Cyclic Contractions and Fixed Point Results via Control Functions on Partial Metric Spaces

Abstract

Cyclic weaker-type contraction conditions involving a generalized control function (with two variables) are used for mappings on 0-complete partial metric spaces to obtain fixed point results, thus generalizing several known results. Various examples are presented showing how the obtained theorems can be used and that they are proper extensions of the known ones.

1. Introduction

The celebrated Banach contraction principle has been generalized in several directions and widely used to obtain various fixed point results, with applications in many branches of mathematics.

Cyclic representations and cyclic contractions were introduced by Kirk et al. [1] and further used by several authors to obtain various fixed point results. See, for example, papers [2–9]. Note that while a classical contraction has to be continuous, cyclic contractions might not be.

On the other hand, Matthews [10] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. In partial metric spaces, self-distance of an arbitrary point need not be equal to zero. Several authors obtained many useful fixed point results in these spaces–-we just mention [11–27]. Several results in ordered partial metric spaces have been obtained as well [28–36]. Some results for cyclic contractions in partial metric spaces have been very recently obtained in [37–41].

Khan et al. [42] addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function. This idea was further used in many papers, such as Choudhury [43] where generalized control functions were used. This approach has been very recently used in [44, 45] to obtain fixed point results in partial metric spaces.

In this paper, we extend these results further, considering cyclic weaker-type contraction conditions involving a generalized control function (with two variables) for mappings on 0-complete partial metric spaces (Romaguera [16]). We obtain fixed point theorems for such mappings, thus generalizing several known results. Various examples are presented showing how the obtained results can be used and that they are proper extensions of the known ones.

2. Preliminaries

In 2003, Kirk et al. introduced the following notion of cyclic representation.

Definition 1 (see [1]). Let 𝑋 be a nonempty set, 𝑚∈ℕ, and let 𝑓∶𝑋→𝑋 be a self-mapping. Then 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑓 if(a)𝐴_𝑖,𝑖=1,…,𝑚 are nonempty subsets of 𝑋;(b)𝑓(𝐴₁)⊂𝐴₂, 𝑓(𝐴₂)⊂𝐴₃,…,𝑓(𝐴_𝑚−1)⊂𝐴_𝑚, 𝑓(𝐴_𝑚)⊂𝐴₁.

They proved the following fixed point result.

Theorem 2 (see [1]). Let (𝑋,𝑑) be a complete metric space, 𝑓∶𝑋→𝑋, and let 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 be a cyclic representation of 𝑋 with respect to 𝑓. Suppose that 𝑓 satisfies the following condition: 𝑑𝑓𝑥,𝑓𝑦≤𝜓𝑑𝑥,𝑦,∀𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑖∈{1,2,…,𝑚}, (1) where 𝐴_𝑚+1=𝐴₁ and 𝜓∶[0,1)→[0,1) is a function, upper semicontinuous from the right and 0≤𝜓(𝑡)<𝑡 for 𝑡>0. Then, 𝑓 has a fixed point 𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖.

In 2010, Păcurar and Rus introduced the following notion of cyclic weaker 𝜑-contraction.

Definition 3 (see [2]). Let (𝑋,𝑑) be a metric space, 𝑚∈𝑁, and let 𝐴₁,𝐴₂,…,𝐴_𝑚 be closed nonempty subsets of 𝑋 and 𝑋=⋃^𝑚_𝑖=1𝐴_𝑖. An operator 𝑓∶𝑋→𝑋 is called a cyclic weaker 𝜑-contraction if(1)𝑋=⋃^𝑚_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑓;(2)there exists a continuous, nondecreasing function 𝜑∶[0,1)→[0,1) with 𝜑(𝑡)>0 for 𝑡∈(0,1) and 𝜑(0)=0 such that 𝑑𝑓𝑥,𝑓𝑦≤𝑑𝑥,𝑦−𝜑𝑑𝑥,𝑦, (2)for any 𝑥∈𝐴_𝑖, 𝑦∈𝐴_𝑖+1,𝑖=1,2,…,𝑚, where 𝐴_𝑚+1=𝐴₁.

They proved the following result.

Theorem 4 (see [2]). Suppose that 𝑓 is a cyclic weaker 𝜑-contraction on a complete metric space (𝑋,𝑑). Then, 𝑓 has a fixed point 𝑧∈⋂^𝑚_𝑖=1𝐴_𝑖.

This was generalized by Karapınar in [3].

Khan et al. introduced the following notion.

Definition 5 (see [42]). A function 𝜑∶[0,+∞)→[0,+∞) is called an altering distance function if the following properties are satisfied: (a)𝜑 is continuous and nondecreasing, (b)𝜑(𝑡)=0⇔𝑡=0.

Choudhury introduced a generalization of Chatterjea type contraction as follows.

Definition 6 (see [43]). 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,𝑇𝑦2𝑑𝑑,𝑥,𝑇𝑦+𝑑𝑦,𝑇𝑥−𝜓𝑥,𝑇𝑦,𝑑𝑦,𝑇𝑥 (3) where 𝜓∶[0,+∞)²→[0,+∞) is a continuous function such that 𝜓𝑥,𝑦=0iff𝑥=𝑦=0. (4)

In [43], the author proved that every weak Chatterjea type contraction on a complete metric space has a unique fixed point.

