Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 495892, 17 pages
http://dx.doi.org/10.1155/2012/495892
Research Article

On Weakly (𝐢,πœ“,πœ™)-Contractive Mappings in Ordered Partial Metric Spaces

1Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey
2Department of Mathematics, Hashemite University, Zarqa, Jordan

Received 12 May 2012; Revised 17 July 2012; Accepted 26 July 2012

Academic Editor: PaulΒ Eloe

Copyright Β© 2012 Erdal KarapΔ±nar and Wasfi Shatanawi. 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

We introduce the notion of weakly (𝐢,πœ“,πœ™)-contractive mappings in ordered partial metric spaces and prove some common fixed point theorems for such contractive mappings in the context of partially ordered partial metric spaces under certain conditions. We give some common fixed point results of integral type as an application of our main theorem. Also, we give an example and an application of integral equation to support the useability of our results.

1. Introduction and Preliminaries

In 1994, Matthews [1] introduced the notion of a partial metric space as a generalization of the usual metric space. In partial metric space self distance, that is. 𝑑(π‘₯,π‘₯) is not necessarily equal a zero. In this interesting paper, Matthews [1] prove the Banach contraction mapping principle in the frame of partial metric spaces. After this initial paper, many authors have studied various type contractions and related fixed point results in partial metric spaces (see, [2–32]).

Definition 1.1 (see [1]). A partial metric on a nonempty set 𝑋 is a function π‘βˆΆπ‘‹Γ—π‘‹β†’β„+ such that for all π‘₯,𝑦,π‘§βˆˆπ‘‹: (𝑝1)π‘₯=𝑦⇔𝑝(π‘₯,π‘₯)=𝑝(π‘₯,𝑦)=𝑝(𝑦,𝑦),  (𝑝2)𝑝(π‘₯,π‘₯)≀𝑝(π‘₯,𝑦),  (𝑝3)𝑝(π‘₯,𝑦)=𝑝(𝑦,π‘₯),  (𝑝4)𝑝(π‘₯,𝑦)≀𝑝(π‘₯,𝑧)+𝑝(𝑧,𝑦)βˆ’π‘(𝑧,𝑧).
A pair (𝑋,𝑝) is called a partial metric space where 𝑋 is a nonempty set and 𝑝 is a partial metric on 𝑋.
Each partial metric 𝑝 on 𝑋 generates a 𝑇0 topology πœπ‘ on 𝑋. The set {𝐡𝑝(π‘₯,πœ€)∢π‘₯βˆˆπ‘‹,β€‰β€‰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 forms the base of πœπ‘.
If 𝑝 is a partial metric on 𝑋, then the function π‘‘π‘βˆΆπ‘‹Γ—π‘‹β†’β„+ given by 𝑑𝑝(π‘₯,𝑦)=2𝑝(π‘₯,𝑦)βˆ’π‘(π‘₯,π‘₯)βˆ’π‘(𝑦,𝑦)(1.1) is a metric on 𝑋.

Definition 1.2 (see [1]). Let (𝑋,𝑝) be a partial metric space. Then one has the following. (1)A sequence {π‘₯𝑛} in a partial metric space (𝑋,𝑝) converges to a point π‘₯βˆˆπ‘‹ if and only if 𝑝(π‘₯,π‘₯)=limπ‘›β†’βˆžπ‘(π‘₯,π‘₯𝑛).(2)A sequence {π‘₯𝑛} in a partial metric space (𝑋,𝑝) is called a Cauchy sequence if there exists (and is finite) lim𝑛,π‘šβ†’βˆžπ‘(π‘₯𝑛,π‘₯π‘š).(3)A partial metric space (𝑋,𝑝) is said to be complete if every Cauchy sequence {π‘₯𝑛} in 𝑋 converges, with respect to πœπ‘, to a point π‘₯βˆˆπ‘‹ such that 𝑝(π‘₯,π‘₯)=lim𝑛,π‘šβ†’βˆžπ‘(π‘₯𝑛,π‘₯π‘š).
The following lemma is crucial in proving our main results.

Lemma 1.3 (see [1]). Let (𝑋,𝑝) be a partial metric space. (1){π‘₯𝑛} is a Cauchy sequence in (𝑋,𝑝) if and only if it is a Cauchy sequence in the metric space (𝑋,𝑑𝑝).(2)A partial metric space (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑑𝑝) is complete. Furthermore, limπ‘›β†’βˆžπ‘‘π‘(π‘₯𝑛,π‘₯)=0 if and only if 𝑝(π‘₯,π‘₯)=limπ‘›β†’βˆžπ‘ξ€·π‘₯𝑛,π‘₯=lim𝑛,π‘šβ†’βˆžπ‘ξ€·π‘₯𝑛,π‘₯π‘šξ€Έ.(1.2)

In 1972, Chatterjea [33] introduced the concept of 𝐢-contraction as follows.

Definition 1.4 (see [33]). Let (𝑋,𝑑) be a metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a mapping. Then 𝑇 is called a 𝐢-contraction if there exists π‘˜βˆˆ[0,1/2) such that 𝑑(𝑇π‘₯,𝑇𝑦)β‰€π‘˜(𝑑(π‘₯,𝑇𝑦)+𝑑(𝑇π‘₯,𝑦))(1.3) holds for all π‘₯,π‘¦βˆˆπ‘‹.
In this interesting paper, Chatterjea [33] proved the following theorem.

Theorem 1.5 (see [33]). Every 𝐢-contraction in a complete metric space has a unique fixed point.

Choudhury [34] introduced the concept of weakly 𝐢-contractive mapping as a generalization of 𝐢-contractive mapping and prove that every weakly 𝐢-contractive mapping in a complete metric space has a unique fixed point.

Definition 1.6 (see [34]). Let (𝑋,𝑑) be a metric space and π‘‡βˆΆπ‘‹β†’π‘‹ be a mapping. Then 𝑇 is called a weakly 𝐢-contractive if there exists a continuous function πœ™βˆΆ[0,+∞)Γ—[0,+∞)β†’[0,+∞) such that for all π‘₯,π‘¦βˆˆπ‘‹, we have 1𝑑(𝑇π‘₯,𝑇𝑦)≀2(𝑑(π‘₯,𝑇𝑦)+𝑑(𝑇π‘₯,𝑦))βˆ’πœ™(𝑑(π‘₯,𝑇𝑦),𝑑(𝑇π‘₯,𝑦)).(1.4) Harjani et al. [35] announced some fixed point results for weakly C-contractive mappings in a complete metric space endowed with a partial order. Meanwhile, Shatanawi [36] proved some fixed point and coupled fixed point theorems for a nonlinear weakly 𝐢-contraction type mapping in metric and ordered metric spaces.
In this paper, we introduce the concept of weakly (𝐢,πœ“,πœ™)-contractive mappings in ordered partial metric spaces, and we prove some existence theorems of common fixed point for such mapping in the context of complete partial metric spaces under certain conditions.

2. The Main Result

We start this section with the following definitions.

