Fixed Point Theory for Cyclic Generalized Weak 𝜙-Contraction on Partial Metric Spaces

Abstract

A new fixed point theorem is obtained for the class of cyclic weak 𝜙-contractions on partially metric spaces. It is proved that a self-mapping 𝑇 on a complete partial metric space 𝑋 has a fixed point if it satisfies the cyclic weak 𝜙-contraction principle.

1. Introduction and Preliminaries

Fixed point theorems analyze the conditions on maps (single or multivalued) under which the existence of a solution for the equation 𝑑(𝑥,𝑇𝑥)=0 can be guaranteed. The Banach contraction mapping principle [1] is one of the earliest, widely known, and important results in this direction. After this pivotal principle, a number of remarkable results started to appear in the literature published by many authors (see, e.g., [2–36]). In this trend, one of the distinguished contributions was announced by Kirk et al. [15] in 2003. They introduced the notion of cyclic contraction, which is defined as follows.

Let 𝐴 and 𝐵 be two subsets of a metric space (𝑋,𝑑). A self-mapping 𝑇 on 𝐴∪𝐵 is called cyclic provided that 𝑇(𝐴)⊂𝐵 and 𝑇(𝐵)⊂𝐴. In addition, if for some 𝑘∈(0,1), the mapping 𝑇 satisfies the inequality 𝑑𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦+(1−𝑘)dist(𝐴,𝐵)∀𝑥∈𝐴,𝑦∈𝐵, (1.1) then it is called cyclic contraction where dist(𝐴,𝐵)=inf{𝑑(𝑥,𝑦)∶𝑥∈𝐴,𝑦∈𝐵}. In their outstanding paper [15], the authors also proved that if 𝐴 and 𝐵 are closed subsets of a complete metric space with 𝐴∩𝐵≠∅, then the cyclic contraction 𝑇 has a unique fixed point in 𝐴∩𝐵.

The notion of Φ-contraction was defined by Boyd and Wong [11]: A self-mapping 𝑇 on a metric space 𝑋 is called Φ-contraction if there exists an upper semi-continuous function Φ∶[0,∞)→[0,∞) such that 𝑑𝑇𝑥,𝑇𝑦≤Φ𝑑𝑥,𝑦∀𝑥,𝑦∈𝑋. (1.2) This concept was generalized by Alber and Guerre-Delabriere [7], by introducing weak 𝜙-contraction. A self-mapping 𝑇 on a metric space 𝑋 is called weak 𝜙-contraction if 𝜙∶[0,∞)→[0,∞) is a strictly increasing map with 𝜙(0)=0 and 𝑑𝑇𝑥,𝑇𝑦≤𝑑𝑥,𝑦−𝜙𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋. (1.3)

Păcurar and Rus [19] gave a characterization of Φ-contraction mappings in the context of cyclic operators and proved some fixed point results for such mappings on a complete metric space. As a natural next step, Karapınar [13] generalized the results in [19] by replacing the notion of cyclic 𝜑-contraction mappings with cyclic weak 𝜙-contraction mappings. For more results for cyclic mapping analysis we refer to [3, 7, 12, 14, 16, 17, 21–23] and the references therein.

Recently, in fixed point theory, one of the celebrated subjects is the partial metric spaces. The notion of a partial metric space was defined by Matthews [37] in 1992 as a generalization of usual metric spaces. The motivation behind the theory of partial metric spaces is to transfer mathematical techniques into computer science to develop the branches of computer science such as domain theory and semantics (see, e.g., [35, 38–46]).

Definition 1.1 (see [37, 47]). A partial metric on a nonempty set 𝑋 is a function 𝑝∶𝑋×𝑋→[0,∞) such that for all 𝑥,𝑦,𝑧∈𝑋(PM1)𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥) (symmetry),(PM2) if 𝑝(𝑥,𝑥)=𝑝(𝑥,𝑦)=𝑝(𝑦,𝑦), then 𝑥=𝑦 (equality),(PM3)𝑝(𝑥,𝑥)≤𝑝(𝑥,𝑦) (small self-distances),(PM4)𝑝(𝑥,𝑧)+𝑝(𝑦,𝑦)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧) (triangularity). The pair (𝑋,𝑝) is called a partial metric space (abbreviated by PMS).

Remark 1.2. If 𝑥=𝑦, then 𝑝(𝑥,𝑦) may not be 0.

Example 1.3 (see [47]). Let 𝑋={[𝑎,𝑏]∶𝑎,𝑏,∈ℝ,𝑎≤𝑏}, and define 𝑝([𝑎,𝑏],[𝑐,𝑑])=max{𝑏,𝑑}−min{𝑎,𝑐}. Then, (𝑋,𝑝) is a partial metric space.

