Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 126205, 11 pages
http://dx.doi.org/10.1155/2011/126205
Research Article

Some Fixed Point Theorems in Ordered 𝐺-Metric Spaces and Applications

Department of Mathematics, The Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

Received 9 January 2011; Revised 22 March 2011; Accepted 19 April 2011

Academic Editor: D.Β Anderson

Copyright Β© 2011 W. 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 study a number of fixed point results for the two weakly increasing mappings 𝑓 and 𝑔 with respect to partial ordering relation βͺ― in generalized metric spaces. An application to integral equation is given.

1. Introduction

The existence of fixed points in partially ordered sets has been at the center of active research. In fact, the existence of fixed point in partially ordered sets has been investigated in [1]. Moreover, Ran and Reurings [1] applied their results to matrix equations. Some generalizations of the results of [1] are given in [2–6]. In [6], O’Regan and Petruşel gave some existence results for the Fredholm and Volterra type.

The notion of 𝐺-metric space was introduced by Mustafa and Sims [7] as a generalization of the notion of metric spaces. Mustafa et al. studied many fixed point results in 𝐺-metric space [8–10] (also see [11–15]). In fact the study of common fixed points of mappings satisfying certain contractive conditions has been at the center of strong research activity. The following definition is introduced by Mustafa and Sims [7].

Definition 1.1 (see [7]). Let 𝑋 be a nonempty set and let πΊβˆΆπ‘‹Γ—π‘‹Γ—π‘‹β†’π‘+ be a function satisfying the following properties: (𝐺1)𝐺(π‘₯,𝑦,𝑧)=0 if π‘₯=𝑦=𝑧, (𝐺2)0<𝐺(π‘₯,π‘₯,𝑦), for all π‘₯,π‘¦βˆˆπ‘‹ with π‘₯≠𝑦, (𝐺3)𝐺(π‘₯,π‘₯,𝑦)≀𝐺(π‘₯,𝑦,𝑧), for all π‘₯,𝑦,π‘§βˆˆπ‘‹ with 𝑧≠𝑦, (𝐺4)𝐺(π‘₯,𝑦,𝑧)=𝐺(π‘₯,𝑧,𝑦)=𝐺(𝑦,𝑧,π‘₯)=β‹―, symmetry in all three variables, (𝐺5)𝐺(π‘₯,𝑦,𝑧)≀𝐺(π‘₯,π‘Ž,π‘Ž)+𝐺(π‘Ž,𝑦,𝑧), for all π‘₯,𝑦,𝑧,π‘Žβˆˆπ‘‹. Then the function 𝐺 is called a generalized metric, or, more specifically, a 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝐺-metric space.

Definition 1.2 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space, and let {π‘₯𝑛} be a sequence of points of 𝑋, a point π‘₯βˆˆπ‘‹ is said to be the limit of the sequence {π‘₯𝑛}, if lim𝑛,π‘šβ†’+∞𝐺(π‘₯,π‘₯𝑛,π‘₯π‘š)=0, and we say that the sequence {π‘₯𝑛} is 𝐺-convergent to π‘₯ or {π‘₯𝑛}𝐺-converges to π‘₯.

Thus, π‘₯𝑛→π‘₯ in a 𝐺-metric space (𝑋,𝐺) if for any πœ€>0, there exists π‘˜βˆˆπ such that 𝐺(π‘₯,π‘₯𝑛,π‘₯π‘š)<πœ€ for all π‘š,𝑛β‰₯π‘˜.

Proposition 1.3 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the following are equivalent: (1){π‘₯𝑛} is 𝐺-convergent to π‘₯; (2)𝐺(π‘₯𝑛,π‘₯𝑛,π‘₯)β†’0 as 𝑛→+∞; (3)𝐺(π‘₯𝑛,π‘₯,π‘₯)β†’0 as 𝑛→+∞; (4)𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯)β†’0 as 𝑛,π‘šβ†’+∞.

Definition 1.4 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space, a sequence {π‘₯𝑛} is called 𝐺-Cauchy if for every πœ€>0, there is π‘˜βˆˆπ such that 𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯𝑙)<πœ€, for all 𝑛,π‘š,𝑙β‰₯π‘˜; that is 𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯𝑙)β†’0 as 𝑛,π‘š,𝑙→+∞.

Proposition 1.5 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the following are equivalent: (1)the sequence {π‘₯𝑛} is 𝐺-Cauchy;(2)for every πœ–>0, there is π‘˜βˆˆπ such that 𝐺(π‘₯𝑛,π‘₯π‘š,π‘₯π‘š)<πœ–, for all 𝑛,π‘šβ‰₯π‘˜.