The following definitions and details can be seen, for example, in [10, 12, 13, 15, 16].

Definition 7. A partial metric on a nonempty set 𝑋 is a function 𝑝∶𝑋×𝑋→ℝ⁺ such that for all 𝑥,𝑦,𝑧∈𝑋:  (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). The pair (𝑋,𝑝) is called a partial metric space.

It is clear that, if 𝑝(𝑥,𝑦)=0, then from (𝑝₁) and (𝑝₂)𝑥=𝑦. But if 𝑥=𝑦, 𝑝(𝑥,𝑦) may not be 0.

Each partial metric 𝑝 on 𝑋 generates a 𝑇₀ topology 𝜏_𝑝 on 𝑋 which has as a base the family of open 𝑝-balls {𝐵_𝑝(𝑥,𝜀)∶𝑥∈𝑋,𝜀>0}, where 𝐵_𝑝(𝑥,𝜀)={𝑦∈𝑋∶𝑝(𝑥,𝑦)<𝑝(𝑥,𝑥)+𝜀} for all 𝑥∈𝑋 and 𝜀>0.

A sequence {𝑥_𝑛} in (𝑋,𝑝) converges to a point 𝑥∈𝑋 (in the sense of 𝜏_𝑝) if lim_𝑛→∞𝑝(𝑥,𝑥_𝑛)=𝑝(𝑥,𝑥). This will be denoted as 𝑥_𝑛→𝑥 (𝑛→∞) or lim_𝑛→∞𝑥_𝑛=𝑥. Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function 𝑝(⋅,⋅) need not be continuous in the sense that 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦 imply 𝑝(𝑥_𝑛,𝑦_𝑛)→𝑝(𝑥,𝑦).

Example 8 (see [10]). (1) A paradigmatic example of a partial metric space is the pair (ℝ⁺,𝑝), where 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for all 𝑥,𝑦∈ℝ⁺.
(2) Let 𝑋={[𝑎,𝑏]∶𝑎,𝑏∈ℝ,𝑎≤𝑏} and let 𝑝([𝑎,𝑏],[𝑐,𝑑])=max{𝑏,𝑑}−min{𝑎,𝑐}. Then (𝑋,𝑝) is a partial metric space.

Definition 9. Let (𝑋,𝑝) be a partial metric space. Then consider the following.(1) A sequence {𝑥_𝑛} in (𝑋,𝑝) is called a Cauchy sequence if lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists (and is finite). The space (𝑋,𝑝) is said to be 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).(2)(see [16]) A sequence {𝑥_𝑛} in (𝑋,𝑝) is called 0-Cauchy if lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚)=0. The space (𝑋,𝑝) is said to be 0-complete if every 0-Cauchy sequence in 𝑋 converges (in 𝜏_𝑝) to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=0.

Lemma 10. Let (𝑋,𝑝) be a partial metric space. (a) (see [46, 47]) 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. (b) (see [16]) If (𝑋,𝑝) is complete, then it is 0-complete.

The converse assertion of (b) does not hold as the following easy example shows.

Example 11 (see [16]). The space 𝑋=[0,+∞)∩ℚ with the partial metric 𝑝(𝑥,𝑦)=max{𝑥,𝑦} is 0-complete, but is not complete. Moreover, the sequence {𝑥_𝑛} with 𝑥_𝑛=1 for each 𝑛∈ℕ is a Cauchy sequence in (𝑋,𝑝), but it is not a 0-Cauchy sequence.

It is easy to see that every closed subset of a 0-complete partial metric space is 0-complete.

3. Main Results

In this section, we will prove some fixed point theorems for self-mappings defined on a 0-complete partial metric space and satisfying certain cyclic weak contractive condition involving a generalized control function. To achieve our goal, we introduce the new notion of a cyclic contraction.

Definition 12. Let (𝑋,𝑝) be a partial metric space, 𝑞∈ℕ, and let 𝐴₁,𝐴₂,…,𝐴_𝑞 be nonempty subsets of 𝑋 and 𝑌=⋃^𝑞_𝑖=1𝐴_𝑖. An operator 𝑇∶𝑌→𝑌 is called a cyclic contraction under weak contractive condition if: (NZ1)𝑌=⋃^𝑞_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇; (NZ2) for any (𝑥,𝑦)∈𝐴_𝑖×𝐴_𝑖+1,𝑖=1,2,…,𝑞 (with 𝐴_𝑞+1=𝐴₁), 𝑝𝑇𝑥,𝑇𝑦≤𝑀𝑥,𝑦−𝜓𝑝𝑥,𝑦,𝑝(𝑥,𝑇𝑥), (5)where 𝑀𝑝(),1𝑥,𝑦=max𝑥,𝑦,𝑝𝑥,𝑇𝑥,𝑝𝑦,𝑇𝑦2𝑝𝑥,𝑇𝑦+𝑝𝑇𝑥,𝑦, (6) and 𝜓∶[0,∞)²→[0,∞) is a lower semicontinuous mapping such that 𝜓(𝑠,𝑡)=0 if and only if 𝑠=𝑡=0.

Our main result is the following.