Example 1.4 (see [47]). Let 𝑋∶=[0,1]∪[2,3], and define 𝑝∶𝑋×𝑋→[0,∞) by𝑝𝑥,𝑦=max𝑥,𝑦if𝑥,𝑦∩[2,3]≠∅,𝑥−𝑦if𝑥,𝑦⊂[0,1]. (1.4) Then, (X, p) is a complete partial metric space.

Example 1.5 (see [48]). Let (𝑋,𝑑) and (𝑋,𝑝) be a metric space and a partial metric space, respectively. Mappings 𝜌_𝑖∶𝑋×𝑋→ℝ⁺ (𝑖∈{1,2,3}) defined by 𝜌₁𝜌𝑥,𝑦=𝑑𝑥,𝑦+𝑝𝑥,𝑦₂𝜔(𝑥)𝑦𝜌𝑥,𝑦=𝑑𝑥,𝑦+max,𝜔₃𝑥,𝑦=𝑑𝑥,𝑦+𝑎 (1.5) induce partial metrics on 𝑋, where 𝜔∶𝑋→ℝ⁺ is an arbitrary function and 𝑎≥0.
Each partial metric 𝑝 on 𝑋 generates a 𝑇₀ topology 𝜏_𝑝 on 𝑋 which has the family of open 𝑝-balls {𝐵_𝑝(𝑥,𝜀),𝑥∈𝑋,𝜀>0} as a base, where 𝐵_𝑝(𝑥,𝜀)={𝑦∈𝑋∶𝑝(𝑥,𝑦)<𝑝(𝑥,𝑥)+𝜀} for all 𝑥∈𝑋 and 𝜀>0.
If 𝑝 is a partial metric on 𝑋, then the function 𝑑_𝑝∶𝑋×𝑋→[0,∞) given by 𝑑_𝑝𝑥,𝑦=2𝑝𝑥,𝑦−𝑝(𝑥,𝑥)−𝑝𝑦,𝑦 (1.6) is a metric on 𝑋. Furthermore, it is possible to observe that the following 𝑑_𝑚𝑥,𝑦=max𝑝𝑥,𝑦−𝑝(𝑥,𝑥),𝑝𝑥,𝑦−𝑝𝑦,𝑦, (1.7) also defines a metric on 𝑋. In fact, 𝑑_𝑝 and 𝑑_𝑚 are equivalent (see, e.g., [9]).

Example 1.6 (see, e.g., [5, 27, 30, 47]). Consider 𝑋=[0,∞) with 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Then, (𝑋,𝑝) is a partial metric space. It is clear that 𝑝 is not a (usual) metric. Note that in this case 𝑑_𝑝(𝑥,𝑦)=|𝑥−𝑦|=𝑑_𝑚(𝑥,𝑦).

For our purposes, we need to recall some basic topological concepts in partial metric spaces (for details see, e.g., [5, 8, 27, 30, 37, 47]).

Definition 1.7. (1) A sequence {𝑥_𝑛} in the PMS (𝑋,𝑝) converges to a limit point 𝑥 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝(𝑥,𝑥_𝑛).
(2) A sequence {𝑥_𝑛} in the PMS (𝑋,𝑝) is called a Cauchy sequence if lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists and is finite.
(3)   A PMS (𝑋,𝑝) is called complete if every Cauchy sequence {𝑥_𝑛} in 𝑋 converges, with respect to 𝜏_𝑝, to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚).
(4) A mapping 𝐹∶𝑋→𝑋 is said to be continuous at 𝑥₀∈𝑋 if, for every 𝜖>0, there exists 𝛿>0 such that 𝐹(𝐵_𝑝(𝑥₀,𝛿))⊆𝐵_𝑝(𝐹𝑥₀,𝜖).

Lemma 1.8. (1) A sequence {𝑥_𝑛} is a Cauchy sequence in the PMS (𝑋,𝑝) if and only if it is a Cauchy sequence in the metric space (𝑋,𝑑_𝑝).
(2) A PMS (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑑_𝑝) is complete. Moreover, lim_𝑛→∞𝑑_𝑝𝑥,𝑥_𝑛=0⟺𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝𝑥,𝑥_𝑛=lim_{𝑛,𝑚→∞}𝑝𝑥_𝑛,𝑥_𝑚. (1.8)

In [32], Romaguera introduced the concepts of a 0-Cauchy sequence in a partial metric space and of a 0-complete partial metric space as follows.

Definition 1.9. A sequence {𝑥_𝑛} in a partial metric space (𝑋,𝑝) is called 0-Cauchy if lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚)=0. A partial metric space (𝑋,𝑝) is said to be 0-complete if every 0-Cauchy sequence in 𝑋 converges, with respect to 𝜏_𝑝, to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=0. In this case, 𝑝 is said to be a 0-complete partial metric on 𝑋.