Definition 2.1. Suppose that (𝑋,𝑝) is a partial metric space. A mapping π‘‡βˆΆπ‘‹β†’π‘‹ is said to be continuous at π‘₯βˆˆπ‘‹ if for every πœ–>0, there exists 𝛿>0 such that 𝑇(𝐡𝑝(π‘₯,𝛿))βŠ†π΅π‘(𝑇π‘₯,πœ–). We say that 𝑇 is continuous if 𝑇 is continuous at all π‘₯βˆˆπ‘‹.
It is easy to see that if (𝑋,𝑝) is a partial metric space, π‘‡βˆΆπ‘‹β†’π‘‹ is continuous, (π‘₯𝑛) is a sequence in 𝑋, π‘₯βˆˆπ‘‹ and lim𝑛→+βˆžπ‘ξ€·π‘₯𝑛,π‘₯=𝑝(π‘₯,π‘₯),thenlim𝑛→+βˆžπ‘ξ€·π‘‡π‘₯𝑛,𝑇π‘₯=𝑝(𝑇π‘₯,𝑇π‘₯).(2.1)
Altun and Simsek [37] introduce the notion of weakly increasing of two mappings 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ in the following way.

Definition 2.2 (see [37]). Let (𝑋,βͺ―) be a partially ordered set. Two mappings 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ are said to be weakly increasing if 𝑇π‘₯βͺ―𝑆𝑇π‘₯ and 𝑆π‘₯βͺ―𝑇𝑆π‘₯ for all π‘₯βˆˆπ‘‹.

For more details on weakly increasing mappings, we refer the reader to [24, 38–41]. Let πœ™ denote all functions πœ™βˆΆ[0,∞)Γ—[0,+∞)β†’[0,∞) such that β€‰πœ™ is continuous, β€‰πœ™(𝑑,𝑠)=0 if and only if 𝑑=𝑠=0. Similarly, we denote by Ξ¨ all functions πœ“βˆΆ[0,+∞)β†’[0,+∞) such that β€‰πœ“ is continuous and nondecreasing, β€‰πœ“(𝑑)=0 if and only if 𝑑=0.

Inspired the definitions above, we introduce the following definition.

Definition 2.3. Let (𝑋,βͺ―,𝑝) be an partially ordered metric space. Then the mappings 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ are said to be weakly (𝐢,πœ“,πœ™)-contractive mappings if 𝑇 and 𝑆 are weakly increasing with respect to βͺ― and for any comparable π‘₯ and 𝑦, we have ξ‚€1πœ“(𝑝(𝑇π‘₯,𝑆𝑦))β‰€πœ“2(𝑝(𝑇π‘₯,𝑦)+𝑝(π‘₯,𝑆𝑦))βˆ’πœ™(𝑝(𝑇π‘₯,𝑦),𝑝(π‘₯,𝑆𝑦)),(2.2) where πœ™βˆˆΞ¦ and πœ“βˆˆΞ¨.

Now, we introduce and prove our first results.

Theorem 2.4. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists a partial metric 𝑝 on 𝑋 such that (𝑋,𝑝) is complete. Suppose that 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ are weakly (𝐢,πœ“,πœ™)-contractive mappings. If 𝑇 and 𝑆 are continuous, then 𝑇 and 𝑆 have a common fixed point; that is, there exists π‘’βˆˆπ‘‹ such that 𝑒=𝑇𝑒=𝑆𝑒.

