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

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. [2–6] 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, [11–14]). 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 𝐺∶𝑋×𝑋×𝑋→𝑅⁺ satisfies the following:𝐆₁:𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧;𝐆₂:0<𝐺(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋, with 𝑥≠𝑦;𝐆₃:𝐺(𝑥,𝑥,𝑦)≤𝐺(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋, with 𝑦≠𝑧;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,𝑎∈𝑋.
Then 𝐺 is called a 𝐺-metric on 𝑋 and (𝑋,𝐺) is called a 𝐺-metric.

Definition 1.2. A sequence {𝑥_𝑛} in a 𝐺-metric space 𝑋 is∶(i)a 𝐺-Cauchy sequence if, for any 𝜀>0, there is an 𝑛₀∈𝑁 (the set of natural numbers) such that for all 𝑛,𝑚,𝑙≥𝑛₀,𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜀,(ii)a 𝐺-convergent sequence if, for any 𝜀>0, there is an 𝑥∈𝑋 and an 𝑛₀∈𝑁, such that for all 𝑛,𝑚≥𝑛₀,𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)<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 {𝑥_𝑛}  𝐺-converges to 𝑥∈𝑋 if and only if 𝐺(𝑥_𝑚,𝑥_𝑛,𝑥)→0 as 𝑛,𝑚→∞.

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

Proposition 1.5. Every 𝐺-metric on 𝑋 will define a metric 𝑑_𝐺 on 𝑋 by 𝑑_𝐺(𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦)+𝐺(𝑦,𝑥,𝑥),∀𝑥,𝑦∈𝑋.(1.1) For a symmetric 𝐺-metric 𝑑_𝐺(𝑥,𝑦)=2𝐺(𝑥,𝑦,𝑦),∀𝑥,𝑦∈𝑋.(1.2) However, if 𝐺 is nonsymmetric, then the following inequality holds: 32𝐺(𝑥,𝑦,𝑦)≤𝑑^𝐺(𝑥,𝑦)≤3𝐺(𝑥,𝑦,𝑦),∀𝑥,𝑦∈𝑋.(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 𝐺∶𝑋×𝑋×𝑋→ℝ⁺ be defined as shown in Table 1.

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, 𝑔∶𝑋→𝑋 are called (i) compatible if, whenever a sequence {𝑥_𝑛} in 𝑋 is such that {𝑓𝑥_𝑛} and {𝑔𝑥_𝑛} are 𝐺-convergent to some 𝑡∈𝑋, then lim_𝑛→∞𝐺(𝑓𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛)=0, (ii) noncompatible if there exists at least one sequence {𝑥_𝑛} in 𝑋 such that {𝑓𝑥_𝑛} and {𝑔𝑥_𝑛} are 𝐺-convergent to some 𝑡∈𝑋, but lim_𝑛→∞𝐺(𝑓𝑔𝑥_𝑛,𝑓𝑔𝑥_𝑛,𝑔𝑓𝑥_𝑛) 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 [20–22]).
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 {𝑥_𝑛} in 𝑋 such that {𝑓𝑥_𝑛} and {𝑔𝑥_𝑛} are 𝐺-convergent to some 𝑡∈𝑋.

