Nonlinear Contractions in 0-Complete Partial Metric Spaces

Abstract

Using the setting of 0-complete partial metric spaces, some common fixed point results of maps that satisfy the nonlinear contractive conditions are obtained. These results generalize and improve the existing fixed point results in the literature in the sense that weaker conditions are used. An example shows how our result can be used when the corresponding result in standard metric cannot.

1. Introduction and Preliminaries

Matthews [1] generalized the concept of a metric space by introducing partial metric spaces. Based on the notion of partial metric spaces, Matthews [1, 2], Oltra and Valero [3], Ilić et al. [4, 5] obtained some fixed point theorems for mappings satisfying different contractive conditions. Recently, Abdeljawad et al. [6] proved one fixed point result for generalized contraction principle with control functions on partial metric spaces. For some new results on partial metric and cone metric spaces, see [1–27].

The aim of this paper is to continue the study of common fixed points of mappings but now in 0-complete partial metric spaces, under nonlinear generalized contractive conditions.

Consistent with Matthews [1, 2], O'Neill [21, 22], and Oltra et al. [23], the following definitions and results will be needed throughout this paper.

Definition 1.1. 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,𝑧)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧)−𝑝(𝑦,𝑦).
A partial metric space is a pair (𝑋,𝑝) such that 𝑋 is a nonempty set and 𝑝 is a partial metric on 𝑋.
For a partial metric 𝑝 on 𝑋, the function 𝑝^𝑠∶𝑋×𝑋→𝑅⁺ given by 𝑝^𝑠(𝑥,𝑦)=2𝑝(𝑥,𝑦)−𝑝(𝑥,𝑥)−𝑝(𝑦,𝑦)(1.1) is a (usual) metric on 𝑋. Each partial metric 𝑝 on 𝑋 generates a 𝑇₀ topology 𝜏_𝑝 on 𝑋 with a base of 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.

Definition 1.2 (see [1, 20]). (i) A sequence {𝑥_𝑛} in a partial metric space (𝑋,𝑝) converges to 𝑥∈𝑋 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→+∞𝑝(𝑥_𝑛,𝑥).(ii) 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.(iii) a partial metric space (𝑋,𝑝) is said to be 0-complete if every 0-Cauchy sequence {𝑥_𝑛} in 𝑋 converges, with respect to 𝜏_𝑝, to a point 𝑥∈𝑋 such that 𝑝(𝑥,𝑥)=0. In this case, 𝑝 is a 0-complete partial metric on 𝑋.

Remark 1.3. (1) (see [20]) Clearly, a limit of a sequence in a partial metric space does not need to be unique. Moreover, the function 𝑝(⋅,⋅) does not need to be continuous in the sense that 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦 implies 𝑝(𝑥_𝑛,𝑦_𝑛)→𝑝(𝑥,𝑦). For example, if 𝑋=[0,+∞) and 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for 𝑥,𝑦∈𝑋, then for {𝑥_𝑛}={1},𝑝(𝑥_𝑛,𝑥)=𝑥=𝑝(𝑥,𝑥) for each 𝑥≥1 and so, for example, 𝑥_𝑛→2 and 𝑥_𝑛→3 when 𝑛→∞.(2) (see [6]) However, if 𝑝(𝑥_𝑛,𝑥)→𝑝(𝑥,𝑥)=0, then 𝑝(𝑥_𝑛,𝑦)→𝑝(𝑥,𝑦) for all 𝑦∈𝑋.
Assertions similar to the following lemma were used (and proved) in the course of proofs of several fixed point results in various papers [8, 9, 20, 28].

Lemma 1.4. Let (𝑋,𝑝) be a partial metric space and let {𝑦_𝑛} be a sequence in 𝑋 such that 𝑝(𝑦_𝑛,𝑦_𝑛+1) is nonincreasing and that lim_𝑛→+∞𝑝𝑦_𝑛,𝑦_𝑛+1=0.(1.2) If {𝑦_2𝑛} is not a 0-Cauchy sequence in (𝑋,𝑝), then there exist 𝜀>0 and two sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that 𝑚_𝑘>𝑛_𝑘>𝑘 and the following four sequences tend to 𝜀⁺ when 𝑘→+∞∶𝑝𝑦_2𝑚𝑘,𝑦_2𝑛𝑘,𝑝𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1,𝑝𝑦_{2𝑚𝑘−1},𝑦_2𝑛𝑘,𝑝𝑦_{2𝑚𝑘−1},𝑦_2𝑛𝑘+1.(1.3)

Definition 1.5 (see [29]). Let 𝑓 and 𝑔 be self-maps of a set 𝑋. If 𝑤=𝑓𝑥=𝑔𝑥 for some 𝑥∈𝑋, then 𝑥 is called a coincidence point of 𝑓 and 𝑔, and 𝑤 is called a point of coincidence of 𝑓 and 𝑔. The pair 𝑓,𝑔 of self-maps is weakly compatible if they commute at their coincidence points.

