 for all 𝑥,𝑦∈𝑋. A finite path of length 𝑛 in 𝐺 from 𝑥 to 𝑦 is a sequence {𝑥_𝑖}^𝑛_𝑖=0 of distinct vertices such that 𝑥₀=𝑥, 𝑥_𝑛=𝑦, and (𝑥_𝑖,𝑥_𝑖+1)∈𝐸(𝐺) for 𝑖=0,1,…,𝑛−1 (see, e.g., [12]). A graph 𝐺 is connected if there is a path between any two vertices. 𝐺 is weakly connected if 𝐺 is connected. We denote by [𝑥]_𝐺 the set of all vertices in 𝐺 that there is a path between 𝑥 and those.

In 2008, Jachymski used the notion of 𝐶-graphs for obtaining the main results of [11]. We say that 𝐺 is a 𝐶-graph whenever for each sequence {𝑥_𝑛}_𝑛≥0 in 𝑋 with 𝑥_𝑛→𝑥 and (𝑥_𝑛,𝑥_𝑛+1)∈𝐸(𝐺) for all 𝑛≥0, there is a subsequence {𝑥_𝑛𝑘}_𝑘≥0 such that (𝑥_𝑛𝑘,𝑥)∈𝐸(𝐺) for all 𝑘≥0 [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 𝐶-graph looks quite strong and in this reason, Aleomraninejad et al. defined the notion of 𝑃-graphs and showed that these notions are independent on infinite graphs (see [12]). We say that 𝐺 is a 𝑃-graph whenever {𝑥_𝑛}_𝑛≥0 is a convergent sequence to a point 𝑥 and 𝑥_𝑛∈[𝑥]_𝐺 for all 𝑛≥0, we have 𝑟(𝑥_𝑛,𝑥)→0 [12]. Here, 𝑟(𝑥,𝑦) is the sum of edges distance between 𝑥 and 𝑦; that is, 𝑟(𝑥,𝑦)=∑^𝑛_𝑖=1𝑑(𝑥_𝑖−1,𝑥_𝑖). They proved the same results for 𝐶-graphs and 𝑃-graphs (see the results of [12]). We will use only 𝐶-graphs in this paper.

In this paper, by combining all of these ideas and introducing 𝐺-type quasi-contractions, we give some results about fixed points of 𝐺-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 (𝑋,𝑑) be a complete metric space and let 𝑇 be a self-map on 𝑋. Define the nonincreasing function 𝜃 from [0,1) onto (1/2,1] by 𝜃(𝑟)=√1𝑖𝑓0≤𝑟≤5−12,(1−𝑟)𝑟⁻²√𝑖𝑓5−12<𝑟<2^−1/2,(1+𝑟)⁻¹𝑖𝑓2^−1/2≤𝑟<1. (1) Assume that there exists 𝑟∈[0,1), such that 𝜃(𝑟)𝑑(𝑥,𝑇𝑥)≤𝑑𝑥,𝑦𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝑡ℎ𝑎𝑡𝑑𝑇𝑥,𝑇𝑦≤𝑟𝑑𝑥,𝑦 (2) for all 𝑥,𝑦∈𝑋. Then, there exists a unique fixed point 𝑧 of 𝑇. Moreover, lim_𝑛→∞𝑇^𝑛𝑥=𝑧 for all 𝑥∈𝑋.

Throughout this paper, suppose that 𝐸=𝐸(𝐺) and 𝐺 is a 𝐶-graph.

