International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 657868 | 6 pages | https://doi.org/10.5402/2011/657868

Fixed Point for Partial Metric Spaces

Academic Editor: X. Wang
Received20 Mar 2011
Accepted01 May 2011
Published06 Jul 2011

Abstract

We consider the partial metric on a set 𝑋, define πœ–-fixed point for maps, and obtain some sufficient and necessary conditions on that, also we obtain some sufficient and necessary theorems on πœ–-fixed point.

1. Introduction

The partial metric spaces were introduced in [1] as a part of the study of denotational semantics of dataflow networks. In particular, he established the precise rela- tionship between partial metric spaces and the so-called weightable quasimetric spaces and proved a partial metric generalization of Banach contraction mapping theorem.

A partial metric [1] on a set 𝑋 is a function π‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞) such that, for all π‘₯,𝑦,π‘§βˆˆπ‘‹(1)π‘₯=𝑦⇔𝑝(π‘₯,π‘₯)=𝑝(π‘₯,𝑦)=𝑝(𝑦,𝑦); (2)𝑝(π‘₯,π‘₯)≀𝑝(π‘₯,𝑦); (3)𝑝(π‘₯,𝑦)≀𝑝(𝑦,π‘₯); (4)𝑝(π‘₯,𝑧)≀𝑝(π‘₯,𝑦)+𝑝(𝑦,𝑧)βˆ’π‘(𝑦,𝑦).

A partial metric space is a pair (𝑋,𝑝), where 𝑝 is a partial metric on 𝑋. Each partial metric 𝑝 on 𝑋 induces a 𝑇0 topology πœπ‘ on 𝑋 which has as a base the family of open balls {𝐡𝑝(π‘₯,πœ–)π‘₯βˆˆπ‘‹,πœ–>0}, where 𝐡𝑝(π‘₯,πœ–)={π‘¦βˆˆπ‘‹βˆΆπ‘(π‘₯,y)<𝑝(π‘₯,π‘₯)+πœ–} for all π‘₯βˆˆπ‘‹ and πœ–>0.

If 𝑝 is a partial metric on 𝑋, then the function π‘π‘ βˆΆπ‘‹Γ—π‘‹β†’[0,∞) given by 𝑝𝑠(π‘₯,𝑦)=2𝑝(π‘₯,𝑦)βˆ’π‘(π‘₯,π‘₯)βˆ’π‘(𝑦,𝑦) is a metric on 𝑋.

A sequence{π‘₯𝑛}π‘›βˆˆπ‘ in a partial metric space (𝑋,𝑝) is called a Cauchy sequence if there exists (and is finite) lim𝑛,π‘šπ‘(π‘₯𝑛,π‘₯π‘š) [1, Definition 5.2].

Note that {π‘₯𝑛}π‘›βˆˆπ‘ is a Cauchy sequence in (𝑋,𝑝) if and only if it is a Cauchy sequence in the metric space (𝑋,𝑝𝑠) [1, page 194].

A partial metric space (𝑋,𝑝) is said to be complete if every Cauchy sequence {π‘₯𝑛}π‘›βˆˆπ‘ in 𝑋 converges, with respect to πœπ‘, to a point π‘₯βˆˆπ‘‹ such that 𝑝(π‘₯,π‘₯)=lim𝑛,π‘šπ‘(π‘₯𝑛,π‘₯π‘š) [1, Definition 5.3].

It is well known and easy to see that a partial metric space (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑝𝑠) is complete.

In [2], S. J. O’Neill proposed one significant change to Matthews definition of the partial metrics, and that was to extend their range from 𝑅+ to 𝑅. In the following, partial metrics in the O’Neill sense will be called dualistic partial metrics and a pair (𝑋,𝑝) such that 𝑋 is a nonempty set and 𝑝 is a dualistic partial metric on 𝑋 will be called a dualistic partial metric space.

