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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 451239, 12 pages
http://dx.doi.org/10.1155/2012/451239
Research Article

Nonlinear Contractions in 0-Complete Partial Metric Spaces

1School of Mathematical Sciences, Faculty of Sciences and Technology, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi, Selangor Darul Ehsan, Malaysia
2Faculty of Economics, University of Belgrade, Kamenička 6, 11000 Beograd, Serbia
3Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11 120 Beograd, Serbia

Received 20 March 2012; Accepted 23 May 2012

Academic Editor: Gaston Mandata N'Guerekata

Copyright © 2012 Abd Ghafur Bin Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [127].

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 𝑝𝑋×𝑋𝑅+ such that for all 𝑥,𝑦,𝑧𝑋(p1)𝑥=𝑦𝑝(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 𝑝𝑠𝑋×𝑋𝑅+ given by 𝑝𝑠(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 𝜏𝑝 on 𝑋 with a base of the family of open 𝑝-balls {𝐵𝑝(𝑥,𝜀)𝑥𝑋,𝜀>0}, where 𝐵𝑝(𝑥,𝜀)={𝑦𝑋𝑝(𝑥,𝑦)<𝑝(𝑥,𝑥)+𝜀} for all 𝑥𝑋 and 𝜀>0.

Definition 1.2 (see [1, 20]). (i) A sequence {𝑥𝑛} in a partial metric space (𝑋,𝑝) converges to 𝑥𝑋 if and only if 𝑝(𝑥,𝑥)=lim𝑛+𝑝(𝑥𝑛,𝑥).(ii) a sequence {𝑥𝑛} in a partial metric space (𝑋,𝑝) is called 0-Cauchy if lim𝑛,𝑚+𝑝(𝑥𝑛,𝑥𝑚)=0.(iii) a partial metric space (𝑋,𝑝) is said to be 0-complete if every 0-Cauchy sequence {𝑥𝑛} in 𝑋 converges, with respect to 𝜏𝑝, to a point 𝑥𝑋 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 𝑥𝑛𝑥 and 𝑦𝑛𝑦 implies 𝑝(𝑥𝑛,𝑦𝑛)𝑝(𝑥,𝑦). For example, if 𝑋=[0,+) and 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for 𝑥,𝑦𝑋, then for {𝑥𝑛}={1},𝑝(𝑥𝑛,𝑥)=𝑥=𝑝(𝑥,𝑥) for each 𝑥1 and so, for example, 𝑥𝑛2 and 𝑥𝑛3 when 𝑛.(2) (see [6]) However, if 𝑝(𝑥𝑛,𝑥)𝑝(𝑥,𝑥)=0, then 𝑝(𝑥𝑛,𝑦)𝑝(𝑥,𝑦) for all 𝑦𝑋.
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 {𝑦𝑛} be a sequence in 𝑋 such that 𝑝(𝑦𝑛,𝑦𝑛+1) is nonincreasing and that lim𝑛+𝑝𝑦𝑛,𝑦𝑛+1=0.(1.2) If {𝑦2𝑛} is not a 0-Cauchy sequence in (𝑋,𝑝), then there exist 𝜀>0 and two sequences {𝑚𝑘} and {𝑛𝑘} of positive integers such that 𝑚𝑘>𝑛𝑘>𝑘 and the following four sequences tend to 𝜀+ when 𝑘+𝑝𝑦2𝑚𝑘,𝑦2𝑛𝑘𝑦,𝑝2𝑚𝑘,𝑦2𝑛𝑘+1𝑦,𝑝2𝑚𝑘1,𝑦2𝑛𝑘𝑦,𝑝2𝑚𝑘1,𝑦2𝑛𝑘+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 𝑥𝑋, 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 lim𝑛+𝜓𝑛(𝑡)=0 for all 𝑡>0, and Φ={𝜓|𝜓[0,)[0,) is a nondecreasing function such that lim𝑛+𝜓𝑛(𝑡)=0 for all 𝑡>0.
It is clear for the function 𝜓1(𝑡)=31𝑡,0𝑡<122,𝑡=13𝑡,𝑡>1,(1.4) that 𝜓Φ but 𝜓Ψ.