Definition 2. Let (𝑋,𝑑) be a metric space, 𝑇 a self map on 𝑋, and 𝐺 a graph with 𝑉(𝐺)=𝑋. We say that 𝑇 is a 𝐺-type quasi-contraction whenever 𝑇 preserves the edges of 𝐺 and there exists 𝑟∈[0,1), such that 𝜃(𝑟)𝑑(𝑥,𝑇𝑥)≤𝑑𝑥,𝑦impliesthat𝑑𝑇𝑥,𝑇𝑦≤𝑟𝑀𝑥,𝑦 (3) for all (𝑥,𝑦)∈𝐸, where 𝑀𝑑(),1𝑥,𝑦=max𝑥,𝑦,𝑑𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦2𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥. (4)

Theorem 3. Let (𝑋,𝑑) be a complete metric space, 𝑇 a 𝐺-type quasi-contraction map with 𝑟²+𝑟<1 such that (𝑥,𝑇𝑥), (𝑥,𝑇²𝑥)∈𝐸(𝐺) for all 𝑥∈𝑋. Then, 𝑇 has a unique fixed point.

Proof. Take 𝑥₀∈𝑋. Since 𝜃(𝑟)𝑑(𝑥₀,𝑇𝑥₀)≤𝑑(𝑥₀,𝑇𝑥₀), we have 𝑑𝑇𝑥₀,𝑇²𝑥₀≤𝑟max𝑑𝑥₀,𝑇𝑥₀,𝑑𝑇𝑥₀,𝑇²𝑥₀,12𝑑𝑥₀,𝑇²𝑥₀≤𝑟max𝑑𝑥₀,𝑇𝑥₀,𝑑𝑇𝑥₀,𝑇²𝑥₀. (5) Since (1/2)𝑑(𝑥₀,𝑇²𝑥₀)≤(1/2)[𝑑(𝑥₀,𝑇𝑥₀)+𝑑(𝑇𝑥₀,𝑇²𝑥₀)], we obtain 𝑑(𝑇²𝑥₀,𝑇𝑥₀)≤𝑟𝑑(𝑥₀,𝑇𝑥₀). Hence, 𝑑𝑇^𝑛𝑥₀,𝑇^𝑛+1𝑥₀≤𝑟^𝑛𝑑𝑥₀,𝑇𝑥₀ (6) for all natural number 𝑛 and so {𝑇^𝑛𝑥₀}_𝑛≥1 is a Cauchy sequence. Since 𝑋 is complete, {𝑇^𝑛𝑥₀}_𝑛≥1 converges to some 𝑥^∗∈𝑋. Since 𝐺 is a 𝐶-Graph, there is a subsequence {𝑇^𝑛𝑘𝑥₀}_𝑘≥1 such that (𝑇^𝑛𝑘𝑥₀,𝑥^∗)∈𝐸 for all 𝑘≥1. Hence, (𝑇^{𝑛𝑘+𝑗}𝑥₀,𝑇^𝑗𝑥^∗)∈𝐸 for all 𝑗≥1. We claim that 𝑇^𝑗0𝑥^∗=𝑥^∗ for some natural number 𝑗₀. Arguing by contradiction, we assume that 𝑇^𝑗𝑥^∗≠𝑥^∗ for all 𝑗. Fix a natural number 𝑗 and put 𝑥_𝑛+1=𝑇^𝑛𝑥₀ for all 𝑛≥1. Choose a natural number 𝑛₀ such that 𝑑(𝑥_𝑛,𝑥^∗)≤𝑑(𝑥^∗,𝑇^𝑗𝑥^∗)/3 