Cyclic Contractions on 𝐺-Metric Spaces

Abstract

Conditions for existence and uniqueness of fixed points of two types of cyclic contractions defined on 𝐺-metric spaces are established and some illustrative examples are given. In addition, cyclic maps satisfying integral type contractive conditions are presented as applications.

1. Introduction

The extensive application potential of fixed point theory in various fields resulted in several generalizations of the metric spaces. Amongst them, one can mention quasimetric spaces, partial metric spaces, rectangular metric spaces, 𝐷-metric spaces, and 𝐺-metric spaces. Perhaps one of the most interesting generalizations is the 𝐺-metric space. Introduced by Mustafa and Sims [1] in 2006, the concept of 𝐺-metric space has drawn the attention of mathematicians and became a very popular subject especially from the point of view of fixed point theory [2–13].

Another attractive topic in fixed point theory is the concept of cyclic maps and best proximity points introduced by Kirk et al. [14] in 2003. Cyclic maps and in particular the fixed points of cyclic maps have been a subject of growing interest recently (see, e.g., [15–27]).

The purpose of this work is to combine these two notions and investigate cyclic maps on 𝐺-metric spaces. We concentrate on two types of cyclic contractions: cyclic type Banach contractions and cyclic weak 𝜙-contractions.

Mustafa and Sims [1] introduced the concept of 𝐺-metric spaces as follows.

Definition 1.1 (see [1]). Let 𝑋 be a nonempty set, 𝐺∶𝑋×𝑋×𝑋→ℝ⁺ be a function satisfying the following properties: (G1) 𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧, (G2) 0<𝐺(𝑥,𝑥,𝑦) for all 𝑥,𝑦∈𝑋 with 𝑥≠𝑦, (G3) 𝐺(𝑥,𝑥,𝑦)≤𝐺(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋 with 𝑦≠𝑧, (G4) 𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑦,𝑧,𝑥)=⋯ (symmetry in all three variables), (G5) 𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧) for all 𝑥,𝑦,𝑧,𝑎∈𝑋 (rectangle inequality). Then the function 𝐺 is called a generalized metric, or, more specifically, a 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝐺-metric space.

Note that every 𝐺-metric on 𝑋 induces a metric 𝑑_𝐺 on 𝑋 defined by 𝑑_𝐺𝑥,𝑦=𝐺𝑥,𝑦,𝑦+𝐺𝑦,𝑥,𝑥,∀𝑥,𝑦∈𝑋. (1.1)

To have better idea about the subject, we give the following examples of 𝐺-metrics.

Example 1.2. Let (𝑋,𝑑) be a metric space. The function 𝐺∶𝑋×𝑋×𝑋→[0,+∞), defined by 𝐺𝑥,𝑦,𝑧=max𝑑𝑥,𝑦,𝑑𝑦,𝑧,𝑑(𝑧,𝑥), (1.2) for all 𝑥,𝑦,𝑧∈𝑋, is a 𝐺-metric on 𝑋.

Example 1.3. Let 𝑋=[0,+∞). The function 𝐺∶𝑋×𝑋×𝑋→[0,+∞), defined by 𝐺𝑥,𝑦,𝑧=𝑥−𝑦+𝑦−𝑧+|𝑧−𝑥|, (1.3) for all 𝑥,𝑦,𝑧∈𝑋, is a 𝐺-metric on 𝑋.

In their initial paper, Mustafa and Sims [1] defined also the basic topological concepts in 𝐺-metric spaces as follows.

Definition 1.4 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space, and let {𝑥_𝑛} be a sequence of points of 𝑋. We say that {𝑥_𝑛} is 𝐺-convergent to 𝑥∈𝑋 if lim_{𝑛,𝑚→+∞}𝐺𝑥,𝑥_𝑛,𝑥_𝑚=0, (1.4) that is, for any 𝜀>0, there exists 𝑁∈ℕ such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)<𝜀, for all 𝑛,𝑚≥𝑁. We call 𝑥 the limit of the sequence and write 𝑥_𝑛→𝑥 or lim_𝑛→+∞𝑥_𝑛=𝑥.

Proposition 1.5 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. The following are equivalent:(1){𝑥_𝑛} is 𝐺-convergent to 𝑥,(2)𝐺(𝑥_𝑛,𝑥_𝑛,𝑥)→0 as 𝑛→+∞,(3)𝐺(𝑥_𝑛,𝑥,𝑥)→0 as 𝑛→+∞,(4)𝐺(𝑥_𝑛,𝑥_𝑚,𝑥)→0 as 𝑛,𝑚→+∞.

Definition 1.6 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. A sequence {𝑥_𝑛} is called a 𝐺-Cauchy sequence if, for any 𝜀>0, there exists 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜀 for all 𝑚,𝑛,𝑙≥𝑁, that is, 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,𝑙→+∞.

Proposition 1.7 (see [1]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the following are equivalent:(1)the sequence {𝑥_𝑛} is 𝐺-Cauchy,(2)for any 𝜀>0, there exists 𝑁∈ℕ such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜀, for all 𝑚,𝑛≥𝑁.