Theorem 13. Let (𝑋,𝑝) be a 0-complete partial metric space, let 𝑞∈ℕ,𝐴₁,𝐴₂,…,𝐴_𝑞 be nonempty closed subsets of (𝑋,𝑝), and let 𝑌=⋃^𝑞_𝑖=1𝐴_𝑖. Suppose that 𝑇∶𝑌→𝑌 is a cyclic contraction as defined in Definition 12. Then, 𝑇 has a unique fixed point 𝑧∈𝑌 such that 𝑝(𝑧,𝑧)=0. Moreover, 𝑧∈⋂^𝑞_𝑖=1𝐴_𝑖. Each Picard sequence 𝑥_𝑛=𝑇^𝑛𝑥₀, 𝑥₀∈𝑌 converges to 𝑧 in topology 𝜏_𝑝.

Proof. Let 𝑥₀ be an arbitrary point of 𝑌. Then there exists some 𝑖₀ such that 𝑥₀∈𝐴_𝑖0. Now 𝑥₁=𝑇𝑥₀∈𝐴_𝑖0+1 and, similarly, 𝑥_𝑛∶=𝑇𝑥_𝑛−1=𝑇^𝑛𝑥₀∈𝐴_𝑖0+𝑛 for 𝑛∈ℕ, where 𝐴_𝑞+𝑘=𝐴_𝑘. In the case 𝑝(𝑥_𝑛0,𝑥_𝑛0+1)=0 for some 𝑛₀∈ℕ₀, it is clear that 𝑥_𝑛0 is a fixed point of 𝑇.
Without loss of the generality, we may assume that 𝑝𝑥_𝑛,𝑥_𝑛+1>0,∀𝑛∈ℕ. (7) From the condition (NZ1), we observe that for all 𝑛, there exists 𝑖=𝑖(𝑛)∈{1,2,…,𝑞} such that (𝑥_𝑛,𝑥_𝑛+1)∈𝐴_𝑖×𝐴_𝑖+1.
Putting 𝑥=𝑥_𝑛 and 𝑦=𝑥_𝑛+1 in (NZ2) condition, we have 𝑝𝑥_𝑛+1,𝑥_𝑛+2=𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1≤𝑀𝑥_𝑛,𝑥_𝑛+1−𝜓𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛,𝑇𝑥_𝑛=max𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛+1,𝑥_𝑛+2,12𝑝𝑥_𝑛,𝑥_𝑛+2+𝑝𝑥_𝑛+1,𝑥_𝑛+1−𝜓𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛,𝑥_𝑛+1. (8) By (𝑝₄), we have 𝑝𝑥_𝑛,𝑥_𝑛+2+𝑝𝑥_𝑛+1,𝑥_𝑛+1≤𝑝𝑥_𝑛,𝑥_𝑛+1+𝑝𝑥_𝑛+1,𝑥_𝑛+2. (9) Therefore, max𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛+1,𝑥_𝑛+2,12𝑝𝑥_𝑛,𝑥_𝑛+2+𝑝𝑥_𝑛+1,𝑥_𝑛+1≤max𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛+1,𝑥_𝑛+2. (10) By (8) and (10), we have 𝑝𝑥_𝑛+1,𝑥_𝑛+2≤max𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛+1,𝑥_𝑛+2−𝜓𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛,𝑥_𝑛+1. (11) If max{𝑝(𝑥_𝑛,𝑥_𝑛+1),𝑝(𝑥_𝑛+1,𝑥_𝑛+2)}=𝑝(𝑥_𝑛+1,𝑥_𝑛+2), then from (11), we have 𝑝𝑥_𝑛+1,𝑥_𝑛+2≤𝑝𝑥_𝑛+1,𝑥_𝑛+2−𝜓𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛,𝑥_𝑛+1<𝑝𝑥_𝑛+1,𝑥_𝑛+2, (12) which is a contradiction (it was used that 𝜓(𝑝(𝑥_𝑛,𝑥_𝑛+1),𝑝(𝑥_𝑛,𝑥_𝑛+1))>0 since 𝑝(𝑥_𝑛,𝑥_𝑛+1)>0). Hence, 𝑝(𝑥_𝑛,𝑥_𝑛+1)=0, and 𝑥_𝑛=𝑥_𝑛+1, which is excluded. Therefore, we have max{𝑝(𝑥_𝑛,𝑥_𝑛+1),𝑝(𝑥_𝑛+1,𝑥_𝑛+2)}=𝑝(𝑥_𝑛,𝑥_𝑛+1) and hence 𝑝𝑥_𝑛+1,𝑥_𝑛+2≤𝑝𝑥_𝑛,𝑥_𝑛+1−𝜓𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛,𝑥_𝑛+1≤𝑝𝑥_𝑛,𝑥_𝑛+1. (13) By (13), we have that {𝑝(𝑥_𝑛,𝑥_𝑛+1)} is a nonincreasing sequence of positive real numbers. Thus, there exists 𝑟≥0 such that lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+1=𝑟. (14) Passing to the limit as 𝑛→∞ in (13) and using (14) and lower semicontinuity of 𝜓, we have 𝑟≤𝑟−liminf_𝑛→∞𝜓𝑝𝑥_𝑛,𝑥_𝑛+1,𝑝𝑥_𝑛,𝑥_𝑛+1≤𝑟−𝜓(𝑟,𝑟), (15) thus, 𝜓(𝑟,𝑟)=0 and hence 𝑟=0. Therefore lim_𝑛→∞𝑝𝑥_𝑛,𝑥_𝑛+1=0. (16)
Next, we claim that {𝑥_𝑛} is a 0-Cauchy sequence in the space (𝑋,𝑝). Suppose that this is not the case. Then there exists 𝜀>0 for which we can find two sequences of positive integers {𝑚(𝑘)} and {𝑛(𝑘)} such that for all positive integers 𝑘𝑛(𝑘)(𝑘)𝑝𝑥>𝑚>𝑘,_𝑚(𝑘),𝑥_𝑛(𝑘)𝑝𝑥≥𝜀,_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}<𝜀. (17) Using (17) and (𝑝₄), we get 𝜀≤𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑝𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}+𝑝𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘)−𝑝𝑥_{𝑛(𝑘)−1},𝑥_{𝑛(𝑘)−1}<𝜀+𝑝𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}. (18) Thus we have 𝜀≤𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)<𝜀+𝑝𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}. (19) Passing to the limit as 𝑘→∞ in the above inequality and using (16), we obtain lim_𝑘→∞𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)=𝜀. (20) On the other hand, for all 𝑘, there exists 𝑗(𝑘)∈{1,…,𝑞} such that 𝑛(𝑘)−𝑚(𝑘)+𝑗(𝑘)≡1[𝑞]. Then 𝑥_{𝑚(𝑘)−𝑗(𝑘)} (for 𝑘 large enough, 𝑚(𝑘)>𝑗(𝑘)) and 𝑥_𝑛(𝑘) lie in different adjacently labelled sets 𝐴_𝑖 and 𝐴_𝑖+1 for certain 𝑖∈{1,…,𝑞}.