Proof. Given π‘₯0βˆˆπ‘‹. Set 𝑇π‘₯0=π‘₯1 and 𝑆π‘₯1=π‘₯2. Continuing this process, we construct sequences (π‘₯𝑛) in 𝑋 such that π‘₯2𝑛+1=𝑇π‘₯2𝑛 and π‘₯2𝑛+2=𝑆π‘₯2𝑛+1. Using the fact that that 𝑆 and 𝑇 are weakly increasing with respect to βͺ―, we obtain that π‘₯1=𝑇π‘₯0ξ€·βͺ―𝑆𝑇π‘₯0ξ€Έ=𝑆π‘₯1=π‘₯2βͺ―β‹…βͺ―π‘₯2𝑛+1βͺ―π‘₯2𝑛+2β‹…(2.3) Now, we will prove that lim𝑛→+βˆžπ‘ξ€·π‘₯𝑛,π‘₯𝑛+1ξ€Έ=0.(2.4) Since π‘₯2𝑛+1 and π‘₯2𝑛+2 are comparable, by (2.2), we have πœ“ξ€·π‘ξ€·π‘₯2𝑛+1,π‘₯2𝑛+2𝑝=πœ“π‘‡π‘₯2𝑛,𝑆π‘₯2𝑛+1ξ‚€1ξ€Έξ€Έβ‰€πœ“2𝑝𝑇π‘₯2𝑛,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑝2𝑛,𝑆π‘₯2𝑛+1ξ‚ξ€·π‘ξ€·ξ€Έξ€Έβˆ’πœ™π‘‡π‘₯2𝑛,π‘₯2𝑛+1ξ€Έξ€·π‘₯,𝑝2𝑛,𝑆π‘₯2𝑛+1ξ‚€1ξ€Έξ€Έ=πœ“2𝑝π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑝2𝑛,π‘₯2𝑛+2𝑝π‘₯ξ€Έξ€Έβˆ’πœ™2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯,𝑝2𝑛,π‘₯2𝑛+2.ξ€Έξ€Έ(2.5) By (𝑝4), we have 𝑝π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑝2𝑛,π‘₯2𝑛+2ξ€Έξ€·π‘₯≀𝑝2𝑛,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑝2𝑛+1,π‘₯2𝑛+2ξ€Έ.(2.6) Thus (2.5) becomes πœ“ξ€·π‘ξ€·π‘₯2𝑛+1,π‘₯2𝑛+2ξ‚€1ξ€Έξ€Έβ‰€πœ“2𝑝π‘₯2𝑛,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑝2𝑛+1,π‘₯2𝑛+2𝑝π‘₯ξ€Έξ€Έβˆ’πœ™2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯,𝑝2𝑛,π‘₯2𝑛+2ξ‚€1ξ€Έξ€Έβ‰€πœ“2𝑝π‘₯2𝑛,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑝2𝑛+1,π‘₯2𝑛+2.ξ€Έξ€Έ(2.7) Using the fact that πœ“ is nondecreasing, we get that 𝑝π‘₯2𝑛+1,π‘₯2𝑛+2≀12𝑝π‘₯2𝑛,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑝2𝑛+1,π‘₯2𝑛+2.ξ€Έξ€Έ(2.8) Hence, we have 𝑝π‘₯2𝑛+1,π‘₯2𝑛+2ξ€Έξ€·π‘₯≀𝑝2𝑛,π‘₯2𝑛+1ξ€Έ.(2.9) Similarly, we may show that 𝑝π‘₯2𝑛+1,π‘₯2𝑛π‘₯≀𝑝2𝑛,π‘₯2π‘›βˆ’1ξ€Έ.(2.10)
From (2.9) and (2.10), we have 𝑝π‘₯𝑛+1,π‘₯𝑛π‘₯≀𝑝𝑛,π‘₯π‘›βˆ’1ξ€Έβˆ€π‘›βˆˆπ.(2.11) By (2.11), we get that {𝑝(π‘₯𝑛+1,π‘₯𝑛)βˆΆπ‘›βˆˆπ} is a non increasing sequence. Hence there is π‘Ÿβ‰₯0 such that lim𝑛→+βˆžπ‘ξ€·π‘₯𝑛,π‘₯𝑛+1ξ€Έ=π‘Ÿ.(2.12)
Letting 𝑛→+∞ in (2.7), we get πœ“(π‘Ÿ)β‰€πœ“(π‘Ÿ)βˆ’liminf𝑛→+βˆžπœ™ξ€·π‘ξ€·π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯,𝑝2𝑛,π‘₯2𝑛+2ξ€Έξ€Έβ‰€πœ“(π‘Ÿ).(2.13) Thus πœ“(π‘Ÿ)βˆ’liminf𝑛→+βˆžπœ™ξ€·π‘ξ€·π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯,𝑝2𝑛,π‘₯2𝑛+2ξ€Έξ€Έ=πœ“(π‘Ÿ),(2.14) and hence liminf𝑛→+βˆžπœ™ξ€·π‘ξ€·π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯,𝑝2𝑛,π‘₯2𝑛+2ξ€Έξ€Έ=0.(2.15) Using the continuity of πœ™, we conclude that liminf𝑛→+βˆžπ‘ξ€·π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έ=0,liminf𝑛→+βˆžπ‘ξ€·π‘₯2𝑛,π‘₯2𝑛+2ξ€Έ=0.(2.16)
Again, on taking limit sup in (2.5), we have πœ“(π‘Ÿ)=0 and hence π‘Ÿ=0. From the definition of 𝑑𝑝, we have lim𝑛→+βˆžπ‘‘π‘ξ€·π‘₯𝑛,π‘₯𝑛+1ξ€Έ=0.(2.17)
Next, we will show that (π‘₯𝑛) is a Cauchy sequence in the metric space (𝑋,𝑑𝑝). It is sufficient to show that (π‘₯2𝑛) is a Cauchy sequence in (𝑋,𝑑𝑝). Suppose to the contrary, that is, (π‘₯2𝑛) is not a Cauchy sequence in (𝑋,𝑑𝑝). Then there exists πœ–>0 for which we can find two subsequences of positive integers (π‘₯2π‘š(𝑖)) and (π‘₯2𝑛(𝑖)) such that 𝑛(𝑖) is the smallest index for which 𝑛(𝑖)>π‘š(𝑖)>𝑖,𝑑𝑝π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)ξ€Έβ‰₯πœ–.(2.18) This means that 𝑑𝑝π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)βˆ’2ξ€Έ<πœ–.(2.19) From (2.18), (2.19), and the triangular inequality, we get that πœ–β‰€π‘‘π‘ξ€·π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)≀𝑑𝑝π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)βˆ’2ξ€Έ+𝑑𝑝π‘₯2𝑛(𝑖)βˆ’2,π‘₯2𝑛(𝑖)βˆ’1ξ€Έ+𝑑𝑝π‘₯2𝑛(𝑖)βˆ’1,π‘₯2𝑛(𝑖)ξ€Έ<πœ–+𝑑𝑝π‘₯2𝑛(𝑖)βˆ’2,π‘₯2𝑛(𝑖)βˆ’1ξ€Έ+𝑑𝑝π‘₯2𝑛(𝑖)βˆ’1,π‘₯2𝑛(𝑖)ξ€Έ.(2.20) Letting 𝑖→+∞ in above inequalities and using (2.17), we have lim𝑖→+βˆžπ‘‘π‘ξ€·π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)ξ€Έ=πœ–.(2.21) Again, from (2.18) and the triangular inequality, we get that πœ–β‰€π‘‘π‘ξ€·π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)≀𝑑𝑝π‘₯2𝑛(𝑖),π‘₯2𝑛(𝑖)βˆ’1ξ€Έ+𝑑𝑝π‘₯2𝑛(𝑖)βˆ’1,π‘₯2π‘š(𝑖)≀𝑑𝑝π‘₯2𝑛(𝑖),π‘₯2𝑛(𝑖)βˆ’1ξ€Έ+𝑑𝑝π‘₯2𝑛(𝑖)βˆ’1,π‘₯2π‘š(𝑖)+1ξ€Έ+𝑑𝑝π‘₯2π‘š(𝑖)+1,π‘₯2π‘š(𝑖)≀𝑑𝑝π‘₯2𝑛(𝑖),π‘₯2𝑛(𝑖)βˆ’1ξ€Έ+𝑑𝑝π‘₯2𝑛(𝑖)βˆ’1,π‘₯2π‘š(𝑖)ξ€Έ+2𝑑𝑝π‘₯2π‘š(𝑖)+1,π‘₯2π‘š(𝑖)≀2𝑑𝑝π‘₯2𝑛(𝑖),π‘₯2𝑛(𝑖)βˆ’1ξ€Έ+𝑑𝑝π‘₯2𝑛(𝑖),π‘₯2π‘š(𝑖)ξ€Έ+2𝑑𝑝π‘₯2π‘š(𝑖)+1,π‘₯2π‘š(𝑖)ξ€Έ.(2.22)
