Fixed Point Theorem for Cyclic Chatterjea Type Contractions

Abstract

We introduce the notion of cyclic weakly Chatterjea type contraction and generalized cyclic weakly Chatterjea type contraction in metric spaces. We discussed the existence of fixed point theorems of (generalized) cyclic weakly Chatterjea type contraction mappings in the context of complete metric spaces. Our main theorems extend and improve some fixed point theorems in the literature.

1. Introduction and Preliminaries

Celebrated Banach’s contraction mapping principle is one of the cornerstones in the development of nonlinear analysis. Fixed point theorems have applications not only in the various branches of mathematics but also in economics, chemistry, biology, computer science, engineering, and others. In particular, such theorems are used to demonstrate the existence and uniqueness of a solution of differential equations, integral equations, functional equations, partial differential equations, and others. Due to the importance, generalizations of Banach fixed point theorem have been investigated heavily by many authors. (see, e.g., [1, 2]).

Following this trend, in 1972, Chatterjea [3] introduced the following definition.

Definition 1.1. Let (𝑋,𝑑) be a metric space. A self-mapping 𝑇∶𝑋→𝑋 is called a Chatterjea type contraction if there exists 𝛼∈(0,1/2) such that for all 𝑥,𝑦∈𝑋 the following inequality holds: 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥. (1.1)

In this interesting paper, Chatterjea [3] proved that every Chatterjea type contraction 𝑇 on a complete metric space has a unique fixed point. Later, Choudhury [4] introduced a generalization of Chatterjea type contraction as follows.

Definition 1.2. A self-mapping 𝑇∶𝑋→𝑋, on a metric space (𝑋,𝑑), is said to be a weakly 𝐶-contractive (or a weak Chatterjea type contraction) if for all 𝑥,𝑦∈𝑋, 𝑑≤1𝑇𝑥,𝑇𝑦2𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥, (1.2) where 𝜓∶[0,+∞)²→[0,+∞) is a continuous function such that 𝜓𝑥,𝑦=0iff𝑥=𝑦=0. (1.3)

In [4], the author proves that every weak Chatterjea type contraction on a complete metric space has a unique fixed point.

One of the interesting generalizations of a Banach’s contraction mapping principle was given by Kirk et al. [5] in 2003 by introducing the following notion of cyclic representation.

Definition 1.3 (see [5]). Let 𝑋 be a nonempty set and let 𝑇∶𝑋→𝑋 be an operator. By definition, 𝑋=⋃^𝑚_𝑖=1𝑋_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇 if(a)𝑋_𝑖,𝑖=1,…,𝑚 are nonempty sets,(b)𝑇(𝑋₁)⊂𝑋₂,…,𝑇(𝑋_𝑚−1)⊂𝑋_𝑚, 𝑇(𝑋_𝑚)⊂𝑋₁.

After the remarkable paper of Kirk et al. [5], some new fixed-point theorems for operators 𝑇 defined on a complete metric space 𝑋 with a cyclic representation of 𝑋 with respect to 𝑇 have appeared in the literature (see, e.g., [1, 2, 4–19]). Very recently, Karapınar [10] introduced the notion of the cyclic weak 𝜑-contraction and proved fixed point theorems for these types of contractions.

Definition 1.4 (see [10]). Let (𝑋,𝑑) be a metric space, 𝑚∈ℕ,𝐴₁,𝐴₂,…,𝐴_𝑚 closed nonempty subsets of 𝑋, and 𝑌=⋃^𝑚_𝑖=1𝐴_𝑖. An operator 𝑇∶𝑌→𝑌 is called a cyclic weak 𝜑-contraction if(1)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇;(2)there exists a continuous, nondecreasing function 𝜑∶[0,∞)→[0,∞) with 𝜑(𝑡)>0 for 𝑡∈(0,∞) and 𝜑(0)=0 such that 𝑑𝑇𝑥,𝑇𝑦≤𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦, (1.4)for any 𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑖=1,2,…,𝑚 where 𝐴_𝑚+1=𝐴₁.

Let 𝐅 denote all the continuous functions 𝜑∶[0,∞)→[0,∞) with 𝜑(𝑡)>0 for 𝑡∈(0,∞) and 𝜑(0)=0.

Theorem 1.5 (see [10]). Let (𝑋,𝑑) be a complete metric space, 𝑚∈ℕ,𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Suppose that 𝑇 is a cyclic weak 𝜑-contraction with 𝜑∈𝐅. Then, 𝑇 has a fixed point 𝑧∈⋂^𝑛_𝑖=1𝐴_𝑖.

In this paper, we introduce the notions of cyclic weakly Chatterjea type contractions and generalized cyclic weakly Chatterjea type contractions and then derive fixed point theorems on these cyclic contractions in the setup of complete metric spaces. Our results generalize fixed point theorems [2, 5, 20] in the sense of metric spaces.

2. Main Results

2.1. Fixed Point Theory for the Cyclic Weakly Chatterjea Type Contractions in Complete Metric Space

First we introduce the notion of cyclic weakly Chatterjea type contraction in metric space.

For convenience, we denote by 𝐅_𝟏 the class of functions 𝜓∶[0,∞)²→[0,∞) lower semicontinuous satisfying 𝜓(𝑥,𝑦)=0 if and only if 𝑥=𝑦=0.

Definition 2.1. Let (𝑋,𝑑) be a metric space, 𝑚∈ℕ,𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋, and 𝑌=⋃^𝑚_𝑖=1𝐴_𝑖. An operator 𝑇∶𝑌→𝑌 is called a cyclic weakly Chatterjea type contraction if(1)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇;(2)for any 𝑥∈𝐴_𝑖,  𝑦∈𝐴_𝑖+1,  𝑖=1,2,…,𝑚𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥, (2.1) where 𝐴_𝑚+1=𝐴₁, 𝜓∈𝐅_𝟏 and 𝛼∈(0,1/2].