Using (𝑝₄) and (20), we get 𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)≤𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑚(𝑘)+𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)−𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)≤^{𝑗(𝑘)−1}_𝑙=0𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)+𝑙},𝑥_{𝑚(𝑘)−𝑗(𝑘)+𝑙+1}+𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤^𝑞−1_𝑙=0𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)+𝑙},𝑥_{𝑚(𝑘)−𝑗(𝑘)+𝑙+1}+𝑝𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)→𝜀as𝑘→∞(from(16)), (21) that is, lim_𝑘→∞𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)=𝜀. (22) Using (16), we have lim_𝑘→∞𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_{𝑚(𝑘)−𝑗(𝑘)}=0,lim_𝑘→∞𝑝𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)=0. (23)(24) Again, using (𝑝₄), we get 𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1≤𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1−𝑝𝑥_𝑛(𝑘),𝑥_𝑛(𝑘). (25) Passing to the limit as 𝑘→∞ in the pervious inequality, and using (24) and (22), we get lim_𝑘→∞𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1=𝜀. (26) Similarly, we have by (𝑝₄) 𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}≤𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}+𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)−𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_{𝑚(𝑘)−𝑗(𝑘)}. (27) Passing to the limit as 𝑘→∞, and using (16) and (22), we obtain lim_𝑘→∞𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}=𝜀. (28) Similarly, we have by (𝑝₄) lim_𝑘→∞𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_𝑛(𝑘)+1=𝜀. (29) Using (NZ2), we obtain 𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_𝑛(𝑘)+1≤max𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘),𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑝𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘),12𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘)+1+𝑝𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)−𝑗(𝑘)+1}−𝜓𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘),𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_{𝑚(𝑘)−𝑗(𝑘)} (30) for all 𝑘. Passing to the limit as 𝑘→∞ in the last inequality (and using the lower semicontinuity of the function 𝜓), we obtain 𝜀≤𝜀−liminf_𝑛→∞𝜓𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)},𝑥_𝑛(𝑘),𝑝𝑥_{𝑚(𝑘)−𝑗(𝑘)+1},𝑥_{𝑚(𝑘)−𝑗(𝑘)}≤𝜀−𝜓(𝜀,0), (31) which implies that 𝜓(𝜀,0)=0; that is a contradiction since 𝜀>0. We deduce that {𝑥_𝑛} is a 0-Cauchy sequence.
Since (𝑋,𝑝) is 0-complete and 𝑌 is closed, it follows that the sequence {𝑥_𝑛} converges to some 𝑢∈𝑌, that is: lim_𝑛→∞𝑥_𝑛=𝑢. (32) We shall prove that 𝑢∈^𝑞_𝑖=1𝐴_𝑖. (33) From condition (NZ1), and since 𝑥₀∈𝐴₁, we have {𝑥_𝑛𝑞}_𝑛≥0⊆𝐴₁. Since 𝐴₁ is closed, from (32), we get that 𝑢∈𝐴₁. Again, from the condition (NZ1), we have {𝑥_𝑛𝑞+1}_𝑛≥0⊆𝐴₂. Since 𝐴₂ is closed, from (32), we get that 𝑢∈𝐴₂. Continuing this process, we obtain (33) and 𝑝(𝑢,𝑢)=0.
Now, we shall prove that 𝑢 is a fixed point of 𝑇. Indeed, from (33), since for all 𝑛, there exists 𝑖(𝑛)∈{1,2,…,𝑞} such that 𝑥_𝑛∈𝐴_𝑖(𝑛); applying (NZ2) with 𝑥=𝑢 and 𝑦=𝑥_𝑛, we obtain 𝑝𝑇𝑢,𝑥_𝑛+1=𝑝𝑇𝑢,𝑇𝑥_𝑛≤max𝑝𝑢,𝑥_𝑛,𝑝(𝑢,𝑇𝑢),𝑝𝑥_𝑛,𝑥_𝑛+1,12𝑝𝑢,𝑥_𝑛+1+𝑝𝑥_𝑛𝑝,𝑇𝑢−𝜓𝑢,𝑥_𝑛,𝑝(𝑢,𝑇𝑢), (34) for all 𝑛. Passing to the limit as 𝑛→∞ in (34), and using (32), we get 𝑝(𝑢,𝑇𝑢)≤lim_𝑛→∞max𝑝𝑢,𝑥_𝑛,𝑝(𝑢,𝑇𝑢),𝑝𝑥_𝑛,𝑥_𝑛+1,12𝑝𝑢,𝑥_𝑛+1+𝑝𝑥_𝑛,𝑇𝑢−liminf_𝑛→∞𝜓𝑝𝑢,𝑥_𝑛(()1,𝑝𝑢,𝑇𝑢)≤max0,𝑝𝑢,𝑇𝑢,0,2𝑝(()()(),𝑢,𝑇𝑢)−𝜓0,𝑝𝑢,𝑇𝑢=𝑝𝑢,𝑇𝑢−𝜓0,𝑝𝑢,𝑇𝑢 (35) which is impossible unless 𝜓(0,𝑝(𝑢,𝑇𝑢))=0, so 𝑢=𝑇𝑢, (36) that is, 𝑢 is a fixed point of 𝑇.
We claim that there is a unique fixed point of 𝑇. Assume on the contrary that 𝑇𝑢=𝑢 and 𝑇𝑣=𝑣 with 𝑝(𝑢,𝑣)>0. By supposition, we can replace 𝑥 by 𝑢 and 𝑦 by 𝑣 in (NZ2) to obtain 𝑝()()𝑝()()(),1𝑢,𝑣=𝑝𝑇𝑢,𝑇𝑣≤max𝑢,𝑣,𝑝𝑢,𝑇𝑢,𝑝𝑣,𝑇𝑣2𝑝()(𝑝()()()𝑝()(),𝑢,𝑇𝑣+𝑝𝑇𝑢,𝑣)−𝜓𝑢,𝑣,𝑝𝑢,𝑇𝑢=𝑝𝑢,𝑣−𝜓𝑢,𝑣,0<𝑝𝑢,𝑣 (37) a contradiction. Hence 𝑝(𝑢,𝑣)=0, that is, 𝑢=𝑣. We conclude that 𝑇 has only one fixed point in 𝑋. The proof is complete.