Lemma 1.8 (see [19]). If 𝜓Φ, 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 𝑇𝑋𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)𝜓max𝑝(𝑓𝑥,𝑓𝑦),𝑝(𝑓𝑥,𝑇𝑥),𝑝(𝑓𝑦,𝑇𝑦),𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥)2(2.1) for all 𝑥,𝑦𝑋, where 𝜓Ψ. 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 𝑦𝑋 with 𝑦=𝑇𝑢=𝑓𝑢 and 𝑧𝑋 with 𝑧=𝑇𝑠=𝑓𝑠, we assume 𝑧𝑦. Applying (2.1) we have 𝑝(𝑦,𝑧)=𝑝(𝑇𝑢,𝑇𝑠)𝜓max𝑝(𝑓𝑢,𝑓𝑠),𝑝(𝑓𝑢,𝑇𝑢),𝑝(𝑓𝑠,𝑇𝑠),𝑝(𝑓𝑢,𝑇𝑠)+𝑝(𝑓𝑠,𝑇𝑢)2𝑝=𝜓max𝑝(𝑦,𝑧),𝑝(𝑦,𝑦),𝑝(𝑧,𝑧),(𝑦,𝑧)+𝑝(𝑧,𝑦)2=𝜓(𝑝(𝑦,𝑧))(by(p2))<𝑝(𝑦,𝑧).(2.2)
This implies a contradiction, so we conclude that 𝑧=𝑦.
Now, consider the sequence {𝑥𝑛}𝑋 defined by 𝑥0𝑋 and 𝑇𝑥𝑛=𝑓𝑥𝑛+1. Consider the two possible cases: (i)𝑝(𝑇𝑥𝑛+1,𝑇𝑥𝑛)=0 for some 𝑛.In this case 𝑇𝑥𝑛=𝑓𝑥𝑛(=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(𝑇𝑥𝑛+1,𝑇𝑥𝑛)>0 for every 𝑛.
Applying (2.1) with 𝑥=𝑥𝑛+1 and 𝑦=𝑥𝑛, we have,

𝑝𝑇𝑥𝑛+1,𝑇𝑥𝑛𝑝𝜓max𝑓𝑥𝑛+1,𝑓𝑥𝑛,𝑝𝑓𝑥𝑛+1,𝑇𝑥𝑛+1,𝑝𝑓𝑥𝑛,𝑇𝑥𝑛,𝑝𝑓𝑥𝑛+1,𝑇𝑥𝑛+𝑝𝑓𝑥𝑛,𝑇𝑥𝑛+12𝑝=𝜓max𝑇𝑥𝑛,𝑇𝑥𝑛1,𝑝𝑇𝑥𝑛,𝑇𝑥𝑛+1,𝑝𝑇𝑥𝑛1,𝑇𝑥𝑛,𝑝𝑇𝑥𝑛,𝑇𝑥𝑛+𝑝𝑇𝑥𝑛1,𝑇𝑥𝑛+12𝑝=𝜓max𝑇𝑥𝑛,𝑇𝑥𝑛1,𝑝𝑇𝑥𝑛,𝑇𝑥𝑛+1,𝑝𝑇𝑥𝑛,𝑇𝑥𝑛+𝑝𝑇𝑥𝑛1,𝑇𝑥𝑛+12𝑝𝜓max𝑇𝑥𝑛,𝑇𝑥𝑛1,𝑝𝑇𝑥𝑛,𝑇𝑥𝑛+1,𝑝𝑇𝑥𝑛1,𝑇𝑥𝑛+𝑝𝑇𝑥𝑛,𝑇𝑥𝑛+12𝑝=𝜓max𝑇𝑥𝑛,𝑇𝑥𝑛1,𝑝𝑇𝑥𝑛,𝑇𝑥𝑛+1.(2.3)