Remark 1.10 (see [32]). Each 0-Cauchy sequence in (𝑋,𝑝) is a Cauchy sequence in (𝑋,𝑝), and each complete partial metric space is 0-complete.

The following example shows that there exists a 0-complete partial metric space that is not complete.

Example 1.11 1.11 (see [32]). Let (ℚ∩[0,∞),𝑝) be a partial metric space, where ℚ denotes the set of rational numbers and the partial metric 𝑝 is given by 𝑝(𝑥,𝑦)=max{𝑥,𝑦}.

Now, we state a number of simple but useful lemmas that can be derived by manipulating the properties (𝑃1)−(𝑃4).

Lemma 1.12. Let (𝑋,𝑝) be a PMS. Then, the following statements hold true: (A) if lim_𝑛→∞𝑝(𝑥_𝑛,𝑧)=𝑝(𝑧,𝑧) and 𝑝(𝑧,𝑧)=0, then lim_𝑛→∞𝑝(𝑥_𝑛,𝑦)=𝑝(𝑧,𝑦) for every 𝑦∈𝑋 (see [5, 27, 30]), (B) if lim_𝑛→∞𝑝(𝑥_𝑛,𝑦)=𝑝(𝑦,𝑦), lim_𝑛→∞𝑝(𝑥_𝑛,𝑧)=𝑝(𝑧,𝑧), and 𝑝(𝑦,𝑦)=𝑝(𝑧,𝑧), then 𝑦=𝑧 (see [31]), (C) if 𝑝(𝑥,𝑦)=0 then 𝑥=𝑦 (see [27, 30]), (D) if 𝑥≠𝑦, then 𝑝(𝑥,𝑦)>0 (see [27, 30]).

One of the implications of Lemma 1.12 can be stated as follows.

Remark 1.13. If {𝑥_𝑛} converges to 𝑥 in (𝑋,𝑝), then lim_𝑛→∞𝑝(𝑥_𝑛,𝑦)≤𝑝(𝑥,𝑦) for all 𝑦∈𝑋.

We would like to point out that the topology induced by a partial metric differs from the topology induced by a metric in certain aspects. The following remark and example highlight one of these aspects.

Remark 1.14. Limit of a sequence {𝑥_𝑛} in a partial metric space (𝑋,𝑝) may not be unique.

Example 1.15. Consider 𝑋=[0,∞) with 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Then, (𝑋,𝑝) is a partial metric space. Clearly, 𝑝 is not a metric. Observe that the sequence {1+(1/1+𝑛)} converges to both 𝑥=5 and 𝑦=6, for example. Therefore, there is no uniqueness of the limit in this partial metric space.

2. Main Results

In this section we aim to state and prove our main results. These results are more general in the sense that they are applicable to chains of cyclic representations and related generalized weak 𝜙-contractions introduced in the following two definitions.

Definition 2.1 (see, e.g., [13]). Let 𝑋 be a nonempty set, 𝑚, a positive integer, and 𝑇∶𝑋→𝑋 an operator. The union 𝑋=∪^𝑚_𝑖=1𝑋_𝑖 is called a cyclic representation of 𝑋 with respect to 𝑇 if (1) each 𝑋_𝑖 is a nonempty set for 𝑖=1,…,𝑚, (2)𝑇(𝑋₁)⊂𝑋₂,…,𝑇(𝑋_𝑚−1)⊂𝑋_𝑚, 𝑇(𝑋_𝑚)⊂𝑋₁.

Definition 2.2 (cf. [13]). Let (𝑋,𝑝) be a partial metric space, 𝑚 be a positive integer, 𝐴₁,…,𝐴_𝑚 closed nonempty subsets of 𝑋 and 𝑌=∪^𝑚_𝑖=1𝐴_𝑖. An operator 𝑇∶𝑌→𝑌 is called a cyclic generalized weak 𝜙-contraction if (1)∪^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇, (2) there exists a continuous, nondecreasing function 𝜙∶[0,∞)→[0,∞) with 𝜙(𝑡)>0 for every 𝑡∈(0,∞) and 𝜙(0)=0, such that 𝑝𝑇𝑥,𝑇𝑦≤𝑀𝑥,𝑦−𝜙𝑀𝑥,𝑦, (2.1)where 𝑀𝑥,𝑦=max𝑝𝑥,𝑦,𝑝(𝑇𝑥,𝑥),𝑝𝑦,𝑇𝑦, (2.2) for any 𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1 for 𝑖=1,2,…,𝑚 with 𝐴_𝑚+1=𝐴₁.

Our main theorem in this work is stated below.