Example 1.9 (see [15]). Let 𝑋=[0,2] be a 𝐺-metric space with 𝐺(𝑥,𝑦,𝑧)=max||𝑥−𝑦||,|𝑥−𝑧|,||𝑦−𝑧||.(1.5) Let 𝑓,𝑔∶𝑋→𝑋 be defined by 𝑓𝑥=22−−𝑥𝑥,𝑥∈[0,1],𝑔𝑥=⎧⎪⎨⎪⎩3−2𝑥,𝑥∈(1,2],𝑥2,𝑥∈[0,1],2,𝑥∈(1,2].(1.6) For a decreasing sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛→1,𝑔𝑥_𝑛→1/2,𝑓𝑥_𝑛→1/2,𝑔𝑓𝑥_𝑛=(4+𝑥_𝑛)/4→5/4 and 𝑓𝑔𝑥_𝑛=(4−𝑥_𝑛)2→3/2. So, 𝑓 and 𝑔 are noncompatible. Note that, there exists a sequence {𝑥_𝑛} in 𝑋 such that lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=1∈𝑋, take 𝑥_𝑛=1 for each 𝑛∈𝑁. 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,𝑇∶𝑋→𝑋 be mappings with 𝑓𝑋⊆𝑇𝑋 and 𝑔𝑋⊆𝑆𝑋 such that where𝑀(𝜓𝑥,(𝑦𝐺,𝑦()𝑓=𝑥,𝑔m[𝑦𝐺a,x𝑔(𝑦𝑓)𝑥),𝐺≤𝑇𝑦(𝜓𝑆,𝑇𝑥(𝑦𝑀,)𝑇+𝑦(,𝑥𝐺𝑇,𝑦𝑦(),𝑆,𝑦𝑥𝐺)),𝑔(−𝑓𝑦𝑥,𝜙𝑔,(𝑆𝑦𝑀)𝑥],(𝑆𝑥𝑥,)𝑦,,𝐺𝑦))(𝑇,𝑦,𝑔𝑦,𝑔𝑦),2(2.1) or where𝑀(𝑥𝜓,𝑥(𝐺,𝑦()𝑓=𝑥,𝑓m[𝐺𝑥a,x(𝑔𝑓𝑦𝑥)),𝐺𝑓≤𝑥(𝑆𝜓,𝑇𝑥(𝑦,𝑀𝑆)+𝑥(,𝑥𝐺𝑇,𝑦𝑥()𝑆,,𝑦𝑥𝐺),)𝑆(−𝑓𝑥𝑥,𝜙,𝑔𝑓(𝑦𝑀𝑥)],𝑆(𝑥𝑥,)𝑥,,𝐺𝑦)()𝑇𝑦,𝑇𝑦,𝑔𝑦),2(2.2) hold for all 𝑥,𝑦∈𝑋, 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 {𝑦_𝑛} in 𝑋, one of the following conditions hold:(a){𝑔𝑦_𝑛} is bounded in case (𝑓,𝑆) satisfies (E.A) property, (b){𝑓𝑦_𝑛} 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 {𝑥_𝑛} in 𝑋 satisfying lim_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑆𝑥_𝑛=𝑞 for some 𝑞∈𝑋. As 𝑓𝑋⊆𝑇𝑋, there exists a sequence {𝑦_𝑛} in 𝑋 such that 𝑓𝑥_𝑛=𝑇𝑦_𝑛. As {𝑔𝑦_𝑛} is bounded, lim_𝑛→∞𝐺(𝑓𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛) and lim_𝑛→∞𝐺(𝑆𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛) are finite numbers. Note that ||𝐺𝑓𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛−𝐺𝑆𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛||≤2𝐺𝑓𝑥_𝑛,𝑆𝑥_𝑛,𝑆𝑥_𝑛.