Definition 1.8 (see [1]). A 𝐺-metric space (𝑋,𝐺) is called 𝐺-complete if every 𝐺-Cauchy sequence is 𝐺-convergent in (𝑋,𝐺).

Definition 1.9. Let (𝑋,𝐺) be a 𝐺-metric space. A mapping 𝐹∶𝑋×𝑋×𝑋→𝑋 is said to be continuous if for any three 𝐺-convergent sequences {𝑥_𝑛}, {𝑦_𝑛}, and {𝑧_𝑛} converging to 𝑥, 𝑦, and 𝑧, respectively, {𝐹(𝑥_𝑛,𝑦_𝑛,𝑧_𝑛)} is 𝐺-convergent to 𝐹(𝑥,𝑦,𝑧).

Note that each 𝐺-metric on 𝑋 generates a topology 𝜏_𝐺 on 𝑋 whose base is a family of open 𝐺-balls {𝐵_𝐺(𝑥,𝜀),𝑥∈𝑋,𝜀>0}, where 𝐵_𝐺(𝑥,𝜀)={𝑦∈𝑋,𝐺(𝑥,𝑦,𝑦)<𝜀} for all 𝑥∈𝑋 and 𝜀>0. A nonempty set 𝐴⊂𝑋 is 𝐺-closed in the the 𝐺-metric space (𝑋,𝐺) if 𝐴=𝐴. Observe that 𝑥∈𝐴⟺𝐵_𝐺(𝑥,𝜀)∩𝐴≠∅,∀𝜀>0. (1.5) Finally, we have the following proposition.

Proposition 1.10. Let (𝑋,𝐺) be a 𝐺-metric space and 𝐴 be a nonempty subset of 𝑋. A is 𝐺-closed if for any 𝐺-convergent sequence {𝑥_𝑛} in 𝐴 with limit 𝑥, one has 𝑥∈𝐴.

2. Banach Contractive Cyclic Maps on 𝐺-Metric Spaces

Our first result is a fixed point theorem which is the Banach contraction mapping analog for cyclic maps on 𝐺-metric spaces.

Theorem 2.1. Let (𝑋,𝐺) be a 𝐺-complete 𝐺-metric space and {𝐴_𝑗}^𝑚_𝑗=1 be a family of nonempty 𝐺-closed subsets of 𝑋. Let 𝑌=∪^𝑚_𝑗=1𝐴_𝑗 and 𝑇∶𝑌→𝑌 be a map satisfying 𝑇𝐴_𝑗⊆𝐴_𝑗+1,𝑗=1,…,𝑚,where𝐴_𝑚+1=𝐴₁. (2.1) If there exists 𝑘∈(0,1) such that 𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝑘𝐺𝑥,𝑦,𝑧 (2.2) holds for all 𝑥∈𝐴_𝑗 and 𝑦,𝑧∈𝐴_𝑗+1, 𝑗=1,…,𝑚  then, 𝑇 has a unique fixed point in ∩^𝑚_𝑗=1𝐴_𝑗.