Letting 𝑖→+∞ in above inequalities and using (2.4) and (2.21), we get that lim𝑖→+βˆžπ‘‘π‘ξ€·π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)ξ€Έ=lim𝑖→+βˆžπ‘‘π‘ξ€·π‘₯2π‘š(𝑖)+1,π‘₯2𝑛(𝑖)βˆ’1ξ€Έ=lim𝑖→+βˆžπ‘‘π‘ξ€·π‘₯2π‘š(𝑖)+1,π‘₯2𝑛(𝑖)ξ€Έ=lim𝑖→+βˆžπ‘‘π‘ξ€·π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)βˆ’1ξ€Έ=πœ–.(2.23) By the fact that 𝑑𝑝(π‘₯,𝑦)=2𝑝(π‘₯,𝑦)βˆ’π‘(π‘₯,π‘₯)βˆ’π‘(𝑦,𝑦),(2.24) for all π‘₯,π‘¦βˆˆπ‘‹, and the expression above, we conclude that lim𝑖→+βˆžπ‘ξ€·π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)ξ€Έ=lim𝑖→+βˆžπ‘ξ€·π‘₯2π‘š(𝑖)+1,π‘₯2𝑛(𝑖)βˆ’1ξ€Έ=lim𝑖→+βˆžπ‘ξ€·π‘₯2π‘š(𝑖)+1,π‘₯2𝑛(𝑖)ξ€Έ=lim𝑖→+βˆžπ‘ξ€·π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)βˆ’1ξ€Έ=lim𝑖→+βˆžπ‘‘π‘ξ€·π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)ξ€Έ=πœ–2.(2.25) By (2.2), we have πœ“ξ€·π‘ξ€·π‘₯2π‘š(𝑖)+1,π‘₯2𝑛(𝑖)𝑝=πœ“π‘‡π‘₯2π‘š(𝑖),𝑆π‘₯2𝑛(𝑖)βˆ’1ξ‚€1ξ€Έξ€Έβ‰€πœ“2𝑝𝑇π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)βˆ’1ξ€Έξ€·π‘₯+𝑝2π‘š(𝑖),𝑆π‘₯2𝑛(𝑖)βˆ’1ξ‚ξ€·π‘ξ€·ξ€Έξ€Έβˆ’πœ™π‘‡π‘₯2π‘š(𝑖),π‘₯2𝑛(𝑖)βˆ’1ξ€Έξ€·π‘₯,𝑝2π‘š(𝑖),𝑆π‘₯2𝑛(𝑖)βˆ’1ξ‚€1ξ€Έξ€Έβ‰€πœ“2𝑝π‘₯2π‘š(𝑖)+1,π‘₯2𝑛(𝑖)βˆ’1ξ€Έξ€·π‘₯+𝑝2π‘š(𝑖),π‘₯2𝑛(𝑖)𝑝π‘₯ξ€Έξ€Έβˆ’πœ™2π‘š(𝑖)+1,π‘₯2𝑛(𝑖)βˆ’1ξ€Έξ€·π‘₯,𝑝2π‘š(𝑖),π‘₯2𝑛(𝑖).ξ€Έξ€Έ(2.26)
Letting 𝑖→+∞ and using the continuity of πœ™ and πœ“, we get that πœ“ξ‚€πœ–2ξ‚ξ‚€πœ–β‰€πœ“2ξ‚ξ‚€πœ–βˆ’πœ™2,πœ–2.(2.27) Therefore, we get that πœ™(πœ–/2,πœ–/2)=0. Hence πœ–=0 which is a contradiction. Thus, {π‘₯𝑛} is a Cauchy sequence in (𝑋,𝑑𝑝). From Lemma 1.3, the sequence (π‘₯𝑛) converges in the metric space (𝑋,𝑑𝑝), say limπ‘›β†’βˆžπ‘‘π‘(π‘₯𝑛,𝑒)=0. Again from Lemma 1.3, we have 𝑝(𝑒,𝑒)=limπ‘›β†’βˆžπ‘ξ€·π‘₯𝑛,𝑒=lim𝑛,π‘šβ†’βˆžπ‘ξ€·π‘₯𝑛,π‘₯π‘šξ€Έ.(2.28) Moreover, since {π‘₯𝑛} is a Cauchy sequence in the metric space (𝑋,𝑑𝑝), we have lim𝑛,π‘šβ†’βˆžπ‘‘π‘ξ€·π‘₯𝑛,π‘₯π‘šξ€Έ=0.(2.29) From the definition of 𝑑𝑝, we have 𝑑𝑝π‘₯𝑛,π‘₯π‘šξ€Έξ€·π‘₯=2𝑝𝑛,π‘₯π‘šξ€Έξ€·π‘₯βˆ’π‘π‘›,π‘₯𝑛π‘₯βˆ’π‘π‘š,π‘₯π‘šξ€Έ.(2.30) Letting 𝑛,π‘šβ†’+∞ in the above equality and using (2.4) and (2.29), we get lim𝑛,π‘šβ†’βˆžπ‘ξ€·π‘₯𝑛,π‘₯π‘šξ€Έ=0.(2.31) Thus by (2.28), we have lim𝑛→+βˆžπ‘ξ€·π‘₯𝑛,𝑒=𝑝(𝑒,𝑒)=0.(2.32) Now, 𝑝π‘₯𝑛π‘₯,𝑇𝑒≀𝑝𝑛π‘₯,𝑒+𝑝(𝑒,𝑇𝑒)βˆ’π‘(𝑒,𝑒)≀𝑝𝑛,𝑒+𝑝𝑒,π‘₯𝑛π‘₯+𝑝𝑛π‘₯,π‘‡π‘’βˆ’π‘π‘›,π‘₯π‘›ξ€Έβˆ’π‘(𝑒,𝑒).(2.33) Letting 𝑛→+∞ in above inequalities and using (2.4) and (2.29), we get that lim𝑛→+βˆžπ‘ξ€·π‘₯𝑛,𝑇𝑒=𝑝(𝑒,𝑇𝑒).(2.34) Similarly, we may show that lim𝑛→+βˆžπ‘ξ€·π‘₯𝑛,𝑆𝑒=𝑝(𝑒,𝑇𝑒).(2.35) Since 𝑇 is continuous and lim𝑛→+βˆžπ‘ξ€·π‘₯2𝑛,𝑒=𝑝(𝑒,𝑒)=0,(2.36) then by (2.34), we have 𝑝(𝑒,𝑇𝑒)=lim𝑛→+βˆžπ‘ξ€·π‘₯2𝑛+1ξ€Έ,𝑇𝑒=lim𝑛→+βˆžπ‘ξ€·π‘‡π‘₯2𝑛,𝑇𝑒=𝑝(𝑇𝑒,𝑇𝑒).(2.37) Similarly, we may show that 𝑝(𝑒,𝑆𝑒)=𝑝(𝑆𝑒,𝑆𝑒). By (𝑝3) and (𝑝4) we derive that 𝑝(𝑒,𝑇𝑒)=𝑝(𝑇𝑒,𝑇𝑒)≀𝑃(𝑇𝑒,𝑆𝑒)+𝑃(𝑆𝑒,𝑇𝑒)βˆ’π‘ƒ(𝑆𝑒,𝑆𝑒)=2𝑝(𝑇𝑒,𝑆𝑒)βˆ’π‘ƒ(𝑒,𝑆𝑒).(2.38) The above inequality yields that 12(𝑝(𝑒,𝑇𝑒)+𝑝(𝑒,𝑆𝑒))≀𝑝(𝑇𝑒,𝑆𝑒).(2.39) Since 𝑒 and 𝑒 are comparable and πœ“ is nondecreasing, then by (2.2), we have πœ“ξ‚€121(𝑝(𝑒,𝑇𝑒)+𝑝(𝑒,𝑆𝑒))β‰€πœ“(𝑝(𝑇𝑒,𝑆𝑒))β‰€πœ“2(𝑃(𝑒,𝑇𝑒)+𝑝(𝑒,𝑆𝑒))βˆ’πœ™(𝑝(𝑒,𝑇𝑒),𝑝(𝑒,𝑆𝑒)).(2.40) Thus, we have πœ™(𝑝(𝑒,𝑇𝑒),𝑝(𝑒,𝑆𝑒))=0 and hence 𝑝(𝑒,𝑇𝑒)=𝑝(𝑒,𝑆𝑒)=0. By using (𝑝1) and (𝑝2) of Definition 1.1, we find that 𝑒=𝑆𝑒=𝑇𝑒. That is, 𝑒 is a common fixed point of 𝑇 and 𝑆.