Notice that the cyclic weak Chatterjea type contractions constitute a strictly larger class of mappings than cyclic weak 𝜑-contractions.

The main result of this section is the following.

Theorem 2.2. Let (𝑋,𝑑) be a complete metric space, 𝑚∈ℕ,  𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty closed subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Suppose that 𝑇 is a cyclic weakly Chatterjea type contraction. Then, 𝑇 has a unique fixed point 𝑧∈⋂^𝑛_𝑖=1𝐴_𝑖.

Proof. Take 𝑥₀∈𝑋 and consider the sequence given by 𝑥_𝑛+1=𝑇𝑥_𝑛,  𝑛=0,1,2,…. If there exists 𝑛₀∈ℕ such that 𝑥_𝑛0+1=𝑥_𝑛0, then the existence of the fixed point is proved. Indeed, 𝑥_𝑛0+1=𝑇𝑥_𝑛0=𝑥_𝑛0 and 𝑥_𝑛0 is the desired point. Suppose that 𝑥_𝑛+1≠𝑥_𝑛 for all 𝑛=0,1,2,…. Since 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖, for any 𝑛>0 there exists 𝑖_𝑛∈{1,2,…,𝑚} such that 𝑥_𝑛−1∈𝐴_𝑖𝑛 and 𝑥_𝑛∈𝐴_𝑖𝑛+1. Due to the fact that 𝑇 is a cyclic weakly Chatterjea type contraction, we have𝑑𝑥_𝑛,𝑥_𝑛+1=𝑑𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝛼𝑑𝑥_𝑛−1,𝑇𝑥_𝑛+𝑑𝑥_𝑛,𝑇𝑥_𝑛−1−𝜓𝑑𝑥_𝑛−1,𝑇𝑥_𝑛,𝑑𝑥_𝑛,𝑇𝑥_𝑛−1=𝛼𝑑𝑥_𝑛−1,𝑥_𝑛+1+𝑑𝑥_𝑛,𝑥_𝑛−𝜓𝑑𝑥_𝑛−1,𝑥_𝑛+1,0≤𝛼𝑑𝑥_𝑛−1,𝑥_𝑛+1≤𝛼𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1. (2.2) Therefore, 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝛼𝑑𝑥1−𝛼^𝑛−1,𝑥_𝑛forany𝑛=1,2,…. (2.3) If we take 𝑘=𝛼/(1−𝛼)<1, then we have 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑘𝑑𝑥_𝑛−1,𝑥_𝑛forany𝑛=1,2,…. (2.4) It follows that 𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝑘^𝑛𝑑(𝑥₀,𝑥₁) and lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0. (2.5) In the sequel, we will prove that {𝑥_𝑛} is a Cauchy sequence. First, we prove the following claim. For every 𝜀>0 there exists 𝑛∈ℕ such that if 𝑟,𝑞≥𝑛 with 𝑟−𝑞≡1(𝑚), then 𝑑(𝑥_𝑟,𝑥_𝑞)<𝜀.
Suppose, to the contrary, that there exists 𝜀>0 such that for any 𝑛∈ℕ we can find 𝑟_𝑛>𝑞_𝑛≥𝑛 with 𝑟_𝑛−𝑞_𝑛≡1(𝑚) satisfying 𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛≥𝜀. (2.6) Now, we take 𝑛>2𝑚. Then, corresponding to 𝑞_𝑛≥𝑛 we can choose 𝑟_𝑛 in such a way that it is the smallest integer with 𝑟_𝑛>𝑞_𝑛 satisfying 𝑟_𝑛−𝑞_𝑛≡1(𝑚) and 𝑑(𝑥_𝑞𝑛,𝑥_𝑟𝑛)≥𝜀. Therefore, 𝑑(𝑥_𝑞𝑛,𝑥_{𝑟𝑛−𝑚})≤𝜀. Using the triangular inequality, we obtain 𝜀≤𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛≤𝑑𝑥_𝑞𝑛,𝑥_{𝑟𝑛−𝑚}+^𝑚_𝑖=1𝑑𝑥_{𝑟𝑛−𝑖},𝑥_{𝑟𝑛−𝑖+1}<𝜀+^𝑚_𝑖=1𝑑𝑥_{𝑟𝑛−𝑖},𝑥_{𝑟𝑛−𝑖+1}. (2.7) Taking (2.5) into account and letting 𝑛→∞ in the inequality above, we find lim_𝑛→∞𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛=𝜀. (2.8) Again, by the triangular inequality, we have 𝜀≤𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛≤𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑑𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1+𝑑𝑥_𝑟𝑛+1,𝑥_𝑟𝑛≤𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑑𝑥_𝑞𝑛+1,𝑥_𝑞𝑛+𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+𝑑𝑥_𝑟𝑛,𝑥_𝑟𝑛+1+𝑑𝑥_𝑟𝑛+1,𝑥_𝑟𝑛=2𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+2𝑑𝑥_𝑟𝑛,𝑥_𝑟𝑛+1. (2.9) Passing to the limit as 𝑛→∞ in (2.8) and taking (2.5) and (2.8) into account, we get lim_𝑛→∞𝑑𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1=𝜀. (2.10) Since 𝑥_𝑞𝑛 and 𝑥_𝑟𝑛 lie in different adjacently