On Cyclic Generalized Weakly 𝐶-Contractions on Partial Metric Spaces

Abstract

We give new results of a cyclic generalized weakly 𝐶-contraction in partial metric space. The results of this paper extend, generalize, and improve some fixed point theorems in the literature.

1. Introduction and Preliminaries

The notion of partial metric space [1], represented by the abbreviation PMS, departs from the usual metric spaces due to removing the assumption of self-distance. In other words, in PMS self-distance needs not to be zero. This interesting distance function is defined by Matthews [1], as a generalization metric to study in computer science, in particular, to get a more efficient programs in computer science. In the remarkable publication of Matthews [1], a characterization of the Banach Contraction Principle was given in the context of PMS. Due to its wide application potential [2–6], PMS and its topological properties are considered by many authors [7–25]. Very recently, Haghi et al. [26] proved that some obtained results in the context of PMS can be deduced from earlier results in the setting of usual metric space.

In the sequel, ℝ⁺, ℕ^∗ will represent the set of all real nonnegative numbers and the set of all positive natural numbers, respectively. Moreover, we use the abbreviations MS, CMS, PMS, and CMPS for metric space, complete metric space, partial metric space, and complete partial metric space, respectively. Let Λ be the collection of function 𝜑∶[0,1)→[0,1) which is nondecreasing, continuous together with the property 𝜑(𝑡)>0 for 𝑡∈(0,1) and 𝜑(0)=0. The following definition introduced by Chatterjea [27] to generalize the Banach contraction principle.

Definition 1. Suppose that (𝑋,𝑑) is an MS. A mapping 𝑇∶𝑋→𝑋 is said to be a 𝐶-contraction if there exists 𝛼∈(0,1/2) such that the following inequality holds: 𝑑𝑇𝑥,𝑇𝑦≤𝛼𝑑𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥∀𝑥,𝑦∈𝑋. (1)
Moreover, Chatterjea [27] reported that every 𝐶-contraction 𝑇∶𝑋→𝑋 has a unique fixed point, where (𝑋,𝑑) is a complete metric space. Recently, Choudhury [28] introduced a generalization of 𝐶-contraction inspired by the notion of weak 𝜙-contraction (see, e.g., [29, 30]).

Definition 2. Suppose that (𝑋,𝑑) is an MS. A self-mapping 𝑇 on 𝑋 is called a weakly 𝐶-contractive if 𝑑≤1𝑇𝑥,𝑇𝑦2𝑑𝑑,𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜑𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥 (2) for all 𝑥,𝑦∈𝑋, where the mapping 𝜑∶[0,+∞)²→[0,+∞) is continuous and has the following property: 𝜑𝑥,𝑦=0iff𝑥=𝑦=0. (3)
The notion of weakly 𝐶-contractive can be also called a weak 𝐶-contraction. In [28], the author proves that on the setting of CMS, every weak 𝐶-contraction possesses a unique fixed point.
On other hand, in 2003, Kirk et al. [31] introduce cyclic contraction and give a characterization of the celebrated fixed-point theorem of Banach (known also as the Banach contraction mapping principle) in the set-up cyclic contraction. The authors [31] introduced the notion of cyclic representation in the following way.

Definition 3 (see [31]). Suppose that (𝑋,𝑑) is an MS and 𝑇 is a self-mapping on 𝑋. Let 𝑚 be a natural number and let 𝑋_𝑖, 𝑖=1,…,𝑚 be nonempty sets. Then, 𝑋=⋃^𝑚_𝑖=1𝑋_𝑖 is called a cyclic representation of 𝑋 with respect to 𝑇 if 𝑋₁⊂𝑋₂,…,𝑇𝑋_𝑚−1⊂𝑋_𝑚,𝑇𝑋_𝑚⊂𝑋₁, (4) where 𝑋_𝑚+1=𝑋₁.
Kirk et al. [31] prove that a self-mapping 𝑇, on a cyclic representation of 𝑋, possesses a fixed point if 𝑑𝑇𝑥,𝑇𝑦≤𝜑𝑑𝑥,𝑦,∀𝑥∈𝑋_𝑖,𝑦∈𝑋_𝑖+1,𝑖∈{1,2,…,𝑚}, (5) where (𝑋,𝑑) is a CMS and 𝜑∶[0,1)→[0,1) is a function, upper semicontinuous from the right and 0≤𝜑(𝑡)<𝑡 for 𝑡>0.
Recently, Păcurar and Rus [32] generalize the result of Kirk et al. [31] via the notion of cyclic 𝜙-contraction. Following the paper of Păcurar and Rus [32], the notion of cyclic weak-𝜙-contraction was introduced by Karapınar [33]. Let Λ be the collection of function 𝜑∶[0,1)→[0,1) which is nondecreasing, continuous together with the property 𝜑(𝑡)>0 for 𝑡∈(0,1) and 𝜑(0)=0.