The continuity of 𝑇 and 𝑆 in Theorem 2.4 can be dropped. For this instance, suppose that 𝑋 satisfies the following property:

(𝑃): if (π‘₯𝑛) is a nondecreasing sequence in 𝑋 such that lim𝑛→+βˆžπ‘(π‘₯𝑛,𝑒)=𝑝(𝑒,𝑒), then π‘₯𝑛βͺ― for all π‘›βˆˆβ„•.

Theorem 2.5. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists a partial metric 𝑝 on 𝑋 such that (𝑋,𝑝) is complete. Suppose that 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ be weakly (𝐢,πœ“,πœ™)-contractive mappings. If 𝑋 satisfies property (𝑃), then 𝑇 and 𝑆 have a common fixed point, that is, there exists π‘’βˆˆπ‘‹ such that 𝑒=𝑇𝑒=𝑆𝑒.

Proof. Following the proof of Theorem 2.4 step by step to construct a nondecreasing sequence (π‘₯𝑛) in 𝑋 such that lim𝑛→+βˆžπ‘(π‘₯𝑛,𝑒)=𝑝(𝑒,𝑒)=0, lim𝑛→+βˆžπ‘ξ€·π‘₯𝑛,𝑇𝑒=𝑝(𝑒,𝑇𝑒),lim𝑛→+βˆžπ‘ξ€·π‘₯𝑛,𝑆𝑒=𝑝(𝑒,𝑇𝑒).(2.41)
By property, we have (𝑃)π‘₯𝑛βͺ―π‘₯ for all π‘›βˆˆβ„•. By (2.2), we have πœ“ξ€·π‘ξ€·π‘₯2𝑛+1𝑝,𝑆𝑒=πœ“π‘‡π‘₯2𝑛1,π‘†π‘’ξ€Έξ€Έβ‰€πœ“2𝑝π‘₯2𝑛,𝑆𝑒+𝑝𝑇π‘₯2𝑛𝑝π‘₯,π‘’ξ€Έξ€Έβˆ’πœ™2𝑛,𝑆𝑒,𝑝𝑇π‘₯2𝑛1,π‘’ξ€Έξ€Έβ‰€πœ“2𝑝π‘₯2𝑛,𝑆𝑒+𝑝𝑇π‘₯2𝑛1,𝑒=πœ“2𝑝π‘₯2𝑛π‘₯,𝑆𝑒+𝑝2𝑛+1.,𝑒(2.42) Letting 𝑛→+∞ in above inequalities, and using (2.41) we get πœ“(𝑝(𝑒,𝑆𝑒))β‰€πœ“((1/2)𝑝(𝑒,𝑆𝑒)). Since πœ“ is nondecreasing we get that 𝑝(𝑒,𝑆𝑒)≀(1/2)𝑝(𝑒,𝑆𝑒). Hence 𝑝(𝑒,𝑆𝑒)=0. By (𝑝1) and (𝑝2), we conclude that 𝑒=𝑆𝑒. By similar arguments, we can show that 𝑒=𝑇𝑒. Thus 𝑒 is a common fixed point of 𝑇 and 𝑆.

By taking πœ“=𝑖 (the identity function on [0,+∞)) and defining πœ™βˆΆ[0,+∞)Γ—[0,+∞)β†’[0,+∞) via πœ™(𝑠,𝑑)=(1/2βˆ’π‘ž/2)(𝑠+𝑑) where π‘žβˆˆ[0,1) in Theorems 2.4 and 2.5, we have the following results.

Corollary 2.6. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists a partial metric 𝑝 on 𝑋 such that (𝑋,𝑝) is complete. Suppose that 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ be weakly increasing mappings with respect to βͺ― such that for any comparable π‘₯ and 𝑦, one has π‘π‘ž(𝑇π‘₯,𝑆𝑦)≀2(𝑝(𝑇π‘₯,𝑦)+𝑝(π‘₯,𝑇𝑦)).(2.43) If 𝑇 and 𝑆 are continuous and π‘ž<1, then 𝑇 and 𝑆 have a common fixed point, that is, there exists π‘’βˆˆπ‘‹ such that 𝑒=𝑇𝑒=𝑆𝑒.

Corollary 2.7. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists a partial metric 𝑝 on 𝑋 such that (𝑋,𝑝) is complete. Suppose that 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ be weakly increasing mappings with respect to βͺ― such that for any comparable π‘₯ and 𝑦, one has π‘π‘ž(𝑇π‘₯,𝑆𝑦)≀2(𝑝(𝑇π‘₯,𝑦)+𝑝(π‘₯,𝑇𝑦)).(2.44) If 𝑋 satisfies property (𝑃) and π‘ž<1, then 𝑇 and 𝑆 have a common fixed point, that is, there exists π‘’βˆˆπ‘‹ such that 𝑒=𝑇𝑒=𝑆𝑒.

Corollary 2.8. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists a partial metric 𝑝 on 𝑋 such that (𝑋,𝑝) is complete. Suppose that π‘‡βˆΆπ‘‹β†’π‘‹ are mapping such that for any comparable π‘₯ and 𝑦 in 𝑋, one has ξ‚€1πœ“(𝑝(𝑇π‘₯,𝑇𝑦))β‰€πœ“2(𝑝(𝑇π‘₯,𝑦)+𝑝(π‘₯,𝑇𝑦))βˆ’πœ™(𝑝(𝑇π‘₯,𝑦),𝑝(π‘₯,𝑇𝑦)),(2.45) where πœ™βˆΆ[0,∞)Γ—[0,+∞)β†’[0,∞) is a continuous function such that πœ™(𝑑,𝑠)=0 if and only if 𝑑=𝑠=0 and πœ“βˆΆ[0,+∞)β†’[0,+∞) is a continuous nondecreasing function such that πœ“(𝑑)=0 if and only if 𝑑=0. Also, suppose that 𝑇π‘₯βͺ―𝑇(𝑇π‘₯) for all π‘₯βˆˆπ‘‹. If 𝑇 is continuous, then 𝑇 has a fixed point, that is, there exists π‘’βˆˆπ‘‹ such that 𝑒=𝑇𝑒.

Proof. It follows from Theorem 2.4 by taking 𝑆=𝑇 and noting that 𝑆 and 𝑇 are weakly (𝐢,πœ“,πœ™)-contractive mappings.

Corollary 2.9. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists a partial metric 𝑝 on 𝑋 such that (𝑋,𝑝) is complete. Suppose that π‘‡βˆΆπ‘‹β†’π‘‹ be mapping such that for any comparable π‘₯ and 𝑦 in 𝑋, one has ξ‚€1πœ“(𝑝(𝑇π‘₯,𝑇𝑦))β‰€πœ“2(𝑝(𝑇π‘₯,𝑦)+𝑝(π‘₯,𝑇𝑦))βˆ’πœ™(𝑝(𝑇π‘₯,𝑦),𝑝(π‘₯,𝑇𝑦)),(2.46) where πœ™βˆΆ[0,∞)Γ—[0,+∞)β†’[0,∞) is a continuous function such that πœ™(𝑑,𝑠)=0 if and only if 𝑑=𝑠=0 and πœ“βˆΆ[0,+∞)β†’[0,+∞) is a continuous nondecreasing function such that πœ“(𝑑)=0 if and only if 𝑑=0. Also, suppose that 𝑇π‘₯βͺ―𝑇(𝑇π‘₯) for all π‘₯βˆˆπ‘‹. If 𝑋 satisfies property (𝑃), then 𝑇 has a fixed point, that is, there exists π‘’βˆˆπ‘‹ such that 𝑒=𝑇𝑒.

Proof. It follows from Theorem 2.5 by taking 𝑆=𝑇 and noting that 𝑆 and 𝑇 are weakly (𝐢,πœ“,πœ™)-contractive mappings.

We present the following common fixed points of integral type as an application of our results.