(2.3) Since 𝐺(𝑓𝑥_𝑛,𝑆𝑥_𝑛,𝑆𝑥_𝑛)→0 as 𝑛→∞, therefore lim_𝑛→∞𝐺(𝑓𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛)=lim_𝑛→∞𝐺(𝑆𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛). Indeed, using that lim_𝑛→∞𝐺(𝑓𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛)=lim_𝑛→∞𝐺(𝑆𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛)=𝑙, we obtain subsequences {𝑥_𝑛𝑘} and {𝑦_𝑛𝑘} such that 𝐺(𝑆𝑥_𝑛𝑘,𝑔𝑦_𝑛𝑘,𝑔𝑦_𝑛𝑘) and 𝐺(𝑓𝑥_𝑛𝑘,𝑔𝑦_𝑛𝑘,𝑔𝑦_𝑛𝑘) are 𝐺-convergent to 𝑙. Replacing 𝑥 by 𝑥_𝑛𝑘 and 𝑦 by 𝑦_𝑛𝑘 in (2.1), we have 𝑀𝑥_𝑛𝑘,𝑦_𝑛𝑘,𝑦_𝑛𝑘=m𝐺ax𝑓𝑥𝐺_𝑛𝑘𝑆𝑥_𝑛𝑘,𝑓𝑥_𝑛𝑘,𝑓𝑥_𝑛𝑘,𝐺𝑓𝑥_𝑛𝑘,𝑆𝑥_𝑛𝑘,𝑆𝑥_𝑛𝑘,𝐺𝑓𝑥_𝑛𝑘,𝑔𝑦_𝑛𝑘,𝑔𝑦_𝑛𝑘,,𝑓𝑥_𝑛𝑘,𝑓𝑥_𝑛𝑘+𝐺𝑆𝑥_𝑛𝑘,𝑔𝑦_𝑛𝑘,𝑔𝑦_𝑛𝑘2(2.4) which on taking limit as 𝑘→∞ implies that lim_𝑛→∞𝑀𝑥_𝑛𝑘,𝑦_𝑛𝑘,𝑦_𝑛𝑘=max𝐺(𝑞,𝑞,𝑞),𝐺(𝑞,𝑞,𝑞),𝑙,𝑙2=𝑙.(2.5) Now 𝜓𝐺𝑓𝑥_𝑛𝑘,𝑔𝑦_𝑛𝑘,𝑔𝑦_𝑛𝑘≤𝜓𝑀𝑥_𝑛𝑘,𝑦_𝑛𝑘,𝑦_𝑛𝑘−𝜙𝑀𝑥_𝑛𝑘,𝑦_𝑛𝑘,𝑦_𝑛𝑘(2.6) which on taking upper limit gives 𝜓(𝑙)≤𝜓(𝑙)−𝜙(𝑙),(2.7) and so 𝑙=0. Hence, lim_𝑛→∞𝐺(𝑓𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛)=lim_𝑛→∞𝐺(𝑆𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝑦_𝑛)=0, and so, lim_𝑛→∞𝑔𝑦_𝑛=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 𝜓𝐺𝑓𝑥_𝑛,𝑔𝑝,𝑔𝑝≤𝜓𝑀𝑥_𝑛,𝑝,𝑝−𝜙𝑀𝑥_𝑛,𝑝,𝑝,(2.8) where 𝑀𝑥_𝑛,𝑝,𝑝=m𝐺ax𝑓𝑥_𝑛𝐺𝑆𝑥_𝑛,𝑇𝑝,𝑇𝑝,𝐺𝑓𝑥_𝑛,𝑆𝑥_𝑛,𝑆𝑥_𝑛,𝐺(𝑇𝑝,𝑔𝑝,𝑔𝑝),,𝑇𝑝,𝑇𝑝+𝐺𝑆𝑥_𝑛,𝑔𝑝,𝑔𝑝=max𝐺𝑆𝑥_𝑛2𝐺𝑓𝑥_𝑛,𝑞,𝑞,𝐺𝑓𝑥_𝑛,𝑆𝑥_𝑛,𝑆𝑥_𝑛,𝐺(𝑞,𝑔𝑝,𝑔𝑝),,𝑞,𝑞+𝐺𝑆𝑥_𝑛,𝑔𝑝,𝑔𝑝lim_𝑛→∞2,𝑀𝑥_𝑛,𝑝,𝑝=max𝐺(𝑞,𝑞,𝑞),𝐺(𝑞,𝑞,𝑞),𝐺(𝑞,𝑔𝑝,𝑔𝑝),[𝐺(𝑞,𝑞,𝑞)+𝐺(𝑞,𝑔𝑝,𝑔𝑝)]=𝐺(𝑞,𝑔𝑝,𝑔𝑝).2(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)=m[𝐺ax(𝑓𝑢,𝐺𝑇𝑝(𝑆,𝑇𝑢𝑝,)𝑇+𝑝,𝐺𝑇𝑝()𝑆,𝑢𝐺,𝑔(𝑓𝑝,𝑢𝑔,𝑝𝑆)𝑢],𝑆𝑢),𝐺(𝑇𝑝,𝑔𝑝,𝑔𝑝),=m[𝐺ax(𝑓𝑢,𝐺𝑆𝑢(𝑆,𝑆𝑢,𝑢𝑆)2𝑢+,𝑆𝐺𝑢()𝑆,𝑢𝐺,𝑆(𝑓𝑢𝑢,,𝑆𝑆𝑢𝑢)],𝑆𝑢),𝐺(𝑆𝑢,𝑆𝑢,𝑆𝑢),=𝐺(𝑓𝑢,𝑆𝑢,𝑆𝑢).2(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)=m[𝐺ax(𝑓𝑞,𝐺𝑇𝑝(𝑆,𝑇𝑞𝑝,)𝑇+𝑝,𝐺𝑇𝑝()𝑆,𝑞𝐺,𝑔(𝑓𝑝,𝑞𝑔,𝑝𝑆)𝑞],𝑆𝑞),𝐺(𝑇𝑝,𝑔𝑝,𝑔𝑝),=m[𝐺ax(𝑓𝑞,𝐺𝑞,(𝑞𝑓)𝑞+,𝑞𝐺,2𝑞()𝑓,𝑞𝐺,𝑞(,𝑓𝑞𝑞),]𝑓𝑞,𝑓𝑞),𝐺(𝑞,𝑞,𝑞),=𝐺(𝑓𝑞,𝑞,𝑞).2(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)=m[𝐺ax(𝑓𝑞,𝐺𝑇𝑝(𝑆,𝑇𝑞𝑝,)𝑇+𝑝,𝐺𝑇𝑝()𝑆,𝑞𝐺,𝑔(𝑓𝑝,𝑞𝑔,𝑝𝑆)𝑞],𝑆𝑞),𝐺(𝑇𝑝,𝑔𝑝,𝑔𝑝),=m[𝐺ax(𝑞,𝑝𝐺,𝑝()𝑞+,𝑝𝐺,𝑝2()𝑞,,𝐺𝑝,(𝑝𝑞),]𝑞,𝑞),𝐺(𝑞,𝑞,𝑞),=𝐺(𝑞,𝑝,𝑝).2(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.