Theorem 2.3. Let (𝑋,𝑝) be a 0-complete partial metric space, 𝑚 a positive integer, and 𝐴₁,…,𝐴_𝑚 be closed non-empty subsets of 𝑋. Let 𝑌=∪^𝑚_𝑖=1𝐴_𝑖 be a cyclic representation of 𝑌 with respect to 𝑇. Suppose that 𝑇 is a cyclic generalized weak 𝜙-contraction. Then, 𝑇 has a unique fixed point 𝑧∈∩^𝑚_𝑖=1𝐴_𝑖.

Proof. First we show that if 𝑇 has a fixed point, then it is unique. Suppose on the contrary that there exist 𝑧,𝑤∈∩^𝑚_𝑖=1𝐴_𝑖 such that 𝑇𝑧=𝑧 and 𝑤=𝑇𝑤 with 𝑧≠𝑤. Due to (2.1), we have 𝑝(𝑧,𝑤)=𝑝(𝑇𝑧,𝑇𝑤)≤𝑀(𝑧,𝑤)−𝜙(𝑀(𝑧,𝑤)), (2.3) which is equivalent to 𝑝(𝑧,𝑤)=𝑝(𝑇𝑧,𝑇𝑤)≤𝑝(𝑧,𝑤)−𝜙𝑝(𝑧,𝑤) (2.4) by (PM3) in Definition 1.1. This is a contradiction because 𝑝(𝑧,𝑤)>0 by Lemma 1.12 and 𝜙(𝑡)>0 for all 𝑡>0. Hence, 𝑇 has a unique fixed point.
Next we prove the existence of a fixed point of 𝑇. Let 𝑥₀∈𝑌=∪^𝑚_𝑖=1𝐴_𝑖 and 𝑥_𝑛+1=𝑇𝑥_𝑛 be the Picard iteration. If there exists 𝑛₀ such that 𝑥_𝑛0=𝑥_𝑛0+1, then the theorem follows. Indeed, we get 𝑥_𝑛0=𝑥_𝑛0+1=𝑇𝑥_𝑛0 and, therefore, 𝑥₀ is a fixed point of 𝑇. Thus, we may assume that 𝑥_𝑛≠𝑥_𝑛+1 for all 𝑛 where we have 𝑝𝑥_𝑛,𝑥_𝑛+1>0, (2.5) by Lemma 1.12.
Since 𝑇 is cyclic, for any 𝑛≥0, there is 𝑖_𝑛∈{1,2,…,𝑚} such that 𝑥_𝑛∈𝐴_𝑖𝑛 and 𝑥_𝑛+1∈𝐴_𝑖𝑛+1. Then, by (2.1), we have 𝑝𝑥_𝑛+1,𝑥_𝑛+2=𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1≤𝑀𝑥_𝑛,x_𝑛+1−𝜙𝑀𝑥_𝑛,𝑥_𝑛+1, (2.6) where 𝑀𝑥_𝑛,𝑥_𝑛+1=max𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑇𝑥_𝑛,𝑥_𝑛,𝑝𝑥_𝑛+1,𝑇𝑥_𝑛+1=max𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛+1,𝑥_𝑛+2. (2.7) If we assume 𝑀(𝑥_𝑛,𝑥_𝑛+1)=𝑝(𝑥_𝑛+1,𝑥_𝑛+2), then (2.6) turns into 𝑝𝑥_𝑛+1,𝑥_𝑛+2≤𝑝𝑥_𝑛+1,𝑥_𝑛+2−𝜙𝑝𝑥_𝑛+1,𝑥_𝑛+2, (2.8) which is a contradiction. Because when we let 𝑡=𝑝(𝑥_𝑛+1,𝑥_𝑛+2), which is positive by (2.5), we get 𝜙(𝑡)=𝜙(𝑝(𝑥_𝑛+1,𝑥_𝑛+2))>0, we need to take 𝑀(𝑥_𝑛,𝑥_𝑛+1)=𝑝(𝑥_𝑛,𝑥_𝑛+1). Define 𝑡_𝑛=𝑝(𝑥_𝑛,𝑥_𝑛+1). Then, the inequality in (2.6) turns into 𝑡_𝑛+1≤𝑡_𝑛−𝜙𝑡_𝑛≤𝑡_𝑛. (2.9) Therefore, {𝑡_𝑛} is