The following lemma is Proposition 1.4 of [29].

Lemma 1.6. Let 𝑓 and 𝑔 be weakly compatible self-maps of a set 𝑋. If 𝑓 and 𝑔 have a unique point of coincidence 𝑤=𝑓𝑥=𝑔𝑥, then 𝑤 is the unique common fixed point of 𝑓 and 𝑔.

Definition 1.7 (see [30]). The following two classes of mappings are defined as Ψ={𝜓|𝜓∶[0,∞)→[0,∞) is a nondecreasing right continuous function such that lim_𝑛→+∞𝜓^𝑛(𝑡)=0 for all 𝑡>0, and Φ={𝜓|𝜓∶[0,∞)→[0,∞) is a nondecreasing function such that lim_𝑛→+∞𝜓^𝑛(𝑡)=0 for all 𝑡>0.
It is clear for the function 𝜓(𝑡)=⎧⎪⎪⎨⎪⎪⎩131𝑡,0≤𝑡<122,𝑡=13𝑡,𝑡>1,(1.4) that 𝜓∈Φ but 𝜓∉Ψ.

Lemma 1.8 (see [19]). If 𝜓∈Φ, then 𝜓(𝑡)<𝑡 for all 𝑡>0 and 𝜓(0)=0.

2. Common Fixed Point Results

In this section, we obtain some common fixed point results defined on 0-complete partial metric spaces.

Theorem 2.1. Let (𝑋,𝑝) be a partial metric space and 𝑇,𝑓 be mappings on 𝑋 with 𝑇𝑋⊂𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜓max𝑝(𝑓𝑥,𝑓𝑦),𝑝(𝑓𝑥,𝑇𝑥),𝑝(𝑓𝑦,𝑇𝑦),𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥)2(2.1) for all 𝑥,𝑦∈𝑋, where 𝜓∈Ψ. If 𝑇𝑋 or 𝑓𝑋 is a 0-complete subspace of 𝑋, then 𝑇 and 𝑓 have a unique point of coincidence. Moreover, if 𝑇 and 𝑓 are weakly compatible, then 𝑇 and 𝑓 have a unique common fixed point.

Proof. First, we prove that 𝑇 and 𝑓 have a unique point of coincidence (if it exists). If 𝑦∈𝑋 with 𝑦=𝑇𝑢=𝑓𝑢 and 𝑧∈𝑋 with 𝑧=𝑇𝑠=𝑓𝑠, we assume 𝑧≠𝑦. Applying (2.1) we have 𝑝(𝑦,𝑧)≤=𝜓𝑝(𝑇𝑢m,𝑇ax𝑠)𝑝(𝑓𝑢,𝑓𝑠),𝑝(𝑓𝑢,𝑇𝑢),𝑝(𝑓𝑠,𝑇𝑠),𝑝(𝑓𝑢,𝑇𝑠)+𝑝(𝑓𝑠,𝑇𝑢)=𝜓max𝑝(𝑦,𝑧),𝑝(𝑦,𝑦),𝑝(𝑧,𝑧),𝑝(𝑦,𝑧)+𝑝(𝑧,𝑦)2=<𝜓𝑝((𝑦𝑝,𝑧(𝑦),.𝑧))(by(p2))2(2.2)
This implies a contradiction, so we conclude that 𝑧=𝑦.
Now, consider the sequence {𝑥_𝑛}⊂𝑋 defined by 𝑥₀∈𝑋 and 𝑇𝑥_𝑛=𝑓𝑥_𝑛+1. Consider the two possible cases: (i)𝑝(𝑇𝑥_𝑛+1,𝑇𝑥_𝑛)=0 for some 𝑛∈ℕ. In this case 𝑇𝑥_𝑛=𝑓𝑥_𝑛(=𝑦 a point of coincidence). Because of Lemma 1.6 (unique point of coincidence and weakly compatible mappings) we have a unique common fixed point.(iii)𝑝(𝑇𝑥_𝑛+1,𝑇𝑥_𝑛)>0 for every 𝑛∈ℕ.
Applying (2.1) with 𝑥=𝑥_𝑛+1 and 𝑦=𝑥_𝑛, we have,