Proof. We prove first the existence part. Take an arbitrary 𝑥₀∈𝑌 and without loss of generality assume that 𝑥₀∈𝐴₁. Define the sequence {𝑥_𝑛} as 𝑥_𝑛=𝑇𝑥_𝑛−1,𝑛=1,2,3,…. (2.3) Since 𝑇 is cyclic, 𝑥₀∈𝐴₁, 𝑥₁=𝑇𝑥₀∈𝐴₂, 𝑥₂=𝑇𝑥₁∈𝐴₃, …, and so on. If 𝑥_𝑛0+1=𝑥_𝑛0 for some 𝑛₀∈ℕ, then obviously, the fixed point of 𝑇 is 𝑥_𝑛0. Assume that 𝑥_𝑛+1≠𝑥_𝑛 for all 𝑛∈ℕ.
Put 𝑥=𝑥_𝑛−1 and 𝑦=𝑧=𝑥_𝑛 in (2.2). Then 0≤𝐺𝑇𝑥_𝑛−1,𝑇𝑥_𝑛,𝑇𝑥_𝑛=𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝑘𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛≤𝑘²𝐺𝑥_𝑛−2,𝑥_𝑛−1,𝑥_𝑛−1⋮≤𝑘^𝑛𝐺𝑥₀,𝑥₁,𝑥₁. (2.4) Then, we have, 0≤𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝑘^𝑛𝐺𝑥₀,𝑥₁,𝑥₁, (2.5) which upon letting 𝑛→∞ implies lim_𝑛→∞𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=0. (2.6) On the other hand, by symmetry (G4) and the rectangle inequality (G5), we have 𝐺𝑥,𝑦,𝑦=𝐺𝑦,𝑦,𝑥≤𝐺𝑦,𝑥,𝑥+𝐺𝑥,𝑦,𝑥=2𝐺𝑦,𝑥,𝑥. (2.7) The inequality (2.7) with 𝑥=𝑥_𝑛 and 𝑦=𝑥_𝑛−1 becomes 𝐺𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−1≤2𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛. (2.8) Letting 𝑛→∞ in (2.8), we get lim_𝑛→∞𝐺𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−1=0. (2.9)
We show next that the sequence {𝑥_𝑛} is a Cauchy sequence in the metric space (𝑋,𝑑_𝐺) where 𝑑_𝐺 is given in (1.1). For 𝑛≥𝑙 we have 𝑑_𝐺𝑥_𝑛,𝑥_𝑙≤𝑑_𝐺𝑥_𝑛,𝑥_𝑛−1+𝑑_𝐺𝑥_𝑛−1,𝑥_𝑛−2+⋯+𝑑_𝐺𝑥_𝑙+1,𝑥_𝑙=𝐺𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−1+𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛+𝐺𝑥_𝑛−1,𝑥_𝑛−2,𝑥_𝑛−2+𝐺𝑥_𝑛−2,𝑥_𝑛−1,𝑥_𝑛−1𝑥+⋯+𝐺_𝑙+1,𝑥_𝑙,𝑥_𝑙+𝐺𝑥_𝑙,𝑥_𝑙+1,𝑥_𝑙+1=^𝑛_𝑖=𝑙+1𝐺𝑥_𝑖,𝑥_𝑖−1,𝑥_𝑖−1+𝐺𝑥_𝑖−1,𝑥_𝑖,𝑥_𝑖, (2.10) and making use of (2.4) and (2.8) we obtain 0≤𝑑_𝐺𝑥_𝑛,𝑥_𝑙≤^𝑛_𝑖=𝑙+13𝑘^𝑖−1𝐺𝑥₀,𝑥₁,𝑥₁≤3𝐺𝑥₀,𝑥₁,𝑥₁^𝑛_𝑖=0𝑘^𝑖−1−^𝑙_𝑖=0𝑘^𝑖−1. (2.11) Hence, 𝑑_𝐺𝑥_𝑛,𝑥_𝑙→0as𝑛,𝑙→∞. (2.12) That is, the sequence {𝑥_𝑛} is Cauchy in (𝑋,𝑑_𝐺). Since the space (𝑋,𝐺) is 𝐺-complete then (𝑋,𝑑_𝐺) is complete (see Proposition 10 in [1]) and hence, {𝑥_𝑛} converges to a number say, 𝑢∈𝑋. Moreover, {𝑥_𝑛} is 𝐺-Cauchy in (𝑋,𝐺) (see Proposition 9 in [1]) and it is easy to see that 𝑢∈∩^𝑚_𝑗=1𝐴_𝑗. Indeed, if 𝑥₀∈𝐴₁, then the subsequence {𝑥_{𝑚(𝑛−1)}}^∞_𝑛=1∈𝐴₁, the subsequence {𝑥_{𝑚(𝑛−1)+1}}^∞_𝑛=1∈𝐴₂, and, continuing in this way, the subsequence {𝑥_𝑚𝑛−1}^∞_𝑛=1∈𝐴_𝑚. All the 𝑚 subsequences are 𝐺-convergent and hence, they all converge to the same limit 𝑢. In addition, the sets 𝐴_𝑗 are 𝐺-closed, thus, the limit 𝑢∈∩^𝑚_𝑗=1𝐴_𝑗.
We show now that 𝑢∈𝑋 is a fixed point of 𝑇, that is, 𝑢=𝑇𝑢. Consider now (1.1) and (2.2) with 𝑥=𝑥_𝑛, 𝑦=𝑧=𝑇𝑢 and suppose that 𝑢≠𝑇𝑢 or 𝑑_𝐺(𝑢,𝑇𝑢)>0, then we have, 0≤𝑑_𝐺𝑥_𝑛,𝑇𝑢=𝐺𝑥_𝑛,𝑇𝑢,𝑇𝑢+𝐺𝑇𝑢,𝑥_𝑛,𝑥_𝑛=𝐺𝑇𝑥_𝑛−1,𝑇𝑢,𝑇𝑢+𝐺𝑇𝑢,𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−1≤𝑘𝐺𝑥_𝑛−1,𝑢,𝑢+𝐺𝑢,𝑥_𝑛−1,𝑥_𝑛−1. (2.13) Passing to limit as 𝑛→∞, we end up with 0≤𝑑_𝐺(𝑢,𝑇𝑢)≤0 which contradicts the assumption 𝑑_𝐺(𝑢,𝑇𝑢)>0. Hence, 𝑢=𝑇𝑢, that is, 𝑢∈ is a fixed point of 𝑇.
To prove the uniqueness, we assume that 𝑣∈𝑋 is another fixed point of 𝑇 such that 𝑣≠𝑢. Both 𝑢 and 𝑣 lie in ∩^𝑚_𝑗=1𝐴_𝑗; thus, we can substitute 𝑥=𝑢 and 𝑦=𝑧=𝑣 in (2.2). This yields 𝐺(𝑇𝑣,𝑇𝑢,𝑇𝑢)≤𝑘𝐺(𝑢,𝑣,𝑣) which is true only for 𝑘=1 but 𝑘∈(0,1) by definition. Thus, the fixed point of 𝑇 is unique.

As a particular case of Theorem 2.1, we give the following corollary.