If we take 𝑞=1 and 𝐴₁=𝑋 in Theorem 13, then we get the following fixed point theorem.

Corollary 14. Let (𝑋,𝑝) be a 0-complete partial metric space, and let 𝑇∶𝑋→𝑋 satisfy the following condition: there exists a lower semicontinuous mapping 𝜓∶[0,∞)²→[0,∞) such that 𝜓(𝑡,𝑠)=0 if and only if 𝑡=𝑠=0 and that 𝑝𝑝(),1𝑇𝑥,𝑇𝑦≤max𝑥,𝑦,𝑝𝑥,𝑇𝑥,𝑝𝑦,𝑇𝑦2𝑝𝑝(,𝑥,𝑇𝑦+𝑝𝑦,𝑇𝑥−𝜓𝑥,𝑦,𝑝𝑥,𝑇𝑥) (38) for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point 𝑧∈𝑋. Moreover, 𝑝(𝑧,𝑧)=0.

Corollary 14 extends and generalizes many existing fixed point theorems in the literature.

By taking 𝜓(𝑠,𝑡)=(1−𝑟)max{𝑠,𝑡} where 𝑟∈[0,1) in Theorem 13, we have the following result.

Corollary 15. Let (𝑋,𝑝) be a 0-complete partial metric space, let 𝑞∈ℕ, 𝐴₁,𝐴₂,…,𝐴_𝑞 be nonempty closed subsets of 𝑋, 𝑌=⋃^𝑞_𝑖=1𝐴_𝑖, and 𝑇∶𝑌→𝑌 such that(NZ1)𝑌=⋃^𝑞_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇;(NZ3) for any (𝑥,𝑦)∈𝐴_𝑖×𝐴_𝑖+1,𝑖=1,2,…,𝑞 (with 𝐴_𝑞+1=𝐴₁), 𝑝𝑝(),1𝑇𝑥,𝑇𝑦≤𝑟max𝑥,𝑦,𝑝𝑥,𝑇𝑥,𝑝𝑦,𝑇𝑦2𝑝𝑥,𝑇𝑦+𝑝𝑇𝑥,𝑦, (39)where 𝑟∈[0,1). Then 𝑇 has a unique fixed point 𝑧 belonging to ⋂^𝑞_𝑖=1𝐴_𝑖; moreover, 𝑝(𝑧,𝑧)=0.

As a special case of Corollary 15, we obtain Matthews’s version of Banach contraction principle [10].