Table 2 

Note that 𝐺 is a nonsymmetric as 𝐺(1,2,2)≠𝐺(1,1,2). Let 𝑓,𝑔,𝑆,𝑇∶𝑋→𝑋 be defined by Table 3.

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, 𝑥_𝑛=0 for each 𝑛∈𝑁 is the required sequence. Note that pair (𝑔,𝑇) is not commuting at 2. The control functions 𝜓,𝜙∶[0,∞)→[0,∞) are defined by 𝜓(𝑡)=3𝑡and𝜙(𝑡)=⎧⎪⎨⎪⎩𝑡4𝑒,if𝑡∈[0,4_4−𝑡],2,if𝑡>4.(2.20) To check contractive conditions (2.1) and (2.2) for all 𝑥,𝑦∈𝑋, 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))=<31𝐺1(0,1,1)=3=141(3)=141𝐺(0,2,2)=𝜓4(𝐺𝑀(𝑆(𝑥𝑥,,𝑦𝑇,𝑦𝑦,)𝑇)𝑦−)𝜙≤(𝑀141𝑀(𝑥,(𝑦𝑥,,𝑦𝑦),)𝑦.)(2.21) Also 𝜓(𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦))=<31𝐺1(0,0,1)=3=141(3)=141𝐺(0,0,2)=𝜓4(𝐺𝑀(𝑆(𝑥𝑥,,𝑥𝑆,𝑥𝑦,))𝑇−𝑦)𝜙≤(𝑀141(𝑀𝑥,(𝑥𝑥,,𝑦𝑥)),𝑦.)(2.22)

