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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 394830, 15 pages
http://dx.doi.org/10.1155/2012/394830
Research Article

Common Fixed Point for Two Pairs of Mappings Satisfying (E.A) Property in Generalized Metric Spaces

1College of Mathematics and Information Science, Jiangxi Normal University, Jiangxi, Nanchang 330022, China
2Department of Mathematics, Lahore University of Management Science, Lahore 54792, Pakistan
3Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia

Received 3 July 2012; Accepted 18 August 2012

Academic Editor: Simeon Reich

Copyright © 2012 Wei Long 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

Recently, Abbas et al. (2012) obtained some unique common fixed-point results for a pair of mappings satisfying (E.A) property under certain generalized strict contractive conditions in the framework of a generalized metric space. In this paper, we present common coincidence and common fixed points of two pairs of mappings when only one pair satisfies (E.A) property in the setup of generalized metric spaces. We present some examples to support our results. We also study well-posedness of common fixed-point problem.

1. Introduction and Preliminaries

Mustafa and Sims [1] generalized the concept of a metric in which the real number is assigned to every triplet of an arbitrary set. Based on the notion of generalized metric spaces, Mustafa et al. [26] obtained some fixed-point theorems for mappings satisfying different contractive conditions. Chugh et al. [7] obtained some fixed-point results for maps satisfying property 𝑝 in 𝐺-metric spaces. Saadati et al. [8] studied fixed-point of contractive mappings in partially ordered 𝐺-metric spaces. Shatanawi [9] obtained fixed-points of Φ-maps in 𝐺-metric spaces. Study of common fixed-point theorems in generalized metric spaces was initiated by Abbas and Rhoades [10] (see also, [1114]). Recently, Abbas et al. [15] obtained some unique common fixed-point results for a pair of mappings satisfying (E.A) property under certain generalized strict contractive conditions in the framework of a generalized metric space.

The aim of this paper is to study common fixed-point of two pairs of mappings for which only one pair needs to satisfy (E.A) property in the framework of 𝐺-metric spaces. Our results do not rely on any commuting or continuity condition of mappings.

Consistent with Mustafa and Sims [2], the following definitions and results will be needed in the sequel.

Definition 1.1. Let 𝑋 be a nonempty set. Suppose that a mapping 𝐺𝑋×𝑋×𝑋𝑅+ satisfies the following:𝐆1:𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧;𝐆2:0<𝐺(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧𝑋, with 𝑥𝑦;𝐆3:𝐺(𝑥,𝑥,𝑦)𝐺(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧𝑋, with 𝑦𝑧;𝐆4:𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑦,𝑧,𝑥)= (symmetry in all three variables);𝐆5:𝐺(𝑥,𝑦,𝑧)𝐺(𝑥,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧) for all 𝑥,𝑦,𝑧,𝑎𝑋.
Then 𝐺 is called a 𝐺-metric on 𝑋 and (𝑋,𝐺) is called a 𝐺-metric.

Definition 1.2. A sequence {𝑥𝑛} in a 𝐺-metric space 𝑋 is(i)a 𝐺-Cauchy sequence if, for any 𝜀>0, there is an 𝑛0𝑁 (the set of natural numbers) such that for all 𝑛,𝑚,𝑙𝑛0,𝐺(𝑥𝑛,𝑥𝑚,𝑥𝑙)<𝜀,(ii)a 𝐺-convergent sequence if, for any 𝜀>0, there is an 𝑥𝑋 and an 𝑛0𝑁, such that for all 𝑛,𝑚𝑛0,𝐺(𝑥,𝑥𝑛,𝑥𝑚)<𝜀.
A 𝐺-metric space on 𝑋 is said to be 𝐺-complete if every 𝐺-Cauchy sequence in 𝑋 is 𝐺-convergent in 𝑋. It is known that {𝑥𝑛}  𝐺-converges to 𝑥𝑋 if and only if 𝐺(𝑥𝑚,𝑥𝑛,𝑥)0 as 𝑛,𝑚.

Proposition 1.3. Let 𝑋 be a 𝐺-metric space. Then the following are equivalent:(1){𝑥𝑛} is 𝐺-convergent to 𝑥.(2)𝐺(𝑥𝑛,𝑥𝑚,𝑥)0 as 𝑛,𝑚.(3)𝐺(𝑥𝑛,𝑥𝑛,𝑥)0 as 𝑛.(4)𝐺(𝑥𝑛,𝑥,𝑥)0 as 𝑛.

Definition 1.4. A 𝐺-metric on 𝑋 is said to be symmetric if 𝐺(𝑥,𝑦,𝑦)=𝐺(𝑦,𝑥,𝑥) for all 𝑥,𝑦𝑋.