Definition 4 (see [33]). Suppose that (𝑋,𝑑) is an MS and 𝑇 is a self-mapping on 𝑋. Let 𝑚 be a natural number and let 𝑋_𝑖, 𝑖=1,…,𝑚, be nonempty closed sets. Assume that 𝑋=⋃^𝑚_𝑖=1𝑋_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇. A mapping 𝑇∶𝑋→𝑋 is said to be a cyclic weaker 𝜑-contraction if there exists 𝜑∈Λ such that 𝑑𝑇𝑥,𝑇𝑦≤𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦, (6) for any 𝑥∈𝑋_𝑖, 𝑦∈𝑋_𝑖+1, 𝑖=1,2,…,𝑚, where 𝑋_𝑚+1=𝑋₁.
The author [33] shows that a self-mapping 𝑇, on a cyclic representation of 𝑋, possesses a fixed point if 𝑇 is a cyclic weaker 𝜑-contraction on a CMS (𝑋,𝑑).
In the last decade, the existence and uniqueness of a fixed point of various cyclic contractions in the context of PMS have been investigated and improved by several authors, see, for example, [7, 11].
In this paper, we derive some fixed-point result on certain cyclic contractions in the setup of complete PMS. Presented results of the paper extend, improve, and generalize some recent results on the topic in the literature. Among them, we list a few of them as follows: [7, 11, 13, 17, 28, 34].
For the sake of completeness, we call up some basic definitions and essential results in PMS. For more details, see, for example, [1, 7, 8, 17, 22].

Definition 5. Let 𝑋 be a nonempty set. A function 𝑝∶𝑋×𝑋→[0,∞) is called partial metric if the following conditions hold:(𝑝₁)𝑥=𝑦⇔𝑝(𝑥,𝑥)=𝑝(𝑥,𝑦)=𝑝(𝑦,𝑦),(𝑝₂)𝑝(𝑥,𝑥)≤𝑝(𝑥,𝑦),(𝑝₃)𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥),(𝑝₄)𝑝(𝑥,𝑦)≤𝑝(𝑥,𝑧)+𝑝(𝑧,𝑦)−𝑝(𝑧,𝑧), for all 𝑥,𝑦,𝑧∈𝑋. A pair (𝑋,𝑝) is called partial metric space.
It is evident that if 𝑝(𝑥,𝑦)=0, then 𝑥=𝑦 due to assumptions (𝑝₁) and (𝑝₂). However, if 𝑥=𝑦, then 𝑝(𝑥,𝑦) need not be 0. It is also known that a PMS generates a topology which is 𝑇₀. We say that a sequence {𝑥_𝑛} is convergent to a point 𝑥∈𝑋 in (𝑋,𝑝) if lim_𝑛→∞𝑝(𝑥,𝑥_𝑛)=𝑝(𝑥,𝑥), denoted as 𝑥_𝑛→𝑥 (𝑛→∞) or lim_𝑛→∞𝑥_𝑛=𝑥, with respect to the corresponding topology. We underline the simple fact that a limit of a sequence in a PMS need not be unique. Notice also that the function 𝑝(⋅,⋅) need not be continuous; that is, 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦 need not yield 𝑝(𝑥_𝑛,𝑦_𝑛)→𝑝(𝑥,𝑦).
There is strong correlation between partial metric and metric. For example, a mapping 𝑑_𝑝∶𝑋×𝑋→ℝ⁺ given by 𝑑_𝑝𝑥,𝑦=2𝑝𝑥,𝑦−𝑝(𝑥,𝑥)−𝑝𝑦,𝑦 (7) forms a metric on 𝑋, where 𝑝 is a partial metric. It is called the corresponding metric of partial metric.