2. 𝝐-Fixed Point

Our basic references for quasi-metric spaces are [3, 4]. In our context, by a quasi-metric on a set 𝑋 we mean a nonnegative real-valued function d on 𝑋×𝑋 such that, for all π‘₯,𝑦,π‘§βˆˆπ‘‹:(i)𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯)=0⇔π‘₯=𝑦, (ii)𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑧)+𝑑(𝑧,𝑦).

A quasi-metric space is a pair (𝑋,𝑑) such that 𝑋 is a (nonempty) set and d is a quasi-metric on 𝑋.

Each quasi-metric 𝑑 on 𝑋 generates a 𝑇0-topology 𝑇(𝑑) on 𝑋 which has as a base the family of open 𝑑-balls 𝐡𝑑(π‘₯,πœ–)={π‘₯βˆˆπ‘‹βˆΆπœ–>0}, where 𝐡𝑑(π‘₯,πœ–)={π‘¦βˆˆπ‘‹βˆΆπ‘‘(π‘₯,𝑦)<πœ–>0} for all π‘₯βˆˆπ‘‹ and πœ–>0.

If 𝑑 is a quasi-metric on 𝑋, then the function 𝑑𝑠 defined on 𝑋×𝑋 by 𝑑𝑠(π‘₯,𝑦)=max{𝑑(π‘₯,𝑦),𝑑(𝑦,π‘₯)} is a metric on 𝑋.

Lemma 2.1. If (𝑋,𝑝) is a dualistic partial metric space, then the function π‘‘π‘βˆΆπ‘‹Γ—π‘‹β†’π‘…+ defined by 𝑑𝑝(π‘₯,𝑦)=𝑝(π‘₯,𝑦)βˆ’π‘(π‘₯,π‘₯), is a quasi-metric on 𝑋 such that 𝜏(𝑝)=𝜏(𝑑𝑝).

Definition 2.2. Let (𝑋,𝑝)be a dualistic partial metric space and let π‘‡βˆΆπ‘‹β†’,𝑋be a map. Then π‘₯0βˆˆπ‘‹ is πœ–-fixed point for 𝑇 if 𝑑𝑝𝑇π‘₯0,π‘₯0ξ€Έβ‰€πœ–.(2.1)

Theorem 2.3. Let (𝑋,𝑝) be a dualistic partial metric space and let π‘‡βˆΆπ‘‹β†’π‘‹ be a map, π‘₯0βˆˆπ‘‹, and πœ–>0. If 𝑑𝑝(𝑇𝑛(π‘₯0),𝑇𝑛+π‘˜(π‘₯0))β†’0 as π‘›β†’βˆž for some π‘˜>0, then π‘‡π‘˜ has an πœ–-fixed point.

Proof. Since 𝑑𝑝(𝑇𝑛(π‘₯0),𝑇𝑛+π‘˜(π‘₯0))β†’0 as π‘›β†’βˆž, πœ–>0, βˆƒπ‘›0>0s.t.βˆ€π‘›β‰₯𝑛0𝑑𝑝𝑇𝑛π‘₯0ξ€Έ,𝑇𝑛+π‘˜ξ€·π‘₯0ξ€Έξ€Έ<πœ–.(2.2) Then 𝑑𝑝(𝑇𝑛0(π‘₯0),π‘‡π‘˜(𝑇𝑛0(π‘₯0)))<πœ–. Therefore 𝑇𝑛0(π‘₯0) is an πœ–-fixed point of π‘‡π‘˜.

Theorem 2.4. Let (𝑋,𝑝) be a dualistic partial metric space and let π‘‡βˆΆπ‘‹β†’π‘‹ be a map also for all π‘₯,π‘¦βˆˆπ‘‹, 𝑑𝑝(𝑇π‘₯,𝑇𝑦)≀𝑐𝑑𝑝(π‘₯,𝑦)0<𝑐<1,(2.3) then 𝑇 has an πœ–-fixed point in partial metric. Moreover, if π‘₯,π‘¦βˆˆπ‘‹ are πœ–-fixed points of 𝑇, then 𝑑𝑝(π‘₯,𝑦)≀2πœ–/(1βˆ’π‘).