(VIII) For 𝑥=1,𝑦=2, then 𝑓𝑥=0,𝑔𝑦=1,𝑆𝑥=2,𝑇𝑦=2. 𝜓(𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦))=<31𝐺1(0,1,1)=3=181(3+4)=141[𝐺(0,2,2)+2𝐺(2,1,1)]𝜓(𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦))=<=3𝜓14𝐺1(𝑀[(𝐺0,((0𝑥𝑓,,𝑥1𝑦,),𝑇=𝑦𝑦),)3𝑇−𝑦)𝜙+2(𝑀𝐺((𝑆𝑥𝑥,𝑦,,𝑔𝑦𝑦),)𝑔,𝑦)]≤141𝑀(𝑥,𝑦,𝑦)=181(3+2)=141[𝐺(0,0,2)+2𝐺(2,2,1)]=𝜓4(𝑀[𝐺((𝑥𝑓,𝑥𝑥,,𝑓𝑦𝑥),)𝑇−𝑦)𝜙+2(𝑀𝐺((𝑆𝑥𝑥,𝑥,𝑆,𝑦𝑥),)𝑔.𝑦)]≤141𝑀(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))=<31𝐺1(0,1,1)=3=141(2)=141𝐺(1,2,2)=𝜓4(𝐺𝑀(𝑆(𝑥𝑥,,𝑦𝑇,𝑦𝑦,)𝑇)𝑦−)𝜙≤(𝑀141𝑀(𝑥,(𝑦𝑥,,𝑦𝑦),)𝑦.)(2.24) Also 𝜓(𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦))=<31𝐺1(0,0,1)=3=141(4)=141𝐺(1,1,2)=𝜓4(𝐺𝑀(𝑆(𝑥𝑥,,𝑥𝑆,𝑥𝑦,))𝑇−𝑦)𝜙≤(𝑀141(𝑀𝑥,(𝑥𝑥,,𝑦𝑥)),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)=(1−𝛾)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,𝑇∶𝑋→𝑋 be mappings with 𝑓(𝑋)⊆𝑇(𝑋) and 𝑔(𝑋)⊆𝑆(𝑋) such that 𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦)≤𝛾[m𝐺a(x𝑓𝑥,𝑇𝐺𝑦,(𝑆𝑇𝑦𝑥),𝑇+𝑦𝐺,𝑇(𝑦𝑆)𝑥,,𝐺𝑔𝑦(𝑓,𝑔𝑥𝑦,)𝑆]𝑥,𝑆𝑥),𝐺(𝑇𝑦,𝑔𝑦,𝑔𝑦),2(2.26) or 𝐺(𝑓𝑥,𝑓𝑥,𝑔𝑦)≤𝛾[m𝐺a(x𝑓𝑥,𝑓𝐺𝑥(,𝑆𝑇𝑦𝑥),𝑆+𝑥𝐺,𝑇(𝑦𝑆)𝑥,,𝐺𝑆(𝑥𝑓,𝑥𝑔,𝑦𝑓)]𝑥,𝑆𝑥),𝐺(𝑇𝑦,𝑇𝑦,𝑔𝑦),2(2.27) hold for all 𝑥,𝑦∈𝑋, where𝛾∈[0,1) hold for all 𝑥,𝑦∈𝑋, 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 {𝑦_𝑛} in 𝑋, one of the following conditions hold:(a){𝑔𝑦_𝑛} is bounded in case (𝑓,𝑆) satisfies (E.A) property (b){𝑓𝑦_𝑛} 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 𝑡∈211𝑑 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,𝑆∶𝑋→𝑋 be mappings with 𝑓𝑋⊆𝑆𝑋 such that where𝑀(𝑥,𝑦,𝐺𝑦)(𝑓=𝑥,m[𝑓𝐺a𝑦x,(𝑓𝑓𝑥𝑦),𝐺𝑆≤(𝑦𝑆𝑀,𝑆𝑥,𝑦(𝑆)𝑥𝑦+,,𝑦𝑆𝐺,𝑦𝑦)()𝑆−,𝑥𝐺𝜙,𝑓((𝑓𝑦𝑀𝑥,𝑓,𝑆(𝑦𝑥)𝑥,],𝑦𝑆,𝑦𝑥))),,𝐺(𝑆𝑦,𝑓𝑦,𝑓𝑦),2(2.28) or where𝑀(𝑥,𝑥,𝐺𝑦)(𝑓=𝑥,m[𝑓𝐺a𝑥x,(𝑓𝑓𝑦𝑥),𝐺𝑓≤𝑥(𝑆𝑀,𝑆𝑥,𝑦(𝑆)𝑥𝑥+,𝑥,𝐺𝑆,𝑦𝑦())𝑆−,𝑥𝐺𝜙,𝑆((𝑓𝑥𝑀𝑥,,𝑓𝑓(𝑦𝑥𝑥),],𝑥𝑆,𝑥𝑦))),,𝐺(𝑆𝑦,𝑆𝑦,𝑓𝑦),2(2.29) hold for all 𝑥,𝑦∈𝑋, 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 [24–27]).

Definition 3.1. Let 𝑋 be a 𝐺-metric space and 𝑓∶𝑋→𝑋 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 {𝑥_𝑛} of points in 𝑋 such that lim_𝑛→∞𝐺(𝑓𝑥_𝑛,𝑥_𝑛,𝑥_𝑛)=0, we have lim_𝑛→∞𝐺(𝑥_𝑛,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)𝑧∈𝑋 is a unique common fixed-point of all mappings in Σ;(b)for any sequence {𝑥_𝑛} of points in 𝑋 such that lim_𝑛→∞𝐺(𝑓𝑥_𝑛,𝑥_𝑛,𝑥_𝑛)=0 for each 𝑓∈Σ, we have lim_𝑛→∞𝐺(𝑥_𝑛,𝑧,𝑧)=0.