Example 6. Let 𝑋=[0,∞). The pair (𝑋,𝑝) is an elementary example of a PMS, where 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for all 𝑥,𝑦∈ℝ⁺. Notice that the corresponding metric is 𝑑_𝑝𝑥,𝑦=2max𝑥,𝑦−𝑥−𝑦=𝑥−𝑦. (8)

Example 7. The mapping 𝑝(𝑥,𝑦)=𝜌(𝑥,𝑦)+𝑐 forms a partial metric on 𝑋. Note also that the corresponding metric is 𝑑_𝑝(𝑥,𝑦)=2𝑑(𝑥,𝑦), where (𝑋,𝜌) is a metric space and 𝑐≥0 is arbitrary.
For the further nontrivial examples of PMS, they can be found in [1–6].

Definition 8. Let (𝑋,𝑝) be a PMS. Then(1)a sequence {𝑥_𝑛} is called a Cauchy if the limit of 𝑝(𝑥_𝑛,𝑥_𝑚) as 𝑛,𝑚→∞ exists (and is finite). If every Cauchy sequence {𝑥_𝑛} in 𝑋 converges to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚), then the space (𝑋,𝑝) is called complete,(2)let {𝑥_𝑛} be a sequence in (𝑋,𝑝). If lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚)=0, then the sequence {𝑥_𝑛} is called 0-Cauchy. Analogously, if every 0-Cauchy sequence in 𝑋 converges to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=0, then the space (𝑋,𝑝) is called 0-complete and denoted by 0-CPMS [22].
This lemma can be found in some recent publication on the topic, see, for example, [2–6].

Lemma 9. Let (𝑋,𝑝) be a PMS. Then (a)a sequence {𝑥_𝑛} is Cauchy in (𝑋,𝑝) if and only if it is a Cauchy sequence in the metric space (𝑋,𝑑_𝑝),(b)a PMS (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑑_𝑝) is complete. Furthermore, lim_𝑛→+∞𝑑_𝑝(𝑥_𝑛,𝑥)=0 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥=lim_{𝑛,𝑚→+∞}𝑝𝑥_𝑛,𝑥_𝑚. (9)

Lemma 10. Let (𝑋,𝑝) be a PMS.(a)If 𝑝(𝑥_𝑛,𝑧)→𝑝(𝑧,𝑧)=0 as 𝑛→∞, then 𝑝(𝑥_𝑛,𝑦)→𝑝(𝑧,𝑦) as 𝑛→∞ for each 𝑦∈𝑋 [8, 9, 18].(b)If (𝑋,𝑝) is complete, then it is 0-complete [22].

The converse assertion of (b) does not hold; for the counter examples, see [22]. Note that every closed subset of a 0-CPMS is 0-complete.

Let Φ be the class of functions 𝜑∶[0,∞)³→[0,∞) which is lower semicontinuous and satisfying 𝜑(𝑥,𝑦,𝑧)=0⇔𝑥=𝑦=𝑧=0.

In what follows we introduce the notion of a cyclic generalized weakly 𝐶-contraction in PMS.

Definition 11. Assume that (𝑋,𝑝) is a PMS and 𝑚 is a natural number. Suppose that 𝑋₁,𝑋₂,…,𝑋_𝑚 are closed nonempty subsets of (𝑋,𝑑_𝑝) and 𝑌=⋃^𝑚_𝑖=1𝑋_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇; a mapping 𝑇∶𝑌→𝑌 is said to be a cyclic generalized weakly 𝐶-contraction if 𝑝≤1𝑇𝑥,𝑇𝑦4𝑝()𝑝()1𝑥,𝑇𝑥+𝑝𝑦,𝑇𝑦+𝑝𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥−𝜑𝑥,𝑇𝑥,𝑝𝑦,𝑇𝑦2𝑝𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥, (10) for any 𝑥∈𝑋_𝑖, 𝑦∈𝑋_𝑖+1, 𝑖=1,2,…,𝑚, where 𝑋_𝑚+1=𝑋₁ and 𝜑∈Φ.

In this paper, we establish a fixed point theorem for cyclic generalized weakly 𝐶-contractions in the frame of CMPS.

2. Main Results

We present the fundamental result of this paper as follows.