𝑝𝑇𝑥_𝑛+1,𝑇𝑥_𝑛≤𝜓max𝑝𝑓𝑥𝑝𝑓_𝑛+1𝑥_𝑛+1,𝑓𝑥_𝑛,𝑝𝑓𝑥_𝑛+1,𝑇𝑥_𝑛+1,𝑝𝑓𝑥_𝑛,𝑇𝑥_𝑛,,𝑇𝑥_𝑛+𝑝𝑓𝑥_𝑛,𝑇𝑥_𝑛+1=𝜓max𝑝𝑇𝑥_𝑛2𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛−1,𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1,𝑝𝑇𝑥_𝑛−1,𝑇𝑥_𝑛,,𝑇𝑥_𝑛+𝑝𝑇𝑥_𝑛−1,𝑇𝑥_𝑛+1=𝜓max𝑝𝑇𝑥_𝑛2,𝑇𝑥_𝑛−1,𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1,𝑝𝑇𝑥^𝑛,𝑇𝑥_𝑛+𝑝𝑇𝑥_𝑛−1,𝑇𝑥_𝑛+1≤𝜓max𝑝𝑇𝑥_𝑛2,𝑇𝑥_𝑛−1,𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1,𝑝𝑇𝑥^𝑛−1,𝑇𝑥_𝑛+𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1=𝜓max𝑝𝑇𝑥_𝑛2,𝑇𝑥_𝑛−1,𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1.(2.3)

If 𝑝(𝑇𝑥_𝑛+1,𝑇𝑥_𝑛) is a maximum, then we have 𝑝𝑇𝑥_𝑛+1,𝑇𝑥_𝑛≤𝜓𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1<𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛+1,(2.4) which is a contradiction. So, we conclude that the maximum is 𝑝(𝑇𝑥_𝑛,𝑇𝑥_𝑛−1). Now, we have the following: 𝑝𝑇𝑥_𝑛+1,𝑇𝑥_𝑛≤𝜓𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛−1<𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛−1.(2.5) It follows from (2.5) that the sequence 𝑝(𝑇𝑥_𝑛+1,𝑇𝑥_𝑛) is monotone decreasing. According to the properties of function 𝜓, it follows 𝑝𝑇𝑥_𝑛+1,𝑇𝑥_𝑛≤≤𝜓𝜓₂𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛−1𝑝𝑇𝑥_𝑛−1,𝑇𝑥_𝑛−2≤⋯≤𝜓^𝑛𝑝𝑇𝑥₁,𝑇𝑥₀→0,(2.6) when 𝑛→+∞. Therefore, lim_𝑛→+∞𝑝(𝑇𝑥_𝑛+1,𝑇𝑥_𝑛)=0.

Next, we prove that {𝑇𝑥_𝑛} is a 0-Cauchy sequence in the space (𝑋,𝑝). It is sufficient to show that {𝑇𝑥_2𝑛} is a 0-Cauchy sequence. Assume the opposite. Then using Lemma 1.4 we get that there exist 𝜀>0 and two sequences {𝑚(𝑘)} and {𝑛(𝑘)} of positive integers and sequences 𝑝𝑦_2𝑚𝑘,𝑦_2𝑛𝑘,𝑝𝑦_2𝑚𝑘,𝑦_2𝑛𝑘+1,𝑝𝑦_{2𝑚𝑘−1},𝑦_2𝑛𝑘,𝑝𝑦_{2𝑚𝑘−1},𝑦_2𝑛𝑘+1,(2.7) all tend to 𝜀⁺ when 𝑘→+∞. Applying condition (2.1) to elements 𝑥=𝑥_2𝑚(𝑘) and 𝑦=𝑥_{2𝑛(𝑘)+1} and putting 𝑦_𝑛=𝑇𝑥_𝑛=𝑓𝑥_𝑛+1 for each 𝑛≥0, we get that 𝑝𝑇𝑥=𝑝_2𝑚(𝑘),𝑦𝑇𝑥_{2𝑛(𝑘)+12𝑚(𝑘)}≤𝜓max,𝑦𝑝_{2𝑛(𝑘)+1}𝑓𝑥_2𝑚(𝑘)𝑝𝑓𝑥_2𝑚(𝑘),𝑓𝑥_{2𝑛(𝑘)+1},𝑝𝑓𝑥_2𝑚(𝑘),𝑇𝑥_2𝑚(𝑘),𝑝𝑓𝑥_{2𝑛(𝑘)+1},𝑇𝑥_{2𝑛(𝑘)+1},,𝑇𝑥_{2𝑛(𝑘)+1}+𝑝𝑓𝑥_{2𝑛(𝑘)+1},𝑇𝑥_2𝑚(𝑘)=𝜓max𝑝𝑦_{2𝑚(𝑘)−1}2𝑝𝑦_{2𝑚(𝑘)−1},𝑦_2𝑛(𝑘),𝑝𝑦_{2𝑚(𝑘)−1},𝑦_2𝑚(𝑘),𝑝𝑦_2𝑛(𝑘),𝑦_{2𝑛(𝑘)+1},,𝑦_{2𝑛(𝑘)+1}+𝑝𝑦_2𝑛(𝑘),𝑦_2𝑚(𝑘)2.(2.8)