If 𝑝(𝑇𝑥𝑛+1,𝑇𝑥𝑛) is a maximum, then we have 𝑝𝑇𝑥𝑛+1,𝑇𝑥𝑛𝑝𝜓𝑇𝑥𝑛,𝑇𝑥𝑛+1<𝑝𝑇𝑥𝑛,𝑇𝑥𝑛+1,(2.4) which is a contradiction. So, we conclude that the maximum is 𝑝(𝑇𝑥𝑛,𝑇𝑥𝑛1). Now, we have the following: 𝑝𝑇𝑥𝑛+1,𝑇𝑥𝑛𝑝𝜓𝑇𝑥𝑛,𝑇𝑥𝑛1<𝑝𝑇𝑥𝑛,𝑇𝑥𝑛1.(2.5) It follows from (2.5) that the sequence 𝑝(𝑇𝑥𝑛+1,𝑇𝑥𝑛) is monotone decreasing. According to the properties of function 𝜓, it follows 𝑝𝑇𝑥𝑛+1,𝑇𝑥𝑛𝑝𝜓𝑇𝑥𝑛,𝑇𝑥𝑛1𝜓2𝑝𝑇𝑥𝑛1,𝑇𝑥𝑛2𝜓𝑛𝑝𝑇𝑥1,𝑇𝑥00,(2.6) when 𝑛+. Therefore, lim𝑛+𝑝(𝑇𝑥𝑛+1,𝑇𝑥𝑛)=0.

Next, we prove that {𝑇𝑥𝑛} is a 0-Cauchy sequence in the space (𝑋,𝑝). It is sufficient to show that {𝑇𝑥2𝑛} 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 𝑝𝑦2𝑚𝑘,𝑦2𝑛𝑘𝑦,𝑝2𝑚𝑘,𝑦2𝑛𝑘+1𝑦,𝑝2𝑚𝑘1,𝑦2𝑛𝑘𝑦,𝑝2𝑚𝑘1,𝑦2𝑛𝑘+1,(2.7) all tend to 𝜀+ when 𝑘+. Applying condition (2.1) to elements 𝑥=𝑥2𝑚(𝑘) and 𝑦=𝑥2𝑛(𝑘)+1 and putting 𝑦𝑛=𝑇𝑥𝑛=𝑓𝑥𝑛+1 for each 𝑛0, we get that 𝑝𝑇𝑥2𝑚(𝑘),𝑇𝑥2𝑛(𝑘)+1𝑦=𝑝2𝑚(𝑘),𝑦2𝑛(𝑘)+1𝑝𝜓max𝑓𝑥2𝑚(𝑘),𝑓𝑥2𝑛(𝑘)+1,𝑝𝑓𝑥2𝑚(𝑘),𝑇𝑥2𝑚(𝑘),𝑝𝑓𝑥2𝑛(𝑘)+1,𝑇𝑥2𝑛(𝑘)+1,𝑝𝑓𝑥2𝑚(𝑘),𝑇𝑥2𝑛(𝑘)+1+𝑝𝑓𝑥2𝑛(𝑘)+1,𝑇𝑥2𝑚(𝑘)2𝑝𝑦=𝜓max2𝑚(𝑘)1,𝑦2𝑛(𝑘)𝑦,𝑝2𝑚(𝑘)1,𝑦2𝑚(𝑘)𝑦,𝑝2𝑛(𝑘),𝑦2𝑛(𝑘)+1,𝑝𝑦2𝑚(𝑘)1,𝑦2𝑛(𝑘)+1𝑦+𝑝2𝑛(𝑘),𝑦2𝑚(𝑘)2.(2.8)

When 𝑘+, we have 𝑝𝑦max2𝑚(𝑘)1,𝑦2𝑛(𝑘)𝑦,𝑝2𝑚(𝑘)1,𝑦2𝑚(𝑘)𝑦,𝑝2𝑛(𝑘),𝑦2𝑛(𝑘)+1,𝑝𝑦2𝑚(𝑘)1,𝑦2𝑛(𝑘)+1𝑦+𝑝2𝑛(𝑘),𝑦2𝑚(𝑘)2𝜀max+1,0,0,2𝜀++𝜀+=𝜀+.(2.9) Using properties of 𝜓, we obtain a contradiction 𝜀𝜓(𝜀)<𝜀, since 𝜀>0.