Theorem 12. Assume that (𝑋,𝑝) is a 0-CPMS and 𝑇∶𝑌→𝑌 is a cyclic generalized weakly 𝐶-contraction. Then, the mapping 𝑇 has a unique fixed point 𝑧∈⋂^𝑛_𝑖=1𝑋_𝑖, and 𝑝(𝑧,𝑧)=0.

Proof. Take 𝑥₀∈𝑌; that is, there is some 𝑖₀ with 𝑥₀∈𝑋_𝑖0. Since 𝑇(𝑋_𝑖0)⊆𝑋_𝑖0+1 implies that 𝑇𝑥₀∈𝑋_𝑖0+1, we find 𝑥₁∈𝑋_𝑖0+1 such that 𝑇𝑥₀=𝑥₁. By using the same argument, we construct the sequence 𝑥_𝑛+1=𝑇𝑥_𝑛, where 𝑥_𝑛∈𝑋_𝑖𝑛. Consequently, for 𝑛≥0, there exists 𝑖_𝑛∈{1,2,…,𝑚} such that 𝑥_𝑛∈𝑋_𝑖𝑛 and 𝑥_𝑛+1∈𝑋_𝑖𝑛+1. We suppose that 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛. Indeed, if 𝑥_𝑛0=𝑥_𝑛0+1 for some 𝑛₀, then we conclude that 𝑇𝑥_𝑛0=𝑥_𝑛0; that is, 𝑥_𝑛0 is the desired fixed point of 𝑇. Consequently, the proof is completed.
Due to (10), we derive that 𝑝𝑥_𝑛,𝑥_𝑛+1=𝑝𝑇𝑥_𝑛−1,𝑇𝑥_𝑛≤14𝑝𝑥_𝑛−1,𝑇𝑥_𝑛−1+𝑝𝑥_𝑛,𝑇𝑥_𝑛+𝑝𝑥_𝑛−1,𝑇𝑥_𝑛+𝑝𝑥_𝑛,𝑇𝑥_𝑛−1−𝜑𝑝𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝑝𝑥_𝑛,𝑇𝑥_𝑛,12𝑝𝑥_𝑛−1,𝑇𝑥_𝑛+𝑝𝑥_𝑛,𝑇𝑥_𝑛−1=14𝑝𝑥_𝑛−1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛+1+𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛−𝜑𝑝𝑥_𝑛−1,𝑥_𝑛,𝑝𝑥_𝑛,𝑥_𝑛+1,12𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛≤14𝑝𝑥_𝑛−1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛+1+𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛≤12𝑝𝑥_𝑛−1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛+1by𝑝₄, (11)(12)(13)(14)(15) for all 𝑛≥1. As a result, we find that 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝑝𝑥_𝑛−1,𝑥_𝑛, (16) for all 𝑛≥1. We set 𝑡_𝑛=𝑝(𝑥_𝑛,𝑥_𝑛−1). On the occasion of the facts above, {𝑡_𝑛} is a nonincreasing sequence of nonnegative real numbers. Consequently, there exists 𝐿≥0 such that lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+1=𝐿. (17) We will prove that 𝐿=0. Suppose, to the contrary, that 𝐿>0. From (14) and (15) we derive that 𝑝𝑥_𝑛,𝑥_𝑛+1≤14𝑝𝑥_𝑛−1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛+1+𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛≤12𝑝𝑥_𝑛−1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛+1, (18) for any 𝑛∈ℕ^∗. Letting 𝑛→∞ in (18), we have 𝐿=lim_𝑛→∞𝑝𝑥_𝑛+1,𝑥_𝑛≤lim_𝑛→∞142𝐿+𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛≤𝐿. (19) This yields that lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+𝑝𝑥_𝑛−1,𝑥_𝑛+1=2𝐿. (20)
On the other hand, by (13) we have 𝑡_𝑛+1≤14𝑡_𝑛+𝑡_𝑛+1+𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛−𝜑𝑡_𝑛,𝑡_𝑛+1,12𝑝𝑥_𝑛−1,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑥_𝑛≤12𝑡_𝑛+𝑡_𝑛+1. (21) Letting 𝑛→∞ in inequality (21), we get that 𝐿≤𝐿−𝜑(𝐿,𝐿,𝐿)≤𝐿. (22) Since 𝜑(𝑥,𝑦,𝑧)=0⇔𝑥=𝑦=𝑧=0, we get 𝐿=0.
Due to (𝑝₂), we have 0≤𝑝(𝑥_𝑛,𝑥_𝑛)≤𝑝(𝑥_𝑛,𝑥_𝑛+1). Hence, lim_𝑛→∞𝑝(𝑥_𝑛,𝑥_𝑛)=0. Then, by (20) we conclude that lim_𝑛→∞𝑝(𝑥_𝑛−1,𝑥_𝑛+1)=0.
Hence, we have lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+1=lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛=lim_𝑛→∞𝑝𝑥_𝑛−1,𝑥_𝑛+1=0. (23)
We assert that the sequence {𝑥_𝑛} is Cauchy. To reach this goal, the standard techniques in the literature will be used (see, e.g., [17]). For the sake of completeness, we explicitly prove that {𝑥_𝑛} is Cauchy. First assert that