for all 𝑛≥𝑛₀. If 𝑛_𝑘≥𝑛₀, then 𝑑𝑥_{𝑛𝑘+𝑗},𝑇𝑥_{𝑛𝑘+𝑗}≤𝑑𝑥_{𝑛𝑘+𝑗},𝑥^∗+𝑑𝑥^∗,𝑥_{𝑛𝑘+𝑗+1}≤23𝑑𝑥^∗,𝑇^𝑗𝑥^∗=𝑑𝑥^∗,𝑇^𝑗𝑥^∗−13𝑑𝑥^∗,𝑇^𝑗𝑥^∗≤𝑑𝑥^∗,𝑇^𝑗𝑥^∗−𝑑𝑥_{𝑛𝑘+𝑗},𝑥^∗≤𝑑𝑥_{𝑛𝑘+𝑗},𝑇^𝑗𝑥^∗. (7) It follows that 𝑑𝑇𝑥_{𝑛𝑘+𝑗},𝑇^𝑗+1𝑥^∗≤𝑟max𝑑𝑥_{𝑛𝑘+𝑗},𝑇^𝑗𝑥^∗,𝑑𝑥_{𝑛𝑘+𝑗},𝑥_{𝑛𝑘+𝑗+1},𝑑𝑇^𝑗𝑥^∗,𝑇^𝑗+1𝑥^∗,12𝑑𝑥_{𝑛𝑘+𝑗},𝑇^𝑗+1𝑥^∗+𝑑𝑇^𝑗𝑥^∗,𝑥_{𝑛𝑘+𝑗+1}, (8) and so 𝑑(𝑥^∗,𝑇^𝑗+1𝑥^∗)≤𝑟max{𝑑(𝑥^∗,𝑇^𝑗𝑥^∗), 𝑑(𝑇^𝑗𝑥^∗,𝑇^𝑗+1𝑥^∗)}. Since 𝑑(𝑇^𝑗𝑥^∗,𝑇^𝑗+1𝑥^∗) ≤ 𝑟^𝑗𝑑(𝑥^∗,𝑇𝑥^∗), we obtain 𝑑𝑥^∗,𝑇^𝑗+1𝑥^∗≤𝑟^𝑗𝑑𝑥^∗,𝑇𝑥^∗ (9) for all 𝑗. Now, we assume that 𝑑(𝑥^∗,𝑇²𝑥^∗)<𝑑(𝑇²𝑥^∗,𝑇³𝑥^∗); then by (6), we have 𝑑𝑥^∗,𝑇𝑥^∗≤𝑑𝑥^∗,𝑇²𝑥^∗+𝑑𝑇𝑥^∗,𝑇²𝑥^∗<𝑑𝑇²𝑥^∗,𝑇³𝑥^∗+𝑑𝑇𝑥^∗,𝑇²𝑥^∗≤𝑟²𝑑𝑥^∗,𝑇𝑥^∗+𝑟𝑑𝑥^∗,𝑇𝑥^∗. (10) This is a contradiction, since 𝑟²+𝑟<1. So, we have 𝑑𝑥^∗,𝑇²𝑥^∗≥𝑑𝑇²𝑥^∗,𝑇³𝑥^∗=𝜃(𝑟)𝑑𝑇²𝑥^∗,𝑇³𝑥^∗, (11) and by (3), we obtain 𝑑𝑇³𝑥^∗,𝑇𝑥^∗≤𝑟max𝑑𝑥^∗,𝑇²𝑥^∗,𝑑𝑇²𝑥^∗,𝑇³𝑥^∗,𝑑𝑥^∗,𝑇𝑥^∗,12𝑑𝑇²𝑥^∗,𝑇𝑥^∗+𝑑𝑥^∗,𝑇³𝑥^∗. (12) By considering the above inequality and (9), we deduce that 𝑑𝑥^∗,𝑇𝑥^∗≤𝑑𝑥^∗,𝑇³𝑥^∗+𝑑𝑇𝑥^∗,𝑇³𝑥^∗≤𝑟²𝑑𝑥^∗,𝑇𝑥^∗+𝑟𝑑𝑥^∗,𝑇𝑥^∗≤𝑟²+𝑟𝑑𝑥^∗,𝑇𝑥^∗<𝑑𝑥^∗,𝑇𝑥^∗, (13) that is a contradiction. Therefore, there exists 𝑗₀∈ℕ such that 𝑇^𝑗0𝑥^∗=𝑥^∗. Since {𝑇^𝑛𝑥^∗}_𝑛≥1 is a Cauchy sequence, we obtain 𝑥^∗=𝑇𝑥^∗. In fact, if 𝑥^∗≠𝑇𝑥^∗, from 𝑑(𝑇^𝑛𝑗0𝑥^∗,𝑇^𝑛𝑗0+1𝑥^∗)=𝑑(𝑥^∗,𝑇𝑥^∗) for all 𝑛≥1, it follows that {𝑇^𝑛𝑥^∗}_𝑛≥1 is not a Cauchy sequence. Thus, 𝑥^∗ is a fixed point of 𝑇. The uniqueness of the fixed point follows easily.