Proof. Suppose π‘₯βˆˆπ‘‹, then 𝑑𝑝𝑇𝑛(π‘₯),𝑇𝑛+1ξ€Έ(π‘₯)=π‘‘π‘ξ€·π‘‡ξ€·π‘‡π‘›βˆ’1ξ€Έ(π‘₯),𝑇(𝑇𝑛(π‘₯))β‰€π‘π‘‘π‘ξ€·π‘‡π‘›βˆ’1(π‘₯),𝑇𝑛(π‘₯)β‰€β‹―β‰€π‘π‘›βˆ’1𝑑𝑝𝑇(π‘₯),𝑇2(ξ€Έπ‘₯)≀𝑐𝑛𝑑𝑝(π‘₯,𝑇π‘₯).(2.4) Therefore 𝑑𝑝(𝑇𝑛(π‘₯),𝑇𝑛+1(π‘₯))β†’0 as π‘›β†’βˆž. From Theorem 2.3, 𝑇 has an πœ–-fixed point and Since 𝑑𝑝(π‘₯,𝑦)≀𝑑𝑝(π‘₯,𝑇π‘₯)+𝑑𝑝(𝑇π‘₯,𝑇𝑦)+𝑑𝑝(𝑦,𝑇𝑦)≀2πœ–+𝑐𝑑𝑝(π‘₯,𝑦).(2.5) Then 𝑑𝑝(π‘₯,𝑦)≀2πœ–/(1βˆ’π‘).

Theorem 2.5. Let 𝑇 be a mapping of a dualistic partial metric space (𝑋,𝑝) into itself such that ||||||||𝑝(𝑇π‘₯,𝑇𝑦)≀𝑐𝑝(π‘₯,𝑦)0<𝑐<1(2.6) for all π‘₯,π‘¦βˆˆπ‘‹. Then π‘‡π‘˜ has an πœ–-fixed point, for all π‘˜.

Proof. Fix π‘₯βˆˆπ‘‹, then it is clear that, for each π‘₯βˆˆπ‘, ||𝑝(𝑇𝑛π‘₯,𝑇𝑛||π‘₯)≀𝑐𝑛||||,||𝑝𝑇𝑝(π‘₯,π‘₯)𝑛π‘₯,𝑇𝑛+1π‘₯ξ€Έ||≀𝑐𝑛||||,𝑝(π‘₯,𝑇π‘₯)(2.7) also 𝑑𝑝𝑇𝑛π‘₯,𝑇𝑛+1π‘₯ξ€Έ+𝑝(𝑇𝑛π‘₯,𝑇𝑛𝑇π‘₯)=𝑝𝑛π‘₯,𝑇𝑛+1π‘₯ξ€Έ.(2.8) We deduce that 𝑑𝑝𝑇𝑛π‘₯,𝑇𝑛+1π‘₯ξ€Έβˆ£+𝑝(𝑇𝑛π‘₯,𝑇𝑛π‘₯)≀𝑐𝑛||||𝑝(π‘₯,𝑇π‘₯).(2.9) Hence 𝑑𝑝𝑇𝑛(π‘₯),𝑇𝑛+1ξ€Έ(π‘₯)=𝑐𝑛||||βˆ’||𝑝(π‘₯,𝑇π‘₯)𝑝(𝑇𝑛π‘₯,𝑇𝑛||π‘₯)≀𝑐𝑛||||+||𝑝(π‘₯,𝑇π‘₯)𝑝(𝑇𝑛π‘₯,𝑇𝑛||π‘₯)≀𝑐𝑛||||βˆ’||||ξ€Έ.𝑝(π‘₯,𝑇π‘₯)𝑝(π‘₯,π‘₯)(2.10)
Therefore, for π‘˜,π‘›βˆˆπ‘, 𝑑𝑝𝑇𝑛(π‘₯),𝑇𝑛+π‘˜ξ€Έ(π‘₯)≀𝑑𝑝𝑇𝑛(π‘₯),𝑇𝑛+1ξ€Έ(π‘₯)+β‹―+𝑑𝑝𝑇𝑛+π‘˜βˆ’1(π‘₯),𝑇𝑛+π‘˜ξ€Έβ‰€ξ€·π‘(π‘₯)𝑛+β‹―+𝑐𝑛+π‘˜βˆ’1||||+||||≀𝑐𝑝(π‘₯,𝑇π‘₯)𝑝(π‘₯,π‘₯)𝑛||||+||||ξ€Έ.1βˆ’π‘π‘(π‘₯,𝑇π‘₯)𝑝(π‘₯,π‘₯)(2.11)
Similarly, we obtain that 𝑑𝑝𝑇𝑛+π‘˜(π‘₯),𝑇𝑛≀𝑐(π‘₯)𝑛||||+||||ξ€Έ.1βˆ’π‘π‘(π‘₯,𝑇π‘₯)𝑝(π‘₯,π‘₯)(2.12) Then limπ‘›β†’βˆžπ‘‘π‘ξ€·π‘‡π‘›+π‘˜(π‘₯),𝑇𝑛(π‘₯)=0.(2.13) Therefore, π‘‡π‘˜ has an πœ–-fixed point.