(𝐾) for each 𝜀>0 there is 𝑛∈ℕ such that if 𝑟,𝑞≥𝑛 with 𝑟−𝑞≡1(𝑚), then 𝑝(𝑥_𝑟,𝑥_𝑞)<𝜀.
Suppose, to the contrary, that there is 𝜀>0 such that for all 𝑛∈ℕ if 𝑟_𝑛>𝑞_𝑛≥𝑛 with 𝑟_𝑛−𝑞_𝑛≡1(𝑚), then 𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛≥𝜀. (24)
We examine the case 𝑛>2𝑚. So, taking 𝑞_𝑛≥𝑛 into account, we can choose 𝑟_𝑛 with 𝑟_𝑛>𝑞_𝑛 in a way that it is the smallest integer satisfying 𝑟_𝑛−𝑞_𝑛≡1(𝑚) and 𝑝(𝑥_𝑞𝑛,𝑥_𝑟𝑛)≥𝜀. Hence, 𝑝(𝑥_𝑞𝑛,𝑥_{𝑟𝑛−𝑚})≤𝜀, by using the triangular inequality 𝜀≤𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛≤𝑝𝑥_𝑞𝑛,𝑥_{𝑟𝑛−𝑚}+^𝑚_𝑖=1𝑝𝑥_{𝑟𝑛−𝑖},𝑥_{𝑟𝑛−𝑖+1}−^𝑚_𝑖=1𝑝𝑥_{𝑟𝑛−𝑖},𝑥_{𝑟𝑛−𝑖}<𝜀+^𝑚_𝑖=1𝑝𝑥_{𝑟𝑛−𝑖},𝑥_{𝑟𝑛−𝑖+1}. (25) Letting 𝑛→∞ in (25) and keeping in mind lim_𝑛→∞𝑝(𝑥_𝑛,𝑥_𝑛+1)=0, we obtain lim_𝑛→∞𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛=𝜀. (26) Again, by (𝑝₄)𝜀≤𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛≤𝑝𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑝𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1+𝑝𝑥_𝑟𝑛+1,𝑥_𝑟𝑛−𝑝𝑥_𝑞𝑛+1,𝑥_𝑞𝑛+1−𝑝𝑥_𝑟𝑛+1,𝑥_𝑟𝑛+1≤𝑝𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑝𝑥_𝑞𝑛+1,𝑥_𝑞𝑛+𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1+𝑝𝑥_𝑟𝑛+1,𝑥_𝑟𝑛−𝑝𝑥_𝑞𝑛,𝑥_𝑞𝑛−𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛≤2𝑝𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+2𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1. (27) Taking (23) and (26) into account, we get lim_𝑛→∞𝑝𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1=𝜀, (28) as 𝑛→∞ in (26).
By (𝑝₄) we have the following inequalities: 𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+1≤𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1−𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛≤𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1,𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛≤𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+1+𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1−𝑝𝑥_𝑟𝑛+1,𝑥_𝑟𝑛+1≤𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+1+𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1. (29) Letting 𝑛→∞ in (29) we derived that lim_𝑛→∞𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+1=𝜀. (30) Again by (𝑝₄) we have 𝑝𝑥_𝑟𝑛,𝑥_𝑞𝑛+1≤𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1+𝑝𝑥_𝑟𝑛+1,𝑥_𝑞𝑛+1−𝑝𝑥_𝑟𝑛+1,𝑥_𝑟𝑛+1≤𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1+𝑝𝑥_𝑟𝑛+1,𝑥_𝑞𝑛+1,𝑝𝑥_𝑟𝑛+1,𝑥_𝑞𝑛+1≤𝑝𝑥_𝑟𝑛+1,𝑥_𝑟𝑛+𝑝𝑥_𝑟𝑛,𝑥_𝑞𝑛+1−𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛≤𝑝𝑥_𝑟𝑛+1,𝑥_𝑟𝑛+𝑝𝑥_𝑟𝑛,𝑥_𝑞𝑛+1. (31)
Letting 𝑛→∞ in (31) we derived that 𝑝𝑥_𝑟𝑛,𝑥_𝑞𝑛+1=𝜀. (32)