a nonnegative nonincreasing sequence. Hence, {𝑡_𝑛} converges to 𝐿≥0. We aim to show that 𝐿=0. Suppose on the contrary that 𝐿>0. Letting 𝑛→∞ in (2.9), we get that 𝐿≤𝐿−lim_𝑛→∞𝜙𝑡_𝑛≤𝐿. (2.10) Thus, we obtain that lim_𝑛→∞𝜙𝑡_𝑛=0. (2.11) Since 𝐿=inf{𝑡_𝑛=𝑝(𝑥_𝑛,𝑥_𝑛+1)∶𝑛∈ℕ}, then 0<𝐿≤𝑡_𝑛 for 𝑛=0,1,2,…. Taking 𝑛→∞ in the previous inequality, we derive 0<𝜙(𝐿)≤lim_𝑛→∞𝜙𝑡_𝑛, (2.12) which contradicts with (2.11). Thus, we have lim_𝑛→∞𝑝(𝑥_𝑛+1,𝑥_𝑛)=0.
We claim that {𝑡_𝑛} is a 0-Cauchy sequence. In order to prove this assertion, we first prove the following statement.(A) For every 𝜖>0, there exists 𝑛₀∈ℕ such that if 𝑠,𝑞≥𝑛₀ with 𝑠−𝑞≡1(𝑚), then 𝑝(𝑥_𝑠,𝑥_𝑞)<𝜖. Assume the contrary that there exists 𝜖>0 such that for any 𝑛∈ℕ we can find 𝑠>𝑞≥𝑛 with 𝑠−𝑞≡1(𝑚) satisfying the inequality 𝑝(𝑥_𝑠,𝑥_𝑞)≥𝜖. Then, we have 𝜖≤𝑝𝑥_𝑠,𝑥_𝑞≤𝑝𝑥_𝑠,𝑥_𝑠+1+𝑝𝑥_𝑠+1,𝑥_𝑞+1+𝑝𝑥_𝑞+1,𝑥_𝑞=𝑝𝑥_𝑠,𝑥_𝑠+1+𝑝𝑇𝑥_𝑠,𝑇𝑥_𝑞+𝑝𝑥_𝑞+1,𝑥_𝑞, (2.13) by the triangular inequality. Since 𝑥_𝑠 and 𝑥_𝑞 lie in different adjacently labeled sets 𝐴_𝑖 and 𝐴_𝑖+1 for certain 1≤𝑖≤𝑚, from (2.13) we get 𝜖≤𝑝𝑥_𝑠,𝑥_𝑞≤𝑝𝑥_𝑠,𝑥_𝑠+1+𝑀𝑥_𝑠,𝑥_𝑞−𝜙𝑀𝑥_𝑠,𝑥_𝑞+𝑝𝑥_𝑞+1,𝑥_𝑞≤𝑝𝑥_𝑠,𝑥_𝑠+1+𝑀𝑥_𝑠,𝑥_𝑞−𝜙𝑝𝑥_𝑠,𝑥_𝑞+𝑝𝑥_𝑞+1,𝑥_𝑞 (2.14) by using the contractive condition in (2.1), where 𝑀𝑥_𝑠,𝑥_𝑞=max𝑝𝑥_𝑠,𝑥_𝑞,𝑝𝑥_𝑠,𝑥_𝑠+1,𝑝𝑥_𝑞+1,𝑥_𝑞. (2.15)
If 𝑀(𝑥_𝑠,𝑥_𝑞) is equal to either 𝑝(𝑥_𝑞+1,𝑥_𝑞) or 𝑝(𝑥_𝑠,𝑥_𝑠+1), then letting 𝑠,𝑞→∞ in (2.14) yields that 𝜖≤lim_{𝑠,𝑞→∞}𝑝𝑥_𝑠,𝑥_𝑞≤−lim_{𝑠,𝑞→∞}𝜙𝑝𝑥_𝑠,𝑥_𝑞 (2.16) is a contradiction. Hence, 𝑀(𝑥_𝑠,𝑥_𝑞)=𝑝(𝑥_𝑠,𝑥_𝑞) and inequality (2.14) give 𝜙𝑝𝑥_𝑠,𝑥_𝑞≤𝑝𝑥_𝑠,𝑥_𝑠+1+𝑝𝑥_𝑞+1,𝑥_𝑞. (2.17) As 𝜖≤𝑝(𝑥_𝑠,𝑥_𝑞) and 𝜙 is nondecreasing, we have 0<𝜙(𝜖)≤𝜙𝑝𝑥_𝑠,𝑥_𝑞 (2.18) Inequalities (2.17) and (2.18) give us0<𝜙(𝜖)≤𝜙𝑝𝑥_𝑠,𝑥_𝑞≤𝑝𝑥_𝑠,𝑥_𝑠+1+𝑝𝑥_𝑞+1,𝑥_𝑞. (2.19) Letting 𝑠,𝑞→∞ with 𝑠−𝑞≡1(𝑚) and since lim_𝑛→∞𝑝(𝑥_𝑛,𝑥_𝑛+1)=0, we have 0<𝜙(𝜖)≤lim_{𝑠,𝑞→∞}𝜙𝑝𝑥_𝑠,𝑥_𝑞(𝑚),≤0𝑠−𝑞≡1 (2.20) which is a contradiction. Therefore, (A) is proved.