This shows that {𝑇𝑥2𝑛} is a 0-Cauchy sequence in the space (𝑋,𝑝) and hence {𝑇𝑥𝑛} is a 0-Cauchy sequence in (𝑋,𝑝). If we suppose that 𝑇𝑋 is a 0-complete subspace of (𝑋,𝑝), then there exists 𝑦𝑇𝑋𝑓𝑋 such that 𝑝(𝑦,𝑦)=lim𝑛+𝑝𝑇𝑥𝑛,𝑦=lim𝑛+𝑝𝑓𝑥𝑛,𝑦=lim𝑛,𝑚+𝑝𝑇𝑥𝑛,𝑇𝑥𝑚=0.(2.10) If 𝑓𝑋 is a 0-complete subspace of (𝑋,𝑝) with 𝑦𝑓𝑋, (2.10) also holds.

Let 𝑢𝑋 and 𝑦=𝑓𝑢. We show that 𝑦 is a point of coincidence of 𝑇 and 𝑓. Suppose that 𝑝(𝑇𝑢,𝑓𝑢)>0. We have 𝑝𝑇𝑥𝑛𝑝,𝑇𝑢𝜓max𝑓𝑥𝑛,𝑓𝑢,𝑝𝑓𝑥𝑛,𝑇𝑥𝑛𝑝,𝑝(𝑓𝑢,𝑇𝑢),𝑓𝑥𝑛,𝑇𝑢+𝑝𝑓𝑢,𝑇𝑥𝑛2.(2.11)