Question 1. Does Theorem 3 hold for each 𝑟∈[0,1)?

Theorem 4. Let (𝑋,𝑑) be a complete metric space. Then, the following statements are equivalent(i)𝐺is weakly connected,(ii)for each 𝐺-type quasi-contraction map 𝑇∶𝑋_𝑇→𝑋_𝑇 and 𝑥,𝑦∈𝑋, the sequences {𝑇^𝑛𝑥}_𝑛≥1 and {𝑇^𝑛𝑦}_𝑛≥1 are Cauchy equivalent, where 𝑋_𝑇={𝑥∈𝑋∶(𝑥,𝑇𝑥)∈𝐸},(iii)for each 𝐺-type quasi-contraction map 𝑇∶𝑋→𝑋, 𝑐𝑎𝑟𝑑(𝐹𝑖𝑥𝑇)≤1.

Proof. (i)⇒(ii) Let 𝑇∶𝑋→𝑋 be a 𝐺-type quasi-contraction map and 𝑥,𝑦∈𝑋. Since 𝐺𝑦∈[𝑥], there is a path {𝑥₀=𝑥,…,𝑥_𝑁=𝑦} in 𝐺 from 𝑥 to 𝑦. Since (𝑥_𝑖−1,𝑥_𝑖)∈𝐸(𝐺), (𝑇^𝑛𝑥_𝑖−1,𝑇^𝑛𝑥_𝑖)∈𝐸(𝐺) for all 𝑛 and 𝑖=1,…,𝑁. Let 1≤𝑖≤𝑁. Put 𝑥_𝑖−1=𝑎 and 𝑥_𝑖=𝑏. If one of the following inequalities holds 𝜃(𝑟)𝑑𝑇^𝑛−1𝑎,𝑇^𝑛𝑎≤𝑑𝑇^𝑛−1𝑎,𝑇^𝑛−1𝑏or𝜃(𝑟)𝑑𝑇^𝑛−1𝑏,𝑇^𝑛𝑏≤𝑑𝑇^𝑛−1𝑏,𝑇^𝑛−1𝑎. (14) Then, we have 𝑑𝑇^𝑛𝑎,𝑇^𝑛𝑏≤𝑟max𝑑𝑇^𝑛−1𝑎,𝑇^𝑛−1𝑏,𝑑𝑇^𝑛−1𝑎,𝑇^𝑛𝑎,𝑑𝑇^𝑛−1𝑏,𝑇^𝑛𝑏,12𝑑𝑇^𝑛−1𝑎,𝑇^𝑛𝑏+𝑑𝑇^𝑛−1𝑏,𝑇^𝑛𝑎∶=𝑟𝑢_𝑛. (15) If 𝑢_𝑛∈{𝑑(𝑇^𝑛−1𝑎,𝑇^𝑛𝑎),𝑑(𝑇^𝑛−1𝑏,𝑇^𝑛𝑏)}, then 𝑑𝑇^𝑛𝑎,𝑇^𝑛𝑏≤𝑟^𝑛max𝑑(𝑎,𝑇𝑎),𝑑(𝑏,𝑇𝑏). (16) If 𝑢_𝑛=(1/2)[𝑑(𝑇^𝑛−1𝑎,𝑇^𝑛𝑏)+𝑑(𝑇^𝑛−1𝑏,𝑇^𝑛𝑎)], then 𝑑𝑇^𝑛𝑎,𝑇^𝑛𝑏≤𝑟𝑑𝑇^𝑛−1𝑎,𝑇^𝑛𝑏or𝑑𝑇^𝑛𝑎,𝑇^𝑛𝑏≤𝑟𝑑𝑇^𝑛−1𝑏,𝑇^𝑛𝑎. (17) Without loss of generality, suppose that 𝑑(𝑇^𝑛𝑎,𝑇^𝑛𝑏)≤𝑟𝑑(𝑇^𝑛−1𝑎,𝑇^𝑛𝑏). Then, 𝑑𝑇^𝑛𝑎,𝑇^𝑛𝑏≤𝑟𝑑𝑇^𝑛−1𝑎,𝑇^𝑛𝑏≤𝑟𝑑𝑇^𝑛−1𝑎,𝑇^𝑛𝑎+𝑟𝑑𝑇^𝑛𝑎,𝑇^𝑛𝑏, (18) and so 𝑑(𝑇^𝑛𝑎,𝑇^𝑛𝑏)≤(𝑟/(1−𝑟))𝑑(𝑇^𝑛−1𝑎,𝑇^𝑛𝑎)≤(𝑟^𝑛/(1−𝑟))𝑑(𝑎,𝑇𝑎). Hence, 