labeled sets 𝐴_𝑖 and 𝐴_𝑖+1 for certain 1≤𝑖≤𝑚, using the fact that 𝑇 is a cyclic weakly Chatterjea type contraction, we have 𝑑𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1=𝑑𝑇𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛≤𝛼𝑑𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛+𝑑𝑥_𝑟𝑛,𝑇𝑥_𝑞𝑛−𝜓𝑑𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛,𝑑𝑥_𝑟𝑛,𝑇𝑥_𝑞𝑛≤𝛼𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+1+𝑑𝑥_𝑟𝑛,𝑥_𝑞𝑛+1−𝜓𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+1,𝑑𝑥_𝑟𝑛,𝑥_𝑞𝑛+1≤𝛼𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+𝑑𝑥_𝑟𝑛,𝑥_𝑟𝑛+1+𝑑𝑥_𝑟𝑛,𝑥_𝑞𝑛+𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1−𝜓𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+1,𝑑𝑥_𝑟𝑛,𝑥_𝑞𝑛+1. (2.11) Taking (2.5) and (2.8) into account together with the lower semicontinuity of 𝜓, the inequalities above yield 𝜀≤𝛼⋅𝜀−liminf_𝑛→+∞𝜓𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+1,𝑑𝑥_𝑟𝑛,𝑥_𝑞𝑛+1≤𝛼⋅𝜀−𝜓(𝜀,𝜀)≤𝛼⋅𝜀 (2.12) as 𝑛→∞. Hence, 𝜓(𝜀,𝜀)=0. From the fact that 𝜓(𝑥,𝑦)=0⇔𝑥=𝑦=0, we have 𝜀=0, which is a contradiction. Therefore, our claim is proved.
We prove that (𝑋,𝑑) is a Cauchy sequence. Fix 𝜀>0. By the claim, we find 𝑛₀∈ℕ such that if 𝑟,𝑞≥𝑛₀ with 𝑟−𝑞≡1(𝑚), 𝑑𝑥_𝑟,𝑥_𝑞≤𝜀2. (2.13) Since lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0, there exists 𝑛₁∈ℕ such that 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜀2𝑚 (2.14) for any 𝑛≥𝑛₁. Suppose that 𝑟,𝑠≥max{𝑛₀,𝑛₁} with 𝑠>𝑟. Then, there exists 𝑘∈{1,2,…,𝑚} such that 𝑠−𝑟≡𝑘(𝑚). Therefore, we have 𝑠−𝑟+𝜑≡1(𝑚) for 𝜑=𝑚−𝑘+1. So, we derive 𝑑𝑥_𝑟,𝑥_𝑠≤𝑑𝑥_𝑟,𝑥_𝑠+𝑗+𝑑𝑥_𝑠+𝑗,𝑥_{𝑠+𝑗−1}+⋯+𝑑𝑥_𝑠+1,𝑥_𝑠. (2.15) By (2.13) and (2.14) and the inequality above, we get 𝑑𝑥_𝑟,𝑥_𝑠≤𝜀2𝜀+𝑗×≤𝜀2𝑚2𝜀+𝑚×2𝑚=𝜀. (2.16) This proves that (𝑥_𝑛) is a Cauchy sequence. Since 𝑋 is a complete metric space, there exists 𝑥∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑥.
Now, we prove that 𝑥 is a fixed point of 𝑇. Since lim_𝑛→∞𝑥_𝑛=𝑥 and, as 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇, the sequence (𝑥_𝑛) has infinite terms in each 𝐴_𝑖 for 𝑖∈{1,2,…,𝑚}. Suppose that 𝑥∈𝐴_𝑖,𝑇𝑥∈𝐴_𝑖+1 and we take a subsequence 𝑥_𝑛𝑘 of (𝑥_𝑛) with 𝑥_𝑛𝑘∈𝐴_𝑖−1 (the existence of this subsequence is guaranteed by the previously mentioned comment). By using the contractive condition, we obtain 𝑑𝑥_𝑛𝑘+1,𝑇𝑥=𝑑𝑇𝑥_𝑛𝑘𝑑𝑥,𝑇𝑥≤𝛼_𝑛𝑘,𝑇𝑥+𝑑𝑥,𝑇𝑥_𝑛𝑘−𝜓𝑑𝑥_𝑛𝑘,𝑇𝑥,𝑑𝑥,𝑇𝑥_𝑛𝑘=𝛼𝑑𝑥_𝑛𝑘,𝑇𝑥+𝑑𝑥,𝑥_𝑛𝑘+1−𝜓𝑑𝑥_𝑛𝑘,𝑇𝑥,𝑑𝑥,𝑥_𝑛𝑘+1. (2.17) Letting 𝑛→∞ and by using 𝑥_𝑛𝑘→𝑥 together with the lower semicontinuity of 𝜓, we have 𝑑(𝑥,𝑇𝑥)≤𝛼𝑑(𝑥,𝑇𝑥)−liminf_𝑛→+∞𝜓𝑑𝑥_𝑛𝑘,𝑇𝑥,𝑑𝑥,𝑥_𝑛𝑘+1≤𝛼𝑑(𝑥,𝑇𝑥)−𝜓(𝑑(𝑥,𝑇𝑥),0)≤𝛼𝑑(𝑥,𝑇𝑥) (2.18) which is a contradiction (since 𝛼<1/2) unless 𝑑(𝑥,𝑇𝑥)=0. Therefore, 𝑥 is a fixed point of 𝑇.
Finally, to prove the uniqueness of the fixed point, we have 𝑦,𝑧∈𝑋 with 𝑦 and 𝑧 being fixed points of 𝑇. The cyclic character of 𝑇 and the fact that 𝑦,𝑧∈𝑋 are fixed points of 𝑇 imply that 𝑦,𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖. Using the contractive condition we obtain 𝑑𝑦,𝑧=𝑑𝑇𝑦,𝑇𝑧≤𝛼𝑑𝑦,𝑇𝑧+𝑑𝑧,𝑇𝑦−𝜓𝑑𝑦,𝑇𝑧,𝑑𝑧,𝑇𝑦. (2.19) That is, 𝑑𝑦,𝑧≤2𝛼𝑑𝑦,𝑧−𝜓𝑑𝑦,𝑧,𝑑𝑧,𝑦≤2𝛼𝑑𝑦,𝑧. (2.20) This gives us 𝜓(𝑑(𝑦,𝑧),𝑑(𝑧,𝑦))=0, and, by our assumption about 𝜓, 𝑑(𝑦,𝑧)=0, that is, 𝑦=𝑧. This finishes the proof.