Now, in order to prove that {𝑥_𝑛} is a Cauchy sequence, we fix an 𝜖>0. Using (A), we find 𝑛₀∈ℕ such that if 𝑠,𝑞≥𝑛₀ with 𝑠−𝑞≡1(𝑚), 𝑝𝑥_𝑠,𝑥_𝑞≤𝜖𝑚. (2.21) On the other hand, since lim_𝑛→∞𝑝(𝑥_𝑛,𝑥_𝑛+1)=0, we also find 𝑛₁∈ℕ such that 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝜖𝑚, (2.22) for any 𝑛≥𝑛₁. We take 𝑘,𝑙≥max{𝑛₀,𝑛₁} with 𝑙>𝑘. Then, there exists 𝐾∈{1,2,…,𝑚} such that 𝑙−𝑘≡𝐾(𝑚). Therefore, 𝑙−𝑘+𝑗≡1(𝑚) for 𝑗=𝑚−𝐾+1, and so 𝑝𝑥_𝑘,𝑥_𝑙≤𝑝𝑥_𝑘,𝑥_𝑙+𝑗+𝑝𝑥_𝑙+𝑗,𝑥_{𝑙+𝑗−1}+⋯+𝑝𝑥_𝑙+1,𝑥_𝑙. (2.23) From (2.21) and (2.22) and the last inequality, 𝑝𝑥_𝑘,𝑥_𝑙≤𝜖𝑚+𝜖𝑗−1𝑚𝜖=𝑗𝑚≤𝜖. (2.24) Hence, lim_{𝑘,𝑙→∞}𝑝𝑥_𝑘,𝑥_𝑙=0. (2.25) This prove that {𝑡_𝑛} is a 0-Cauchy sequence.
Since 𝑋 is a 0-complete partial metric space, there exists 𝑥∈𝑋 such that lim_𝑛→∞𝑝(𝑥_𝑛,𝑥)=𝑝(𝑥,𝑥). Now, we will prove that 𝑥 is a fixed point of 𝑇. In fact, since {𝑥_𝑛} converges to 𝑥 and 𝑌=∪^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇, the sequence {𝑥_𝑛} has infinite number of terms in each 𝐴_𝑖 for 𝑖∈{1,2,…,𝑚}. Regarding that 𝐴_𝑖 is closed for 𝑖∈{1,2,…,𝑚}, we have 𝑥∈∩^𝑚_𝑖=1𝐴_𝑖. Now fix 𝑖∈{1,2,…,𝑚} such that 𝑥∈𝐴_𝑖 and 𝑇𝑥∈𝐴_𝑖+1. We take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑚} with 𝑥_𝑛𝑘∈𝐴_𝑖−1 (the existence of this subsequence is guaranteed by the mentioned comment above). Using the triangular inequality and the contractive conditions, we can obtain 𝑝(𝑥,𝑇𝑥)≤𝑝𝑥,𝑥_𝑛𝑘+1+𝑝𝑥_𝑛𝑘+1,𝑇𝑥=𝑝𝑥,𝑥_𝑛𝑘+1+𝑝𝑇𝑥_𝑛𝑘,𝑇𝑥≤𝑝𝑥,𝑥_𝑛𝑘+1+𝑀𝑥_𝑛𝑘,𝑥−𝜙𝑀𝑥_𝑛𝑘,𝑥, (2.26) where 𝑀(𝑥,𝑥_𝑛𝑘)={𝑝(𝑥,𝑇𝑥),𝑝(𝑥_𝑛𝑘,𝑥_𝑛𝑘+1),𝑝(𝑥,𝑥_𝑛𝑘)}. If 𝑀(𝑥,𝑥_𝑛𝑘) is equal to either 𝑝(𝑥_𝑛𝑘,𝑥_𝑛𝑘+1) or 𝑝(𝑥,𝑥_𝑛𝑘), then by letting 𝑘→∞, inequality (2.26) implies 𝑝(𝑥,𝑇𝑥)≤0. (2.27) Therefore, 𝑝(𝑥,𝑇𝑥)=0, that is, 𝑥 is a fixed point of 𝑇 by Lemma 1.12. If 𝑀(𝑥,𝑥_𝑛𝑘)=𝑝(𝑥,𝑇𝑥) then (2.26) turns into 𝑝(𝑥,𝑇𝑥)≤𝑝𝑥,𝑥_𝑛𝑘+1+𝑝(𝑥,𝑇𝑥)−𝜙𝑝(𝑥,𝑇𝑥)⟺𝜙𝑝(𝑥,𝑇𝑥)≤𝑝𝑥,𝑥_𝑛𝑘+1. (2.28) Letting 𝑘→∞, we get that lim_𝑘→∞𝜙(𝑝(𝑥,𝑇𝑥))=0. Hence, by the properties of 𝜙 we have 𝑝(𝑥,𝑇𝑥)=0.