When 𝑘→+∞, we have m𝑝ax𝑦𝑝𝑦_{2𝑚(𝑘)−12𝑚(𝑘)−1},𝑦_2𝑛(𝑘),𝑝𝑦_{2𝑚(𝑘)−1},𝑦_2𝑚(𝑘),𝑝𝑦_2𝑛(𝑘),𝑦_{2𝑛(𝑘)+1},,𝑦_{2𝑛(𝑘)+1}+𝑝𝑦_2𝑛(𝑘),𝑦_2𝑚(𝑘)→max𝜀₊2,0,0,12𝜀⁺+𝜀⁺=𝜀⁺.(2.9) Using properties of 𝜓, we obtain a contradiction 𝜀≤𝜓(𝜀)<𝜀, since 𝜀>0.

This shows that {𝑇𝑥_2𝑛} is a 0-Cauchy sequence in the space (𝑋,𝑝) and hence {𝑇𝑥_𝑛} is a 0-Cauchy sequence in (𝑋,𝑝). If we suppose that 𝑇𝑋 is a 0-complete subspace of (𝑋,𝑝), then there exists 𝑦∈𝑇𝑋⊂𝑓𝑋 such that 𝑝(𝑦,𝑦)=lim_𝑛→+∞𝑝𝑇𝑥_𝑛,𝑦=lim_𝑛→+∞𝑝𝑓𝑥_𝑛,𝑦=lim_{𝑛,𝑚→+∞}𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑚=0.(2.10) If 𝑓𝑋 is a 0-complete subspace of (𝑋,𝑝) with 𝑦∈𝑓𝑋, (2.10) also holds.

Let 𝑢∈𝑋 and 𝑦=𝑓𝑢. We show that 𝑦 is a point of coincidence of 𝑇 and 𝑓. Suppose that 𝑝(𝑇𝑢,𝑓𝑢)>0. We have 𝑝𝑇𝑥_𝑛,𝑇𝑢≤𝜓max𝑝𝑓𝑥_𝑛,𝑓𝑢,𝑝𝑓𝑥_𝑛,𝑇𝑥_𝑛,𝑝(𝑓𝑢,𝑇𝑢),𝑝𝑓𝑥^𝑛,𝑇𝑢+𝑝𝑓𝑢,𝑇𝑥_𝑛2.(2.11)