Corollary 2.3. Let (𝑋,𝑑) be a complete metric space, 𝑚∈ℕ,𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Suppose that 𝑇∶𝑋→𝑋 be an operator such that(1)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇;(2)there exists 𝛽∈[0,1/2) such that 𝑑𝑇𝑥,𝑇𝑦≤𝛽𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥 (2.21)for any 𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑖=1,2,…,𝑚 where 𝐴_𝑚+1=𝐴₁. Then, 𝑇 has a fixed point 𝑧∈⋂^𝑛_𝑖=1𝐴_𝑖.

Proof. Let 𝛽∈[0,1/2). Here, it suffices to take the function 𝜓∶[0,+∞)²→[0,+∞) as 𝜓(𝑎,𝑏)=(𝛼−𝛽)(𝑎+𝑏). It is clear that 𝜓 satisfies the following conditions:(1)𝜓(𝑎,𝑏)=0 if and only if 𝑎=𝑏=0,(2)𝜓(𝑥,𝑦)=(𝛼−𝛽)(𝑥+𝑦)=𝜓(𝑥+𝑦,0). Hence, we apply Theorems 2.2 for 𝛼∈[1/4,1/2) and get the desired result.

The following corollary gives us a fixed point theorem with a contractive condition of integral type for cyclic contractions.

Corollary 2.4. Let (𝑋,𝑑) be a complete metric space, 𝑚∈ℕ,  𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty closed subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Suppose that 𝑇∶𝑋→𝑋 be an operator such that(i)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇;(ii)there exists 𝛽∈[0,1/2) such that ₀^{𝑑(𝑇𝑥,𝑇𝑦)}𝜌(𝑡)𝑑𝑡≤𝛽₀^{𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)}𝜌(𝑡)𝑑𝑡 (2.22)for any 𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑖=1,2,…,𝑚 where 𝐴_𝑚+1=𝐴₁ and 𝜌∶[0,∞)→[0,∞) is a Lebesgue-integrable mapping satisfying ∫^𝜀₀𝜌(𝑡)𝑑𝑡 for 𝜀>0. Then 𝑇 has unique fixed point 𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖.

Proof. It is easily proved that the function 𝜑∶[0,∞)→[0,∞) given by 𝜑(𝑡)=∫^𝑡₀𝜌(𝑠)𝑑𝑠 satisfies that 𝜑∈𝐅_𝟏. Therefore, Corollary 2.3 is obtained from Theorem 2.2, taking as 𝜑 the previously defined function and as 𝜓 the function 𝜓(𝑥,𝑦)=(𝛼−𝛽)(𝑥+𝑦)=𝜑(𝑥+𝑦,0).

If in Corollary 2.4, we take 𝐴_𝑖=𝑋 for 𝑖=1,2,…,𝑚, we obtain the following result.

Corollary 2.5. Let (𝑋,𝑑) be a complete metric space and let 𝑇∶𝑋→𝑋 be a mapping such that for any 𝑥,𝑦∈𝑋, ₀^{𝑑(𝑇𝑥,𝑇𝑦)}𝜌(𝑡)𝑑𝑡≤𝛽₀^{𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)}𝜌(𝑡)𝑑𝑡, (2.23) where 𝜌∶[0,∞)→[0,∞) is a Lebesgue-integrable mapping satisfying ∫^𝜀₀𝜌(𝑡)𝑑𝑡 for 𝜀>0 and the constant 𝛽∈[0,1/2). Then 𝑇 has unique fixed point.

If in Theorem 2.2 we put 𝐴_𝑖=𝑋 for 𝑖=1,2,…,𝑚 we get the main result of [4].

Corollary 2.6. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 a mapping such that for any 𝑥,𝑦∈𝑋, 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥, (2.24) where 𝜓∈𝐅_𝟏 and 𝛼∈(0,1/2). Then 𝑇 has unique fixed point.