Proposition 1.5. Every 𝐺-metric on 𝑋 will define a metric 𝑑𝐺 on 𝑋 by 𝑑𝐺(𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦)+𝐺(𝑦,𝑥,𝑥),𝑥,𝑦𝑋.(1.1) For a symmetric 𝐺-metric 𝑑𝐺(𝑥,𝑦)=2𝐺(𝑥,𝑦,𝑦),𝑥,𝑦𝑋.(1.2) However, if 𝐺 is nonsymmetric, then the following inequality holds: 32𝐺(𝑥,𝑦,𝑦)𝑑𝐺(𝑥,𝑦)3𝐺(𝑥,𝑦,𝑦),𝑥,𝑦𝑋.(1.3) It is also obvious that 𝐺(𝑥,𝑥,𝑦)2𝐺(𝑥,𝑦,𝑦).(1.4) Now, we give an example of a nonsymmetric 𝐺-metric.

Example 1.6. Let 𝑋={1,2} and a mapping 𝐺𝑋×𝑋×𝑋+ be defined as shown in Table 1.

tab1
Table 1

Note that 𝐺 satisfies all the axioms of a generalized metric but 𝐺(𝑥,𝑥,𝑦)𝐺(𝑥,𝑦,𝑦) for distinct𝑥,𝑦 in 𝑋. Therefore, 𝐺is a nonsymmetric 𝐺-metric on 𝑋.

Sessa [16] introduced the notion of the weak commutativity of mappings in metric spaces.

Definition 1.7 (see [13]). Let 𝑋 be a 𝐺-metric space. Mappings 𝑓, 𝑔𝑋𝑋 are called (i) compatible if, whenever a sequence {𝑥𝑛} in 𝑋 is such that {𝑓𝑥𝑛} and {𝑔𝑥𝑛} are 𝐺-convergent to some 𝑡𝑋, then lim𝑛𝐺(𝑓𝑔𝑥𝑛,𝑓𝑔𝑥𝑛,𝑔𝑓𝑥𝑛)=0, (ii) noncompatible if there exists at least one sequence {𝑥𝑛} in 𝑋 such that {𝑓𝑥𝑛} and {𝑔𝑥𝑛} are 𝐺-convergent to some 𝑡𝑋, but lim𝑛𝐺(𝑓𝑔𝑥𝑛,𝑓𝑔𝑥𝑛,𝑔𝑓𝑥𝑛) is either nonzero or does not exist.
Jungck [17] defined 𝑓 and 𝑔 to be weakly compatible if 𝑓𝑥=𝑔𝑥 implies 𝑓𝑔𝑥=𝑔𝑓𝑥.
In 2002, Aamri and Moutaawakil [18] introduced (E.A) property to obtain common fixed-point of two mappings. Recently, Babu and Negash [19] employed this concept to obtain some new common fixed-point results (see also [2022]).
Recently, Abbas et al. [15] studied (E.A) property in the frame work of 𝐺-metric space.

Definition 1.8 (see [15]). Let 𝑋 be a 𝐺-metric space. Self-maps 𝑓 and 𝑔 on 𝑋 are said to satisfy the (E.A) property if there exists a sequence {𝑥𝑛} in 𝑋 such that {𝑓𝑥𝑛} and {𝑔𝑥𝑛} are 𝐺-convergent to some 𝑡𝑋.

Example 1.9 (see [15]). Let 𝑋=[0,2] be a 𝐺-metric space with 𝐺||||,||||(𝑥,𝑦,𝑧)=max𝑥𝑦|𝑥𝑧|,𝑦𝑧.(1.5) Let 𝑓,𝑔𝑋𝑋 be defined by [],𝑓𝑥=2𝑥,𝑥0,12𝑥2],,𝑥(1,2𝑔𝑥=3𝑥2[],𝑥,𝑥0,12].,𝑥(1,2(1.6) For a decreasing sequence {𝑥𝑛} in 𝑋 such that 𝑥𝑛1,𝑔𝑥𝑛1/2,𝑓𝑥𝑛1/2,𝑔𝑓𝑥𝑛=(4+𝑥𝑛)/45/4 and 𝑓𝑔𝑥𝑛=(4𝑥𝑛)23/2. So, 𝑓 and 𝑔 are noncompatible. Note that, there exists a sequence {𝑥𝑛} in 𝑋 such that lim𝑛𝑓𝑥𝑛=lim𝑛𝑔𝑥𝑛=1𝑋, take 𝑥𝑛=1 for each 𝑛𝑁. Hence 𝑓 and 𝑔 satisfy (E.A) property.

Definition 1.10 (see [23]). The control functions 𝜓 and 𝜙 are defined as follows:(a)𝜓[0,)[0,) is a continuous nondecreasing function with 𝜓(𝑡)=0 if and only if 𝑡=0,(b)𝜙[0,)[0,) is a lower semicontinuous function with 𝜙(𝑡)=0 if and only if 𝑡=0.

2. Common Fixed-Point Theorems

In this section, we obtain some common fixed-point results for two pairs of mappings satisfying certain contractive conditions in the frame work of a generalized metric space. It is worth mentioning to note that, one needs (E.A) property of only one pair to prove the existence of coincidence point of mappings involved therein. We start with the following result.