Since 𝑥_𝑞𝑛 and 𝑥_𝑟𝑛 lie in distinct adjacently labeled sets 𝑋_𝑖 and 𝑋_𝑖+1 for certain 1≤𝑖≤𝑚, keeping in mind that 𝑇 is a cyclic generalized weakly 𝐶-contraction, we have 𝑝𝑥_𝑞𝑛+1,𝑥_𝑟𝑛+1=𝑝𝑇𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛≤14𝑝𝑥_𝑞𝑛,𝑇𝑥_𝑞𝑛+𝑝𝑥_𝑟𝑛,𝑇𝑥_𝑟𝑛+𝑝𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛+𝑝𝑥_𝑟𝑛,𝑇𝑥_𝑞𝑛−𝜑𝑝𝑥_𝑞𝑛,𝑇𝑥_𝑞𝑛,𝑝𝑥_𝑟𝑛,𝑇𝑥_𝑟𝑛,12𝑝𝑥_𝑞𝑛,𝑇𝑥_𝑟𝑛+𝑝𝑥_𝑟𝑛,𝑇𝑥_𝑞𝑛≤14𝑝𝑥_𝑞𝑛,𝑥_𝑞𝑛+1+𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1+𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+1+𝑝𝑥_𝑟𝑛,𝑥_𝑞𝑛+1−𝜑𝑝𝑥_𝑞𝑛,𝑥_𝑞𝑛+1,𝑝𝑥_𝑟𝑛,𝑥_𝑟𝑛+1,12𝑝𝑥_𝑞𝑛,𝑥_𝑟𝑛+1+𝑝𝑥_𝑟𝑛,𝑥_𝑞𝑛+1. (33) Taking into account (23), (26), (28), (30), (32), and the lower semicontinuity of 𝜑, letting 𝑛→∞ in the inequality above, we find that 1𝜀≤4[]()≤10+0+𝜀+𝜀−𝜑0,0,𝜀2𝜀, (34) which is a contradiction. Hence, (𝐾) holds.
We are ready to show that the sequence {𝑥_𝑛} is Cauchy. Fix 𝜀>0. Due to the assumptions, one can find 𝑛₀∈ℕ such that if 𝑟,𝑞≥𝑛₀ with 𝑟−𝑞≡1(𝑚), 𝑝𝑥_𝑟,𝑥_𝑞≤𝜀2. (35) Since lim_𝑛→∞𝑝(𝑥_𝑛,𝑥_𝑛+1)=0, we also find 𝑛₁∈ℕ such that 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜀,2𝑚 (36) for any 𝑛≥𝑛₁. Assume that 𝑟,𝑠≥max{𝑛₀,𝑛₁} and 𝑠>𝑟. Consequently, there is a 𝑘∈{1,2,…,𝑚} with 𝑠−𝑟≡𝑘(𝑚). Therefore, 𝑠−𝑟+𝛼≡1(𝑚) for 𝛼=𝑚−𝑘+1. Thus, we obtain for 𝑗∈{1,…,𝑚}, 𝑠+𝑗−𝑟≡1(𝑚)𝑝𝑥_𝑟,𝑥_𝑠≤𝑝𝑥_𝑟,𝑥_𝑠+𝑗+𝑝𝑥_𝑠+𝑗,𝑥_{𝑠+𝑗−1}+⋯+𝑝𝑥_𝑠+1,𝑥_𝑠−𝑝𝑥_𝑠+𝑗,𝑥_𝑠+𝑗+⋯+𝑝𝑥_𝑠+1,𝑥_𝑠+1≤𝑝𝑥_𝑟,𝑥_𝑠+𝑗+𝑝𝑥_𝑠+𝑗,𝑥_{𝑠+𝑗−1}𝑥+⋯+𝑝_𝑠+1,𝑥_𝑠. (37) By (35) and (36) together with the last inequality, we find that 𝑝𝑥_𝑟,𝑥_𝑠≤𝜀2𝜀+𝑗×≤𝜀2𝑚2𝜀+𝑚×2𝑚=𝜀, (38) which yields that the sequence {𝑥_𝑛} is Cauchy. Regarding that 𝜀 is arbitrary, we conclude that {𝑥_𝑛} is a 0-Cauchy sequence.
Taking into account that 𝑌 is closed in (𝑋,𝑝), we observe that (𝑌,𝑝) is also 0-complete. Thus, there exists 𝑥∈𝑌=⋃^𝑚_𝑖=1𝑋_𝑖 such that lim_𝑛→∞𝑥_𝑛=𝑥 in (𝑌,𝑝); equivalently 𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝𝑥,𝑥_𝑛=lim_{𝑛,𝑚→∞}𝑝𝑥_𝑛,𝑥_𝑚=0. (39)
Now, we assert that 𝑥 is a fixed point of 𝑇. First, we observed that the sequence (𝑥_𝑛) has infinite terms in each 𝑋_𝑖 for 𝑖∈{1,2,…,𝑚}, since lim_𝑛→∞𝑥_𝑛=𝑥 and as 𝑌=⋃^𝑚_𝑖=1𝑋_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇. Assume that 𝑥∈𝑋_𝑖 and 𝑇𝑥∈𝑋_𝑖+1. We consider a subsequence 𝑥_𝑛𝑘 of (𝑥_𝑛) with 𝑥_𝑛𝑘∈𝑋_𝑖−1. Notice that such subsequence exists due to the above-mentioned comment. By applying the contractive condition, we find 𝑝𝑥_𝑛𝑘+1,𝑇𝑥=𝑝𝑇𝑥_𝑛𝑘≤1,𝑇𝑥4𝑝𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘()𝑥+𝑝𝑥,𝑇𝑥+𝑝_𝑛𝑘,𝑇𝑥+𝑝𝑥,𝑇𝑥_𝑛𝑘−𝜑𝑝𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘(),1,𝑝𝑥,𝑇𝑥2𝑝𝑥_𝑛𝑘,𝑇𝑥+𝑝𝑥,𝑇𝑥_𝑛𝑘=14𝑝𝑥_𝑛𝑘,𝑥_𝑛𝑘+1()𝑥+𝑝𝑥,𝑇𝑥+𝑝_𝑛𝑘,𝑇𝑥+𝑝𝑥,𝑥_𝑛𝑘+1−𝜑𝑝𝑥_𝑛𝑘,𝑥_𝑛𝑘+1(),1,𝑝𝑥,𝑇𝑥2𝑝𝑥_𝑛𝑘,𝑇𝑥+𝑝𝑥,𝑥_𝑛𝑘+1. (40) Letting 𝑛→∞ and by using 𝑥_𝑛𝑘→𝑥, together with the lower semicontinuity of 𝜑, we get 𝑝()≤1𝑥,𝑇𝑥2𝑝()(),1𝑥,𝑇𝑥−𝜑0,𝑝𝑥,𝑇𝑥2𝑝(≤1𝑥,𝑇𝑥)2𝑝(𝑥,𝑇𝑥). (41) So 𝑝(𝑥,𝑇𝑥)=0 which yields that 𝑇𝑥=𝑥. We will prove the uniqueness of 𝑥 to complete the proof. Suppose, on the contrary, that 𝑦,𝑧∈𝑋 are distinct fixed points of 𝑇. We observe that 𝑦,𝑧∈⋂^𝑚_𝑖=1𝑋_𝑖, since 𝑇 is cyclic mapping and 𝑦,𝑧∈𝑋 are fixed points of 𝑇. Due to mentioned contractive condition, we derive that 𝑝≤1𝑦,𝑧=𝑝𝑇𝑦,𝑇𝑥4𝑝()𝑝(),1𝑦,𝑇𝑦+𝑝𝑧,𝑇𝑧+𝑝𝑦,𝑇𝑧+𝑝𝑧,𝑇𝑦−𝜑𝑦,𝑇𝑦,𝑝𝑧,𝑇𝑧2𝑝𝑦,𝑇𝑧+𝑝𝑧,𝑇𝑦, (42) that is, 𝑝≤1𝑦,𝑧2𝑝1𝑦,𝑧−𝜑0,0,2𝑝≤1𝑦,𝑧+𝑝𝑧,𝑦2𝑝𝑦,𝑧. (43) This gives us 𝑝(𝑦,𝑧)=0; that is, 𝑦=𝑧.