As 𝑛→+∞, we have 𝑝(𝑓𝑥_𝑛,𝑓𝑢)→0,𝑝(𝑓𝑥_𝑛,𝑇𝑥_𝑛)→0 and by Remark 1.3 (2) (𝑝(𝑓𝑥_𝑛,𝑇𝑢)+𝑝(𝑓𝑢,𝑇𝑥_𝑛))/2→𝑝(𝑓𝑢,𝑇𝑢)/2. Since 𝑝(𝑓𝑢,𝑇𝑢)>0, there exists 𝑛₀∈ℕ such that for every 𝑛>𝑛₀ we have 𝑝(𝑓𝑥_𝑛,𝑓𝑢)<(1/2)𝑝(𝑓𝑢,𝑇𝑢) and 𝑝(𝑓𝑥_𝑛,𝑇𝑥_𝑛)<(1/2)𝑝(𝑓𝑢,𝑇𝑢). Now, for 𝑛>𝑛₀, we obtain 𝑝(𝑓𝑢,𝑇𝑢)≤𝑝𝑓𝑢,𝑓𝑥_𝑛+1+𝑝𝑇𝑥_𝑛,𝑇𝑢≤𝑝𝑓𝑢,𝑓𝑥_𝑛+1+𝜓(𝑝(𝑓𝑢,𝑇𝑢))(2.12) or 𝑝(𝑓𝑢,𝑇𝑢)≤𝑝𝑓𝑢,𝑓𝑥_𝑛+1+𝜓(𝑝(𝑓𝑢,𝑇𝑢)).(2.13) As 𝑛→+∞ we have a contradiction: 𝑝(𝑓𝑢,𝑇𝑢)≤𝜓(𝑝(𝑓𝑢,𝑇𝑢))<𝑝(𝑓𝑢,𝑇𝑢).(2.14) This implies 𝑝(𝑇𝑢,𝑓𝑢)=0, that is, 𝑇𝑢=𝑓𝑢. Hence, 𝑇 and 𝑓 have a (unique) point of coincidence. From Lemma 1.6 follows that this is a unique common fixed point of 𝑇 and 𝑓.

Remark 2.2. Assumption about right continuity of the function is only used in the proof that {𝑇𝑥_𝑛} is a 0-Cauchy sequence.
In the following theorem we consider a weaker assumption for the function 𝜓. As a compensation we assume a bit stronger contractive condition.

Theorem 2.3. Let (𝑋,𝑝) be a partial metric space and 𝑇,𝑓 be mappings on 𝑋 with 𝑇𝑋⊂𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜓max𝑝(𝑓𝑥,𝑓𝑦),𝑝(𝑓𝑥,𝑇𝑥),𝑝(𝑓𝑦,𝑇𝑦),𝑝(𝑓𝑥,𝑇𝑦)2(2.15) for all 𝑥,𝑦∈𝑋, where 𝜓∈Φ. If 𝑇𝑋 or 𝑓𝑋 is a 0-complete subspace of 𝑋, then 𝑇 and 𝑓 have a unique point of coincidence. Moreover, if 𝑇 and 𝑓 are weakly compatible, then 𝑇 and 𝑓 have a unique common fixed point.

Proof. We can prove that 𝑇 and 𝑓 have a unique point of coincidence in a similar way like in Theorem 2.1. If we consider the sequence {𝑥_𝑛}⊂𝑋 defined by 𝑥₀∈𝑋 and 𝑇𝑥_𝑛=𝑓𝑥_𝑛+1, we used Theorem 2.1 to show lim_𝑛→+∞𝑝(𝑇𝑥_𝑛+1,𝑇𝑥_𝑛)=0.
Here, we only prove that {𝑇𝑥_𝑛} is a 0-Cauchy sequence in the space (𝑋,𝑝) by using induction. Let us denote 𝑦_𝑛=𝑇𝑥_𝑛=𝑓𝑥_𝑛+1 for each 𝑛≥0, We have 𝑝(𝑦_𝑛,𝑦_𝑛+1)→0, as 𝑛→+∞. For 𝜀>0 there exists 𝑛(𝜀) such that for every 𝑛>𝑛(𝜀) we have 𝑝(𝑦_𝑛,𝑦_𝑛+1)<𝜀−𝜓(𝜀). Assuming that for some 𝑛>𝑛(𝜀) and 𝑘∈ℕ it holds that 𝑝(𝑦_𝑛,𝑦_𝑛+𝑘)<𝜀 and we need to prove that 𝑝(𝑦_𝑛,𝑦_𝑛+𝑘+1)<𝜀.
We have 𝑝𝑦_𝑛+1𝑝𝑦_𝑛,𝑦_𝑛+𝑘+1≤𝑝𝑦_𝑛,𝑦_𝑛+1+𝑝𝑦_𝑛+1,𝑦_𝑛+𝑘+1,=𝑝𝑇,𝑥𝑦_{𝑛+𝑘+1𝑛+1}≤𝜓max,𝑇𝑥𝑝𝑓_𝑛+𝑘+1𝑥_𝑛+1,𝑓𝑥_𝑛+𝑘+1,𝑝𝑓𝑥_𝑛+1,𝑇𝑥_𝑛+1,𝑝𝑓𝑥_𝑛+𝑘+1,𝑇𝑥_𝑛+𝑘+1,𝑝𝑓𝑥^𝑛+1,𝑇𝑥_𝑛+𝑘+1=𝜓max𝑝𝑦_𝑛2,𝑦_𝑛+𝑘,𝑝𝑦_𝑛,𝑦_𝑛+1,𝑝𝑦_𝑛+𝑘,𝑦_𝑛+𝑘+1,𝑝𝑦^𝑛,𝑦_𝑛+𝑘+1≤𝜓max𝑝𝑦_𝑛2,𝑦_𝑛+𝑘,𝑝𝑦_𝑛,𝑦_𝑛+1,𝑝𝑦_𝑛+𝑘,𝑦_𝑛+𝑘+1,𝑝𝑦^𝑛,𝑦_𝑛+𝑘+𝑝𝑦_𝑛+𝑘,𝑦_𝑛+𝑘+1≤𝜓max𝜀,𝜀−𝜓(𝜀),𝜀−𝜓(𝜀),𝜀+𝜀−𝜓(𝜀)22=𝜓(𝜀).(2.16)
Then (2.16) becomes 𝑝(𝑦_𝑛,𝑦_𝑛+𝑘+1)≤𝑝(𝑦_𝑛,𝑦_𝑛+1)+𝑝(𝑦_𝑛+1,𝑦_𝑛+𝑘+1)<𝜀−𝜓(𝜀)+𝜓(𝜀)=𝜀. This shows that {𝑇𝑥_𝑛} is a 0-Cauchy sequence in (𝑋,𝑝). Further, proof is similar as in Theorem 2.1, so we omit it.