Corollary 2.2. Let (𝑋,𝐺) be a 𝐺-complete 𝐺-metric space and {𝐴_𝑗}^𝑚_𝑗=1 be a family of nonempty 𝐺-closed subsets of 𝑋. Let 𝑌=∪^𝑚_𝑗=1𝐴_𝑗 and 𝑇∶𝑌→𝑌 be a map satisfying 𝑇𝐴_𝑗⊆𝐴_𝑗+1,𝑗=1,…,𝑚,where𝐴_𝑚+1=𝐴₁. (2.14) If there exists 𝑘∈(0,1) such that 𝐺𝑇𝑥,𝑇𝑦,𝑇𝑦≤𝑘𝐺𝑥,𝑦,𝑦 (2.15) holds for all 𝑥∈𝐴_𝑗 and 𝑦∈𝐴_𝑗+1, 𝑗=1,…,𝑚 then, 𝑇 has a unique fixed point in ∩^𝑚_𝑗=1𝐴_𝑗.

3. Generalized Cyclic Weak 𝜙-Contractions on 𝐺-Metric Spaces

The main goal of a number of studies regarding fixed points is to weaken the contractive conditions on the map under consideration. Inspired by this idea, in 1969, Boyd and Wong [28] defined the concept of Φ-contraction. Later, in 1997, Alber and Guerre-Delabriere [29] defined the weak 𝜙-contractions on Hilbert spaces and proved fixed point theorem regarding such contractions. A map 𝑇∶𝑋→𝑋 on a metric space (𝑋,𝑑) is called a weak 𝜙-contraction if there exists a strictly increasing function 𝜙∶[0,∞)→[0,∞) with 𝜙(0)=0 such that 𝑑𝑇𝑥,𝑇𝑦≤𝑑𝑥,𝑦−𝜙𝑑𝑥,𝑦, (3.1) for all 𝑥,𝑦∈𝑋. These types of contractions have also been a subject of extensive research (see, e.g., [30–32]). In what follows, we discuss cyclic weak 𝜙-contractions on 𝐺-metric spaces.

Consider the set Ψ of continuous functions 𝜙∶[0,∞)→[0,∞) with 𝜙(0)=0 and 𝜙(𝑡)>0 for 𝑡>0. We have the following fixed point theorem.

Theorem 3.1. Let (𝑋,𝐺) be a 𝐺-complete 𝐺-metric space and {𝐴_𝑗}^𝑚_𝑗=1 be a family of nonempty 𝐺-closed subsets of 𝑋 with 𝑌=∪^𝑚_𝑗=1𝐴_𝑗. Let 𝑇∶𝑌→𝑌 be a map satisfying 𝑇𝐴_𝑗⊆𝐴_𝑗+1,𝑗=1,…,𝑚,where𝐴_𝑚+1=𝐴₁. (3.2) Suppose that there exists a function 𝜙∈Ψ such that the map 𝑇 satisfies 𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝑀𝑥,𝑦,𝑧−𝜙𝑀𝑥,𝑦,𝑧, (3.3) for all 𝑥∈𝐴_𝑗 and 𝑦,𝑧∈𝐴_𝑗+1, 𝑗=1,…,𝑚 where 𝑀𝑥,𝑦,𝑧=max𝐺𝑥,𝑦,𝑧,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺(𝑧,𝑇𝑧,𝑇𝑧). (3.4) Then 𝑇 has a unique fixed point in ∩^𝑚_𝑗=1𝐴_𝑗.