Denote by Ξ© the set of functions πœ‡βˆΆ[0,+∞)β†’[0,+∞) satisfying the following hypotheses: (1)πœ‡ is a Lebesgue integrable function on each compact subset of [0,+∞), (2)for every πœ–>0, we have βˆ«πœ–0πœ‡(𝑠)𝑑𝑠>0.

It is easy to see that the mapping ξ€œπœ“(𝑑)=𝑑0ξ€·π‘‘πœ‡(𝑠)π‘‘π‘ βˆˆΞ¦,πœ™1,𝑑2ξ€Έ=ξ€œ(𝑑1+𝑑20)/2πœ‡(𝑠)π‘‘π‘ βˆˆΞ¨.(2.47) We have the following result.

Corollary 2.10. Let (𝑋,βͺ―,𝑝) be an partially ordered metric space. Suppose that 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ are weakly increasing mappings with respect to βͺ― and for any comparable π‘₯ and 𝑦, one has ξ€œ0𝑝(𝑇π‘₯,𝑆𝑦)ξ€œπœ‡(𝑠)𝑑𝑠≀01/2(𝑑(π‘₯,𝑆𝑦)+𝑑(𝑇π‘₯,𝑦))ξ€œπœ‡(𝑠)π‘‘π‘ βˆ’0(1/4)((𝑑(π‘₯,𝑆𝑦)+𝑑(𝑦,𝑓π‘₯))/4)πœ‡(𝑠)𝑑𝑠.(2.48) If 𝑇 and 𝑆 are continuous, then 𝑇 and 𝑆 have a common fixed point.

Proof. It follows from Theorem 2.4 by defining πœ“βˆΆ[0,+∞)β†’[0,+∞) via βˆ«πœ“(𝑑)=𝑑0πœ‡(𝑠)𝑑𝑠 and πœ™βˆΆ[0,+∞)Γ—[0,+∞)β†’[0,+∞) via πœ™ξ€·π‘‘1,𝑑2ξ€Έ=ξ€œ(𝑑1+𝑑20)/2πœ‡(𝑠)𝑑𝑠.(2.49)

Corollary 2.11. Let (𝑋,βͺ―,𝑝) be an partially ordered metric space. Suppose that 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ are weakly increasing mappings with respect to βͺ― and for any comparable π‘₯ and 𝑦, one has ξ€œ0𝑝(𝑇π‘₯,𝑆𝑦)ξ€œπœ‡(𝑠)𝑑𝑠≀0(1/2)(𝑑(π‘₯,𝑆𝑦)+𝑑(𝑇π‘₯,𝑦))ξ€œπœ‡(𝑠)π‘‘π‘ βˆ’0(1/4)((𝑑(π‘₯,𝑆𝑦)+𝑑(𝑦,𝑓π‘₯))/4)πœ‡(𝑠)𝑑𝑠.(2.50) If 𝑋 satisfies property (𝑃), then 𝑇 and 𝑆 have a common fixed point.

Proof. It follows from Theorem 2.5 by defining πœ“βˆΆ[0,+∞)β†’[0,+∞) via βˆ«πœ“(𝑑)=𝑑0πœ‡(𝑠)𝑑𝑠 and πœ™βˆΆ[0,+∞)Γ—[0,+∞)β†’[0,+∞) via πœ™ξ€·π‘‘1,𝑑2ξ€Έ=ξ€œ(𝑑1+𝑑20)/2πœ‡(𝑠)𝑑𝑠.(2.51)

Example 2.12. Let 𝑋=[0,1]. Define the partial metric space on 𝑋 by 𝑝(π‘₯,𝑦)=max{π‘₯,𝑦} and the relation βͺ― on 𝑋 by 𝑦βͺ―π‘₯ if and only if π‘₯≀𝑦. Also, define the mappings 𝑇,π‘†βˆΆπ‘‹β†’π‘‹ by 𝑇π‘₯=1/4π‘₯2, 𝑆π‘₯=1/5π‘₯2 and the functions πœ“βˆΆ[0,+∞)β†’[0,+∞) by πœ“(𝑑)=𝑑3 and πœ™βˆΆ[0,+∞)Γ—[0,+∞)β†’[0,+∞)πœ™(𝑠,𝑑)=(7/64)(𝑠+𝑑)3. Then one has the following. (1)(𝑋,𝑝,βͺ―) is a complete ordered partial metric space. (2)𝑇 and 𝑆 are continuous. (3)𝑆 and 𝑇 are weakly increasing. (4)For any two comparable elements π‘₯ and 𝑦 in 𝑋, we have ξ‚€1πœ“(𝑝(𝑇π‘₯,𝑆𝑦))β‰€πœ“2(𝑝(𝑦,𝑇π‘₯)+𝑝(π‘₯,𝑆𝑦))βˆ’πœ™(𝑝(𝑦,𝑇π‘₯),𝑝(π‘₯,𝑆𝑦)).(2.52)

Proof. The proof of (1) and (2) is clear. To prove (3), given π‘₯βˆˆπ‘‹. Since ξ‚€1𝑆(𝑇π‘₯)=𝑆4π‘₯2=1π‘₯804≀14π‘₯2=𝑇π‘₯,(2.53) we have 𝑇π‘₯βͺ―𝑆(𝑇π‘₯). Similarly, we can show that 𝑆π‘₯βͺ―𝑇(𝑆π‘₯). Thus 𝑇 and 𝑆 are weakly increasing mappings. To prove (4), given two comparable elements π‘₯ and 𝑦 in 𝑋. Without loss of generality, we assume that π‘₯βͺ―𝑦, that is, 𝑦≀π‘₯. So, 𝑝1πœ“(𝑝(𝑇π‘₯,𝑆𝑦))=πœ“4π‘₯2,15𝑦21=πœ“max4π‘₯2,15𝑦21ξ‚‡ξ‚β‰€πœ“max4π‘₯2,14𝑦2ξ‚€1=πœ“4π‘₯2=1π‘₯646≀1π‘₯643≀164(𝑝(𝑦,𝑇π‘₯)+π‘₯)3=18(𝑝(𝑦,𝑇π‘₯)+π‘₯)3βˆ’764(𝑝(𝑦,𝑇π‘₯)+π‘₯)3ξ‚€1=πœ“21(𝑝(𝑦,𝑇π‘₯)+π‘₯)βˆ’πœ™(𝑝(𝑦,𝑇π‘₯),π‘₯)=πœ“2(𝑝(𝑇π‘₯,𝑦)+𝑝(π‘₯,𝑆𝑦))βˆ’πœ™(𝑝(𝑇π‘₯,𝑦),𝑝(π‘₯,𝑆𝑦)).(2.54)

From (3) and (4), we conclude that 𝑇 and 𝑆 are weakly (𝐢,πœ“,πœ™)-contractive mappings. Note that Example 2.12 satisfies all the hypotheses of Theorem 2.4. Thus 𝑇 and 𝑆 have a common fixed point. Here 0 is a common fixed point of 𝑇 and 𝑆.