Theorem 2.1. Let 𝑋 be a 𝐺-metric space and 𝑓,𝑔,𝑆,𝑇𝑋𝑋 be mappings with 𝑓𝑋𝑇𝑋 and 𝑔𝑋𝑆𝑋 such that [𝐺]𝜓(𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦))𝜓(𝑀(𝑥,𝑦,𝑦))𝜙(𝑀(𝑥,𝑦,𝑦)),where𝑀(𝑥,𝑦,𝑦)=max𝐺(𝑆𝑥,𝑇𝑦,𝑇𝑦),𝐺(𝑓𝑥,𝑆𝑥,𝑆𝑥),𝐺(𝑇𝑦,𝑔𝑦,𝑔𝑦),(𝑓𝑥,𝑇𝑦,𝑇𝑦)+𝐺(𝑆𝑥,𝑔𝑦,𝑔𝑦)2(2.1) or [𝐺]𝜓(𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦))𝜓(𝑀(𝑥,𝑥,𝑦))𝜙(𝑀(𝑥,𝑥,𝑦))where𝑀(𝑥,𝑥,𝑦)=max𝐺(𝑆𝑥,𝑆𝑥,𝑇𝑦),𝐺(𝑓𝑥,𝑓𝑥,𝑆𝑥),𝐺(𝑇𝑦,𝑇𝑦,𝑔𝑦),(𝑓𝑥,𝑓𝑥,𝑇𝑦)+𝐺(𝑆𝑥,𝑆𝑥,𝑔𝑦)2(2.2) hold for all 𝑥,𝑦𝑋, where 𝜓 and 𝜙 are control functions. Suppose that one of the pairs (𝑓,𝑆) and (𝑔,𝑇) satisfies (E.A) property and one of the subspace 𝑓(𝑋),𝑔(𝑋),𝑆(𝑋),𝑇(𝑋) is closed in 𝑋. If for every sequence {𝑦𝑛} in 𝑋, one of the following conditions hold:(a){𝑔𝑦𝑛} is bounded in case (𝑓,𝑆) satisfies (E.A) property, (b){𝑓𝑦𝑛} is bounded in case (𝑔,𝑇) satisfies (E.A) property.
Then, the pairs (𝑓,𝑆) and (𝑔,𝑇) have a common point of coincidence in 𝑋. Moreover, if the pairs (𝑓,𝑆) and (𝑔,𝑇) are weakly compatible, then 𝑓,𝑔,𝑆, and 𝑇 have a unique common fixed-point.