Theorem 2.4. Let 𝑇∶𝑌→𝑌 be a self-mapping as in Theorem 2.3. Then, the fixed point problem for 𝑇 is well posed, that is, if there exists a sequence {𝑦_𝑛} in 𝑌 with 𝑝(𝑦_𝑛,𝑇𝑦_𝑛)→0, as 𝑛→∞, then 𝑝(𝑦_𝑛,𝑧)→0 as 𝑛→∞.

Proof. Due to Theorem 2.3, we know that for any initial value 𝑦∈𝑌, 𝑧∈∩^𝑚_𝑖=1𝐴_𝑖 is the unique fixed point of 𝑇. Thus, 𝑝(𝑦_𝑛,𝑧) is well defined. Consider 𝑝𝑦_𝑛,𝑧≤𝑝𝑦_𝑛,𝑇𝑦_𝑛+𝑝𝑇𝑦_𝑛,𝑇𝑧≤𝑝𝑦_𝑛,𝑇𝑦_𝑛+𝑀𝑦_𝑛,𝑧−𝜙𝑀𝑦_𝑛,𝑧, (2.29) where 𝑀(𝑦_𝑛,𝑧)={𝑝(𝑦_𝑛,𝑧),𝑝(𝑦_𝑛,𝑇𝑦_𝑛),𝑝(𝑧,𝑇𝑧)}. There are three cases to consider: if 𝑀(𝑦_𝑛,𝑧)=𝑝(𝑦_𝑛,𝑧), then (2.29) is equivalent to 0≤𝑝𝑦_𝑛,𝑇𝑦_𝑛−𝜙𝑝𝑦_𝑛,𝑧⟺𝜙𝑝𝑦_𝑛,𝑧≤𝑝𝑦_𝑛,𝑇𝑦_𝑛. (2.30) Since we are given 𝑝(𝑦_𝑛,𝑇𝑦_𝑛)→0 as 𝑛→∞, we derive that 𝜙(𝑝(𝑦_𝑛,𝑧))→0. Regarding the definition of 𝜙, we obtain that 𝑝(𝑦_𝑛,𝑧)→0. The theorem follows.
If it is the case that 𝑀(𝑦_𝑛,𝑧)=𝑝(𝑦_𝑛,𝑇𝑦_𝑛), then the right-hand side of (2.29) tends to 0 as 𝑛 tends to infinity. Hence, the theorem follows again.
As the last case, we have 𝑀(𝑦_𝑛,𝑧)=𝑝(𝑧,𝑇𝑧). Similarly we conclude that the right-hand side of (2.29) tends to 0 as 𝑛 tends to infinity because we know that 𝑝(𝑧,𝑇𝑧)=0 by Theorem 2.3 and 𝑝(𝑦_𝑛,𝑇𝑦_𝑛)→0, which completes the proof.

Theorem 2.5. Let 𝑇∶𝑌→𝑌 be a self-mapping as in Theorem 2.3. Then, 𝑇 has the limit shadowing property, that is, if there exists a convergent sequence {𝑦_𝑛} in 𝑌 with 𝑝(𝑦_𝑛+1,𝑇𝑦_𝑛)→0 and 𝑝(𝑦_𝑛,𝑇𝑦_𝑛)→0, as 𝑛→∞, then there exists 𝑥∈𝑌 such that 𝑝(𝑦_𝑛,𝑇^𝑛𝑥)→0 as 𝑛→∞.