Corollary 13. Suppose that (𝑋,𝑝) is a 0-CPMS, 𝑚∈ℕ, 𝑋₁,𝑋₂,…,𝑋_𝑚 are nonempty closed subsets of 𝑋. Let 𝑇∶𝑌→𝑌 be and let 𝑌=⋃^𝑚_𝑖=1𝑋_𝑖. 𝑌=⋃^𝑚_𝑖=1𝑋_𝑖 be a cyclic representation of 𝑋 with respect to 𝑇.
If there exists 𝛽∈[0,1/4) such that 𝑝𝑝(),𝑇𝑥,𝑇𝑦≤𝛽𝑥,𝑇𝑥+𝑝𝑦,𝑇𝑦+𝑝𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥 (44) for any 𝑥∈𝑋_𝑖, 𝑦∈𝑋_𝑖+1, 𝑖=1,2,…,𝑚, where 𝑋_𝑚+1=𝑋₁, then, 𝑇 has a fixed point 𝑧∈⋂^𝑛_𝑖=1𝑋_𝑖 and 𝑝(𝑧,𝑧)=0.

Proof. Let 𝛽∈[0,1/4). Hence, it suffices to take the function 𝜑∶[0,+∞)³→[0,+∞) defined by 𝜑(𝑎,𝑏,𝑐)=(1/4−𝛽)(𝑎+𝑏+2𝑐). It is evident that 𝜑 satisfies the following conditions: (1)𝜑(𝑎,𝑏,𝑐)=0 if and only if 𝑎=𝑏=𝑐=0, and (2)𝜑 is lower semi-continuous. The results follow when we apply Theorem 12.