Corollary 16. Let (𝑋,𝑝) be a 0-complete partial metric space, let 𝑝∈ℕ, 𝐴₁,𝐴₂,…,𝐴_𝑞 be nonempty closed subsets of 𝑋, 𝑌=⋃^𝑞_𝑖=1𝐴_𝑖, and 𝑇∶𝑌→𝑌 such that(NZ1)𝑋=⋃^𝑞_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑋 with respect to 𝑇;(NZ4) for any (𝑥,𝑦)∈𝐴_𝑖×𝐴_𝑖+1,𝑖=1,2,…,𝑞 (with 𝐴_𝑞+1=𝐴₁), 𝑝𝑇^𝑛𝑥,𝑇^𝑛𝑦≤max𝑝𝑥,𝑦,𝑝𝑥,𝑇^𝑛𝑥,𝑝𝑦,𝑇^𝑛𝑦,12𝑝𝑥,𝑇^𝑛𝑦+𝑝𝑦,𝑇^𝑛𝑥−𝜓𝑝𝑥,𝑦,𝑝𝑥,𝑇^𝑛𝑥, (40)where 𝑛 is a positive integer and 𝜓∶[0,∞)²→[0,∞) is a lower semi-continuous mapping such that 𝜓(𝑡,𝑠)=0 if and only if 𝑡=𝑠=0. Then 𝑇 has a unique fixed point belonging to ⋂^𝑞_𝑖=1𝐴_𝑖.

4. Examples

The following example shows how Theorem 13 can be used. It is adapted from [38, Example 2.9].

Example 17. Consider the partial metric space (𝑋,𝑑) of Example 8 (2). It is easy to see that it is 0-complete. Consider the following closed subsets of 𝑋: 𝐴₁=1−2^−𝑛∪1,𝐴,1∶𝑛∈ℕ{{}}₂=1,1+2^−𝑛∶𝑛∈ℕ∪{{1}}, (41)𝑌=𝐴₁∪𝐴₂, and define a mapping 𝑇∶𝑌→𝑌 by 𝑇𝑥=1,1+2^−(𝑛+1),if𝑥=1−2^−𝑛,1,1−2^−(𝑛+1),1,if𝑥=1,1+2^−𝑛,{1},if𝑥={1}. (42) Obviously, 𝑌=𝐴₁∪𝐴₂ is a cyclic representation of 𝑌 with respect to 𝑇. We will show that 𝑇 satisfies the contractive condition (NZ2) of Definition 12 with the control function 𝜓∶[0,+∞)²→[0,+∞) given by 𝜓(𝑠,𝑡)=(1/2)max{𝑠,𝑡}.
Let (𝑥,𝑦)∈𝐴₁×𝐴₂ (the other possibility is treated similarly) and consider the following cases:(1)𝑥=[1−2^−𝑛,1], 𝑦=[1,1+2^−𝑘] and 𝑛<𝑘, that is, 𝑛+1≤𝑘. Then 𝑝(𝑇𝑥,𝑇𝑦)=𝑝([1,1+2^−(𝑛+1)],[1−2^−(𝑘+1),1])=(1/2)(2^−𝑛+2^−𝑘)≤(3/4)⋅2^−𝑛, 𝑀𝑥,𝑦=max2^−𝑛+2^−𝑘,32⋅2^−𝑛,32⋅2^−𝑘,122^−𝑛−2^−(𝑘+1)+2^−𝑘−2^−(𝑛+1)=32⋅2^−𝑛, (43) and 𝜓(𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥))=(1/2)⋅(3/2)⋅2^−𝑛=(3/4)⋅2^−𝑛. Hence, the condition (NZ2) reduces to (3/4)⋅2^−𝑛≤(3/2)⋅2^−𝑛−(3/4)⋅2^−𝑛 and holds true.(2)𝑥=[1−2^−𝑛,1], 𝑦=[1,1+2^−𝑘] and 𝑛=𝑘. Then 𝑝(𝑇𝑥,𝑇𝑦)=2^−𝑛, 𝑀(𝑥,𝑦)=2⋅2^−𝑛, and 𝜓(𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥))=(1/2)⋅2⋅2^−𝑛=2^−𝑛; hence (NZ2) reduces to 2^−𝑛≤2⋅2^−𝑛−2^−𝑛.(3)𝑥=[1−2^−𝑛,1], 𝑦=[1,1+2^−𝑘] and 𝑛>𝑘, that is, 𝑛≥𝑘+1. Then 𝑝(𝑇𝑥,𝑇𝑦)≤(3/4)⋅2^−𝑘, 𝑀(𝑥,𝑦)=(3/2)⋅2^−𝑘 and 𝜓(𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥))=(1/2)⋅(3/2)⋅2^−𝑘=(3/4)⋅2^−𝑘. Hence, (NZ2) reduces to (3/4)⋅2^−𝑘≤(3/2)⋅2^−𝑘−(3/4)⋅2^−𝑘 and holds true.(4)𝑥=[1−2^−𝑛,1], 𝑦={1}. Then 𝑝(𝑇𝑥,𝑇𝑦)=𝑝([1,1+2^−(𝑛+1)],{1})=(1/2)⋅2^−𝑛, 𝑀(𝑥,𝑦)=𝑝(𝑥,𝑇𝑥)=(3/2)⋅2^−𝑛 and 𝜓(𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥))=(1/2)⋅(3/2)⋅2^−𝑛=(3/4)⋅2^−𝑛. (NZ2) reduces to (1/2)⋅2^−𝑛≤(3/2)⋅2^−𝑛−(3/4)⋅2^−𝑛.(5)The case 𝑥={1}, 𝑦=[1,1+2^−𝑘] is treated symmetrically.(6)The case 𝑥=𝑦={1} is trivial.
We conclude that all conditions of Theorem 13 are satisfied. The mapping 𝑇 has a unique fixed point {1}∈𝐴₁∩𝐴₂.