3. Application

In this section, we apply our results to prove an existence solution of the following integral equation: ξ€œπ‘₯(𝑑)=10[].𝐺(𝑑,𝑠)𝑓(𝑠,π‘₯(𝑠))𝑑𝑠,βˆ€π‘‘βˆˆ0,1(3.1)

Let 𝑋=𝐢([0,1]) be the space of all continuous functions defined on 𝐼=[0,1]. Define a partial metric space: [π‘βˆΆπΆ(𝐼)×𝐢(𝐼)β†’0,+∞),(3.2) by 𝑝||π‘₯||[]ξ€Ύ.(π‘₯,𝑦)=β€–π‘₯βˆ’π‘¦β€–βˆΆ=sup(𝑑)βˆ’π‘¦(𝑑)βˆΆπ‘‘βˆˆ0,1(3.3) Also, define a relation βͺ― on 𝑋 by π‘₯βͺ―𝑦if,andonlyifπ‘₯(𝑑)≀𝑦(𝑑)βˆ€π‘‘βˆˆπΌ.(3.4) Then (𝑋,𝑝,≀) is an ordered complete partial metric space.

Now, we will give an existence theorem for the solution of the integral equation (3.1).

Theorem 3.1. Suppose the following hypotheses hold. (i)βˆ«π‘“(𝑠,π‘₯(𝑠))≀𝑓(𝑠,10𝐺(𝑠,𝑑)𝑓(𝑑,π‘₯(𝑑))𝑑𝑑) for all π‘ βˆˆπΌ. (ii)There exists a continuous function 𝐻∢[0,1]β†’[0,+∞) such that ||||||||.𝑓(𝑑,π‘Ž)βˆ’π‘“(𝑑,𝑏)≀𝐻(𝑑)π‘Žβˆ’π‘(3.5)(iii)There exists π‘˜<1 such that sup[]π‘‘βˆˆ0,1ξ€œ10ξ‚€1𝐻(𝑑)𝑑𝑑≀3ξ‚π‘˜.(3.6)(iv)𝐺(𝑠,𝑑)≀1 for all 𝑠,π‘‘βˆˆπΌ. Then the integral equation (3.1) has a solution π‘₯βˆ—βˆˆπΆ2(𝐼).

Proof. Define the operators: 𝑇,π‘†βˆΆπΆ(𝐼)→𝐢(𝐼),(3.7) by ξ€œπ‘‡π‘₯(𝑑)=𝑆π‘₯(𝑑)=10𝐺(𝑑,𝑠)𝑓(𝑠,π‘₯(𝑠))π‘‘π‘ βˆ€π‘‘βˆˆπΌ.(3.8) Given π‘₯∈𝐢(𝐼). Then from (i), we have ξ€œπ‘‡π‘₯(𝑑)=10β‰€ξ€œπΊ(𝑑,𝑠)𝑓(𝑠,π‘₯(𝑠))𝑑𝑠10πΊξ‚΅ξ€œ(𝑑,𝑠)𝑓𝑠,10𝐺(𝑠,𝜏)𝑓(𝜏,π‘₯(𝜏))π‘‘πœπ‘‘π‘ =𝑇(𝑇π‘₯(𝑑)).(3.9) Thus 𝑇 and 𝑆 are weakly increasing mappings with respect to βͺ―. Again, for π‘₯,π‘¦βˆˆπΆ(𝐼) with π‘₯βͺ―𝑦, we have 𝑝(𝑇π‘₯,𝑇𝑦)=‖𝑇π‘₯βˆ’π‘‡π‘¦β€–=supπ‘‘βˆˆπΌ||||𝑇π‘₯(𝑑)βˆ’π‘‡π‘¦(𝑑)=supπ‘‘βˆˆπΌ||||ξ€œ10||||𝐺(𝑑,𝑠)(𝑓(𝑠,π‘₯(𝑠))βˆ’π‘“(𝑠,𝑦(𝑠)))𝑑𝑠≀supπ‘‘βˆˆπΌξ€œ10||||𝐺(𝑑,𝑠)𝑓(𝑠,π‘₯(𝑠))βˆ’π‘“(𝑠,𝑦(𝑠))𝑑𝑠≀supπ‘‘βˆˆπΌξ€œ10||||𝐺(𝑑,𝑠)𝐻(𝑠)π‘₯(𝑠)βˆ’π‘¦(𝑠)𝑑𝑠≀supπ‘‘βˆˆπΌξ€œ10𝐺(𝑑,𝑠)𝐻(𝑠)β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ .(3.10) By (iv), we have 𝑝(𝑇π‘₯,𝑇𝑦)≀supπ‘‘βˆˆπΌξ€œ10𝐻(𝑠)β€–π‘₯βˆ’π‘¦β€–π‘‘π‘ .(3.11) By using (iii), we get 1𝑝(𝑇π‘₯,𝑇𝑦)≀3π‘˜β‰€1β€–π‘₯βˆ’π‘¦β€–3π‘˜ξ€·||||||||+||||||||+||||||||ξ€Έ=1π‘₯βˆ’π‘‡π‘¦π‘‡π‘¦βˆ’π‘‡π‘₯𝑇π‘₯βˆ’π‘¦3π‘˜(𝑝(π‘₯,𝑇𝑦)+𝑝(𝑇π‘₯,𝑇𝑦)+𝑝(𝑦,𝑇π‘₯)).(3.12) Hence π‘˜π‘(𝑇π‘₯,𝑇𝑦)≀3βˆ’π‘˜(𝑝(π‘₯,𝑇𝑦)+𝑝(𝑦,𝑇π‘₯)).(3.13) Take π‘ž=2π‘˜/(3βˆ’π‘˜). Since π‘˜<1, we have π‘ž<1. Also, we have π‘π‘ž(𝑇π‘₯,𝑇𝑦)≀2(𝑝(π‘₯,𝑇𝑦)+𝑝(𝑇π‘₯,𝑦)).(3.14) Moreover if (𝑓𝑛) is a nondecreasing sequence in 𝐢(𝐼) such that 𝑓𝑛→𝑓 as 𝑛→+∞, then 𝑓𝑛≀𝑓 for all π‘›βˆˆβ„• (see [25]). Thus, all the required hypotheses of Corollary 2.7 are satisfied. Therefore, 𝑇 has a fixed point and hence the integral equation (3.1) has a solution.

Acknowledgments

The authors thank the editor and the referees for their useful comments and suggestions.