Example 2.7. Let 𝑋=ℝ with the usual metric. Suppose 𝐴₁=[0,1],𝐴₂=[0,1/2], and 𝑌=⋃²_𝑖=1𝐴_𝑖. Define 𝑇∶𝑌→𝑌 such that 𝑇𝑥=𝑥/9 for all 𝑥∈𝑌. It is clear that ⋃²_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇. Furthermore, if 𝜓∶[0,+∞)²→[0,+∞) is defined by 𝜓(𝑥,𝑦)=(𝑥+𝑦)/9, then 𝜓∈𝐅_𝟐. Here 𝑇 is a generalized cyclic weakly Chatterjea type contraction for 𝛼=1/2. To see this fact we examine three cases.(i)Suppose that 𝑥≥𝑦. Then, the inequality (2.1) turns into 𝑑=𝑥𝑇𝑥,𝑇𝑦7−𝑦7=𝑥−𝑦7≤12𝑑𝑑=1𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥2𝑦𝑥−7+𝑥𝑦−7−19𝑦𝑥−7+𝑥𝑦−7=12𝑦𝑥−7+𝑥𝑦−7−19𝑦𝑥−7+𝑥𝑦−7=7𝑦18𝑥−7+𝑥𝑦−7. (2.25) If 𝑦<𝑥/7, then (2.25) becomes 𝑥−𝑦7≤7𝑦18𝑥−7+𝑥7=4−𝑦9𝑥−𝑦. (2.26) Hence, (2.1) is satisfied. If 𝑦≥𝑥/7, then (2.25) becomes 𝑥−𝑦7≤7𝑦18𝑥−7𝑥+𝑦−7=13𝑥+𝑦. (2.27) Thus, (2.1) is true.(ii)Suppose that 𝑦/7≤𝑥≤𝑦. Then, the inequality (2.1) yields that 𝑑=𝑥𝑇𝑥,𝑇𝑦7−𝑦7=𝑦−𝑥7≤12𝑑𝑑=1𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥2𝑦𝑥−7+𝑥𝑦−7−19𝑦𝑥−7+𝑥𝑦−7=7𝑦18𝑥−7+𝑥𝑦−7=7𝑦18𝑥−7𝑥+𝑦−7=13𝑥+𝑦. (2.28) So, (2.1) holds.(iii)Finally, suppose that 𝑦/7≥𝑥. Then, the inequality (2.1) yields that 𝑑=𝑥𝑇𝑥,𝑇𝑦7−𝑦7=𝑦−𝑥7≤12𝑑𝑑=1𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥2𝑦𝑥−7+𝑥𝑦−7−19𝑦𝑥−7+𝑥𝑦−7=7𝑦18𝑥−7+𝑥𝑦−7=7𝑦18𝑥−7𝑥+𝑦−7=13𝑦+𝑥. (2.29) Hence, (2.1) is true. Therefore, all conditions of Theorem 1.5 are satisfied, and so 𝑇 has a fixed point (which is 𝑧=0∈⋂²_𝑖=1𝐴_𝑖.)

2.2. Fixed Point Theory for the Generalized Cyclic Weakly Chatterjea Type Contractions in Complete Metric Space

In this section we derive fixed point theorems for self-maps satisfying certain generalized cyclic weakly Chatterjea type contractions in a complete metric space.

For convenience, we denote by 𝐅_𝟐 the class of functions 𝜓∶[0,∞)⁴→[0,∞) lower semicontinuous satisfying 𝜓(𝑥,𝑦,𝑧,𝑡)=0 if and only if 𝑥=𝑦=𝑧=𝑡=0.

We introduce the notion of generalized cyclic weakly Chatterjea type contraction in metric space.

Definition 2.8. Let (𝑋,𝑑) be a metric space, 𝑚∈ℕ,𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋 and 𝑌=⋃^𝑚_𝑖=1𝐴_𝑖. An operator 𝑇∶𝑌→𝑌 is called a generalized cyclic weakly Chatterjea type contraction if(1)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇; (2)𝑑𝑑()𝑑(),𝑇𝑥,𝑇𝑦≤𝛼𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦+𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑥,𝑇𝑥,𝑑𝑥,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥 (2.30) for any 𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑖=1,2,…,𝑚 where 𝐴_𝑚+1=𝐴₁ and 𝜓𝑑(𝑥,𝑇𝑥),𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥∈𝐅_𝟐1and𝛼∈0,4. (2.31)

We state the main result of this section as follows.

Theorem 2.9. Let (𝑋,𝑑) be a complete metric space, 𝑚∈ℕ,  𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty closed subsets of 𝑋 and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Suppose that 𝑇 is a generalized cyclic weakly C-contraction. Then, 𝑇 has a unique fixed point 𝑧∈⋂^𝑛_𝑖=1𝐴_𝑖.