Corollary 2.4. Let (𝑋,𝑝) be a partial metric space and let 𝑇∶𝑋→𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜓max𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥),𝑝(𝑦,𝑇𝑦),𝑝(𝑥,𝑇𝑦)+𝑝(𝑦,𝑇𝑥)2(2.17) for all 𝑥,𝑦∈𝑋, where 𝜓∈Ψ. If 𝑇𝑋 is a 0-complete subspace of 𝑋, then 𝑇 has a unique fixed point.

Proof. It follows from Theorem 2.1 by taking 𝑓=𝑖_𝑋 (the identity map).

Corollary 2.5. Let (𝑋,𝑝) be a partial metric space and let 𝑇∶𝑋→𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜓max𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥),𝑝(𝑦,𝑇𝑦),𝑝(𝑥,𝑇𝑦)2(2.18) for all 𝑥,𝑦∈𝑋, where 𝜓∈Φ. If 𝑇𝑋 is a 0-complete subspace of 𝑋, then 𝑇 has a unique fixed point.

Proof. It follows from Theorem 2.3 by taking 𝑓=𝑖_𝑋 (the identity map).

Corollary 2.6. Let (𝑋,𝑝) be a partial metric space and let 𝑇,𝑓 be mappings on 𝑋 with 𝑇𝑋⊂𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜆max𝑝(𝑓𝑥,𝑓𝑦),𝑝(𝑓𝑥,𝑇𝑥),𝑝(𝑓𝑦,𝑇𝑦),𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥)2(2.19) for all 𝑥,𝑦∈𝑋, where 𝜆∈[0,1). If 𝑇𝑋 or 𝑓𝑋 is a 0-complete subspace of 𝑋, then 𝑇 and 𝑓 have a unique point of coincidence. Moreover, if 𝑇 and 𝑓 are weakly compatible, then 𝑇 and 𝑓 have a unique fixed point.

Proof. It follows from Theorem 2.1 by taking 𝜓(𝑡)=𝜆𝑡, where 𝜆∈[0,1).

Corollary 2.7. Let(𝑋,𝑝) be a partial metric space and let 𝑇,𝑓 be mappings on 𝑋 with 𝑇𝑋⊂𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)≤𝑎+₁𝑝𝑎(₄𝑓𝑥,𝑓𝑦)+𝑎₂𝑝(𝑓𝑥,𝑇𝑥)+𝑎₃𝑝(𝑓𝑦,𝑇𝑦)(𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥))(2.20) for all 𝑥,𝑦∈𝑋, where 𝑎₁,𝑎₂,𝑎₃,𝑎₄≥0 and 𝑎₁+𝑎₂+𝑎₃+2𝑎₄∈[0,1). If 𝑇𝑋 or 𝑓𝑋 is a 0-complete subspace of 𝑋, then 𝑇 and 𝑓 have a unique point of coincidence. Moreover, if 𝑇 and 𝑓 are weakly compatible, then 𝑇 and 𝑓 have a unique fixed point.