Proof. Suppose that the pair (𝑓,𝑆) satisfies (E.A) property, there exists a sequence {𝑥𝑛} in 𝑋 satisfying lim𝑛𝑓𝑥𝑛=lim𝑛𝑆𝑥𝑛=𝑞 for some 𝑞𝑋. As 𝑓𝑋𝑇𝑋, there exists a sequence {𝑦𝑛} in 𝑋 such that 𝑓𝑥𝑛=𝑇𝑦𝑛. As {𝑔𝑦𝑛} is bounded, lim𝑛𝐺(𝑓𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛) and lim𝑛𝐺(𝑆𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛) are finite numbers. Note that ||𝐺𝑓𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛𝐺𝑆𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛||2𝐺𝑓𝑥𝑛,𝑆𝑥𝑛,𝑆𝑥𝑛.(2.3) Since 𝐺(𝑓𝑥𝑛,𝑆𝑥𝑛,𝑆𝑥𝑛)0 as 𝑛, therefore lim𝑛𝐺(𝑓𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛)=lim𝑛𝐺(𝑆𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛). Indeed, using that lim𝑛𝐺(𝑓𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛)=lim𝑛𝐺(𝑆𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛)=𝑙, we obtain subsequences {𝑥𝑛𝑘} and {𝑦𝑛𝑘} such that 𝐺(𝑆𝑥𝑛𝑘,𝑔𝑦𝑛𝑘,𝑔𝑦𝑛𝑘) and 𝐺(𝑓𝑥𝑛𝑘,𝑔𝑦𝑛𝑘,𝑔𝑦𝑛𝑘) are 𝐺-convergent to 𝑙. Replacing 𝑥 by 𝑥𝑛𝑘 and 𝑦 by 𝑦𝑛𝑘 in (2.1), we have 𝑀𝑥𝑛𝑘,𝑦𝑛𝑘,𝑦𝑛𝑘𝐺=max𝑆𝑥𝑛𝑘,𝑓𝑥𝑛𝑘,𝑓𝑥𝑛𝑘,𝐺𝑓𝑥𝑛𝑘,𝑆𝑥𝑛𝑘,𝑆𝑥𝑛𝑘,𝐺𝑓𝑥𝑛𝑘,𝑔𝑦𝑛𝑘,𝑔𝑦𝑛𝑘,𝐺𝑓𝑥𝑛𝑘,𝑓𝑥𝑛𝑘,𝑓𝑥𝑛𝑘+𝐺𝑆𝑥𝑛𝑘,𝑔𝑦𝑛𝑘,𝑔𝑦𝑛𝑘2(2.4) which on taking limit as 𝑘 implies that lim𝑛𝑀𝑥𝑛𝑘,𝑦𝑛𝑘,𝑦𝑛𝑘𝑙=max𝐺(𝑞,𝑞,𝑞),𝐺(𝑞,𝑞,𝑞),𝑙,2=𝑙.(2.5) Now 𝜓𝐺𝑓𝑥𝑛𝑘,𝑔𝑦𝑛𝑘,𝑔𝑦𝑛𝑘𝑀𝑥𝜓𝑛𝑘,𝑦𝑛𝑘,𝑦𝑛𝑘𝑀𝑥𝜙𝑛𝑘,𝑦𝑛𝑘,𝑦𝑛𝑘(2.6) which on taking upper limit gives 𝜓(𝑙)𝜓(𝑙)𝜙(𝑙),(2.7) and so 𝑙=0. Hence, lim𝑛𝐺(𝑓𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛)=lim𝑛𝐺(𝑆𝑥𝑛,𝑔𝑦𝑛,𝑔𝑦𝑛)=0, and so, lim𝑛𝑔𝑦𝑛=𝑞.
If 𝑇(𝑋) is a closed subspace of 𝑋. Then, there exist a 𝑝 in 𝑋 such that 𝑞=𝑇𝑝. From (2.1), we have 𝜓𝐺𝑓𝑥𝑛𝑀𝑥,𝑔𝑝,𝑔𝑝𝜓𝑛𝑀𝑥,𝑝,𝑝𝜙𝑛,𝑝,𝑝,(2.8) where 𝑀𝑥𝑛𝐺,𝑝,𝑝=max𝑆𝑥𝑛,𝑇𝑝,𝑇𝑝,𝐺𝑓𝑥𝑛,𝑆𝑥𝑛,𝑆𝑥𝑛𝐺,𝐺(𝑇𝑝,𝑔𝑝,𝑔𝑝),𝑓𝑥𝑛,𝑇𝑝,𝑇𝑝+𝐺𝑆𝑥𝑛,𝑔𝑝,𝑔𝑝2𝐺=max𝑆𝑥𝑛,𝑞,𝑞,𝐺𝑓𝑥𝑛,𝑆𝑥𝑛,𝑆𝑥𝑛𝐺,𝐺(𝑞,𝑔𝑝,𝑔𝑝),𝑓𝑥𝑛,𝑞,𝑞+𝐺𝑆𝑥𝑛,𝑔𝑝,𝑔𝑝2,lim𝑛𝑀𝑥𝑛[𝐺],𝑝,𝑝=max𝐺(𝑞,𝑞,𝑞),𝐺(𝑞,𝑞,𝑞),𝐺(𝑞,𝑔𝑝,𝑔𝑝),(𝑞,𝑞,𝑞)+𝐺(𝑞,𝑔𝑝,𝑔𝑝)2=𝐺(𝑞,𝑔𝑝,𝑔𝑝).(2.9) Hence, we have 𝜓(𝐺(𝑞,𝑔𝑝,𝑔𝑝))𝜓(𝐺(𝑞,𝑔𝑝,𝑔𝑝))𝜙(𝐺(𝑞,𝑔𝑝,𝑔𝑝))(2.10) and 𝜙(𝐺(𝑞,𝑔𝑝,𝑔𝑝))0. Hence 𝑔𝑝=𝑞, 𝑝 is the coincidence point of pair (𝑔,𝑇). As 𝑔(𝑋)𝑆(𝑋), there exist a point 𝑢 in 𝑋 such that 𝑞=𝑆𝑢. We claim that 𝑆𝑢=𝑓𝑢. From (2.1), we get 𝜓(𝐺(𝑓𝑢,𝑔𝑝,𝑔𝑝))𝜓(𝑀(𝑢,𝑝,𝑝))𝜙(𝑀(𝑢,𝑝,𝑝)),(2.11) where []𝑀(𝑢,𝑝,𝑝)=max𝐺(𝑆𝑢,𝑇𝑝,𝑇𝑝),𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢),𝐺(𝑇𝑝,𝑔𝑝,𝑔𝑝),𝐺(𝑓𝑢,𝑇𝑝,𝑇𝑝)+𝐺(𝑆𝑢,𝑔𝑝,𝑔𝑝)2[]=max𝐺(𝑆𝑢,𝑆𝑢,𝑆𝑢),𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢),𝐺(𝑆𝑢,𝑆𝑢,𝑆𝑢),𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢)+𝐺(𝑆𝑢,𝑆𝑢,𝑆𝑢)2=𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢).(2.12) Hence, we have 𝜓(𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢))𝜓(𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢))𝜙(𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢))(2.13) which implies 𝜙(𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢))0. Hence 𝑓𝑢=𝑆𝑢, so 𝑢 is the coincidence point of pair (𝑓,𝑆). Thus 𝑓𝑢=𝑆𝑢=𝑇𝑝=𝑔𝑝=𝑞. Now, weakly compatibility of pairs (𝑓,𝑆) and (𝑔,𝑇) give that 𝑓𝑞=𝑆𝑞and 𝑇𝑞=𝑔𝑞. From (2.1), we have 𝜓(𝐺(𝑓𝑞,𝑞,𝑞))=𝜓(𝐺(𝑓𝑞,𝑔𝑝,𝑔𝑝))𝜓(𝑀(𝑞,𝑝,𝑝))𝜙(𝑀(𝑞,𝑝,𝑝)),(2.14) where []𝑀(𝑞,𝑝,𝑝)=max𝐺(𝑆𝑞,𝑇𝑝,𝑇𝑝),𝐺(𝑓𝑞,𝑆𝑞,𝑆𝑞),𝐺(𝑇𝑝,𝑔𝑝,𝑔𝑝),𝐺(𝑓𝑞,𝑇𝑝,𝑇𝑝)+𝐺(𝑆𝑞,𝑔𝑝,𝑔𝑝)2[]=max𝐺(𝑓𝑞,𝑞,𝑞),𝐺(𝑓𝑞,𝑓𝑞,𝑓𝑞),𝐺(𝑞,𝑞,𝑞),𝐺(𝑓𝑞,𝑞,𝑞)+𝐺(𝑓𝑞,𝑞,𝑞)2=𝐺(𝑓𝑞,𝑞,𝑞).(2.15) From (2.14), we obtain 𝜓(𝐺(𝑓𝑞,𝑞,𝑞))𝜓(𝐺(𝑓𝑞,𝑞,𝑞))𝜙(𝐺(𝑓𝑞,𝑞,𝑞)),(2.16) and so 𝜙(𝐺(𝑓𝑞,𝑞,𝑞))0. Therefore 𝑓𝑞=𝑆𝑞=𝑞. Similarly, it can be shown that 𝑔𝑞=𝑞. Therefore 𝑔𝑞=𝑇𝑞=𝑞. To prove uniqueness of 𝑞, suppose that 𝑓𝑝=𝑔𝑝=𝑆𝑝=𝑇𝑝=𝑝. From (2.1) we have the following: 𝜓(𝐺(𝑞,𝑝,𝑝))=𝜓(𝐺(𝑓𝑞,𝑔𝑝,𝑔𝑝))𝜓(𝑀(𝑞,𝑝,𝑝))𝜙(𝑀(𝑞,𝑝,𝑝)),(2.17) where []𝑀(𝑞,𝑝,𝑝)=max𝐺(𝑆𝑞,𝑇𝑝,𝑇𝑝),𝐺(𝑓𝑞,𝑆𝑞,𝑆𝑞),𝐺(𝑇𝑝,𝑔𝑝,𝑔𝑝),𝐺(𝑓𝑞,𝑇𝑝,𝑇𝑝)+𝐺(𝑆𝑞,𝑔𝑝,𝑔𝑝)2[]=max𝐺(𝑞,𝑝,𝑝),𝐺(𝑞,𝑞,𝑞),𝐺(𝑞,𝑞,𝑞),𝐺(𝑞,𝑝,𝑝)+𝐺(𝑞,𝑝,𝑝)2=𝐺(𝑞,𝑝,𝑝).(2.18) Thus from (2.17), we obtain 𝜓(𝐺(𝑞,𝑝,𝑝))𝜓(𝐺(𝑞,𝑝,𝑝))𝜙(𝐺(𝑞,𝑝,𝑝)),(2.19) which implies that 𝐺(𝑞,𝑝,𝑝)0, and so 𝑞=𝑝. The proof using (2.2) is similar.