𝑑𝑇^𝑛𝑎,𝑇^𝑛𝑏≤𝑟^𝑛𝑑()(.1−𝑟max𝑎,𝑇𝑎,𝑑𝑏,𝑇𝑏) (19) Now, suppose that both of the inequalities (14) do not hold. If 𝜃(𝑟)𝑑𝑇^𝑛−1𝑎,𝑇^𝑛𝑎>𝑑𝑇^𝑛−1𝑎,𝑇^𝑛−1𝑏, (20) then 𝑑𝑇^𝑛−1𝑎,𝑇^𝑛−1𝑏<𝜃(𝑟)𝑑𝑇^𝑛−1𝑎,𝑇^𝑛𝑎≤𝜃(𝑟)𝑟^𝑛−1𝑑(𝑎,𝑇𝑎), (21) and so 𝑑𝑇^𝑛𝑎,𝑇^𝑛𝑏≤𝑑𝑇^𝑛𝑎,𝑇^𝑛−1𝑎+𝑑𝑇^𝑛−1𝑎,𝑇^𝑛−1𝑏+𝑑𝑇^𝑛−1𝑏,𝑇^𝑛𝑏≤3𝑟^𝑛−1max𝑑(𝑎,𝑇𝑎),𝑑(𝑏,𝑇𝑏). (22) If 𝑢_𝑛=𝑑(𝑇^𝑛−1𝑎,𝑇^𝑛−1𝑏), then we can continue in a similar process for 𝑛−1. In the general case, we get 𝑑(𝑇^𝑛𝑎,𝑇^𝑛𝑏)≤(3𝑟^𝑛−1/(1−𝑟))max{𝑑(𝑎,𝑇𝑎),𝑑(𝑏,𝑇𝑏)} and so 𝑑(𝑇^𝑛𝑎,𝑇^𝑛𝑏)→0. Thus, 𝑑𝑇^𝑛𝑥,𝑇^𝑛𝑦≤^𝑁_𝑖=1𝑑𝑇^𝑛𝑥_𝑖−1,𝑇^𝑛𝑥_𝑖≤3𝑟^𝑛−1≤1−𝑟^𝑁_𝑖=1max𝑑𝑥_𝑖−1,𝑇𝑥_𝑖−1,𝑑𝑥_𝑖,𝑇𝑥_𝑖≤3𝑁3𝑟^𝑛−11−𝑟max_{1≤𝑖≤𝑁}𝑑𝑥_𝑖−1,𝑇𝑥_𝑖−1. (23) Therefore, {𝑇^𝑛𝑥}_𝑛≥1 and {𝑇^𝑛𝑦}_𝑛≥1 are Cauchy equivalent.
(ii)⇒(iii) Let 𝑥,𝑦∈Fix𝑇. By using (ii) and the above process, we obtain easily that 𝑥=𝑦.
(iii)⇒(i) If 𝐺 is not weakly connected, then there exists 𝑥₀ such that 𝑋⧵[𝑥₀]𝐺 is not empty. Take 𝑦₀∈𝑋⧵[𝑥₀]𝐺 and define 𝑥𝑇𝑥=⁰,𝑥∈𝑥₀𝐺,𝑦₀,𝑥∈𝑋⧵𝑥₀𝐺. (24) Clearly, Fix𝑇={𝑥₀,𝑦₀}. Now, we show that 𝑇 is a 𝐺-type quasi-contraction. For this reason, let (𝑥,𝑦)∈𝐸. Since 𝐺𝐺[𝑥]=[𝑦], either 𝑥,𝑦∈[𝑥₀]𝐺 or 𝑥,𝑦∈𝑋⧵[𝑥₀]𝐺. In both cases, we get 𝑇𝑥=𝑇𝑦. Thus, 𝑇 is a 𝐺-type quasi-contraction which has two fixed points. This contradiction completes the proof.