Definition 1.6 (see [7]). Let (𝑋,𝐺) and (π‘‹ξ…ž,πΊξ…ž) be 𝐺-metric spaces, and let π‘“βˆΆ(𝑋,𝐺)β†’(π‘‹ξ…ž,πΊξ…ž) be a function. Then 𝑓 is said to be 𝐺-continuous at a point π‘Žβˆˆπ‘‹ if and only if for every πœ€>0, there is 𝛿>0 such that π‘₯,π‘¦βˆˆπ‘‹ and 𝐺(π‘Ž,π‘₯,𝑦)<𝛿 implies πΊξ…ž(𝑓(π‘Ž),𝑓(π‘₯),𝑓(𝑦))<πœ€. A function 𝑓 is 𝐺-continuous at 𝑋 if and only if it is 𝐺-continuous at all π‘Žβˆˆπ‘‹.

Proposition 1.7 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the function 𝐺(π‘₯,𝑦,𝑧) is jointly continuous in all three of its variables.

Every 𝐺-metric on 𝑋 will define a metric 𝑑𝐺 on 𝑋 by𝑑𝐺(π‘₯,𝑦)=𝐺(π‘₯,𝑦,𝑦)+𝐺(𝑦,π‘₯,π‘₯),βˆ€π‘₯,π‘¦βˆˆπ‘‹.(1.1) For a symmetric 𝐺-metric space,𝑑𝐺(π‘₯,𝑦)=2𝐺(π‘₯,𝑦,𝑦),βˆ€π‘₯,π‘¦βˆˆπ‘‹.(1.2) However, if 𝐺 is not symmetric, then the following inequality holds:32𝐺(π‘₯,𝑦,𝑦)≀𝑑𝐺(π‘₯,𝑦)≀3𝐺(π‘₯,𝑦,𝑦),βˆ€π‘₯,π‘¦βˆˆπ‘‹.(1.3)

The following are examples of 𝐺-metric spaces.

Example 1.8 (see [7]). Let (𝐑,𝑑) be the usual metric space. Define 𝐺𝑠 by 𝐺𝑠(π‘₯,𝑦,𝑧)=𝑑(π‘₯,𝑦)+𝑑(𝑦,𝑧)+𝑑(π‘₯,𝑧),(1.4) for all π‘₯,𝑦,π‘§βˆˆπ‘. Then it is clear that (𝐑,𝐺𝑠) is a 𝐺-metric space.

Example 1.9 (see [7]). Let 𝑋={π‘Ž,𝑏}. Define 𝐺 on 𝑋×𝑋×𝑋 by 𝐺(π‘Ž,π‘Ž,π‘Ž)=𝐺(𝑏,𝑏,𝑏)=0,𝐺(π‘Ž,π‘Ž,𝑏)=1,𝐺(π‘Ž,𝑏,𝑏)=2(1.5) and extend 𝐺 to 𝑋×𝑋×𝑋 by using the symmetry in the variables. Then it is clear that (𝑋,𝐺) is a 𝐺-metric space.

Definition 1.10 (see [7]). A 𝐺-metric space (𝑋,𝐺) is called 𝐺-complete if every 𝐺-Cauchy sequence in (𝑋,𝐺) is 𝐺-convergent in (𝑋,𝐺).

The notion of weakly increasing mappings was introduced in by Altun and Simsek [16].

Definition 1.11 (see [16]). Let (𝑋,βͺ―) be a partially ordered set. Two mappings 𝐹,πΊβˆΆπ‘‹β†’π‘‹ are said to be weakly increasing if 𝐹π‘₯βͺ―𝐺𝐹π‘₯ and 𝐺π‘₯βͺ―𝐹𝐺π‘₯, for all π‘₯βˆˆπ‘‹.

Two weakly increasing mappings need not be nondecreasing.

Example 1.12 (see [16]). Let 𝑋=𝐑, endowed with the usual ordering. Let 𝐹,πΊβˆΆπ‘‹β†’π‘‹ defined by ξƒ―βŽ§βŽͺ⎨βŽͺ⎩√𝐹π‘₯=π‘₯,0≀π‘₯≀1,0,1<π‘₯<+∞,𝑔π‘₯=π‘₯,0≀π‘₯≀1,0,1<π‘₯<+∞.(1.6) Then 𝐹 and 𝐺 are weakly increasing mappings. Note that 𝐹 and 𝐺 are not nondecreasing.

The aim of this paper is to study a number of fixed point results for two weakly increasing mappings 𝑓 and 𝑔 with respect to partial ordering relation (βͺ―) in a generalized metric space.