Example 2.2. Let 𝑋={0,1,2} be a set with 𝐺-metric defined by Table 2.

tab2
Table 2

Note that 𝐺 is a nonsymmetric as 𝐺(1,2,2)𝐺(1,1,2). Let 𝑓,𝑔,𝑆,𝑇𝑋𝑋 be defined by Table 3.

tab3
Table 3

Clearly, 𝑓(𝑋)𝑇(𝑋) and 𝑔(𝑋)𝑆(𝑋)with the pairs (𝑓,𝑆) and (𝑔,𝑇) being weakly compatible. Also a pair (𝑓,𝑆)satisfy (E.A) property, indeed, 𝑥𝑛=0 for each 𝑛𝑁 is the required sequence. Note that pair (𝑔,𝑇) is not commuting at 2. The control functions 𝜓,𝜙[0,)[0,) are defined by 𝑡𝜓(𝑡)=3𝑡and𝜙(𝑡)=4[],𝑒,if𝑡0,44𝑡2,if𝑡>4.(2.20) To check contractive conditions (2.1) and (2.2) for all 𝑥,𝑦𝑋, we consider the following cases:

Note that for cases (I) 𝑥=𝑦=0, (II) 𝑥=0,𝑦=1, (III) 𝑥=1,𝑦=0, (IV) 𝑥=1,𝑦=1, (V) 𝑥=2,𝑦=0, and (VI) 𝑥=2,𝑦=1,

We have 𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦)=0,𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦)=0, and hence (2.1) and (2.2) are obviously satisfied now.