Theorem 5. Let (𝑋,𝑑) be a complete metric space and let 𝑇 be a 𝐺-type quasi-contraction and orbitally 𝐺-continuous self-map on 𝑋. Then,(i)for each 𝑥∈𝑋_𝑇, 𝑇|_[𝑥]𝐺 is a Picard operator,(ii)𝐺𝑐𝑎𝑟𝑑(𝐹𝑖𝑥𝑇)=𝑐𝑎𝑟𝑑{[𝑥]∶𝑥∈𝑋_𝑇}.

Proof. Let 𝑥∈𝑋_𝑇. Then, 𝐺𝑇𝑥∈[𝑥]. It is easy to check that {𝑇^𝑛}_𝑛≥1 is a Cauchy sequence. Let lim_𝑛→∞𝑇^𝑛𝑥=𝑥^∗. Since 𝐺 is a 𝐶-Graph, there exists a subsequence {𝑇^𝑛𝑘𝑥}_𝑛≥1 such that (𝑇^𝑛𝑘𝑥,𝑥^∗)∈𝐸(𝐺) for all 𝑘. Thus, (𝑇^𝑛𝑘+1𝑥,𝑇𝑥^∗)∈𝐸(𝐺) for all 𝑘. Since (𝑇^𝑛𝑘𝑥,𝑇^𝑛𝑘+1𝑥)∈𝐸(𝐺), 𝑇𝑥^∗𝐺∈[𝑥]. Since 𝑇 is orbitally 𝐺-continuous, lim_𝑛→∞𝑇^𝑛𝑘+1𝑥=𝑥^∗ which yields 𝑥^∗=𝑇𝑥^∗. To prove (ii), define the mapping 𝜋 by 𝐺𝜋(𝑥)=[𝑥] for all 𝑥∈Fix𝑇. It is sufficient to show that 𝜋 is a bijection from Fix𝑇 onto 𝐺𝒞={[𝑥]∶𝑥∈𝑋_𝑇}. Since Δ⊆𝐸(𝐺), we get Fix𝑇⊆𝑋_𝑇 which yields 𝜋(Fix𝑇)⊆𝒞. On the other hand, if 𝑥∈𝑋_𝑇, then lim_𝑛→∞𝑇^𝑛𝐺𝑥∈[𝑥]⋂Fix𝑇 which implies that 𝜋(lim_𝑛→∞𝑇^𝑛𝐺𝑥)=[𝑥]. Thus, 𝜋 is a surjection from Fix𝑇 onto 𝒞. Now, if 𝑥₁,𝑥₂∈Fix𝑇 with 𝜋(𝑥₁)=𝜋(𝑥₂), then 𝑥₂∈[𝑥₁]𝐺 and so by using (i) we obtain lim_𝑛→∞𝑇^𝑛𝑥₂∈𝑥₁𝐺Fix𝑇=𝑥₁, (25) which implies that 𝑥₂=𝑥₁. Therefore, 𝑇 is an injective and this completes the proof.

We need the following results for our last result.