Example 2.6. Let 𝑋=(βˆ’βˆž,βˆ’2], and let p be the dualistic metric on 𝑋 given by 𝑝(π‘₯,𝑦)=π‘₯βˆ¨π‘¦,(2.14) for all π‘₯,π‘¦βˆˆπ‘‹.

Let 𝑇 be the mapping from 𝑋 into itself defined by 𝑇(π‘₯)=π‘₯+1, for all 𝑋=(βˆ’βˆž,βˆ’2]. It is immediate to see that 1𝑝(𝑇(π‘₯),𝑇(𝑦))≀2𝑝(π‘₯,𝑦),(2.15) for all π‘₯β‹…π‘¦βˆˆπ‘‹. However, 𝑇 has no fixed point, of course. But by the Theorem 2.5, for every πœ–>0, 𝑇 has an πœ–-fixed point. That is, there exists π‘₯π‘œβˆˆπ‘‹ such that 𝑑𝑝𝑇π‘₯0ξ€Έ,π‘₯0ξ€Έβ‰€πœ–,πœ–>0,(2.16) since 𝑑𝑝𝑇π‘₯0ξ€Έ,π‘₯0𝑇π‘₯=𝑝0ξ€Έ,π‘₯0𝑇π‘₯βˆ’π‘0ξ€Έ,𝑇π‘₯0ξ€Έξ€·π‘₯=𝑝0+1,π‘₯0ξ€Έξ€·π‘₯βˆ’π‘0+1,π‘₯0ξ€Έ+1=π‘₯0ξ€·π‘₯+1βˆ’0ξ€Έ+1=0β‰€πœ–.(2.17)

Theorem 2.7. Let 𝑇 be a mapping of a dualistic partial metric space (𝑋,𝑝) into itself such that 𝑑𝑝(𝑇π‘₯,𝑇𝑦)βˆ£β‰€π›½(𝑑𝑝(π‘₯,𝑇π‘₯)+𝑑𝑝(𝑦,𝑇𝑦)) where 2𝛽<1.
If π‘₯0 is an πœ–-fixed point for 𝑇, then 𝑇π‘₯0 is an πœ–-fixed point for 𝑇2.