(VII) If 𝑥=0,𝑦=2, then 𝑓𝑥=0,𝑔𝑦=1, 𝑆𝑥=0,𝑇𝑦=2. <𝜓(𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦))=3𝐺(0,1,1)=3114(3)=114=𝐺(0,2,2)114𝐺(𝑆𝑥,𝑇𝑦,𝑇𝑦)114𝑀(𝑥,𝑦,𝑦)=𝜓(𝑀(𝑥,𝑦,𝑦))𝜙(𝑀(𝑥,𝑦,𝑦)).(2.21) Also <𝜓(𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦))=3𝐺(0,0,1)=3114(3)=114=𝐺(0,0,2)114𝐺(𝑆𝑥,𝑆𝑥,𝑇𝑦)114𝑀(𝑥,𝑥,𝑦)=𝜓(𝑀(𝑥,𝑥,𝑦))𝜙(𝑀(𝑥,𝑥,𝑦)).(2.22)

(VIII) For 𝑥=1,𝑦=2, then 𝑓𝑥=0,𝑔𝑦=1,𝑆𝑥=2,𝑇𝑦=2. <𝜓(𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦))=3𝐺(0,1,1)=3118(3+4)=114[]𝐺(0,2,2)+𝐺(2,1,1)2=114[]𝐺(𝑓𝑥,𝑇𝑦,𝑇𝑦)+𝐺(𝑆𝑥,𝑔𝑦,𝑔𝑦)2114<𝑀(𝑥,𝑦,𝑦)=𝜓(𝑀(𝑥,𝑦,𝑦))𝜙(𝑀(𝑥,𝑦,𝑦)),𝜓(𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦))=3𝐺(0,0,1)=3118(3+2)=114[]𝐺(0,0,2)+𝐺(2,2,1)2=114[𝐺](𝑓𝑥,𝑓𝑥,𝑇𝑦)+𝐺(𝑆𝑥,𝑆𝑥,𝑔𝑦)2114𝑀(𝑥,𝑥,𝑦)=𝜓(𝑀(𝑥,𝑥,𝑦))𝜙(𝑀(𝑥,𝑥,𝑦)).(2.23)

(IX) Now, when 𝑥=2,𝑦=2, then 𝑓𝑥=0,𝑔𝑦=1,𝑆𝑥=1,𝑇𝑦=2. <𝜓(𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦))=3𝐺(0,1,1)=3114(2)=114=𝐺(1,2,2)114𝐺(𝑆𝑥,𝑇𝑦,𝑇𝑦)114𝑀(𝑥,𝑦,𝑦)=𝜓(𝑀(𝑥,𝑦,𝑦))𝜙(𝑀(𝑥,𝑦,𝑦)).(2.24) Also <𝜓(𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦))=3𝐺(0,0,1)=3114(4)=114=𝐺(1,1,2)114𝐺(𝑆𝑥,𝑆𝑥,𝑇𝑦)114𝑀(𝑥,𝑥,𝑦)=𝜓(𝑀(𝑥,𝑥,𝑦))𝜙(𝑀(𝑥,𝑥,𝑦)).(2.25)

Hence, all of the conditions of Theorem 2.1 are satisfied. Moreover, 0 is the unique common fixed-point of 𝑓, 𝑔, 𝑆, and 𝑇.

As two noncompatible selfmappings on 𝐺-metric space 𝑋 satisfy the (E.A) property, so above result remains true if any one of the pair of mapping is noncompatible.

Above theorem is true for any choice of control functions, for example if we take 𝜓(𝑡)=𝑡 and 𝜙(𝑡)=(1𝛾)𝑡 for 𝛾[0,1) in Theorem 2.1, we have the following corollary.