Proof. As in the proof of Theorem 2.2, we take 𝑥₀∈𝑋 and construct a sequence by defining 𝑥_𝑛+1=𝑇𝑥_𝑛,𝑛=0,1,2,…. Suppose that there exists 𝑛₀∈ℕ such that 𝑥_𝑛0+1=𝑥_𝑛0. Then, since 𝑥_𝑛0+1=𝑇𝑥_𝑛0=𝑥_𝑛0, the existence of the fixed point is proved. So, we assume that 𝑥_𝑛+1≠𝑥_𝑛 for any 𝑛=0,1,2,…. Since 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖, for any 𝑛>0, there exists 𝑖_𝑛∈{1,2,…,𝑚} such that 𝑥_𝑛−1∈𝐴_𝑖𝑛 and 𝑥_𝑛∈𝐴_𝑖𝑛+1. Regarding that 𝑇 is a generalized cyclic weakly Chatterjea type contraction, we have 𝑑𝑥_𝑛,𝑥_𝑛+1=𝑑𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤𝛼𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1+𝑑𝑥_𝑛,𝑇𝑥_𝑛+𝑑𝑥_𝑛−1,𝑇𝑥_𝑛+𝑑𝑥_𝑛,𝑇𝑥_𝑛−1−𝜓𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑑𝑥_𝑛,𝑇𝑥_𝑛,𝑑𝑥_𝑛−1,𝑇𝑥_𝑛,𝑑𝑥_𝑛,𝑇𝑥_𝑛−1=𝛼𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛−1,𝑥_𝑛+1−𝜓𝑑𝑥_𝑛−1,𝑥_𝑛,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛−1,𝑥_𝑛+1𝑑𝑥,0≤𝛼_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛−1,𝑥_𝑛+1≤2𝛼𝑑𝑥_𝑛−1,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑛+1. (2.32) Therefore, 𝑑𝑥_𝑛,𝑥_𝑛+1≤2𝛼𝑑𝑥1−2𝛼^𝑛−1,𝑥_𝑛forany𝑛=1,2,…. (2.33) If we take 𝑘=𝛼/(1−𝛼)<1, then we have 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝑘𝑑𝑥_𝑛−1,𝑥_𝑛forany𝑛=1,2,…. (2.34) It follows that 𝑑(𝑥_𝑛,𝑥_𝑛+1)≤𝑘^𝑛𝑑(𝑥₀,𝑥₁) and lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=0. (2.35) We show that {𝑥_𝑛} is a Cauchy sequence. For this purpose, first, we prove the following claim. For every 𝜀>0 there exists 𝑛∈ℕ such that if 𝑟,𝑞≥𝑛 with 𝑟−𝑞≡1(𝑚), then 𝑑(𝑥_𝑟,𝑥_𝑞)<𝜀.
Assume the contrary. So there exists 𝜀>0 such that for any 𝑛∈𝐍 we can find 𝑟_𝑛>𝑞_𝑛≥𝑛 with 𝑟_𝑛−𝑞_𝑛≡1(𝑚) satisfying 𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛≥𝜀. (2.36) Now, we take 𝑛>2𝑚. Then, corresponding to 𝑞_𝑛≥𝑛 we can choose 𝑟_𝑛 in such a way that it is the smallest integer with 𝑟_𝑛>𝑞_𝑛 satisfying 𝑟_𝑛−𝑞_𝑛≡1(𝑚) and 𝑑(𝑥_𝑞𝑛,𝑥_𝑟𝑛)≥𝜀. Therefore, 𝑑(𝑥_𝑞𝑛,𝑥_{𝑟𝑛−𝑚})≤𝜀 by using the triangular inequality 𝜀≤𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛≤𝑑𝑥_𝑞𝑛,𝑥_{𝑟𝑛−𝑚}+^𝑚_𝑖=1𝑑𝑥_{𝑟𝑛−𝑖},𝑥_{𝑟𝑛−𝑖+1}<𝜀+^𝑚_𝑖=1𝑑𝑥_{𝑟𝑛−𝑖},𝑥_{𝑟𝑛−𝑖+1}. (2.37) Passing to the limit as 𝑛→∞ in the last inequality and taking into account that lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0, we obtain lim_𝑛→∞𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛=𝜀. (2.38) Again, by the triangular inequality 𝜀≤𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛≤𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑑𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1+𝑑𝑥_𝑟𝑛+1,𝑥_𝑟𝑛≤𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑑𝑥_𝑞𝑛+1,𝑥_𝑞𝑛+𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+𝑑𝑥_𝑟𝑛,𝑥_𝑟𝑛+1+𝑑𝑥_𝑟𝑛+1,𝑥_𝑟𝑛=2𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+2𝑑𝑥_𝑟𝑛,𝑥_𝑟𝑛+1. (2.39) Taking (2.35) and (2.38) into account, we get lim_𝑛→∞𝑑𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1=𝜀 (2.40) as 𝑛→∞ in (2.38).