Proof. To prove the existence part, we construct a sequence of Picard iterations as usual. Take an arbitrary 𝑥₀∈𝐴₁ and define the sequence {𝑥_𝑛} as 𝑥_𝑛=𝑇𝑥_𝑛−1,𝑛=1,2,3,…. (3.5) Since 𝑇 is cyclic, 𝑥₀∈𝐴₁, 𝑥₁=𝑇𝑥₀∈𝐴₂, 𝑥₂=𝑇𝑥₁∈𝐴₃,…, and so on. If 𝑥_𝑛0+1=𝑥_𝑛0 for some 𝑛₀∈ℕ, then, obviously, the fixed point of 𝑇 is 𝑥_𝑛0. Assume that 𝑥_𝑛+1≠𝑥_𝑛 for all 𝑛∈ℕ.
Let 𝑥=𝑥_𝑛 and 𝑦=𝑧=𝑥_𝑛+1 in (3.3) 𝐺𝑇𝑥_𝑛,𝑇𝑥_𝑛+1,𝑇𝑥_𝑛+1=𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2≤𝑀𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1−𝜙𝑀𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1, (3.6) where 𝑀𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=max𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝐺𝑥_𝑛+1,𝑇𝑥_𝑛+1,𝑇𝑥_𝑛+1,𝐺𝑥_𝑛+1,𝑇𝑥_𝑛+1,𝑇𝑥_𝑛+1=max𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2. (3.7) If 𝑀(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)=𝐺(𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2), then (3.6) yields 𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2≤𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2−𝜙𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2, (3.8) which implies 𝜙(𝐺(𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2))=0 and hence 𝐺(𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2)=0. This contradicts the assumption 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛∈ℕ. Then we must have 𝑀(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)=𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1), in (3.6), so that 𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2≤𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1−𝜙𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1. (3.9) Thus, the sequence {𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)} is a nonnegative nonincreasing sequence which converges to 𝐿≥0. Letting 𝑛→∞ in (3.9) we get 𝐿≤𝐿−𝜙(𝐿). (3.10) It follows that 𝜙(𝐿)=0; therefore, 𝐿=0, that is, lim_𝑛→∞𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1=0. (3.11) The equation (2.7) in the proof of Theorem 2.1 with 𝑥=𝑥_𝑛 and 𝑦=𝑥_𝑛−1 yields 𝐺𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−1≤2𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛, (3.12) and hence, lim_𝑛→∞𝐺𝑥_𝑛,𝑥_𝑛−1,𝑥_𝑛−1=0. (3.13)
We claim that {𝑥_𝑛} is a 𝐺-Cauchy sequence in (𝑋,𝐺). Assume the contrary, that is, {𝑥_𝑛} is not 𝐺-Cauchy. Then, according to Proposition 1.7 there exist 𝜀>0 and corresponding subsequences {𝑛(𝑘)} and {𝑙(𝑘)} of ℕ satisfying 𝑛(𝑘)>𝑙(𝑘)>𝑘 for which 𝐺𝑥_𝑙(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≥𝜀, (3.14) where 𝑛(𝑘) is chosen as the smallest integer satisfying (3.14), that is, 𝐺𝑥_𝑙(𝑘),𝑥_{𝑛(𝑘)−1},𝑥_{𝑛(𝑘)−1}<𝜀. (3.15) It is easy to see from (3.14) and (3.15) and the rectangle inequality (G5) that 𝜀≤𝐺𝑥_𝑙(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑙(𝑘),𝑥_{𝑛(𝑘)−1},𝑥_{𝑛(𝑘)−1}+𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)<𝜀+𝐺𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘),𝑥_𝑛(𝑘). (3.16) Taking limit as 𝑘→∞ in (3.16) and using (3.11) we obtain lim_𝑘→∞𝐺𝑥_𝑙(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)=𝜀. (3.17) Observe that for every 𝑘∈ℕ there exists 𝑠(𝑘) satisfying 0≤𝑠(𝑘)≤𝑚 such that 𝑛(𝑘)−𝑙(𝑘)+𝑠(𝑘)≡1(𝑚). (3.18) Therefore, for large enough values of 𝑘 we have 𝑟(𝑘)=𝑙(𝑘)−𝑠(𝑘)>0 and 𝑥_𝑟(𝑘) and 𝑥_𝑛(𝑘) lie in the consecutive sets 𝐴_𝑗 and 𝐴_𝑗+1, respectively, for