2. Main Results

Theorem 2.1. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists 𝐺-metric in 𝑋 such that (𝑋,𝐺) is 𝐺-complete. Let 𝑓,π‘”βˆΆπ‘‹β†’π‘‹ be two weakly increasing mappings with respect to βͺ―. Suppose there exist nonnegative real numbers π‘Ž, 𝑏, and 𝑐 with π‘Ž+2𝑏+2𝑐<1 such that [][𝐺],𝐺(𝑓π‘₯,𝑔𝑦,𝑔𝑦)β‰€π‘ŽπΊ(π‘₯,𝑦,𝑦)+𝑏𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+𝐺(𝑦,𝑔𝑦,𝑔𝑦)+𝑐(π‘₯,𝑔𝑦,𝑔𝑦)+𝐺(𝑦,𝑓π‘₯,𝑓π‘₯)(2.1)[][],𝐺(𝑔π‘₯,𝑓𝑦,𝑓𝑦)β‰€π‘ŽπΊ(π‘₯,𝑦,𝑦)+𝑏𝐺(π‘₯,𝑔π‘₯,𝑔π‘₯)+𝐺(𝑦,𝑓𝑦,𝑓𝑦)+𝑐𝐺(π‘₯,𝑓𝑦,𝑓𝑦)+𝐺(𝑦,𝑔π‘₯,𝑔π‘₯)(2.2) for all comparative π‘₯,π‘¦βˆˆπ‘‹. If 𝑓 or 𝑔 is continuous, then 𝑓 and 𝑔 have a common fixed point π‘’βˆˆπ‘‹.