Corollary 2.3. Let 𝑋 be a 𝐺-metric space and 𝑓,𝑔,𝑆,𝑇𝑋𝑋 be mappings with 𝑓(𝑋)𝑇(𝑋) and 𝑔(𝑋)𝑆(𝑋) such that []𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦)𝛾max𝐺(𝑆𝑥,𝑇𝑦,𝑇𝑦),𝐺(𝑓𝑥,𝑆𝑥,𝑆𝑥),𝐺(𝑇𝑦,𝑔𝑦,𝑔𝑦),𝐺(𝑓𝑥,𝑇𝑦,𝑇𝑦)+𝐺(𝑆𝑥,𝑔𝑦,𝑔𝑦)2(2.26) or []𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦)𝛾max𝐺(𝑆𝑥,𝑆𝑥,𝑇𝑦),𝐺(𝑓𝑥,𝑓𝑥,𝑆𝑥),𝐺(𝑇𝑦,𝑇𝑦,𝑔𝑦),𝐺(𝑓𝑥,𝑓𝑥,𝑇𝑦)+𝐺(𝑆𝑥,𝑆𝑥,𝑔𝑦)2(2.27) hold for all 𝑥,𝑦𝑋, where𝛾[0,1) hold for all 𝑥,𝑦𝑋, where 𝜓 and 𝜙 are control functions. Suppose that one of the pairs (𝑓,𝑆) and (𝑔,𝑇) satisfies (E.A) property and one of the subspace 𝑓(𝑋),𝑔(𝑋),𝑆(𝑋), and 𝑇(𝑋) is closed in 𝑋. If for every sequence {𝑦𝑛} in 𝑋, one of the following conditions hold:(a){𝑔𝑦𝑛} is bounded in case (𝑓,𝑆) satisfies (E.A) property (b){𝑓𝑦𝑛} is bounded in case (𝑔,𝑇) satisfies (E.A) property.
Then, the pairs (𝑓,𝑆) and (𝑔,𝑇) have a point of coincidence in 𝑋. Moreover, if the pairs (𝑓,𝑆) and (𝑔,𝑇) are weakly compatible, then 𝑓,𝑔,𝑆, and 𝑇 have a unique common fixed-point.
If we take 𝑓=𝑔 and 𝑆=𝑇 with 𝜓(𝑡)=𝑡 for all 𝑡211𝑑 in Theorem 2.1, we obtain the following corollary which extends Theorem 3.1 of [19] to generalized metric space.

Corollary 2.4. Let 𝑋 be a 𝐺-metric space and 𝑓,𝑆𝑋𝑋 be mappings with 𝑓𝑋𝑆𝑋 such that [𝐺]𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑦)𝑀(𝑥,𝑦,𝑦)𝜙(𝑀(𝑥,𝑦,𝑦)),where𝑀(𝑥,𝑦,𝑦)=max𝐺(𝑆𝑥,𝑆𝑦,𝑆𝑦),𝐺(𝑓𝑥,𝑆𝑥,𝑆𝑥),𝐺(𝑆𝑦,𝑓𝑦,𝑓𝑦),(𝑓𝑥,𝑆𝑦,𝑆𝑦)+𝐺(𝑆𝑥,𝑓𝑦,𝑓𝑦)2(2.28) or [𝐺]𝐺(𝑓𝑥,𝑓𝑥,𝑓𝑦)𝑀(𝑥,𝑥,𝑦)𝜙(𝑀(𝑥,𝑥,𝑦)),where𝑀(𝑥,𝑥,𝑦)=max𝐺(𝑆𝑥,𝑆𝑥,𝑆𝑦),𝐺(𝑓𝑥,𝑓𝑥,𝑆𝑥),𝐺(𝑆𝑦,𝑆𝑦,𝑓𝑦),(𝑓𝑥,𝑓𝑥,𝑆𝑦)+𝐺(𝑆𝑥,𝑆𝑥,𝑓𝑦)2(2.29) hold for all 𝑥,𝑦𝑋, where 𝜙 are control functions. Suppose that the pair (𝑓,𝑆) satisfy (E.A) property and one of the subspaces 𝑓(𝑋),𝑆(𝑋) is closed in 𝑋. Then, the pair (𝑓,𝑆) has a common point of coincidence in 𝑋. Moreover, if the pair (𝑓,𝑆) is weakly compatible, then 𝑓and 𝑆 have a unique common fixed-point.

3. Well-Posedness

The notion of well-posedness of a fixed-point problem has evoked much interest of several mathematicians, (see [2427]).

Definition 3.1. Let 𝑋 be a 𝐺-metric space and 𝑓𝑋𝑋 be a mapping. The fixed-point problem of 𝑓 is said to be well-posed if:(a)𝑓 has a unique fixed-point 𝑧 in 𝑋;(b)for any sequence {𝑥𝑛} of points in 𝑋 such that lim𝑛𝐺(𝑓𝑥𝑛,𝑥𝑛,𝑥𝑛)=0, we have lim𝑛𝐺(𝑥𝑛,𝑧,𝑧)=0.

Definition 3.2. Let 𝑋 be a 𝐺-metric space and Σbe a set of mappings on 𝑋. Common fixed-point problem 𝐶𝐹(Σ) is said to be well-posed if:(a)𝑧𝑋 is a unique common fixed-point of all mappings in Σ;(b)for any sequence {𝑥𝑛} of points in 𝑋 such that lim𝑛𝐺(𝑓𝑥𝑛,𝑥𝑛,𝑥𝑛)=0 for each 𝑓Σ, we have lim𝑛𝐺(𝑥𝑛,𝑧,𝑧)=0.