Since 𝑥_𝑞𝑛 and 𝑥_𝑟𝑛 lie in different adjacently labeled sets 𝐴_𝑖 and 𝐴_𝑖+1 for certain 1≤𝑖≤𝑚, using the fact that 𝑇 is a generalized cyclic weakly Chatterjea type contraction, we have 𝑑𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1=𝑑𝑇𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛≤𝛼𝑑𝑥_𝑞𝑛,𝑇𝑥_𝑞𝑛+𝑑𝑥_𝑟𝑛,𝑇𝑥_𝑟𝑛+𝑑𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛+𝑑𝑥_𝑟𝑛,𝑇𝑥_𝑞𝑛−𝜓𝑑𝑥_𝑞𝑛,𝑇𝑥_𝑞𝑛,𝑑𝑥_𝑟𝑛,𝑇𝑥_𝑟𝑛,𝑑𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛,𝑑𝑥_𝑟𝑛,𝑇𝑥_𝑞𝑛≤𝛼𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑑𝑥_𝑟𝑛,𝑥_𝑟𝑛+1+𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+1+𝑑𝑥_𝑟𝑛,𝑥_𝑞𝑛+1−𝜓𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1,𝑑𝑥_𝑟𝑛,𝑥_𝑟𝑛+1,𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+1,𝑑𝑥_𝑟𝑛,𝑥_𝑞𝑛+1. (2.41) Taking into account (2.35) and (2.38) and the lower semicontinuity of 𝜓, letting 𝑛→∞ in the last inequality, we obtain 𝜀≤𝛼[𝜀+0+𝜀+0]−liminf_𝑛→+∞𝜓𝑑𝑥_𝑞𝑛,𝑥_𝑞𝑛+1,𝑑𝑥_𝑟𝑛,𝑥_𝑟𝑛+1,𝑑𝑥_𝑞𝑛,𝑥_𝑟𝑛+1,𝑑𝑥_𝑟𝑛,𝑥_𝑞𝑛+1≤𝛼[𝜀+0+𝜀+0]−𝜓(𝜀,0,𝜀,0)≤2𝛼⋅𝜀 (2.42) and from the last inequality, 𝜓(𝜀,0,𝜀,0)≤−2𝛼𝜀≤0. Therefore 𝜑(𝜀,0,𝜀,0)=0. From the fact that 𝜓(𝑥,𝑦,𝑧,𝑡)=0⇔𝑥=𝑦=𝑧=𝑡=0, we have 𝜀=0, which is a contradiction. Therefore, our claim is proved.
In the sequel, we will prove that (𝑋,𝑑) is a Cauchy sequence. Fix 𝜀>0. By the claim, we find 𝑛₀∈ℕ such that if 𝑟,𝑞≥𝑛₀ with 𝑟−𝑞≡1(𝑚)𝑑𝑥_𝑟,𝑥_𝑞≤𝜀2. (2.43) Since lim_𝑛→∞𝑑(𝑥_𝑛,𝑥_𝑛+1)=0, we also find 𝑛₁∈ℕ such that 𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜀2𝑚 (2.44) for any 𝑛≥𝑛₁. Suppose that 𝑟,𝑠≥max{𝑛₀,𝑛₁} and 𝑠>𝑟. Then there exists 𝑘∈{1,2,…,𝑚} such that 𝑠−𝑟≡𝑘(𝑚). Therefore, 𝑠−𝑟+𝜑≡1(𝑚) for 𝜑=𝑚−𝑘+1. So, we have 𝑑𝑥_𝑟,𝑥_𝑠≤𝑑𝑥_𝑟,𝑥_𝑠+𝑗+𝑑𝑥_𝑠+𝑗,𝑥_{𝑠+𝑗−1}+⋯+𝑑𝑥_𝑠+1,𝑥_𝑠. (2.45) By (2.43) and (2.44) and from the last inequality, we get 𝑑𝑥_𝑟,𝑥_𝑠≤𝜀2𝜀+𝑗×≤𝜀2𝑚2𝜀+𝑚×2𝑚=𝜀. (2.46) This proves that (𝑥_𝑛) is a Cauchy sequence. Since 𝑋 is a complete metric space, there exists 𝑥∈𝑋 such that lim_𝑛→∞𝑥_𝑛=𝑥. In what follows, we prove that 𝑥 is a fixed point of 𝑇. In fact, since lim_𝑛→∞𝑥_𝑛=𝑥 and as 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇, the sequence (𝑥_𝑛) has infinite terms in each 𝐴_𝑖 for 𝑖∈{1,2,…,𝑚}. Suppose that 𝑥∈𝐴_𝑖,𝑇𝑥∈𝐴_𝑖+1 and we take a subsequence 𝑥_𝑛𝑘 of (𝑥_𝑛) with 𝑥_𝑛𝑘∈𝐴_𝑖−1 (the existence of this subsequence is guaranteed by the previously mentioned comment). By using the contractive condition, we can obtain 𝑑𝑥_𝑛𝑘+1,𝑇𝑥=𝑑𝑇𝑥_𝑛𝑘𝑑𝑥,𝑇𝑥≤𝛼_𝑛𝑘,𝑇𝑥_𝑛𝑘+𝑑(𝑥,𝑇𝑥)+𝑑𝑥_𝑛𝑘,𝑇𝑥+𝑑𝑥,𝑇𝑥_𝑛𝑘−𝜓𝑑𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘,𝑑(𝑥,𝑇𝑥),𝑑𝑥_𝑛𝑘,𝑇𝑥,𝑑𝑥,𝑇𝑥_𝑛𝑘=𝛼𝑑𝑥_𝑛𝑘,𝑥_𝑛𝑘+1+𝑑(𝑥,𝑇𝑥)+𝑑𝑥_𝑛𝑘,𝑇𝑥+𝑑𝑥,𝑥_𝑛𝑘+1−𝜓𝑑𝑥_𝑛𝑘,𝑥_𝑛𝑘+1,𝑑(𝑥,𝑇𝑥),𝑑𝑥_𝑛𝑘,𝑇𝑥,𝑑𝑥,𝑥_𝑛𝑘+1. (2.47) Passing to the limit as 𝑛→∞ and using 𝑥_𝑛𝑘→𝑥, lower semicontinuity of 𝜓, we have 𝑑()≤1𝑥,𝑇𝑥2𝑑(𝑥,𝑇𝑥)−liminf_𝑛→+∞𝜓𝑑𝑥_𝑛𝑘,𝑥_𝑛𝑘+1,𝑑(𝑥,𝑇𝑥),𝑑𝑥_𝑛𝑘,𝑇𝑥,𝑑𝑥,𝑥_𝑛𝑘+1≤12𝑑()(()())≤1𝑥,𝑇𝑥−𝜓0,𝑑𝑥,𝑇𝑥,𝑑𝑥,𝑇𝑥,02𝑑(𝑥,𝑇𝑥), (2.48) which is a contradiction unless 𝑑(𝑥,𝑇𝑥)=0, and, therefore, 𝑥 is a fixed point of 𝑇. Finally, to prove the uniqueness of the fixed point, we have 𝑦,𝑧∈𝑋 with 𝑦 and 𝑧 being fixed points of 𝑇. The cyclic character of 𝑇 and the fact that 𝑦,𝑧∈𝑋 are fixed points of 𝑇 imply that 𝑦,𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖. Using the contractive condition we obtain 𝑑𝑑()𝑑().𝑦,𝑧=𝑑𝑇𝑦,𝑇𝑧≤𝛼𝑦,𝑇𝑦+𝑑𝑧,𝑇𝑧+𝑑𝑦,𝑇𝑧+𝑑𝑧,𝑇𝑦−𝜓𝑦,𝑇𝑦,𝑑𝑧,𝑇𝑧,𝑑𝑦,𝑇𝑧,𝑑𝑧,𝑇𝑦 (2.49) that is, 𝑑𝑦,𝑧≤2𝛼𝑑𝑦,𝑧−𝜓0,0,𝑑𝑦,𝑧,𝑑𝑧,𝑦≤𝑑𝑦,𝑧. (2.50) This gives us 𝜓(0,0,𝑑(𝑦,𝑧),𝑑(𝑧,𝑦))≤(1−2𝛼)𝑑(𝑦,𝑧)≤0, and, by our assumption about 𝜓, 𝑑(𝑦,𝑧)=0, that is, 𝑦=𝑧. This finishes the proof.