Proof. As in the proof of Theorem 2.4, we observe that for any initial value 𝑥∈𝑌, 𝑧∈∩^𝑚_𝑖=1𝐴_𝑖 is the unique fixed point of 𝑇. Thus, 𝑝(𝑦_𝑛,𝑧),𝑝(𝑦_𝑛+1,𝑧) are well defined. Set 𝑦 as a limit of a convergent sequence {𝑦_𝑛} in 𝑌. Consider 𝑝𝑦_𝑛+1,𝑧≤𝑝𝑦_𝑛+1,𝑇𝑦_𝑛+𝑝𝑇𝑦_𝑛,𝑇𝑧≤𝑝𝑦_𝑛+1,𝑇𝑦_𝑛+𝑀𝑦_𝑛,𝑧−𝜙𝑀𝑦_𝑛,𝑧, (2.31) where 𝑀(𝑦_𝑛,𝑧)={𝑝(𝑦_𝑛,𝑧),𝑝(𝑦_𝑛,𝑇𝑦_𝑛),𝑝(𝑧,𝑇𝑧)}. There are three cases to consider: If 𝑀(𝑦_𝑛,𝑧)=𝑝(𝑦_𝑛,𝑧), then (2.31) is equivalent to 𝑝𝑦,𝑧≤𝑝𝑦,𝑧−𝜙𝑝𝑦,𝑧, (2.32) as 𝑛→∞. This is possible only if 𝜙(𝑝(𝑦,𝑧))=0, which implies that 𝑝(𝑦,𝑧)=0 and thus 𝑦=𝑧. Thus, we have 𝑝(𝑦_𝑛,𝑇^𝑛𝑥)→𝑝(𝑦,𝑧)=0, as 𝑛→∞.
If it is the case that 𝑀(𝑦_𝑛,𝑧)=𝑝(𝑦_𝑛,𝑇𝑦_𝑛), then the right-hand side of (2.31) tends to 0 as 𝑛 tends to infinity. Hence, the theorem follows again.
As the last case, we have 𝑀(𝑦_𝑛,𝑧)=𝑝(𝑧,𝑇𝑧). Similarly we conclude that the right-hand side of (2.31) tends to 0 as 𝑛 tends to infinity because we know that 𝑝(𝑧,𝑇𝑧)=0 by Theorem 2.3 and 𝑝(𝑦_𝑛+1,𝑇𝑦_𝑛)→0, which completes the proof.

Theorem 2.6. Let 𝑋 be a non-empty set, (𝑋,𝑝) and (𝑋,𝜌) partial metric spaces, 𝑚 a positive integer, and 𝐴₁,𝐴₂,…,𝐴_𝑚 closed non-empty subsets of 𝑋 and 𝑌=∪^𝑚_𝑖=1𝐴_𝑖. Suppose that (1)𝑌=∪^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇. (2)𝑝(𝑥,𝑦)≤𝜌(𝑥,𝑦) for all 𝑥,𝑦∈𝑌, (3)(𝑌,𝑝) is a complete partial metric space, (4)𝑇∶(𝑌,𝑝)→(𝑌,𝜌) is continuous, (5)𝑇∶(𝑌,𝑝)→(𝑌,𝜌) is a cyclic weak 𝜙-contraction where 𝜙∶[0,∞)→[0,∞) with 𝜙(𝑡)>0 is a lower semicontinuous function for 𝑡∈(0,∞) and 𝜙(0)=0. Then, {𝑇^𝑛𝑥₀} converges to 𝑧 in (𝑌,𝑝) for any 𝑥₀∈𝑌 and 𝑧 is the unique fixed point of 𝑇.

Proof. Let 𝑥₀∈𝑌. As in Theorem 2.3, assumption (5) implies that {𝑇^𝑛𝑥₀} is a Cauchy sequence in (𝑌,𝜌). Taking (2) into account, {𝑇^𝑛𝑥₀} is a Cauchy sequence in (𝑌,𝑑), and due to (3) it converges to 𝑧 in (𝑌,𝜌) for any 𝑥₀∈𝑌. Condition (4) implies the uniqueness of 𝑧.

Definition 2.7 (cf. [2]). Let (𝑋,𝑝) be a partial metric space, 𝑚 a positive integer, and 𝐴₁,…,𝐴_𝑚 closed non-empty subsets of 𝑋 and 𝑌=∪^𝑚_𝑖=1𝐴_𝑖. An operator 𝑇∶𝑌→𝑌 is called a cyclic generalized 𝜑-contraction if (1)∪^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇, and (2) there exists a continuous, non-decreasing function 𝜑∶[0,∞)→[0,∞) with 𝜑(𝑡)>0 for every 𝑡∈(0,∞) and 𝜑(0)=0, such that 𝑝𝑇𝑥,𝑇𝑦≤𝜑𝑀𝑥,𝑦, (2.33) where 𝑀𝑥,𝑦=max𝑝𝑥,𝑦,𝑝(𝑇𝑥,𝑥),𝑝𝑦,𝑇𝑦 (2.34) for any 𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1 for 𝑖=1,2,…,𝑚 with 𝐴_𝑚+1=𝐴₁.

The following is an analog of the main theorem in [2].

Corollary 2.8. Let (𝑋,𝑝) be a 0-complete partial metric space, 𝑚 a positive integer, and 𝐴₁,…,𝐴_𝑚 closed non-empty subsets of 𝑋. Let 𝑌=∪^𝑚_𝑖=1𝐴_𝑖 be a cyclic representation of 𝑌 with respect to 𝑇. Suppose that 𝑇 is a cyclic generalized 𝜑-contraction. Then, 𝑇 has a unique fixed point 𝑧∈∩^𝑚_𝑖=1𝐴_𝑖.

Proof. It follows from Theorem 2.3 by taking 𝜙(𝑡)=𝑡−𝜑(𝑡).