References

  1. S. G. Matthews, β€œPartial metric topology,” in Proceedings of the 8th Summer Conference on General Topology and its Applications, Queen's College (1992), vol. 728, pp. 183–197, Annals of the New York Academy of Sciences, 1994.
  2. T. Abdeljawad, E. Karapınar, and K. Taş, β€œExistence and uniqueness of a common fixed point on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1900–1904, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. Th. Abdeljawad, E. Karapınar, and K. Taş, β€œA generalized contraction principle with control functions on partial metric spaces,” Computers & Mathematics with Applications, vol. 63, no. 3, pp. 716–719, 2012. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. Th. Abdeljawad, β€œFixed points for generalized weakly contractive mappings in partial metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2923–2927, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. I. Altun and A. Erduran, β€œFixed point theorems for monotone mappings on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 508730, 10 pages, 2011. View at Google Scholar Β· View at Zentralblatt MATH
  6. I. Altun and H. Simsek, β€œSome fixed point theorems on dualistic partial metric spaces,” Journal of Advanced Mathematical Studies, vol. 1, no. 1-2, pp. 1–8, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. I. Altun, F. Sola, and H. Simsek, β€œGeneralized contractions on partial metric spaces,” Topology and its Applications, vol. 157, no. 18, pp. 2778–2785, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. I. Altun and K. Sadarangani, β€œCorrigendum to “Generalized contractions on partial metric spaces” [Topology Appl. 157 (2010) 2778–2785],” Topology and its Applications, vol. 158, no. 13, pp. 1738–1740, 2011. View at Publisher Β· View at Google Scholar
  9. H. Aydi, β€œFixed point results for weakly contractive mappings in ordered partial metric spaces,” Journal of Advanced Mathematical Studies, vol. 4, no. 2, pp. 1–12, 2011. View at Google Scholar Β· View at Zentralblatt MATH
  10. H. Aydi, β€œSome coupled fixed point results on partial metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 647091, 11 pages, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. H. Aydi, β€œFixed point theorems for generalized weakly contractive condition in partially ordered metric spaces,” Journal of Nonlinear Analysis and Optimization, vol. 2, no. 2, pp. 33–48, 2011. View at Google Scholar
  12. H. Aydi, E. Karapınar, and W. Shatanawi, β€œCoupled fixed point results for (ψ,ϕ)-weakly contractive condition in ordered partial metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4449–4460, 2011. View at Publisher Β· View at Google Scholar
  13. A. G. Bin Ahmad, Z. M. Fadail, V. Ć. Rajić, and S. Radenović, β€œNonlinear contractions in 0-complete partial metric spaces,” Abstract and Applied Analysis. In press.
  14. L. Ćirić, B. Samet, H. Aydi, and C. Vetro, β€œCommon fixed points of generalized contractions on partial metric spaces and an application,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2398–2406, 2011. View at Publisher Β· View at Google Scholar
  15. Z. Golubović, S. Kadelburg, and Radenović, β€œCoupled coincidence points of mappings in partially ordered metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 192581, 18 pages, 2012. View at Publisher Β· View at Google Scholar
  16. E. Karapınar, β€œGeneralizations of Caristi Kirk's theorem on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, p. 4, 2011. View at Publisher Β· View at Google Scholar
  17. E. Karapınar and U. Yüksel, β€œSome common fixed point theorems in partial metric spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 263621, 16 pages, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. E. Karapınar, β€œA note on common fixed point theorems in partial metric spaces,” Miskolc Mathematical Notes, vol. 12, no. 2, pp. 185–191, 2011. View at Google Scholar
  19. E. Karapınar, N. Shobkolaei, S. Sedghi, and S. M. Vaezpour, β€œA common fixed point theorem for cyclic operators on partial metric spaces,” Filomat, vol. 26, no. 2, pp. 407–414, 2012. View at Google Scholar
  20. K. P. Chi, E. Karapinar, and T. D. Thanh, β€œA generalized contraction principle in partial metric spaces,” Mathematical and Computer Modelling, vol. 55, no. 5-6, pp. 1673–1681, 2012. View at Publisher Β· View at Google Scholar
  21. E. Karapınar, β€œWeak ϕ-contraction on partial contraction,” Journal of Computational Analysis and Applications, vol. 14, no. 2, pp. 206–210, 2012. View at Google Scholar
  22. E. Karapınar, β€œSome fixed point theorems on the class of comparable partial metric spaces,” Applied General Topology, vol. 12, no. 2, pp. 187–192, 2011. View at Google Scholar Β· View at Zentralblatt MATH
  23. E. Karapınar and M. Erhan, β€œFixed point theorems for operators on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1894–1899, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  24. H. K. Nashine, Z. Kadelburg, and S. Radenović, β€œCommon fixed point theorems for weakly isotone increasing mappings in partially ordered metric spaces,” Mathematical and Computer Modelling. In press.
  25. J. J. Nieto and R. Rodríguez-López, β€œContractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  26. S. Oltra and O. Valero, β€œBanach's fixed point theorem for partial metric spaces,” Rendiconti dell'Istituto di Matematica dell'Università di Trieste, vol. 36, no. 1-2, pp. 17–26, 2004. View at Google Scholar Β· View at Zentralblatt MATH
  27. S. Romaguera, β€œA Kirk type characterization of completeness for partial metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 493298, 6 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  28. B. Samet, M. Rajović, R. Lazović, and R. Stoiljković, β€œCommon fixed point results for nonlinear contractions in partially ordered metric spaces,” Fixed Point Theory and Applications. In press.
  29. W. Shatanawi and H. K. Nashine, β€œA generalization of Banach's contraction principle for nonlinear contraction in a partial metric space,” Journal of Nonlinear Science and Applications, vol. 5, pp. 37–43, 2012. View at Google Scholar
  30. W. Shatanawi, H. K. Nashine, and N. Tahat, β€œGeneralization of some coupled fixed point results on partial metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 686801, 10 pages, 2012. View at Publisher Β· View at Google Scholar
  31. W. Shatanawi, B. Samet, and M. Abbas, β€œCoupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 680–687, 2012. View at Google Scholar
  32. O. Valero, β€œOn Banach fixed point theorems for partial metric spaces,” Applied General Topology, vol. 6, no. 2, pp. 229–240, 2005. View at Google Scholar Β· View at Zentralblatt MATH
  33. S. K. Chatterjea, β€œFixed-point theorems,” Comptes Rendus de l'Académie Bulgare des Sciences, vol. 25, pp. 727–730, 1972. View at Google Scholar Β· View at Zentralblatt MATH
  34. B. S. Choudhury, β€œUnique fixed point theorem for weakly C-contractive mappings,” Kathmandu University Journal of Science, Engineering and Technology, vol. 5, no. 1, pp. 6–13, 2009. View at Publisher Β· View at Google Scholar
  35. J. Harjani, B. López, and K. Sadarangani, β€œFixed point theorems for weakly C-contractive mappings in ordered metric spaces,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 790–796, 2011. View at Publisher Β· View at Google Scholar
  36. W. Shatanawi, β€œFixed point theorems for nonlinear weakly C-contractive mappings in metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2816–2826, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  37. I. Altun and H. Simsek, β€œSome fixed point theorems on ordered metric spaces and application,” Fixed Point Theory and Applications, vol. 2010, Article ID 621469, 17 pages, 2010. View at Publisher Β· View at Google Scholar
  38. I. Altun, β€œCommon fixed point theorems for weakly increasing mappings on ordered uniform spaces,” Miskolc Mathematical Notes, vol. 12, no. 1, pp. 3–10, 2011. View at Google Scholar Β· View at Zentralblatt MATH
  39. I. Altun and V. Rakočević, β€œOrdered cone metric spaces and fixed point results,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1145–1151, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  40. W. Shatanawi, β€œSome fixed point theorems in ordered G-metric spaces and applications,” Abstract and Applied Analysis, Article ID 126205, 11 pages, 2011. View at Publisher Β· View at Google Scholar
  41. W. Shatanawi, β€œSome common coincidence point theorems on weakly increasing mappings in partially ordered cone metric spaces,” Gazi University Journal of Science, vol. 25, pp. 89–95, 2012. View at Google Scholar