Proof. By inequality (2.2), we have [][𝐺].𝐺(𝑔𝑦,𝑓π‘₯,𝑓π‘₯)β‰€π‘ŽπΊ(𝑦,π‘₯,π‘₯)+𝑏𝐺(𝑦,𝑔𝑦,𝑔𝑦)+𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+𝑐(𝑦,𝑓π‘₯,𝑓π‘₯)+𝐺(π‘₯,𝑔𝑦,𝑔𝑦)(2.3) If 𝑋 is a symmetric 𝐺-metric space, then by adding inequalities (2.1) and (2.3), we obtain [𝐺][𝐺][],𝐺(𝑓π‘₯,𝑔𝑦,𝑔𝑦)+𝐺(𝑔𝑦,𝑓π‘₯,𝑓π‘₯)β‰€π‘Ž(π‘₯,𝑦,𝑦)+𝐺(𝑦,π‘₯,π‘₯)+2𝑏(π‘₯,𝑓π‘₯,𝑓π‘₯)+𝐺(𝑦,𝑔𝑦,𝑔𝑦)+2𝑐𝐺(π‘₯,𝑔𝑦,𝑔𝑦)+𝐺(𝑦,𝑓π‘₯,𝑓π‘₯)(2.4) which further implies that 𝑑𝐺(𝑓π‘₯,𝑓𝑦)β‰€π‘Žπ‘‘πΊξ€Ίπ‘‘(π‘₯,𝑦)+𝑏𝐺(π‘₯,𝑓π‘₯)+𝑑𝐺𝑑(𝑦,𝑔𝑦)+𝑐𝐺(π‘₯,𝑔𝑦)+𝑑𝐺,(𝑦,𝑓π‘₯)(2.5) for all π‘₯,π‘¦βˆˆπ‘‹ with 0β‰€π‘Ž+2𝑏+2𝑐<1 and the fixed point of 𝑓 and 𝑔 follows from [2].
Now if 𝑋 is not a symmetric 𝐺-metric space. Then by the definition of metric (𝑋,𝑑𝐺) and inequalities (2.1) and (2.3), we obtain𝑑𝐺[𝐺][𝐺][](𝑓π‘₯,𝑔𝑦)=𝐺(𝑓π‘₯,𝑔𝑦,𝑔𝑦)+𝐺(𝑔𝑦,𝑓π‘₯,𝑓π‘₯)β‰€π‘Ž(π‘₯,𝑦,𝑦)+𝐺(π‘₯,π‘₯,𝑦)+2𝑏(π‘₯,𝑓π‘₯,𝑓π‘₯)+𝐺(𝑦,𝑔𝑦,𝑔𝑦)+2𝑐𝐺(π‘₯,𝑔𝑦,𝑔𝑦)+𝐺(𝑦,𝑓π‘₯,𝑓π‘₯)β‰€π‘Žπ‘‘πΊξ‚ƒ2(π‘₯,𝑦)+2𝑏3𝑑𝐺2(π‘₯,𝑓π‘₯)+3𝑑𝐺2(𝑦,𝑔𝑦)+2𝑐3𝑑𝐺2(π‘₯,𝑔𝑦)+3𝑑𝐺(𝑦,𝑓π‘₯)=π‘Žπ‘‘πΊ4(π‘₯,𝑦)+3𝑏𝑑𝐺(π‘₯,𝑓π‘₯)+𝑑𝐺+4(𝑦,𝑔𝑦)3𝑐𝑑𝐺(π‘₯,𝑔𝑦)+𝑑𝐺,(𝑦,𝑓π‘₯)(2.6) for all π‘₯βˆˆπ‘‹. Here, the contractivity factor π‘Ž+(8/3)𝑏+(8/3)𝑐 may not be less than 1.
Therefore metric gives no information. In this case, for given π‘₯0βˆˆπ‘‹, choose π‘₯1βˆˆπ‘‹ such that π‘₯1=𝑓π‘₯0. Again choose π‘₯2βˆˆπ‘‹ such that 𝑔π‘₯1=π‘₯2. Also, we choose π‘₯3βˆˆπ‘‹ such that π‘₯3=𝑓π‘₯2. Continuing as above process, we can construct a sequence {π‘₯𝑛} in 𝑋 such that π‘₯2𝑛+1=𝑓π‘₯2𝑛, π‘›βˆˆπβˆͺ{𝟎} and π‘₯2𝑛+2=𝑔π‘₯2𝑛+1, π‘›βˆˆπβˆͺ{𝟎}. Since 𝑓 and 𝑔 are weakly increasing with respect to βͺ―, we have π‘₯1=𝑓π‘₯0ξ€·βͺ―𝑔𝑓π‘₯0ξ€Έ=𝑔π‘₯1=π‘₯2ξ€·βͺ―𝑓𝑔π‘₯1ξ€Έ=𝑓π‘₯2=π‘₯3ξ€·βͺ―𝑔𝑓π‘₯2ξ€Έ=𝑔π‘₯3=π‘₯4βͺ―β‹―.(2.7) Thus from (2.1), we have 𝐺π‘₯2𝑛+1,π‘₯2𝑛+2,π‘₯2𝑛+2ξ€Έξ€·=𝐺𝑓π‘₯2𝑛,𝑔π‘₯2𝑛+1,𝑔π‘₯2𝑛+1ξ€Έξ€·π‘₯β‰€π‘ŽπΊ2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1𝐺π‘₯+𝑏2𝑛,𝑓π‘₯2𝑛,𝑓π‘₯2𝑛π‘₯+𝐺2𝑛+1,𝑔π‘₯2𝑛+1,𝑔π‘₯2𝑛+1𝐺π‘₯ξ€Έξ€»+𝑐2𝑛,𝑔π‘₯2𝑛+1,𝑔π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝐺2𝑛+1,𝑓π‘₯2𝑛,𝑓π‘₯2𝑛π‘₯ξ€Έξ€»=π‘ŽπΊ2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1𝐺π‘₯+𝑏2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝐺2𝑛+1,π‘₯2𝑛+2,π‘₯2𝑛+2𝐺π‘₯ξ€Έξ€»+𝑐2𝑛,π‘₯2𝑛+2,𝑔π‘₯2𝑛+2ξ€Έξ€·π‘₯+𝐺2𝑛+1,π‘₯2𝑛+1,π‘₯2𝑛+1ξ€·π‘₯ξ€Έξ€»=(π‘Ž+𝑏)𝐺2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑏𝐺2𝑛+1,π‘₯2𝑛+2,π‘₯2𝑛+2ξ€Έξ€·π‘₯+𝑐𝐺2𝑛,π‘₯2𝑛+2,π‘₯2𝑛+2ξ€Έ.(2.8) By (𝐺5), we have 𝐺π‘₯2𝑛+1,π‘₯2𝑛+2,π‘₯2𝑛+2ξ€Έβ‰€π‘Ž+𝑏+𝑐𝐺π‘₯1βˆ’π‘βˆ’π‘2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έ.(2.9) Also, we have 𝐺π‘₯2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·=𝐺𝑔π‘₯2π‘›βˆ’1,𝑓π‘₯2𝑛,𝑓π‘₯2𝑛π‘₯β‰€π‘ŽπΊ2π‘›βˆ’1,π‘₯2𝑛,π‘₯2𝑛𝐺π‘₯+𝑏2π‘›βˆ’1,𝑔π‘₯2π‘›βˆ’1,𝑔π‘₯2π‘›βˆ’1ξ€Έξ€·π‘₯+𝐺2𝑛,𝑓π‘₯2𝑛,𝑓π‘₯2𝑛𝐺π‘₯ξ€Έξ€»+𝑐2π‘›βˆ’1,𝑓π‘₯2𝑛,𝑓π‘₯2𝑛π‘₯+𝐺2𝑛,𝑔π‘₯2π‘›βˆ’1,𝑔π‘₯2π‘›βˆ’1ξ€·π‘₯ξ€Έξ€»=π‘ŽπΊ2π‘›βˆ’1,π‘₯2𝑛,π‘₯2𝑛𝐺π‘₯+𝑏2π‘›βˆ’1,π‘₯2𝑛,π‘₯2𝑛π‘₯+𝐺2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1𝐺π‘₯ξ€Έξ€»+𝑐2π‘›βˆ’1,π‘₯2𝑛+1,𝑔π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝐺2𝑛,π‘₯2𝑛,π‘₯2𝑛π‘₯ξ€Έξ€»=(π‘Ž+𝑏)𝐺2π‘›βˆ’1,π‘₯2𝑛,π‘₯2𝑛π‘₯+𝑏𝐺2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έξ€·π‘₯+𝑐𝐺2π‘›βˆ’1,π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έ.(2.10) By (𝐺5), we get 𝐺π‘₯2𝑛,π‘₯2𝑛+1,π‘₯2𝑛+1ξ€Έβ‰€π‘Ž+𝑏+𝑐𝐺π‘₯1βˆ’π‘βˆ’π‘2π‘›βˆ’1,π‘₯2𝑛,π‘₯2𝑛.(2.11) Let π‘˜=π‘Ž+𝑏+𝑐.1βˆ’π‘βˆ’π‘(2.12) Then by (2.9) and (2.11), we have 𝐺π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯β‰€π‘˜πΊπ‘›βˆ’1,π‘₯𝑛,π‘₯𝑛,βˆ€π‘›βˆˆπ.(2.13) Thus, if π‘₯0=π‘₯1, we get 𝐺(π‘₯𝑛,π‘₯𝑛+1,π‘₯𝑛+1)=0 for each π‘›βˆˆπ. Hence π‘₯𝑛=π‘₯0 for each π‘›βˆˆπ. Therefore {π‘₯𝑛} is 𝐺-Cauchy. So we may assume that π‘₯0β‰ π‘₯1. Let 𝑛,π‘šβˆˆπ with π‘š>𝑛. By axiom (𝐺5) of the definition of 𝐺-metric space, we have 𝐺π‘₯𝑛,π‘₯π‘š,π‘₯π‘šξ€Έξ€·π‘₯≀𝐺𝑛,π‘₯𝑛+1,π‘₯𝑛+1ξ€Έξ€·π‘₯+𝐺𝑛+1,π‘₯𝑛+2,π‘₯𝑛+2ξ€Έξ€·π‘₯+β‹―+πΊπ‘šβˆ’1,π‘₯π‘š,π‘₯π‘šξ€Έ.(2.14) By (2.13), we get 𝐺π‘₯𝑛,π‘₯π‘š,π‘₯π‘šξ€Έβ‰€π‘˜π‘›πΊξ€·π‘₯0,π‘₯1,π‘₯1ξ€Έ+π‘˜π‘›+1𝐺π‘₯0,π‘₯1,π‘₯1ξ€Έ+β‹―+π‘˜π‘šβˆ’1𝐺π‘₯0,π‘₯1,π‘₯1ξ€Έβ‰€π‘˜π‘›πΊξ€·π‘₯1βˆ’π‘˜0,π‘₯1,π‘₯1ξ€Έ.(2.15) On taking limit π‘š,π‘›β†’βˆž, we have limπ‘š,π‘›β†’βˆžπΊξ€·π‘₯𝑛,π‘₯π‘š,π‘₯π‘šξ€Έ=0.(2.16) So we conclude that (π‘₯𝑛) is a Cauchy sequence in 𝑋. Since 𝑋 is 𝐺-complete, then it yields that (π‘₯𝑛) and hence any subsequence of (π‘₯𝑛) converges to some π‘’βˆˆπ‘‹. So that, the subsequences (π‘₯2𝑛+1)=(𝑓π‘₯2𝑛) and (π‘₯2𝑛+2)=(𝑔π‘₯2𝑛+1) converge to 𝑒. First suppose that 𝑓 is 𝐺-continuous. Since (π‘₯2𝑛) converges to 𝑒, we get (𝑓π‘₯2𝑛) converges 𝑓𝑒. By the uniqueness of limit we get 𝑓𝑒=𝑒. Claim: 𝑔𝑒=𝑒.
Since 𝑒βͺ―𝑒, by inequality (2.1), we have [𝐺][]𝐺(𝑒,𝑔𝑒,𝑔𝑒)=𝐺(𝑓𝑒,𝑔𝑒,𝑔𝑒)β‰€π‘ŽπΊ(𝑒,𝑒,𝑒)+𝑏(𝑒,𝑓𝑒,𝑓𝑒)+𝐺(𝑒,𝑔𝑒,𝑔𝑒)+𝑐𝐺(𝑒,𝑔𝑒,𝑔𝑒)+𝐺(𝑒,𝑓𝑒,𝑓𝑒)≀(𝑏+𝑐)𝐺(𝑒,𝑔𝑒,𝑔𝑒).(2.17) Since 𝑏+𝑐<1, we get 𝐺(𝑒,𝑔𝑒,𝑔𝑒)=0. Hence 𝑔𝑒=𝑒. If 𝑔 is 𝐺-continuous, by similar argument as above we show that 𝑔 and 𝑓 have a common fixed point.