Theorem 14. Suppose that (𝑋,𝑑) is a 0-CPMS. If the mapping 𝑇∶𝑋→𝑋 satisfies 𝑝≤1𝑇𝑥,𝑇𝑦4𝑝()𝑝(),1𝑥,𝑇𝑥+𝑝𝑦,𝑇𝑦+𝑝𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥−𝜑𝑥,𝑇𝑥,𝑝𝑥,𝑇𝑦2𝑝𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥, (45) for any 𝑥,𝑦∈𝑋, where 𝜙∈Φ, then, it has a unique fixed point 𝑧∈𝑋 with 𝑝(𝑧,𝑧)=0.

Proof. It is sufficient to take 𝑋_𝑖=𝑋 for 𝑖=1,…,𝑚 in Theorem 12.

Remark 15. Let us remark that if in Definition 11 we consider the following condition 𝑝𝑝(),1𝑇𝑥,𝑇𝑦≤max𝑥,𝑥,𝑝𝑦,𝑦4𝑝()𝑝(),1𝑥,𝑇𝑥+𝑝𝑦,𝑇𝑦+𝑝𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥−𝜑𝑥,𝑇𝑥,𝑝𝑥,𝑇𝑦2𝑝𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥, (46) instead of (10), then by following the lines in the proof of Theorem 12, we obtain the same conclusions in our results.

Example 16. Let 𝑋=[0,1] and 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. It is clear that (𝑋,𝑝) is a 0-complete partial metric space. Fix 𝑚∈ℕ and define 𝑌=∪^𝑚_𝑖=1𝑋_𝑖, where 𝑋_𝑖=[0,(1/2^𝑖)] for 𝑖=1,2,…,𝑚. Let 𝑇∶𝑋→𝑋 and let 𝜑∶[0,∞)³→[0,∞)³ be defined as 𝜑=0𝑝,𝑞,𝑟if𝑝=𝑞=𝑟=0,𝑝+𝑞+𝑟36otherwise, (47) respectively. Then, all the conditions of Theorem 12 are satisfied. Hence, 𝑇 has a unique fixed point, namely, 0.

Remark 17. Notice that we get the same results if we replace 0-CPMS with CPMS.

Acknowledgment

Vladimir Rakocević is supported by Grant no. 174025 of the Ministry of Science, Technology, and Development, Serbia.