Corollary 2.10. Let (𝑋,𝑑) be a complete metric space, 𝑚∈ℕ, 𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty subsets of 𝑋, and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Suppose that 𝑇∶𝑋→𝑋 be an operator such that(1)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇;(2)there exists 𝛽∈[0,1/4) such that 𝑑𝑇𝑥,𝑇𝑦≤𝛽𝑑(𝑥,𝑇𝑥)+𝑑𝑦,𝑇𝑦+𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥 (2.51) for any 𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑖=1,2,…,𝑚, where 𝐴_𝑚+1=𝐴₁. Then, 𝑇 has a fixed point 𝑧∈⋂^𝑛_𝑖=1𝐴_𝑖.

Proof. Let 𝛽∈[0,1/4). Here, it suffices to take the function 𝜓∶[0,+∞)⁴→[0,+∞) defined by 𝜓(𝑎,𝑏,𝑐,𝑒)=(𝛼−𝛽)(𝑎+𝑏+𝑐+𝑒). Obviously, 𝜓 satisfies that 𝜓(𝑎,𝑏,𝑐,𝑒)=0 if and only if 𝑎=𝑏=𝑐=𝑒=0, and 𝜓(𝑥,𝑦,𝑧,𝑡)=(𝛼−𝛽)(𝑥+𝑦+𝑧+𝑡)=𝜓(𝑥+𝑦+𝑧+𝑡,0). Then, we can apply Theorems 2.9.

The following corollary gives us a fixed point theorem with a contractive condition of integral type for cyclic contractions.

Corollary 2.11. Let (𝑋,𝑑) be a complete metric space, 𝑚∈ℕ,𝐴₁,𝐴₂,…,𝐴_𝑚 nonempty closed subsets of 𝑋 and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. Suppose that 𝑇∶𝑋→𝑋 be an operator such that(1)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇;(2)there exists 𝛽∈[0,1/4) such that ₀^{𝑑(𝑇𝑥,𝑇𝑦)}𝜌(𝑡)𝑑𝑡≤𝛽₀^{𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)+𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)}𝜌(𝑡)𝑑𝑡 (2.52)for any 𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑖=1,2,…,𝑚, where 𝐴_𝑚+1=𝐴₁ and 𝜌∶[0,∞)→[0,∞) is a Lebesgue-integrable mapping satisfying ∫^𝜀₀𝜌(𝑡)𝑑𝑡 for 𝜀>0. Then 𝑇 has unique fixed point 𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖.

Proof. It is easily proved that the function 𝜑∶[0,∞)→[0,∞) given by 𝜑(𝑡)=∫^𝑡₀𝜌(𝑠)𝑑𝑠 satisfies that 𝜑∈𝐅_𝟏. Therefore, Corollary 2.10 is obtained from Theorem 2.9, taking as 𝜑 the perviously defined function and as 𝜓 the function 𝜓(𝑥,𝑦,𝑧,𝑡)=(1/4−𝛼)(𝑥+𝑦+𝑧+𝑡)=𝜑(𝑥+𝑦+𝑧+𝑡,0).

If in Corollary 2.11, we take 𝐴_𝑖=𝑋 for 𝑖=1,2,…,𝑚, we obtain the following result.

Corollary 2.12. Let (𝑋,𝑑) be a complete metric space and let 𝑇∶𝑋→𝑋 be a mapping such that for any 𝑥,𝑦∈𝑋, ₀^{𝑑(𝑇𝑥,𝑇𝑦)}𝜌(𝑡)𝑑𝑡≤𝛽₀^{𝑑(𝑥,𝑇𝑥)+𝑑(𝑦,𝑇𝑦)+𝑑(𝑥,𝑇𝑦)+𝑑(𝑦,𝑇𝑥)}𝜌(𝑡)𝑑𝑡, (2.53) where 𝜌∶[0,∞)→[0,∞) is a Lebesgue-integrable mapping satisfying ∫^𝜀₀𝜌(𝑡)𝑑𝑡 for 𝜀>0 and the constant 𝛽∈[0,1/4). Then 𝑇 has unique fixed point.

If in Theorem 2.9 we put 𝐴_𝑖=𝑋 for 𝑖=1,2,…,𝑚, we have the generalized result of [4].

Corollary 2.13. Let (𝑋,𝑑) be a complete metric space and let 𝑇∶𝑋→𝑋 be a mapping such that for any 𝑥,𝑦∈𝑋, 𝑑𝑑()𝑑(),𝑇𝑥,𝑇𝑦≤𝛼𝑥,𝑇𝑥+𝑑𝑦,𝑇𝑦+𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑥,𝑇𝑥,𝑑𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥 (2.54) where 𝜓∈𝐅_𝟐 and 𝛼∈(0,1/4]. Then 𝑇 has unique fixed point.

Example 2.14. Let 𝑋=ℝ with the usual metric. Suppose 𝐴₁=[0,1], 𝐴₂=[0,1/3], 𝐴₃=[0,1/9], 𝐴₄=[0,1/27] and 𝑌=⋃⁴_𝑖=1𝐴_𝑖. Define 𝑇∶𝑌→𝑌 such that 𝑇𝑥=𝑥/4 for all 𝑥∈𝑌. It is clear that ⋃⁴_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇. Furthermore, if 𝜓∶[0,+∞)⁴→[0,+∞) is defined by 𝜓(𝑥,𝑦,𝑧,𝑡)=(𝑥+𝑦+𝑧+𝑡)/16, then 𝜓∈𝐅_𝟐. As in Example 2.7, it can be easily shown that 𝑇 is a generalized cyclic weakly Chatterjea type contraction 𝛼=1/4. Therefore, all conditions of Theorem 2.9 are satisfied, and so 𝑇 has a fixed point (which is 𝑧=0∈⋂⁴_𝑖=1𝐴_𝑖).