some 0≤𝑗≤𝑚. We next substitute 𝑥=𝑥_𝑟(𝑘) and 𝑦=𝑧=𝑥_𝑛(𝑘) in (3.3) to obtain 𝐺𝑇𝑥_𝑟(𝑘),𝑇𝑥_𝑛(𝑘),𝑇𝑥_𝑛(𝑘)≤𝑀𝑥_𝑟(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)−𝜙𝑀𝑥_𝑟(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘), (3.19) where 𝑀𝑥_𝑟(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)=max𝐺𝑥_𝑟(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘),𝐺𝑥_𝑟(𝑘),𝑥_𝑟(𝑘)+1,𝑥_𝑟(𝑘)+1,𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1. (3.20) Employing rectangle inequality (G5) repeatedly we see that 𝐺𝑥_𝑟(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑟(𝑘),𝑥_𝑟(𝑘)+1,𝑥_𝑟(𝑘)+1+𝐺𝑥_𝑟(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤𝐺𝑥_𝑟(𝑘),𝑥_𝑟(𝑘)+1,𝑥_𝑟(𝑘)+1+𝐺𝑥_𝑟(𝑘)+1,𝑥_𝑟(𝑘)+2,𝑥_𝑟(𝑘)+2+𝐺𝑥_𝑟(𝑘)+2,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤≤⋯_{𝑙−1𝑖=𝑟}𝐺𝑥_𝑖(𝑘),𝑥_𝑖(𝑘)+1,𝑥_𝑖(𝑘)+1+𝐺𝑥_𝑙(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘), (3.21) or equivalently 0≤𝐺𝑥_𝑟(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)−𝐺𝑥_𝑙(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤^𝑙−1_𝑖=𝑟𝐺𝑥_𝑖(𝑘),𝑥_𝑖(𝑘)+1,𝑥_𝑖(𝑘)+1. (3.22) Note that the sum on the right-hand side of (3.22) consists of finite 𝑠−1≤𝑚 number of terms, and due to (3.11) each term of this sum tends to 0 as 𝑘→∞. Therefore, lim_𝑘→∞𝐺𝑥_𝑟(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)=lim_𝑘→∞𝐺𝑥_𝑙(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)=𝜀. (3.23) Using rectangle inequality (G5) again, we have 0≤𝐺𝑥_𝑟(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑟(𝑘)+1,𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1≤𝐺𝑥_𝑟(𝑘)+1,𝑥_𝑟(𝑘),𝑥_𝑟(𝑘)+𝐺𝑥_𝑟(𝑘),𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+𝐺𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1, (3.24) from which we deduce lim_𝑘→∞𝐺𝑥_𝑟(𝑘)+1,𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+1=𝜀 (3.25) upon letting 𝑘→∞ and using (3.23). Now, passing to limit as 𝑘→∞ in (3.19) and using (3.11), (3.23), and (3.25) we get 𝜀≤max{𝜀,0,0}−𝜙(max{𝜀,0,0})=𝜀−𝜙(𝜀), (3.26) and hence 𝜙(𝜀)=0. We conclude that 𝜀=0 which contradicts the assumption that {𝑥_𝑛} is not 𝐺-Cauchy. Thus, the sequence {𝑥_𝑛} is 𝐺-Cauchy and since (𝑋,𝐺) is 𝐺-complete; it is 𝐺-convergent to a limit, say 𝑤∈𝑋. It can be easily seen that 𝑤∈∩^𝑚_𝑗=1𝐴_𝑗. Since 𝑥₀∈𝐴₁, then the subsequence {𝑥_{𝑚(𝑛−1)}}^∞_𝑛=1∈𝐴₁, the subsequence {𝑥_{𝑚(𝑛−1)+1}}^∞_𝑛=1∈𝐴₂, and, continuing in this way, the subsequence {𝑥_𝑚𝑛−1}^∞_𝑛=1∈𝐴_𝑚. All the 𝑚 subsequences are 𝐺-convergent in the 𝐺-closed sets 𝐴_𝑗 and hence, they all converge to the same limit 𝑤∈∩^𝑚_𝑗=1𝐴_𝑗.
To show that the limit of the Picard sequence is the fixed point of 𝑇, that is, 𝑤=𝑇𝑤 we employ (3.3) with 𝑥=𝑥_𝑛,𝑦=𝑧=𝑤. This leads to 𝐺𝑇𝑥_𝑛,𝑇𝑤,𝑇𝑤=𝑀𝑥_𝑛,𝑤,𝑤−𝜙𝑀𝑥_𝑛,𝑤,𝑤, (3.27) where 𝑀𝑥_𝑛,𝑤,𝑤=max𝐺𝑥_𝑛,𝑤,𝑤,𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1,𝐺(𝑤,𝑇𝑤,𝑇𝑤). (3.28) Passing to limit as 𝑛→∞, we get 𝐺(𝑤,𝑇𝑤,𝑇𝑤)=𝐺(𝑤,𝑇𝑤,𝑇𝑤)−𝜙(𝐺(𝑤,𝑇𝑤,𝑇𝑤)). (3.29) Thus, 𝜙(𝐺(𝑤,𝑇𝑤,𝑇𝑤))=0 and hence, 𝐺(𝑤,𝑇𝑤,𝑇𝑤)=0, that is, 𝑤=𝑇𝑤.
Finally, we prove that the fixed point is unique. Assume that 𝑣∈𝑋 is another fixed point of 𝑇 such that 𝑣≠𝑤. Then, since both 𝑣 and 𝑤 belong to ∩^𝑚_𝑗=1𝐴_𝑗, we set 𝑥=𝑣 and 𝑦=𝑧=𝑤 in (3.3) which yields 𝐺(𝑇𝑣,𝑇𝑤,𝑇𝑤)≤𝑀(𝑣,𝑤,𝑤)−𝜙(𝑀(𝑣,𝑤,𝑤)), (3.30) where, 𝑀(𝑣,𝑤,𝑤)=max{𝐺(𝑣,𝑤,𝑤),𝐺(𝑣,𝑇𝑣,𝑇𝑣),𝐺(𝑤,𝑇𝑤,𝑇𝑤)}=𝐺(𝑣,𝑤,𝑤). (3.31) Then (3.30) becomes 𝐺(𝑣,𝑤,𝑤)≤𝐺(𝑣,𝑤,𝑤)−𝜙(𝐺(𝑣,𝑤,𝑤)), (3.32) and clearly, 𝐺(𝑣,𝑤,𝑤)=0, so we conclude that 𝑣=𝑤, that is, the fixed point of 𝑇 is unique.