Lemma 6 (see [18]). Let 𝑋 be a nonempty set and let 𝑇∶𝑋→𝑋 be a mapping. Then, there exists a subset 𝑌⊆𝑋 such that 𝑇𝑌=𝑇𝑋 and 𝑇∶𝑌→𝑋 is one-to-one.

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

Theorem 8. Let (𝑋,𝑑) be a metric space, and let 𝑓 and 𝑇 be two self-maps on 𝑋 such that 𝑇𝑋⊆𝑓𝑋 and 𝑓𝑋 is complete. Suppose that 𝑓 and 𝑇 satisfy the following conditions:(i)(𝑓𝑥,𝑓𝑦)∈𝐸(𝐺) implies that (𝑇𝑥,𝑇𝑦)∈𝐸(𝐺),(ii)if (𝑓𝑥,𝑇𝑥)∈𝐸(𝐺) and 𝑇𝑥=𝑓𝑦 for some 𝑦∈𝑋, then (𝑓𝑥,𝑇𝑦)∈𝐸(𝐺),(iii)there exists 𝑟∈[0,1) such that 𝑟²+𝑟<1 and 𝜃(𝑟)𝑑(𝑓𝑥,𝑇𝑥)≤𝑑(𝑓𝑥,𝑓𝑦) implies that 𝑑𝑑,1𝑇𝑥,𝑇𝑦≤𝑟max𝑓𝑥,𝑓𝑦,𝑑𝑓𝑥,𝑇𝑥,𝑑𝑓𝑦,𝑇𝑦2𝑑𝑓𝑥,𝑇𝑦+𝑑𝑓𝑦,𝑇𝑥. (26)Then, 𝑇 and 𝑓 have a unique coincidence point. Moreover, if 𝑇 and 𝑓 are weakly compatible, then 𝑇 and 𝑓 have a unique fixed point.

Proof. By using Lemma 6, there exists 𝑌⊂𝑋 such that 𝑓∶𝑌→𝑋 is one-to-one and 𝑓𝑌=𝑓𝑋. Define the self-map ℎ∶𝑓𝑌→𝑓𝑌 by ℎ(𝑓𝑥)=𝑇𝑥. Clearly, ℎ is well defined and ℎ preserves the edges of 𝐺. In fact, (𝑓𝑥,𝑓𝑦)∈𝐸(𝐺) implies that (ℎ𝑓𝑥,ℎ𝑓𝑦)∈𝐸(𝐺). Note that 𝜃(𝑟)𝑑(𝑓𝑥,ℎ𝑓𝑥)≤𝑑(𝑓𝑥,𝑓𝑦) implies that 𝑑𝑑,1ℎ𝑓𝑥,ℎ𝑓𝑦≤𝑟max𝑓𝑥,𝑓𝑦,𝑑𝑓𝑥,ℎ𝑓𝑥,𝑑𝑓𝑦,ℎ𝑓𝑦2𝑑𝑓𝑥,ℎ𝑓𝑦+𝑑𝑓𝑦,ℎ𝑓𝑥. (27) Also, (𝑓𝑥,ℎ𝑓𝑥) and (𝑓𝑥,ℎ²𝑓𝑥) lie in 𝐸(𝐺) for all 𝑥∈𝑌. To see this, take ℎ𝑓𝑥=𝑇𝑥. Then, 𝑇𝑥=𝑓𝑦 for some 𝑦∈𝑌 and so ℎ²𝑓𝑥=𝑇𝑦. By using (ii), (𝑓𝑥,𝑇𝑦)∈𝐸(𝐺). Since 𝑓𝑌 is complete, by using Theorem 3, ℎ has a unique fixed point in 𝑓𝑌, namely, ℎ𝑓𝑥^∗=𝑓𝑥^∗. Thus, 𝑥^∗ is a coincidence point of 𝑓 and 𝑇. Note that the assumption (iii) shows the uniqueness of the coincidence point of 𝑓 and 𝑇. Now, by using Lemma 7, it is easy to see that if 𝑓 and 𝑇 are weakly compatible, then 𝑓 and 𝑇 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.