As 𝑛+, we have 𝑝(𝑓𝑥𝑛,𝑓𝑢)0,𝑝(𝑓𝑥𝑛,𝑇𝑥𝑛)0 and by Remark 1.3 (2) (𝑝(𝑓𝑥𝑛,𝑇𝑢)+𝑝(𝑓𝑢,𝑇𝑥𝑛))/2𝑝(𝑓𝑢,𝑇𝑢)/2. Since 𝑝(𝑓𝑢,𝑇𝑢)>0, there exists 𝑛0 such that for every 𝑛>𝑛0 we have 𝑝(𝑓𝑥𝑛,𝑓𝑢)<(1/2)𝑝(𝑓𝑢,𝑇𝑢) and 𝑝(𝑓𝑥𝑛,𝑇𝑥𝑛)<(1/2)𝑝(𝑓𝑢,𝑇𝑢). Now, for 𝑛>𝑛0, we obtain 𝑝(𝑓𝑢,𝑇𝑢)𝑝𝑓𝑢,𝑓𝑥𝑛+1+𝑝𝑇𝑥𝑛,𝑇𝑢𝑝𝑓𝑢,𝑓𝑥𝑛+1+𝜓(𝑝(𝑓𝑢,𝑇𝑢))(2.12) or 𝑝(𝑓𝑢,𝑇𝑢)𝑝𝑓𝑢,𝑓𝑥𝑛+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 {𝑇𝑥𝑛} 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 𝑇𝑋𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)𝜓max𝑝(𝑓𝑥,𝑓𝑦),𝑝(𝑓𝑥,𝑇𝑥),𝑝(𝑓𝑦,𝑇𝑦),𝑝(𝑓𝑥,𝑇𝑦)2(2.15) for all 𝑥,𝑦𝑋, where 𝜓Φ. 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 {𝑥𝑛}𝑋 defined by 𝑥0𝑋 and 𝑇𝑥𝑛=𝑓𝑥𝑛+1, we used Theorem 2.1 to show lim𝑛+𝑝(𝑇𝑥𝑛+1,𝑇𝑥𝑛)=0.
Here, we only prove that {𝑇𝑥𝑛} is a 0-Cauchy sequence in the space (𝑋,𝑝) by using induction. Let us denote 𝑦𝑛=𝑇𝑥𝑛=𝑓𝑥𝑛+1 for each 𝑛0, We have 𝑝(𝑦𝑛,𝑦𝑛+1)0, as 𝑛+. For 𝜀>0 there exists 𝑛(𝜀) such that for every 𝑛>𝑛(𝜀) we have 𝑝(𝑦𝑛,𝑦𝑛+1)<𝜀𝜓(𝜀). Assuming that for some 𝑛>𝑛(𝜀) and 𝑘 it holds that 𝑝(𝑦𝑛,𝑦𝑛+𝑘)<𝜀 and we need to prove that 𝑝(𝑦𝑛,𝑦𝑛+𝑘+1)<𝜀.
We have 𝑝𝑦𝑛,𝑦𝑛+𝑘+1𝑦𝑝𝑛,𝑦𝑛+1𝑦+𝑝𝑛+1,𝑦𝑛+𝑘+1,𝑝𝑦𝑛+1,𝑦𝑛+𝑘+1=𝑝𝑇𝑥𝑛+1,𝑇𝑥𝑛+𝑘+1𝑝𝜓max𝑓𝑥𝑛+1,𝑓𝑥𝑛+𝑘+1,𝑝𝑓𝑥𝑛+1,𝑇𝑥𝑛+1,𝑝𝑓𝑥𝑛+𝑘+1,𝑇𝑥𝑛+𝑘+1,𝑝𝑓𝑥𝑛+1,𝑇𝑥𝑛+𝑘+12𝑝𝑦=𝜓max𝑛,𝑦𝑛+𝑘𝑦,𝑝𝑛,𝑦𝑛+1𝑦,𝑝𝑛+𝑘,𝑦𝑛+𝑘+1,𝑝𝑦𝑛,𝑦𝑛+𝑘+12𝑝𝑦𝜓max𝑛,𝑦𝑛+𝑘𝑦,𝑝𝑛,𝑦𝑛+1𝑦,𝑝𝑛+𝑘,𝑦𝑛+𝑘+1,𝑝𝑦𝑛,𝑦𝑛+𝑘𝑦+𝑝𝑛+𝑘,𝑦𝑛+𝑘+12𝜓max𝜀,𝜀𝜓(𝜀),𝜀𝜓(𝜀),𝜀+𝜀𝜓(𝜀)2=𝜓(𝜀).(2.16)
Then (2.16) becomes 𝑝(𝑦𝑛,𝑦𝑛+𝑘+1)𝑝(𝑦𝑛,𝑦𝑛+1)+𝑝(𝑦𝑛+1,𝑦𝑛+𝑘+1)<𝜀𝜓(𝜀)+𝜓(𝜀)=𝜀. This shows that {𝑇𝑥𝑛} 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 𝑇𝑋𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)𝜓max𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥),𝑝(𝑦,𝑇𝑦),𝑝(𝑥,𝑇𝑦)+𝑝(𝑦,𝑇𝑥)2(2.17) for all 𝑥,𝑦𝑋, where 𝜓Ψ. 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=𝑖𝑋 (the identity map).

Corollary 2.5. Let (𝑋,𝑝) be a partial metric space and let 𝑇𝑋𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)𝜓max𝑝(𝑥,𝑦),𝑝(𝑥,𝑇𝑥),𝑝(𝑦,𝑇𝑦),𝑝(𝑥,𝑇𝑦)2(2.18) for all 𝑥,𝑦𝑋, where 𝜓Φ. 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=𝑖𝑋 (the identity map).