Theorem 3.3. Let 𝑋 be a 𝐺-metric space and 𝑓,𝑔,𝑆,𝑇𝑋𝑋 be mappings such that [𝐺]𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦)𝐺(𝑆𝑥,𝑇𝑦,𝑇𝑦)𝜓(𝑀(𝑥,𝑦,𝑦)),where𝑀(𝑥,𝑦,𝑦)=max𝐺(𝑆𝑥,𝑇𝑦,𝑇𝑦),𝐺(𝑓𝑥,𝑆𝑥,𝑆𝑥),𝐺(𝑇𝑦,𝑔𝑦,𝑔𝑦),(𝑓𝑥,𝑇𝑦,𝑇𝑦)+𝐺(𝑆𝑥,𝑔𝑦,𝑔𝑦)2(3.1) or [𝐺]𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦)𝐺(𝑆𝑥,𝑆𝑥,𝑇𝑦))𝜓(𝑀(𝑥,𝑥,𝑦)),where𝑀(𝑥,𝑥,𝑦)=max𝐺(𝑆𝑥,𝑆𝑥,𝑇𝑦),𝐺(𝑓𝑥,𝑓𝑥,𝑆𝑥),𝐺(𝑇𝑦,𝑇𝑦,𝑔𝑦),(𝑓𝑥,𝑓𝑥,𝑇𝑦)+𝐺(𝑆𝑥,𝑆𝑥,𝑔𝑦)2(3.2) hold for all 𝑥,𝑦𝑋, where 𝜓 is a control function. Suppose that one of the pairs (𝑓,𝑆) and (𝑔,𝑇) satisfies (E.A) property and one of the subspace 𝑓(𝑋),𝑔(𝑋),𝑆(𝑋),𝑇(𝑋) is closed in 𝑋. If for every sequence {𝑦𝑛} in 𝑋, one of the following conditions hold:(a){𝑔𝑦𝑛} is bounded in case (𝑓,𝑆) satisfies (E.A) property; (b){𝑓𝑦𝑛} is bounded in case (𝑔,𝑇) satisfies (E.A) property.
If pairs (𝑓,𝑆) and (𝑔,𝑇) are weakly compatible, then 𝐶𝐹({𝑓,𝑔,𝑆,𝑇}) is well-posed.

Proof. From Theorem 2.1, the mappings 𝑓,𝑔,𝑆,𝑇𝑋𝑋 have a unique common fixed-point (say) 𝑧 in 𝑋. Let {𝑥𝑛} be a sequence in 𝑋 such that lim𝑛𝐺𝑓𝑥𝑛,𝑥𝑛,𝑥𝑛=lim𝑛𝐺𝑔𝑥𝑛,𝑥𝑛,𝑥𝑛=lim𝑛𝐺𝑆𝑥𝑛,𝑥𝑛,𝑥𝑛=lim𝑛𝐺𝑇𝑥𝑛,𝑥𝑛,𝑥𝑛=0.(3.3) Now by using (3.1), we have 𝐺𝑧,𝑥𝑛,𝑥𝑛𝐺𝑓𝑧,𝑔𝑥𝑛,𝑔𝑥𝑛+𝐺𝑔𝑥𝑛,𝑥𝑛,𝑥𝑛𝐺𝑆𝑧,𝑇𝑥𝑛,𝑇𝑥𝑛𝑀𝜓𝑧,𝑥𝑛,𝑥𝑛+𝐺𝑔𝑥𝑛,𝑥𝑛,𝑥𝑛𝐺𝑧,𝑥𝑛,𝑥𝑛𝑥+𝐺𝑛,𝑇𝑥𝑛,𝑇𝑥𝑛𝐺𝜓𝑓𝑧,𝑇𝑥𝑛,𝑇𝑥𝑛+𝐺𝑔𝑥𝑛,𝑆𝑧,𝑆𝑧2+𝐺𝑔𝑥𝑛,𝑥𝑛,𝑥𝑛(3.4) which further implies 𝜓𝐺𝑧,𝑇𝑥𝑛,𝑇𝑥𝑛𝑥+𝐺𝑛,𝑧,𝑧2𝑥𝐺𝑛,𝑇𝑥𝑛,𝑇𝑥𝑛+𝐺𝑔𝑥𝑛,𝑥𝑛,𝑥𝑛2𝐺𝑇𝑥𝑛,𝑥𝑛,𝑥𝑛+𝐺𝑔𝑥𝑛,𝑥𝑛,𝑥𝑛.(3.5) On taking limit as 𝑛 implies that lim𝑛𝜓𝐺𝑧,𝑇𝑥𝑛,𝑇𝑥𝑛𝑥+𝐺𝑛,𝑧,𝑧2=0,(3.6) and by the property of 𝜓, we have lim𝑛𝐺𝑧,𝑇𝑥𝑛,𝑇𝑥𝑛=lim𝑛𝐺𝑥𝑛,𝑧,𝑧=0.(3.7) Hence the result follows.

Remark 3.4. A 𝐺-metric naturally induces a metric 𝑑𝐺 given by 𝑑𝐺(𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦)+𝐺(𝑥,𝑥,𝑦). If 𝐺-metric is not symmetric, either of the inequalities (2.1) or (2.2) does not reduce to any metric inequality with the metric 𝑑𝐺. Hence our theorems do not reduce to fixed-point problems in the corresponding metric space (𝑋,𝑑𝐺).

Acknowledgments

Wei Long acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province, and the Foundation of Jiangxi Provincial Education Department (GJJ12205).

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