Proof. We have 𝑑𝑝𝑇π‘₯,𝑇2π‘₯𝑑≀𝛽𝑝(π‘₯,𝑇π‘₯)+𝑑𝑝𝑇π‘₯,𝑇2π‘₯,ξ€Έξ€Έ(2.18) therefore 𝑑𝑝𝑇π‘₯,𝑇2π‘₯≀𝛽𝑑1βˆ’π›½π‘(π‘₯,𝑇π‘₯).(2.19) Since 2𝛽<1, 𝑑𝑝𝑇π‘₯,𝑇2π‘₯≀𝑑𝑝(π‘₯,𝑇π‘₯).(2.20) Since π‘₯0 is an πœ–-fixed point for 𝑇, then 𝑑𝑝(𝑇π‘₯0,𝑇2π‘₯0)β‰€πœ–. So 𝑇π‘₯0 is an πœ–-fixed point for 𝑇2.

Theorem 2.8. Let (𝑋,𝑝) be a dualistic partial metric, and let π‘‡βˆΆπ‘‹β†’π‘‹ be a mapping and πœ–>0. If 𝑑𝑝(𝑇π‘₯,𝑇𝑦)≀𝛼𝑑𝑝(π‘₯,𝑇π‘₯)+𝛽𝑑𝑝(𝑦,𝑇𝑦) and 𝛼+𝛽<1. then 𝑇 has πœ–-fixed point. Moreover, if π‘₯,π‘¦βˆˆπ‘‹ are πœ–-fixed points of 𝑇, then 𝑑𝑝(π‘₯,𝑦)≀(2+𝛼+𝛽)πœ–.

Proof. We have 𝑑𝑝𝑇π‘₯,𝑇2π‘₯≀𝛼𝑑𝑝(π‘₯,𝑇π‘₯)+𝛽𝑑𝑝𝑇π‘₯,𝑇2π‘₯ξ€Έ.(2.21) Therefore 𝑑𝑝𝑇π‘₯,𝑇2π‘₯≀𝛼𝑑1βˆ’π›½π‘(π‘₯,𝑇π‘₯),(2.22) also 𝑑𝑝𝑇2π‘₯,𝑇3π‘₯≀𝛼𝑑𝑝𝑇π‘₯,𝑇2π‘₯ξ€Έ+𝛽𝑑𝑝𝑇2π‘₯,𝑇3π‘₯ξ€Έ,(2.23) so 𝑑𝑝𝑇2π‘₯,𝑇3π‘₯≀𝛼1βˆ’π›½2𝑑𝑝𝑇π‘₯,𝑇2π‘₯ξ€Έ(2.24) and for every 𝑛β‰₯1, we have 𝑑𝑝𝑇𝑛π‘₯,𝑇𝑛+1π‘₯≀𝛼1βˆ’π›½π‘›π‘‘π‘π›Ό(π‘₯,𝑇π‘₯),1βˆ’π›½<1.(2.25) Thus, since 𝛼/(1βˆ’π›½)<1,   𝑑𝑝(𝑇𝑛π‘₯,𝑇𝑛+1π‘₯)β†’0 as π‘›β†’βˆž. Now, by Theorem 2.3, 𝑇 has an πœ–-fixed point and since 𝑑𝑝(π‘₯,𝑦)≀𝑑𝑝(π‘₯,𝑇π‘₯)+𝑑𝑝(𝑇π‘₯,𝑇𝑦)+𝑑𝑝(𝑦,𝑇𝑦)≀(1+𝛼)𝑑𝑝(π‘₯,𝑇π‘₯)+(1+𝛽)𝑑𝑝(𝑦,𝑇𝑦).(2.26) Then 𝑑𝑝(π‘₯,𝑦)≀(2+𝛼+𝛽)πœ–.

Corollary 2.9. Let (𝑋,𝑝) be a dualistic partial metric, and let π‘‡βˆΆπ‘‹β†’π‘‹ be a mapping and πœ–>0. If 𝑑𝑝(𝑇π‘₯,𝑇𝑦)βˆ£β‰€π›½(𝑑𝑝(π‘₯,𝑇π‘₯)+𝑑𝑝(𝑦,𝑇𝑦)) and 2𝛽<1, then 𝑇 has an πœ–-fixed point.

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Copyright © 2011 S. A. M. Mohsenalhosseini 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.

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