For particular choices of the function 𝜙 we obtain the following corollaries.

Corollary 3.2. Let (𝑋,𝐺) be a 𝐺-complete 𝐺-metric space and {𝐴_𝑗}^𝑚_𝑗=1 be a family of nonempty 𝐺-closed subsets of 𝑋 with 𝑌=∪^𝑚_𝑗=1𝐴_𝑗. Let 𝑇∶𝑌→𝑌 be a map satisfying 𝑇𝐴_𝑗⊆𝐴_𝑗+1,𝑗=1,…,𝑚,where𝐴_𝑚+1=𝐴₁. (3.33) Suppose that there exists 𝑘∈(0,1) such that the map 𝑇 satisfies 𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝑘𝑀𝑥,𝑦,𝑧, (3.34) for all 𝑥∈𝐴_𝑗 and 𝑦,𝑧∈𝐴_𝑗+1, 𝑗=1,…,𝑚 where 𝑀𝑥,𝑦,𝑧=max𝐺𝑥,𝑦,𝑧,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺(𝑧,𝑇𝑧,𝑇𝑧). (3.35) Then 𝑇 has a unique fixed point in ∩^𝑚_𝑗=1𝐴_𝑗.

Proof. The proof is obvious by choosing the function 𝜙 in Theorem 3.1 as 𝜙(𝑡)=(1−𝑘)𝑡.

Corollary 3.3. Let (𝑋,𝐺) be a 𝐺-complete 𝐺-metric space and {𝐴_𝑗}^𝑚_𝑗=1 be a family of nonempty 𝐺-closed subsets of 𝑋 with 𝑌=∪^𝑚_𝑗=1𝐴_𝑗. Let 𝑇∶𝑌→𝑌 be a map satisfying 𝑇𝐴_𝑗⊆𝐴_𝑗+1,𝑗=1,…,𝑚,where𝐴_𝑚+1=𝐴₁. (3.36) Suppose that there exist constants 𝑎,  𝑏,  𝑐,  and 𝑑 with 0<𝑎+𝑏+𝑐+𝑑<1 such that the map 𝑇 satisfies 𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝑎𝐺𝑥,𝑦,𝑧+𝑏𝐺(𝑥,𝑇𝑥,𝑇𝑥)+𝑐𝐺𝑦,𝑇𝑦,𝑇𝑦+𝑑𝐺(𝑧,𝑇𝑧,𝑇𝑧), (3.37) for all 𝑥∈𝐴_𝑗 and 𝑦,𝑧∈𝐴_𝑗+1, 𝑗=1,…,𝑚. Then, 𝑇 has a unique fixed point in ∩^𝑚_𝑗=1𝐴_𝑗.

Proof. Clearly we have, 𝑎𝐺𝑥,𝑦,𝑧+𝑏𝐺(𝑥,𝑇𝑥,𝑇𝑥)+𝑐𝐺𝑦,𝑇𝑦,𝑇𝑦+𝑑𝐺(𝑧,𝑇𝑧,𝑇𝑧)≤(𝑎+𝑏+𝑐+𝑑)𝑀𝑥,𝑦,𝑧, (3.38) where 𝑀𝑥,𝑦,𝑧=max𝐺𝑥,𝑦,𝑧,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺(𝑧,𝑇𝑧,𝑇𝑧). (3.39) By Corollary 3.2, the map 𝑇 has a unique fixed point.

4. An Example and Applications

To illustrate the cyclic weak 𝜙-contractions on 𝐺-metric spaces we give the following example.

Example 4.1. Let 𝑋=[−1,1] and let 𝑇∶𝑋→𝑋 be given as 𝑇𝑥=−𝑥/3. Let 𝐴=[−1,0] and 𝐵=[0,1]. Define the function 𝐺∶𝑋×𝑋×𝑋→[0,∞) as 𝐺𝑥,𝑦,𝑧=𝑥³−𝑦³+𝑦³−𝑧³+𝑧³−𝑥³. (4.1) Clearly, the function 𝐺 is a 𝐺-metric on 𝑋. Define also 𝜙∶[0,∞)→[0,∞) as 𝜙(𝑡)=2𝑡/3. Obviously, the map 𝑇 has a unique fixed point 𝑥=0∈𝐴∩𝐵.
It can be easily shown that the map 𝑇 satisfies the condition (3.3). Indeed, note that 𝐺=1𝑇𝑥,𝑇𝑦,𝑇𝑧𝑥27³−𝑦³+𝑦³−𝑧³+𝑧³−𝑥³,𝑀𝑥,𝑦,𝑧=max𝑥³−𝑦³+𝑦³−𝑧³+𝑧³−𝑥³,56𝑥27³,56𝑦27³,56𝑧27³. (4.2) Then, 𝑀𝑀=1𝑥,𝑦,𝑧−𝜙𝑥,𝑦,𝑧3𝑀𝑥,𝑦,𝑧, (4.3) and clearly, 𝐺=1𝑇𝑥,𝑇𝑦,𝑇𝑧𝑥27³−𝑦³+𝑦³−𝑧³+𝑧³−𝑥³<13𝑥³−𝑦³+𝑦³−𝑧³+𝑧³−𝑥³≤13𝑀𝑥,𝑦,𝑧. (4.4) Hence, 𝑇 has a unique fixed point by Theorem 3.1.

Cyclic maps satisfying integral type contractive conditions are amongst common applications of fixed point theorems. In this context, we consider the following applications.

Corollary 4.2. Let (𝑋,𝐺) be a 𝐺-complete 𝐺-metric space and {𝐴_𝑗}^𝑚_𝑗=1 be a family of nonempty 𝐺-closed subsets of 𝑋 with 𝑌=∪^𝑚_𝑗=1𝐴_𝑗. Let 𝑇∶𝑌→𝑌 be a map satisfying 𝑇𝐴_𝑗⊆𝐴_𝑗+1,𝑗=1,…,𝑚,where𝐴_𝑚+1=𝐴₁. (4.5) Suppose also that ₀^{𝐺(𝑇𝑥,𝑇𝑦,𝑇𝑧)}𝑑𝑠≤₀^{𝑀(𝑥,𝑦,𝑧)}𝑑𝑠−𝜙₀^{𝑀(𝑥,𝑦,𝑧)}𝑑𝑠, (4.6) where 𝜙∈Ψ and 𝑀𝑥,𝑦,𝑧=max𝐺𝑥,𝑦,𝑧,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺(𝑧,𝑇𝑧,𝑇𝑧), (4.7) for all 𝑥∈𝐴_𝑗 and 𝑦,𝑧∈𝐴_𝑗+1, 𝑗=1,…,𝑚. Then 𝑇 has a unique fixed point in ∩^𝑚_𝑖=1𝐴_𝑖.