Theorem 2.2. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists 𝐺-metric in 𝑋 such that (𝑋,𝐺) is 𝐺-complete. Let 𝑓,π‘”βˆΆπ‘‹β†’π‘‹ be two weakly increasing mappings with respect to βͺ―. Suppose there exist nonnegative real numbers π‘Ž, 𝑏, and 𝑐 with π‘Ž+2𝑏+2𝑐<1 such that [][𝐺],[][],𝐺(𝑓π‘₯,𝑔𝑦,𝑔𝑦)β‰€π‘ŽπΊ(π‘₯,𝑦,𝑦)+𝑏𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+𝐺(𝑦,𝑔𝑦,𝑔𝑦)+𝑐(π‘₯,𝑔𝑦,𝑔𝑦)+𝐺(𝑦,𝑓π‘₯,𝑓π‘₯)𝐺(𝑔π‘₯,𝑓𝑦,𝑓𝑦)β‰€π‘ŽπΊ(π‘₯,𝑦,𝑦)+𝑏𝐺(π‘₯,𝑔π‘₯,𝑔π‘₯)+𝐺(𝑦,𝑓𝑦,𝑓𝑦)+𝑐𝐺(π‘₯,𝑓𝑦,𝑓𝑦)+𝐺(𝑦,𝑔π‘₯,𝑔π‘₯)(2.18) for all comparative π‘₯,π‘¦βˆˆπ‘‹. Assume that 𝑋 has the following property: (𝑃)If (π‘₯𝑛) is an increasing sequence converges to π‘₯ in 𝑋, then π‘₯𝑛βͺ―π‘₯ for all π‘›βˆˆπ.Then 𝑓 and 𝑔 have a common fixed point π‘’βˆˆπ‘‹.