Corollary 2.6. Let (𝑋,𝑝) be a partial metric space and let 𝑇,𝑓 be mappings on 𝑋 with 𝑇𝑋𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)𝜆max𝑝(𝑓𝑥,𝑓𝑦),𝑝(𝑓𝑥,𝑇𝑥),𝑝(𝑓𝑦,𝑇𝑦),𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥)2(2.19) for all 𝑥,𝑦𝑋, 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 𝑇𝑋𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)𝑎1𝑝(𝑓𝑥,𝑓𝑦)+𝑎2𝑝(𝑓𝑥,𝑇𝑥)+𝑎3𝑝(𝑓𝑦,𝑇𝑦)+𝑎4(𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥))(2.20) for all 𝑥,𝑦𝑋, where 𝑎1,𝑎2,𝑎3,𝑎40 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(𝑓𝑥,𝑓𝑦)+𝑎2𝑝(𝑓𝑥,𝑇𝑥)+𝑎3𝑝(𝑓𝑦,𝑇𝑦)+𝑎4𝑎(𝑝(𝑓𝑥,𝑇𝑦)+𝑝(𝑓𝑦,𝑇𝑥))1+𝑎2+𝑎3+2𝑎4max𝑝(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 𝑇𝑋𝑓𝑋. Assume that 𝑝(𝑇𝑥,𝑇𝑦)𝜆𝑝(𝑓𝑥,𝑓𝑦),(2.22) for all 𝑥,𝑦𝑋, 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 𝑇𝑋𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)𝜓(𝑝(𝑥,𝑦))(2.23) for all 𝑥,𝑦𝑋, where 𝜓Φ. 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=𝑖𝑋 (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 𝑇𝑋𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)𝜆𝑝(𝑥,𝑦)(2.24) for all 𝑥,𝑦𝑋, 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 𝑇𝑋𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)𝜆(𝑝(𝑥,𝑇𝑥)+𝑝(𝑦,𝑇𝑦))(2.25) for all 𝑥,𝑦𝑋, 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 𝑇𝑋𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)𝑎1𝑝(𝑥,𝑦)+𝑎2𝑝(𝑥,𝑇𝑥)+𝑎3𝑝(𝑦,𝑇𝑦)(2.26) for all 𝑥,𝑦𝑋, 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 𝑇𝑋𝑋 be a map such that 𝑝(𝑇𝑥,𝑇𝑦)𝜆(𝑝(𝑥,𝑇𝑦)+𝑝(𝑦,𝑇𝑥))(2.27) for all 𝑥,𝑦𝑋, 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 𝑇𝑋𝑋 be a map such that at least one of the following is true 𝑝(𝑇𝑥,𝑇𝑦)𝑎1𝑝𝑝(𝑥,𝑇𝑥)(𝑇𝑥,𝑇𝑦)𝑎2(𝑝(𝑥,𝑇𝑥)+𝑝(𝑦,𝑇𝑦))𝑝(𝑇𝑥,𝑇𝑦)𝑎3(𝑝(𝑥,𝑇𝑦)+𝑝(𝑦,𝑇𝑥))(2.28) for all 𝑥,𝑦𝑋, 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)=([0,1),𝑝), where 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,𝑓𝑋𝑋 are such that 𝑇𝑥=𝑥2/(1+𝑥),𝑓𝑥=𝑥 for all 𝑥𝑋 and 𝑡𝜓(𝑡)=21,𝑡>5151𝑡,𝑡5.(2.29) Then 𝜓Ψ. 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 𝑥𝐿=max2,𝑦1+𝑥2=𝑦1+𝑦2,𝑥1+𝑦(2.30)𝑅=𝜓max𝑝(𝑥,𝑦),𝑝𝑥,2𝑦1+𝑥,𝑝𝑦,2,𝑝1+𝑦𝑥,𝑦2/(1+𝑦)+𝑝𝑦,𝑥2/(1+𝑥)2=𝜓max𝑦,𝑥,𝑦,max𝑥,𝑦2/(1+𝑦)+max𝑦,𝑥2/(1+𝑥)2𝜓(𝑦)(2.31) and so 𝑦2/(1+𝑦)𝜓(𝑦). If 𝑦1/5, then we have 𝑦2/(1+𝑦)𝑦/5 (this is true because 𝑦2/(1+𝑦)𝑦2𝑦/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)=|𝑥𝑦| and take 𝑥=0.10,𝑦=0.30. Then, from (2.1) follows 𝐿=0.0090.069=0.06 and 𝑅=𝜓max0.100.30,0.100.009,0.300.069,0.100.069+0.300.0092=𝜓(0.231)=0.053.(2.32) Hence 𝐿𝑅 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.

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