Here is another example showing the use of Theorem 13.

Example 18. Let 𝑋=[0,1] be equipped with the partial metric given as 𝑝=[),𝑥,𝑦𝑥−𝑦,if𝑥,𝑦∈0,11,if𝑥=1∨𝑦=1. (44) Then, (𝑋,𝑝) is a 0-complete partial metric space. Let 𝐴₁=[0,1/2], 𝐴₂=[1/2,1], and 𝑇∶𝑋→𝑋 be given as 1𝑇𝑥=2[),1,if𝑥∈0,16,if𝑥=1. (45) Obviously, 𝑋=𝐴₁∪𝐴₂ is a cyclic representation of 𝑋 with respect to 𝑇. We will check the contractive condition (NZ2) of Definition 12 with the control function 𝜓∶[0,+∞)²→[0,+∞) given by 𝜓(𝑠,𝑡)=(𝑠+𝑡)/(2+𝑠+𝑡). Let (𝑥,𝑦)∈𝐴₁×𝐴₂ (the other possibility is treated symmetrically). Consider the following possible cases.(1)𝑥∈[0,1/2], 𝑦∈[1/2,1). Then 𝑝(𝑇𝑥,𝑇𝑦)=𝑝(1/2,1/2)=0, 𝑀(𝑥,𝑦)=max{𝑦−𝑥,1/2−𝑥,𝑦−1/2,(1/2)(1/2−𝑥+𝑦−1/2)}=𝑦−𝑥. 𝜓(𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥))=(𝑦+1/2−2𝑥)/(5/2+𝑦−2𝑥). It is easy to check that 𝑝𝑇𝑥,𝑇𝑦=0≤𝑦−𝑥−𝑦+1/2−2𝑥𝑝()5/2+𝑦−2𝑥=𝑀𝑥,𝑦−𝜓𝑥,𝑦,𝑝𝑥,𝑇𝑥 (46) holds for the given values of 𝑥 and 𝑦.(2)𝑥∈[0,1/2], 𝑦=1. Then 𝑝(𝑇𝑥,𝑇𝑦)=𝑝(1/2,1/6)=1/3, 𝑀(𝑥,𝑦)=1 and 𝜓(𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥))=𝜓(1,1/2−𝑥)=(3/2−𝑥)/(7/2−𝑥). The condition (NZ2) reduces to 13≤1−3−2𝑥7−2𝑥 (47)and can be checked directly.
Thus, all the conditions of Theorem 13 are fulfilled, and we conclude that 𝑇 has a unique fixed point 1/2∈𝐴₁∩𝐴₂.

We state a more involved example that is inspired with the one from [48].