Proof. As in the proof of Theorem 3.1, we construct an increasing sequence (π‘₯𝑛) in 𝑋 such that π‘₯2𝑛+1=𝑓π‘₯2𝑛 and π‘₯2𝑛+2=𝑔π‘₯2𝑛+1. Also, we can show (π‘₯𝑛) is 𝐺-Cauchy. Since 𝑋 is 𝐺-complete, there is π‘’βˆˆπ‘‹ such that (π‘₯𝑛) is converges to π‘’βˆˆπ‘‹. Thus (π‘₯2𝑛),(π‘₯2𝑛+1),(𝑓π‘₯2𝑛) and (𝑔π‘₯2𝑛+1) converge to 𝑒. Since 𝑋 satisfies property (𝑃), we get that π‘₯𝑛βͺ―𝑒, for all π‘›βˆˆπ. Thus π‘₯2𝑛 and 𝑒 are comparative. Hence by inequality (2.1), we have 𝐺𝑓π‘₯2𝑛π‘₯,𝑔𝑒,π‘”π‘’β‰€π‘ŽπΊ2𝑛𝐺π‘₯,𝑒,𝑒+𝑏2𝑛,𝑓π‘₯2𝑛,𝑓π‘₯2𝑛𝐺π‘₯+𝐺(𝑒,𝑔𝑒,𝑔𝑒)+𝑐2𝑛,𝑔𝑒,𝑔𝑒+𝐺𝑒,𝑓π‘₯2𝑛,𝑓π‘₯2𝑛.ξ€Έξ€»(2.19) On letting 𝑛→+∞, we get 𝐺(𝑒,𝑔𝑒,𝑔𝑒)≀(𝑏+𝑐)𝐺(𝑒,𝑔𝑒,𝑔𝑒).(2.20) Since 𝑏+𝑐<1, we get 𝐺(𝑒,𝑔𝑒,𝑔𝑒)=0. Hence 𝑔𝑒=𝑒. By similar argument, we may show that 𝑒=𝑓𝑒.

Corollary 2.3. Let (𝑋,βͺ―) be a partially ordered set, and suppose that (𝑋,𝐺) is a 𝐺-complete metric space. Let π‘“βˆΆπ‘‹β†’π‘‹ be a continuous mapping such that 𝑓π‘₯βͺ―𝑓(𝑓π‘₯), for all π‘₯βˆˆπ‘‹. Suppose there exist nonnegative real numbers π‘Ž, 𝑏 and 𝑐 with π‘Ž+2𝑏+2𝑐<1 such that [][𝐺],𝐺(𝑓π‘₯,𝑓𝑦,𝑓𝑦)β‰€π‘ŽπΊ(π‘₯,𝑦,𝑦)+𝑏𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+𝐺(𝑦,𝑓𝑦,𝑓𝑦)+𝑐(π‘₯,𝑓𝑦,𝑓𝑦)+𝐺(𝑦,𝑓π‘₯,𝑓π‘₯)(2.21) for all comparative π‘₯,π‘¦βˆˆπ‘‹. Then 𝑓 has a fixed point π‘’βˆˆπ‘‹.

Proof. It follows from Theorem 2.1 by taking 𝑔=𝑓.

Corollary 2.4. Let (𝑋,βͺ―) be a partially ordered set and suppose that there exists 𝐺-metric in 𝑋 such that (𝑋,𝐺) is 𝐺-complete. Let π‘“βˆΆπ‘‹β†’π‘‹ be a mapping such that 𝑓π‘₯βͺ―𝑓(𝑓π‘₯) for all π‘₯βˆˆπ‘‹. Suppose there exist nonnegative real numbers π‘Ž, 𝑏 and 𝑐 with π‘Ž+2𝑏+2𝑐<1 such that [][𝐺]𝐺(𝑓π‘₯,𝑓𝑦,𝑓𝑦)β‰€π‘ŽπΊ(π‘₯,𝑦,𝑦)+𝑏𝐺(π‘₯,𝑓π‘₯,𝑓π‘₯)+𝐺(𝑦,𝑓𝑦,𝑓𝑦)+𝑐(π‘₯,𝑓𝑦,𝑓𝑦)+𝐺(𝑦,𝑓π‘₯,𝑓π‘₯)(2.22) for all comparative π‘₯,π‘¦βˆˆπ‘‹. Assume that 𝑋 has the following property: (𝑃)If (π‘₯𝑛) is an increasing sequence converges to π‘₯ in 𝑋, then π‘₯𝑛βͺ―π‘₯ for all π‘›βˆˆπ.Then 𝑓 has fixed point π‘’βˆˆπ‘‹.