Theorem 3.3. Let 𝑋 be a 𝐺-metric space and 𝑓,𝑔,𝑆,𝑇∶𝑋→𝑋 be mappings such that where𝑀(𝑥,𝐺𝑦,(𝑦𝑓)𝑥=,𝑔𝑦m[,𝐺a𝑔x𝑦(𝑓)𝑥≤,𝐺𝑇𝐺𝑦(𝑆(,𝑆𝑇𝑥𝑥𝑦,)𝑇,𝑇+𝑦𝑦,𝐺𝑇,𝑇𝑦(𝑦)𝑆),𝑥𝐺−,𝑔(𝜓𝑓𝑦𝑥,(𝑀𝑔,𝑆𝑦)𝑥(]𝑥,,𝑆𝑦𝑥,)𝑦,)𝐺),(𝑇𝑦,𝑔𝑦,𝑔𝑦),2(3.1) or where𝑀(𝑥𝐺,𝑥(,𝑓𝑦𝑥),=𝑓𝑥m,[𝑔𝐺a𝑦x()𝑓≤𝑥,𝐺𝑓𝐺𝑥(𝑆(,𝑆𝑇𝑥𝑥𝑦,,𝑆)𝑆+𝑥𝑥,𝐺,𝑇𝑇𝑦(𝑦)𝑆),)𝑥𝐺−,𝑆(𝑓𝜓𝑥𝑥,(,𝑔𝑀𝑓𝑦𝑥)](,𝑥𝑆,𝑥𝑥),,𝑦𝐺))(,𝑇𝑦,𝑇𝑦,𝑔𝑦),2(3.2) hold for all 𝑥,𝑦∈𝑋, 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 {𝑦_𝑛} in 𝑋, one of the following conditions hold:(a){𝑔𝑦_𝑛} is bounded in case (𝑓,𝑆) satisfies (E.A) property; (b){𝑓𝑦_𝑛} 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,𝑇∶𝑋→𝑋 have a unique common fixed-point (say) 𝑧 in 𝑋. Let {𝑥_𝑛} be a sequence in 𝑋 such that lim_𝑛→∞𝐺𝑓𝑥_𝑛,𝑥_𝑛,𝑥_𝑛==lliimm_{𝑛→∞𝑛→∞}𝐺𝑔𝑥_𝑛,𝑥_𝑛,𝑥_𝑛𝐺𝑆𝑥_𝑛,𝑥_𝑛,𝑥_𝑛=lim_𝑛→∞𝐺𝑇𝑥_𝑛,𝑥_𝑛,𝑥_𝑛=0.(3.3) Now by using (3.1), we have 𝐺𝑧,𝑥_𝑛,𝑥_𝑛≤≤𝐺𝐺𝑓𝑆𝑧𝑧,,𝑔𝑇𝑥𝑥_𝑛𝑛,𝑔𝑥_𝑛+𝐺𝑔𝑥_𝑛,𝑥_𝑛,𝑥_𝑛≤𝐺𝑧,𝑥_𝑛,𝑇𝑥_𝑛−𝜓𝑀𝑧,𝑥_𝑛,𝑥_𝑛+𝐺𝑔𝑥_𝑛,𝑥_𝑛,𝑥_𝑛−𝜓,𝐺𝑥_𝑛𝑓𝑧+,𝑇𝐺𝑥_𝑛𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛,𝑇𝑥_𝑛+𝐺𝑔𝑥_𝑛,𝑆𝑧,𝑆𝑧2+𝐺𝑔𝑥^𝑛,𝑥_𝑛,𝑥_𝑛(3.4) which further implies 𝜓𝐺𝑧,𝑇𝑥^𝑛,𝑇𝑥_𝑛+𝐺𝑥_𝑛,𝑧,𝑧2≤𝐺𝑥^𝑛≤2𝐺𝑇,𝑥𝑇_𝑛𝑥_𝑛,𝑇𝑥_𝑛+𝐺𝑔𝑥_𝑛,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛,𝑥_𝑛+𝐺𝑔𝑥_𝑛,𝑥_𝑛,𝑥_𝑛.(3.5) On taking limit as 𝑛→∞ implies that lim_𝑛→∞𝜓𝐺𝑧,𝑇𝑥^𝑛,𝑇𝑥_𝑛+𝐺𝑥_𝑛,𝑧,𝑧2=0,(3.6) and by the property of 𝜓, we have lim_𝑛→∞𝐺𝑧,𝑇𝑥_𝑛,𝑇𝑥_𝑛=lim_𝑛→∞𝐺𝑥_𝑛,𝑧,𝑧=0.(3.7) Hence the result follows.

Remark 3.4. A 𝐺-metric naturally induces a metric 𝑑_𝐺 given by 𝑑_𝐺(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 𝑑_𝐺. Hence our theorems do not reduce to fixed-point problems in the corresponding metric space (𝑋,𝑑_𝐺).

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).