Example 19. Let 𝑋⊂ℓ¹, 𝑋∋𝑥=(𝑥_𝑛)^∞_𝑛=1 if and only if 𝑥_𝑛≥0 for each 𝑛∈ℕ. Define a partial metric 𝑝 on 𝑋 by 𝑝𝑥_𝑛,𝑦_𝑛=^∞_𝑛=1max𝑥_𝑛,𝑦_𝑛 (48) (it is easy to check that axioms (𝑝₁)–(𝑝₄) hold true). Let 𝛼∈(0,1) be fixed, denote 𝟎=(0)^∞_𝑛=1, and consider the subsets 𝐴₁ and 𝐴₂ of 𝑋 defined by 𝐴₁=𝐴^′∪{𝟎},𝐴₂=𝐴^′′∪{𝟎}, where 𝐴^′∋𝑥^𝑙=𝑥^𝑙_𝑛^∞_𝑛=1iff𝑥^𝑙_𝑛=𝛼0,𝑛<2𝑙∨𝑛=2𝑘−1,𝑘∈ℕ,_𝑛𝐴,𝑛=2𝑘≥2𝑙𝑙=1,2,…,_′′∋𝑥^𝑙=𝑥^𝑙_𝑛^∞_𝑛=1iff𝑥^𝑙_𝑛=𝛼0,𝑛<2𝑙−1∨𝑛=2𝑘,𝑘∈ℕ,_𝑛,𝑛=2𝑘−1≥2𝑙−1𝑙=1,2,…. (49) Denote 𝑌=𝐴₁∪𝐴₂ (obviously 𝐴₁∩𝐴₂={𝟎}).
Consider the mapping 𝑇∶𝑌→𝑌 given by. 𝑇(𝟎)𝑇=𝟎,0,…,0_2𝑙−1,𝛼^2𝑙,0,𝛼^2𝑙+2=,0,…0,…,0_2𝑙,𝛼^2𝑙+1,0,𝛼^2𝑙+3,𝑇,0,…0,…,0_2𝑙,𝛼^2𝑙+1,0,𝛼^2𝑙+3=,0,…0,…,0_2𝑙+1,𝛼^2𝑙+2,0,𝛼^2𝑙+4,0,…. (50) Obviously, 𝑇(𝐴₁)⊂𝐴₂ and 𝑇(𝐴₂)⊂𝐴₁; hence 𝑌=𝐴₁∪𝐴₂ is a cyclic representation of 𝑌 with respect to 𝑇. Take 𝜓∶[0,+∞)²→[0,+∞) defined by 𝜓(𝑠,𝑡)=(1−𝛼)max{𝑠,𝑡}.
Let us check the contractive condition (NZ2) of Theorem 13. Take 𝑥=(0,…,0_2𝑙−1,𝛼^2𝑙,0,𝛼^2𝑙+2,0,…)∈𝐴₁ and 𝑦=(0,…,0_2𝑚,𝛼^2𝑚+1,0,𝛼^2𝑚+3,0,…)∈𝐴₂ and assume, for example, that 𝑙≤𝑚 (the case 𝑙>𝑚 is treated similarly, as well as the case when 𝑥 or 𝑦 is equal to 𝟎). Then 𝑝𝑥,𝑦=𝛼^2𝑙+⋯+𝛼^2𝑚−2+𝛼^2𝑚,𝑝1−𝛼𝑇𝑥,𝑇𝑦=𝛼^2𝑙+1+⋯+𝛼^2𝑚−1+𝛼^2𝑚+1,𝑝()=𝛼1−𝛼𝑥,𝑇𝑥^2𝑙=𝛼1−𝛼,𝑝𝑦,𝑇𝑦^2𝑚+1.1−𝛼 (51) Hence, 𝑝𝛼𝑇𝑥,𝑇𝑦=𝛼𝑝𝑥,𝑦≤𝛼⋅^2𝑙=𝛼1−𝛼_2𝑙−()𝛼1−𝛼1−𝛼^2𝑙𝑝().1−𝛼=𝑀𝑥,𝑦−𝜓𝑥,𝑦,𝑝𝑥,𝑇𝑥 (52) Obviously, 𝑇 has a unique fixed point 𝟎∈𝐴₁∩𝐴₂.

Finally, we present an example showing that in certain situations the existence of a fixed point can be concluded under partial metric conditions, while the same cannot be obtained using the standard metric.

Example 20. Let 𝑋=ℝ⁺ be equipped with the usual partial metric 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Suppose 𝐴₁=[0,1], 𝐴₂=[0,1/2], 𝐴₃=[0,1/6], 𝐴₄=[0,1/42] and 𝑌=⋃⁴_𝑖=1𝐴_𝑖. Consider the mapping 𝑇∶𝑌→𝑌 defined by 𝑥𝑇𝑥=².1+𝑥 (53) It is clear that ⋃⁴_𝑖=1𝐴_𝑖 is a cyclic representation of 𝑌 with respect to 𝑇. Further, consider the function 𝜓∶[0,+∞)²→[0,+∞) given by 𝜓(𝑠,𝑡)=𝑠+𝑡.2+𝑠+𝑡 (54) Take an arbitrary pair (𝑥,𝑦)∈𝐴_𝑖×𝐴_𝑖+1 with, say, 𝑦≤𝑥 (the other possibility can be treated in a similar way). Then 𝑝𝑥𝑇𝑥,𝑇𝑦=max²,𝑦1+𝑥²=𝑥1+𝑦².1+𝑥 (55) On the other hand, 𝑀𝑝𝑥𝑥,𝑦=max𝑥,𝑦,𝑝𝑥,²𝑦1+𝑥,𝑝𝑦,²,11+𝑦2𝑝𝑦𝑥,²𝑥1+𝑦+𝑝𝑦,²11+𝑥=max𝑥,𝑥,𝑦,2𝑥𝑥+max𝑦,²𝑀𝑝()1+𝑥=𝑥,𝑥,𝑦−𝜓𝑥,𝑦,𝑝𝑥,𝑇𝑥=𝑥−2𝑥=𝑥2+2𝑥².1+𝑥 (56) Hence, condition (NZ2) is satisfied, as well as other conditions of Theorem 13 (with 𝑞=4). We deduce that 𝑇 has a unique fixed point 𝑧=0∈𝐴₁∩𝐴₂∩𝐴₃∩𝐴₄.
On the other hand, consider the same problem in the standard metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| and take 𝑥=1 and 𝑦=1/2. Then 𝑑=1𝑇𝑥,𝑇𝑦2−16=13,𝑀1𝑥,𝑦=max2,12,13,1256=1+02 (57) and hence 𝑀𝑑()=1𝑥,𝑦−𝜓𝑥,𝑦,𝑑𝑥,𝑇𝑥2−13=16. (58) Thus, condition (NZ2) for 𝑝=𝑑 does not hold and the existence of a fixed point of 𝑇 cannot be derived using the standard metric.

Remark 21. The results of this paper are obtained under the assumption that the given partial metric space is 0-complete. Taking into account Lemma 10 and Example 11, it follows that they also hold if the space is complete, but that our assumption is weaker.

Acknowledgments

The authors thank the referees for valuable suggestions that helped them to improve the text. The second author is thankful to the Ministry of Science and Technological Development of Serbia.