Proof. It follows from Theorem 2.2 by taking 𝑔=𝑓.

3. Application

Consider the integral equation:ξ€œπ‘’(𝑑)=𝑇0[],𝐾(𝑑,𝑠,𝑒(𝑠))𝑑𝑠+𝑔(𝑑),π‘‘βˆˆ0,𝑇(3.1) where 𝑇>0. The aim of this section is to give an existence theorem for a solution of the above integral equation using Corollary 2.4 This section is related to those [16–19].

Let 𝑋=𝐢([0,𝑇]) be the set of all continuous functions defined on [0,𝑇]. Define πΊβˆΆπ‘‹Γ—π‘‹Γ—π‘‹β†’π‘+(3.2) by 𝐺(π‘₯,𝑦,𝑧)=sup[]π‘‘βˆˆ0,𝑇||||π‘₯(𝑑)βˆ’π‘¦(𝑑)+sup[]π‘‘βˆˆ0,𝑇||||π‘₯(𝑑)βˆ’π‘§(𝑑)+sup[]π‘‘βˆˆ0,𝑇||||.𝑦(𝑑)βˆ’π‘§(𝑑)(3.3) Then (𝑋,𝐺) is a 𝐺-complete metric space. Define an ordered relation βͺ― on 𝑋 by π‘₯βͺ―𝑦iff[].π‘₯(𝑑)≀𝑦(𝑑),βˆ€π‘‘βˆˆ0,𝑇(3.4) Then (𝑋,βͺ―) is a partially ordered set. The purpose of this section is to give an existence theorem for solution of integral equation on (3.1). This section is inspired in [17–19].

Theorem 3.1. Suppose the following hypotheses hold: (1)𝐾∢[0,𝑇]Γ—[0,𝑇]×𝐑→𝐑 and π‘”βˆΆπ‘β†’π‘ are continuous, (2)for each 𝑑,π‘ βˆˆ[0,𝑇], one has ξ‚΅ξ€œπΎ(𝑑,𝑠,𝑒(𝑑))≀𝐾𝑑,𝑠,𝑇0ξ‚Ά,𝐾(𝑠,𝜏,𝑒(𝜏))π‘‘πœ+𝑔(𝑠)(3.5)(3)there exists a continuous function 𝐺∢[0,𝑇]Γ—[0,𝑇]β†’[0,+∞] such that ||||𝐾(𝑑,𝑠,𝑒)βˆ’πΎ(𝑑,𝑠,𝑣)≀𝐺(𝑑,𝑠)|π‘’βˆ’π‘£|,(3.6) for each comparable 𝑒,π‘£βˆˆπ‘ and each 𝑑,π‘ βˆˆ[0,𝑇], (4)supπ‘‘βˆˆ[0,𝑇]βˆ«π‘‡0𝐺(𝑑,𝑠)π‘‘π‘ β‰€π‘Ÿ for some π‘Ÿ<1. Then the integral equation (3.1) has a solution π‘’βˆˆπΆ([0,𝑇]).