Proof. It follows from Corollary 2.6 by noting that 𝑎₁𝑝≤(𝑓𝑥𝑎,𝑓𝑦)+𝑎₂𝑝(𝑓𝑥,𝑇𝑥)+𝑎₃𝑝(𝑓𝑦,𝑇𝑦)+𝑎₄(𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥))₁+𝑎₂+𝑎₃+2𝑎₄max𝑝(𝑓𝑥,𝑓𝑦),𝑝(𝑓𝑥,𝑇𝑥),𝑝(𝑓𝑦,𝑇𝑦),𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥)2(2.21)

Corollary 2.8. Let (𝑋,𝑝) be a partial metric space and let 𝑇,𝑓 be mappings on 𝑋 with 𝑇𝑋⊂𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜆𝑝(𝑓𝑥,𝑓𝑦),(2.22) for all 𝑥,𝑦∈𝑋, where 𝜆∈[0,1). If 𝑇𝑋 or 𝑓𝑋 is a 0-complete subspace of 𝑋, then 𝑇 and 𝑓 have a unique point of coincidence. Moreover, if 𝑇 and 𝑓 are weakly compatible, then 𝑇 and 𝑓 have a unique fixed point.

Proof. It follows from Corollary 2.7.

From Corollary 2.8 follows the theorem proved by Jungck [31], where we consider a 0-complete partial metric space instead of a complete metric space, under the assumption that 𝑇 and 𝑓 are commuting mappings.

Corollary 2.9. Let (𝑋,𝑝) be a 0-complete partial metric space and 𝑇∶𝑋→𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜓(𝑝(𝑥,𝑦))(2.23) for all 𝑥,𝑦∈𝑋, where 𝜓∈Φ. If 𝑇𝑋 is a closed subspace of 𝑋, then 𝑇 has a unique fixed point.

Proof. It follows from Theorem 2.3 and Corollary 2.5. by taking 𝑓=𝑖_𝑋 (the identity map).

Under a weaker assumption for the function 𝜓, from Corollary 2.9 follows the theorem proved by Boyd and Wong [32] where we consider a partial metric instead of a standard metric.

From Corollary 2.7 we deduce the following corollaries, which are extensions of some well-known theorems.

Corollary 2.10. Let (𝑋,𝑝) be a 0-complete partial metric space and let 𝑇∶𝑋→𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜆𝑝(𝑥,𝑦)(2.24) for all 𝑥,𝑦∈𝑋, where 𝜆∈[0,1). Then 𝑇 has a unique fixed point.

This corollary is an extension of Banach contraction theorem on a 0-complete partial metric space. This corollary is already mentioned in Matthews [2] for a complete partial metric space, but is true for a 0-complete partial metric spaces.

Corollary 2.11. Let (𝑋,𝑝) be a 0-complete partial metric space and let 𝑇∶𝑋→𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜆(𝑝(𝑥,𝑇𝑥)+𝑝(𝑦,𝑇𝑦))(2.25) for all 𝑥,𝑦∈𝑋, where 𝜆∈[0,(1/2)). Then 𝑇 has a unique fixed point.

This corollary is extension of Kannan theorem [33] (4) on a 0-complete partial metric spaces.

Corollary 2.12. Let (𝑋,𝑝) be a 0-complete partial metric space and let 𝑇∶𝑋→𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)≤𝑎₁𝑝(𝑥,𝑦)+𝑎₂𝑝(𝑥,𝑇𝑥)+𝑎₃𝑝(𝑦,𝑇𝑦)(2.26) for all 𝑥,𝑦∈𝑋, where a₁,𝑎₂,𝑎₃≥ and 𝑎₁+𝑎₂+𝑎₂∈[0,1). Then 𝑇 has a unique fixed point.

This corollary is extension of Reich theorem [33] (8) on a 0-complete partial metric spaces.

Corollary 2.13. Let (𝑋,𝑝) be a 0-complete partial metric space and let 𝑇∶𝑋→𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)≤𝜆(𝑝(𝑥,𝑇𝑦)+𝑝(𝑦,𝑇𝑥))(2.27) for all 𝑥,𝑦∈𝑋, where 𝜆∈[0,(1/2)). Then 𝑇 has a unique fixed point.

This corollary is extension of Chatterjea theorem [33] (11) on a 0-complete partial metric spaces.

Corollary 2.14. Let (𝑋,𝑝) be a 0-complete partial metric space and let 𝑇∶𝑋→𝑋 be a map such that at least one of the following is true 𝑝𝑝((𝑇𝑇𝑥𝑥,,𝑇𝑇𝑦𝑦))≤≤𝑎𝑎₁₂𝑝(𝑥,𝑇𝑥)𝑝(𝑇𝑥,𝑇𝑦)≤𝑎₃(𝑝(𝑥,𝑇𝑥)+𝑝(𝑦,𝑇𝑦))(𝑝(𝑥,𝑇𝑦)+𝑝(𝑦,𝑇𝑥))(2.28) for all 𝑥,𝑦∈𝑋, where 𝑎₁∈[0,1),𝑎₂,𝑎₃∈[0,(1/2)). Then 𝑇 has a unique fixed point.

This corollary is extension of Zamfirescu theorem [33] (19) on a 0-complete partial metric spaces.

We demonstrate the use of Theorem 2.1 with the help of the following example.

Example 2.15. Let (𝑋,𝑝)=(ℚ∩[0,1),𝑝), where ℚ denotes the set of rational numbers and 𝑝 is given by 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Then (𝑋,𝑝) is a 0-complete partial metric space which is not complete. Suppose that 𝑇,𝑓∶𝑋→𝑋 are such that 𝑇𝑥=𝑥²/(1+𝑥),𝑓𝑥=𝑥 for all 𝑥∈𝑋 and 𝜓(𝑡)=⎧⎪⎨⎪⎩𝑡²,𝑡>1155𝑡,𝑡≤51.(2.29) Then 𝜓∈Ψ. Without loss of generality assume that 𝑥<𝑦. Then 𝑥²/(1+𝑥)<𝑦/(1+𝑦),𝑥²/(1+𝑥)≤𝑥 and (𝑦²/(1+𝑦))≤𝑦. From (2.1) follows 𝐿=max𝑥²𝑅=𝜓max𝑝(𝑥,𝑦),𝑝𝑥,𝑥₂1+𝑥,1𝑦+²𝑦=1𝑦+²𝑦,(2.30)1+𝑥,𝑝𝑦,1𝑦+²𝑦,𝑝𝑥,𝑦²/(1+𝑦)+𝑝𝑦,𝑥²/(1+𝑥)=𝜓max𝑦,𝑥,𝑦,max𝑥,𝑦₂2/(1+𝑦)+max𝑦,𝑥²/(1+𝑥)2≤𝜓(𝑦)(2.31) and so 𝑦²/(1+𝑦)≤𝜓(𝑦). If 𝑦≤1/5, then we have 𝑦²/(1+𝑦)≤𝑦/5 (this is true because 𝑦²/(1+𝑦)≤𝑦²≤𝑦/5). If 𝑦>1/5 then we have 𝑦²/(1+𝑦)≤𝑦², which is true. It follows that 𝑇 has a unique fixed point.
On the other hand, consider the same problem in the standard metric 𝑑(𝑥,𝑦)=|𝑥−𝑦| and take 𝑥=0.10,𝑦=0.30. Then, from (2.1) follows 𝐿=0.009−0.069=0.06 and 𝑅=𝜓max0.10−0.30,0.10−0.009,0.30−0.069,0.10−0.069+0.30−0.009=𝜓(0.231)=0.053.2(2.32) Hence 𝐿≤𝑅 does not hold and the existence of a unique fixed point cannot be obtained.

Remark 2.16. Note that Theorem 2.1 improves [12, Theorem 1 and Corollary 1], [25, Theorem 3, Corollaries 1 and 2 and Theorem 4], and [13, Corollary 2.3] since our assumptions are weaker than the assumptions from [12, 13, 25] in several places.

Finally, it is worth to notice that all new results in recently papers [7, 10, 15, 20, 27] are true if partial metric space (𝑋,𝑝) is 0-complete instead complete.

Acknowledgments

The authors are thankful to the referees for their remarks which helped to improve the presentation of the paper. The authors (first and second) would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grant OUP-UKM-FST-2012. The third and the fourth authors are thankful to the Ministry of Science and Technological Development of Serbia.