Corollary 4.3. Let (𝑋,𝐺) be a 𝐺-complete 𝐺-metric space and {𝐴_𝑗}^𝑚_𝑗=1 be a family of nonempty 𝐺-closed subsets of 𝑋 with 𝑌=∪^𝑚_𝑗=1𝐴_𝑗. Let 𝑇∶𝑌→𝑌 be a map satisfying 𝑇𝐴_𝑗⊆𝐴_𝑗+1,𝑗=1,…,𝑚,where𝐴_𝑚+1=𝐴₁. (4.8) Suppose also that ₀^{𝐺(𝑇𝑥,𝑇𝑦,𝑇𝑧)}𝑑𝑠≤𝑘₀^{𝑀(𝑥,𝑦,𝑧)}𝑑𝑠, (4.9) where 𝑘∈(0,1) and 𝑀𝑥,𝑦,𝑧=max𝐺𝑥,𝑦,𝑧,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺(𝑧,𝑇𝑧,𝑇𝑧), (4.10) for all 𝑥∈𝐴_𝑗 and 𝑦,𝑧∈𝐴_𝑗+1, 𝑗=1,…,𝑚. Then 𝑇 has a unique fixed point in ∩^𝑚_𝑖=1𝐴_𝑖.

Very recently, Jachymski proved the equivalence of auxiliary functions (see Lemma, in [33]). Inspired by the results from the remarkable paper of Jachymski, we state the following theorem.

Theorem 4.4 (see [33]). Let 𝑇 be a self-map on a 𝐺-complete 𝐺-metric space (𝑋,𝐺) and {𝐴_𝑗}^𝑚_𝑗=1 be a family of nonempty 𝐺-closed subsets of 𝑋 with 𝑌=∪^𝑚_𝑗=1𝐴_𝑗. Let 𝑇∶𝑌→𝑌 be a map satisfying 𝑇𝐴_𝑗⊆𝐴_𝑗+1,𝑗=1,…,𝑚,where𝐴_𝑚+1=𝐴₁. (4.11) Assume that 𝑀𝑥,𝑦,𝑧=max𝐺𝑥,𝑦,𝑧,𝐺(𝑥,𝑇𝑥,𝑇𝑥),𝐺𝑦,𝑇𝑦,𝑇𝑦,𝐺(𝑧,𝑇𝑧,𝑇𝑧). (4.12) Then the following statements are equivalent: (i)there exist functions 𝜓,𝜂∈Ψ such that 𝜓𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝜓𝑀𝑥,𝑦,𝑧−𝜂𝑀𝑥,𝑦,𝑧, (4.13)forany𝑥∈𝐴_𝑖,𝑦,𝑧∈𝐴_𝑖+1,𝑖=1,2,…,𝑚,(ii)there exists a function 𝛽∶[0,∞)→[0,1] such that for any bounded sequence {𝑡_𝑛} of positive reals, 𝛽(𝑡_𝑛)→1 implies 𝑡_𝑛→0 and 𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝛽𝑀𝑥,𝑦,𝑧𝜓𝑀𝑥,𝑦,𝑧, (4.14)forany𝑥∈𝐴_𝑖,𝑦,𝑧∈𝐴_𝑖+1,𝑖=1,2,…,𝑚,(iii)there exists a continuous function 𝜂∶[0,∞)→[0,∞) such that 𝜂⁻¹({0})=0 and 𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝑀𝑥,𝑦,𝑧−𝜂𝑀𝑥,𝑦,𝑧 (4.15)forany𝑥∈𝐴_𝑖,𝑦,𝑧∈𝐴_𝑖+1,𝑖=1,2,…,𝑚,(iv)there exists function 𝜓∈Ψ and a nondecreasing, right continuous function 𝜑∶[0,∞)→[0,∞) with 𝜑(𝑡)<𝑡 and for all 𝑡>0 with 𝜓𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝜑𝜓𝑀𝑥,𝑦,𝑧, (4.16)forany𝑥∈𝐴_𝑖,𝑦,𝑧∈𝐴_𝑖+1,𝑖=1,2,…,𝑚,(v)there exists a continuous and nondecreasing function 𝜑∶[0,∞)→[0,∞) such that 𝜑(𝑡)<𝑡 and for all 𝑡>0 with 𝜓𝐺𝑇𝑥,𝑇𝑦,𝑇𝑧≤𝜑𝑀𝑥,𝑦,𝑧, (4.17)forany𝑥∈𝐴_𝑖,𝑦,𝑧∈𝐴_𝑖+1,𝑖=1,2,…,𝑚.