Proof. Define π‘†βˆΆπΆ([0,𝑇])→𝐢([0,𝑇]) by ξ€œπ‘†π‘₯(𝑑)=𝑇0[].𝐾(𝑑,𝑠,π‘₯(𝑠))𝑑𝑠+𝑔(𝑑),π‘‘βˆˆ0,𝑇(3.7) Now, we have ξ€œπ‘†π‘₯(𝑑)=𝑇0β‰€ξ€œπΎ(𝑑,𝑠,π‘₯(𝑠))𝑑𝑠+𝑔(𝑑)𝑇0πΎξ‚΅ξ€œπ‘‘,𝑠,𝑇0ξ‚Ά=ξ€œπΎ(𝑠,𝜏,π‘₯(𝜏))π‘‘πœ+𝑔(𝑠)𝑑𝑠+𝑔(𝑑)𝑇0𝐾(𝑑,𝑠,𝑆π‘₯(𝑠))𝑑𝑠+𝑔(𝑑)=𝑆(𝑆π‘₯(𝑑)).(3.8) Thus, we have 𝑆π‘₯βͺ―𝑆(𝑆π‘₯), for all π‘₯∈𝐢([0,𝑇]).
For π‘₯,π‘¦βˆˆπΆ([0,𝑇]) with π‘₯βͺ―𝑦, we have 𝐺(𝑆π‘₯,𝑆𝑦,𝑆𝑦)=2supπ‘‘βˆˆ[0,𝑇]||||𝑆π‘₯(𝑑)βˆ’π‘†π‘¦(𝑑)=2supπ‘‘βˆˆ[0,𝑇]||||ξ€œπ‘‡0||||𝐾(𝑑,𝑠,π‘₯(𝑠))βˆ’πΎ(𝑑,𝑠,𝑦(𝑠))𝑑𝑠≀2supπ‘‘βˆˆ[0,𝑇]ξ€œπ‘‡0||||𝐾(𝑑,𝑠,π‘₯(𝑠))βˆ’πΎ(𝑑,𝑠,𝑦(𝑠))𝑑𝑠≀2sup[]π‘‘βˆˆ0,π‘‡ξ€œπ‘‡0||||𝐺(𝑑,𝑠)π‘₯(𝑠)βˆ’π‘¦(𝑠)𝑑𝑠≀2supπ‘‘βˆˆ[0,𝑇]||||π‘₯(𝑑)βˆ’π‘¦(𝑑)sup[]π‘‘βˆˆ0,π‘‡ξ€œπ‘‡0𝐺(𝑑,𝑠)𝑑𝑠=𝐺(π‘₯,𝑦,𝑦)sup[]π‘‘βˆˆ0,π‘‡ξ€œπ‘‡0𝐺(𝑑,𝑠)𝑑𝑠.(3.9) By using hypotheses (4), there is π‘Ÿβˆˆ[0,1) such that supπ‘‘βˆˆ[0,𝑇]ξ€œπ‘‡0𝐺(𝑑,𝑠)𝑑𝑠<π‘Ÿ.(3.10) Thus, we have 𝐺(𝑆π‘₯,𝑆𝑦,𝑆𝑦)β‰€π‘ŸπΊ(π‘₯,𝑦,𝑦). Thus all the required hypotheses of Corollary 2.4 are satisfied. Thus there exist a solution π‘’βˆˆπΆ([0,𝑇]) of the integral equation (3.1).

Acknowledgments

The author would like to thank the editor of the paper and the referees for their precise remarks to correct and improve the paper. Also, the author would like to thank the hashemite university for the financial assistant of this paper.

References

  1. A. C. M. Ran and M. C. B. Reurings, β€œA fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan, β€œGeneralized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. 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
  4. J. J. Nieto and R. Rodríguez-López, β€œExistence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. J. J. Nieto, R. L. Pouso, and R. Rodríguez-López, β€œFixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2505–2517, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. D. O'Regan and A. Petruşel, β€œFixed point theorems for generalized contractions in ordered metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 2505–2517, 2007. View at Publisher Β· View at Google Scholar
  7. Z. Mustafa and B. Sims, β€œA new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006. View at Google Scholar Β· View at Zentralblatt MATH
  8. Z. Mustafa and B. Sims, β€œSome remarks concerning D-metric spaces,” in Proceedings of the International Conference on Fixed Point Theory and Applications, pp. 189–198, Valencia, Spain, July 2003.
  9. Z. Mustafa, W. Shatanawi, and M. Bataineh, β€œExistence of fixed point results in G-metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 283028, 10 pages, 2009. View at Publisher Β· View at Google Scholar
  10. Z. Mustafa, H. Obiedat, and F. Awawdeh, β€œSome fixed point theorem for mapping on complete G-metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 189870, 12 pages, 2008. View at Publisher Β· View at Google Scholar
  11. M. Abbas and B. E. Rhoades, β€œCommon fixed point results for noncommuting mappings without continuity in generalized metric spaces,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 262–269, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. R. Chugh, T. Kadian, A. Rani, and B. E. Rhoades, β€œProperty p in G-metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 401684, 12 pages, 2010. View at Publisher Β· View at Google Scholar
  13. R. Saadati, S. M. Vaezpour, P. Vetro, and B. E. Rhoades, β€œFixed point theorems in generalized partially ordered G-metric spaces,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 797–801, 2010. View at Publisher Β· View at Google Scholar
  14. W. Shatanawi, β€œFixed point theory for contractive mappings satisfying Φ-maps in Gmetric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 181650, 9 pages, 2010. View at Publisher Β· View at Google Scholar
  15. H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, β€œCoupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces,” Mathematical and Computer Modelling. In press. View at Publisher Β· View at Google Scholar
  16. 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 Google Scholar Β· View at Zentralblatt MATH
  17. B. Ahmed and J. J. Nieto, β€œThe monotone iterative technique for three-point secondorder integrodifferential boundary value problems with p-Laplacian,” Boundary Value Problems, vol. 2007, Article ID 57481, 9 pages, 2007. View at Publisher Β· View at Google Scholar
  18. A. Cabada and J. J. Nieto, β€œFixed points and approximate solutions for nonlinear operator equations,” Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 17–25, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. J. J. Nieto, β€œAn abstract monotone iterative technique,” Nonlinear Analysis: Theory Method and Applications, vol. 28, no. 12, pp. 1923–1933, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH