Common Fixed Points for Weak ψ-Contractive Mappings in Ordered Metric Spaces with Applications

Abstract

We obtain some new common fixed point theorems satisfying a weak contractive condition in the framework of partially ordered metric spaces. The main result generalizes and extends some known results given by some authors in the literature.

1. Introduction and Preliminaries

Fixed point and common fixed point theorems for different types of nonlinear contractive mappings have been investigated extensively by various researchers (see [1–41]). Fixed point problems involving weak contractions and the mappings satisfying weak contractive type inequalities have been studied by many authors (see [10–20] and references cited therein).

Recently, many researchers have obtained fixed point, common fixed point, coupled fixed point, and coupled common fixed point results in partially ordered metric spaces (see [3, 6–8, 10–12, 29, 30, 32, 36]) and other spaces (see [5, 15, 31, 35, 38, 40, 41]).

Let (𝑋,≤) be a partially ordered set and 𝑓,𝑔 two self-mappings on 𝑋. A pair (𝑓,𝑔) of self-mappings of 𝑋 is said to be weakly increasing [4] if 𝑓𝑥≤𝑔𝑓𝑥 and 𝑔𝑥≤𝑓𝑔𝑥 for any 𝑥∈𝑋. An ordered pair (𝑓,𝑔) is said to be partially weakly increasing if 𝑓𝑥≤𝑔𝑓𝑥 for all 𝑥∈𝑋.

Note that a pair (𝑓,𝑔) is weakly increasing if and only if the ordered pairs (𝑓,𝑔) and (𝑔,𝑓) are partially weakly increasing.

Example 1 (see [3]). Let 𝑋=[0,1] be endowed with usual ordering and 𝑓,𝑔∶𝑋→𝑋 two mappings given by 𝑓𝑥=𝑥² and 𝑔𝑥=√𝑥. Clearly, the pair (𝑓,𝑔) is partially weakly increasing. But 𝑔𝑥=√𝑥≰𝑥=𝑓𝑔𝑥 for any 𝑥∈(0,1) implies that the pair (𝑔,𝑓) is not partially weakly increasing.
Let (𝑋,≤) be a partially ordered set. A mapping 𝑓∶𝑋→𝑋 is called a weak annihilator of a mapping 𝑔∶𝑋→𝑋 if 𝑓𝑔𝑥≤𝑥 for all 𝑥∈𝑋.

Example 2 (see [3]). Let 𝑋=[0,1] be endowed with usual ordering and 𝑓,𝑔∶𝑋→𝑋 be two mappings given by 𝑓𝑥=𝑥² and 𝑔𝑥=𝑥³. It is clear that 𝑓𝑔𝑥=𝑥⁶≤𝑥 for 𝑥∈𝑋 implies that 𝑓 is a weak annihilator of 𝑔.

Let (𝑋,≤) be a partially ordered set. A mapping 𝑓 is called a dominating if 𝑥≤𝑓𝑥 for any 𝑥∈𝑋.

Example 3 (see [3]). Let 𝑋=[0,1] be endowed with usual ordering and 𝑓∶𝑋→𝑋 a mapping defined by 𝑓𝑥=𝑥^1/3, since 𝑥≤𝑥^1/3=𝑓𝑥 for 𝑥∈𝑋 implies that 𝑓 is a dominating mapping.

A subset 𝑊 of a partially ordered set 𝑋 is said to be well ordered if every two elements of 𝑊 are comparable.

Let 𝑀 be a nonempty subset of a metric space (𝑋,𝑑). Let 𝑆 and 𝑇 be mappings from a metric space (𝑋,𝑑) into itself. A point 𝑥∈𝑀 is a common fixed (resp., coincidence) point of 𝑆 and 𝑇 if 𝑥=𝑆𝑥=𝑇𝑥(resp.,𝑆𝑥=𝑇𝑥). The set of fixed points (resp., coincidence points) of 𝑆 and 𝑇 is denoted by 𝐹(𝑆,𝑇) (resp., 𝐶(𝑆,𝑇)).

In 1986, Jungck [24] introduced the more generalized commuting mappings in metric spaces, called compatible mappings, which also are more general than the concept of weakly commuting mappings (that is, the mappings 𝑆,𝑇∶𝑋→𝑋 are said to be weakly commuting if 𝑑(𝑆𝑇𝑥,𝑇𝑆𝑥)≤𝑑(𝑆𝑥,𝑇𝑥) for all 𝑥∈𝑋) introduced by Sessa [34] as follows.

Definition 4. Let 𝑆 and 𝑇 be mappings from a metric space (𝑋,𝑑) into itself. The mappings 𝑆 and 𝑇 are said to be compatible if lim_𝑛→∞𝑑𝑆𝑇𝑥_𝑛,𝑇𝑆𝑥_𝑛=0, (1) whenever {𝑥_𝑛} is a sequence in 𝑋 such that lim_𝑛→∞𝑆𝑥_𝑛=lim_𝑛→∞𝑇𝑥_𝑛=𝑧 for some 𝑧∈𝑋.

In general, commuting mappings are weakly commuting and weakly commuting mappings are compatible, but the converses are not necessarily true and some examples can be found in [24–26].

In [27], Jungck and Rhoades introduced the concept of weakly compatible mappings and proved some common fixed point theorems for these mappings.

Definition 5. The mappings 𝑆 and 𝑇 are said to be weakly compatible if they commute at coincidence points of 𝑆 and 𝑇.

In Djoudi and Nisse [21], we can find an example to show that there exists weakly compatible mappings which are not compatible mappings in metric spaces.

Let Ψ denote the set of all functions 𝜓∶[0,∞)⁵→[0,∞) such that(a)𝜓is continuous;(b)𝜓 is strictly increasing in all the variables;(c)for all 𝑡∈[0,∞)⧵{0}, 𝜓()()𝜓()()𝜓()𝑡,𝑡,𝑡,0,2𝑡<𝑡,𝜓𝑡,𝑡,𝑡,2𝑡,0<𝑡,0,0,𝑡,𝑡,0<𝑡,𝜓0,𝑡,0,0,𝑡<𝑡,𝑡,0,0,𝑡,𝑡<𝑡. (2)It is easy to verify that the following functions are from the class Ψ, see [18]: 𝜓𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅=𝑘max𝑡₁,𝑡₂,𝑡₃,𝑡₄2,𝑡₅2,();𝜓𝑡for𝑘∈0,1₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅=𝑘max𝑡₁,𝑡₂,𝑡₃,𝑡₄+𝑡₅2,for𝑘∈(0,1). (3)

Definition 6 (see [18]). Let (𝑋,≤) be a partially ordered set and suppose that there exists a metric 𝑑 in 𝑋 such that (𝑋,𝑑) is a metric space. The mapping 𝑓∶𝑋→𝑋 is said to be a 𝜓- contractive mapping, if 𝑑𝑑,𝑑,𝑓𝑥,𝑓𝑦≤𝜓𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑓𝑦𝑥,𝑓𝑦,𝑑𝑦,𝑓𝑥 (4) for 𝑥≥𝑦.

Recently, Chen introduced 𝜓-contractive mappings. The purpose of this paper is to extend the results of Chen for four mappings, in the framework of ordered metric spaces.

2. Main Results

Now, we give the main results in this paper.

Theorem 7. Let (𝑋,≤) be a partially ordered set, and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose that 𝑇, 𝑓, 𝑔, and 𝑆 are self-mappings on 𝑋, the pairs (𝑇,𝑓) and (𝑆,𝑔) are partially weakly increasing with 𝑓(𝑋)⊆𝑇(𝑋) and 𝑔(𝑋)⊆𝑆(𝑋), and the dominating mappings 𝑓 and 𝑔 are weak annihilators of 𝑇 and 𝑆, respectively. Further, suppose that for any two comparable elements 𝑥,𝑦∈𝑋, and 𝜓∈Ψ, 𝑑𝑑,𝑑,𝑓𝑥,𝑔𝑦≤𝜓𝑆𝑥,𝑇𝑦,𝑑𝑆𝑥,𝑓𝑥,𝑑𝑇𝑦,𝑔𝑦𝑆𝑥,𝑔𝑦,𝑑𝑇𝑦,𝑓𝑥 (5) holds. If, for a nondecreasing sequence {𝑥_𝑛} with 𝑥_𝑛≤𝑦_𝑛 for all 𝑛≥1, 𝑦_𝑛→𝑢 implies that 𝑥_𝑛≤𝑢 and either(a)𝑓 and 𝑆 are compatible, 𝑓 or 𝑆 is continuous, and 𝑔,𝑇 are weakly compatible or(b)𝑔 and 𝑇 are compatible, 𝑔 or 𝑇 is continuous, and 𝑓,𝑆 are weakly compatible, then 𝑓,𝑔,𝑆, and 𝑇 have a common fixed point in 𝑋. Moreover, the set of common fixed points of 𝑓,𝑔,𝑆, and 𝑇 is well ordered if and only if 𝑓,𝑔,𝑆, and 𝑇 have one and only one common fixed point in 𝑋.

Proof. Let 𝑥₀∈𝑋 be an arbitrary point. Since 𝑓(𝑋)⊆𝑇(𝑋) and 𝑔(𝑋)⊆𝑆(𝑋), we can construct the sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that 𝑦_2𝑛−1=𝑓𝑥_2𝑛−2=𝑇𝑥_2𝑛−1,𝑦_2𝑛=𝑔𝑥_2𝑛−1=𝑆𝑥_2𝑛, (6) for each 𝑛≥1. By assumptions, we have 𝑥_2𝑛−2≤𝑓𝑥_2𝑛−2=𝑇𝑥_2𝑛−1≤𝑓𝑇𝑥_2𝑛−1≤𝑥_2𝑛−1,𝑥_2𝑛−1≤𝑔𝑥_2𝑛−1=𝑆𝑥_2𝑛≤𝑔𝑆𝑥_2𝑛≤𝑥_2𝑛, (7) for each 𝑛≥1. Thus, for each 𝑛≥1, we have 𝑥_𝑛≤𝑥_𝑛+1. Without loss of generality, we assume that 𝑦_2𝑛≠𝑦_2𝑛+1 for each 𝑛≥1.
Now, we claim that for all 𝑛∈ℕ, we have 𝑑𝑦_𝑛+1,𝑦_𝑛+2<𝑑𝑦_𝑛,𝑦_𝑛+1. (8)
Suppose to the contrary that 𝑑(𝑦_2𝑛,𝑦_2𝑛+1)≤𝑑(𝑦_2𝑛+1,𝑦_2𝑛+2) for some 𝑛∈ℕ. Since 𝑦_2𝑛 and 𝑦_2𝑛+1 are comparable, from (5), we have 𝑑𝑦_2𝑛+1,𝑦_2𝑛+2=𝑑𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1≤𝜓𝑑𝑆𝑥_2𝑛,𝑇𝑥_2𝑛+1,𝑑𝑆𝑥_2𝑛,𝑓𝑥_2𝑛,𝑑𝑇𝑥_2𝑛+1,𝑔𝑥_2𝑛+1,𝑑𝑆𝑥_2𝑛,𝑔𝑥_2𝑛+1,𝑑𝑇𝑥_2𝑛+1,𝑓𝑥_2𝑛=𝜓𝑑𝑦_2𝑛,𝑦_2𝑛+1,𝑑𝑦_2𝑛,𝑦_2𝑛+1,𝑑𝑦_2𝑛+1,𝑦_2𝑛+2,𝑑𝑦_2𝑛,𝑦_2𝑛+2,𝑑𝑦_2𝑛+1,𝑦_2𝑛+1=𝜓𝑑𝑦_2𝑛,𝑦_2𝑛+1,𝑑𝑦_2𝑛,𝑦_2𝑛+1,𝑑𝑦_2𝑛+1,𝑦_2𝑛+2,𝑑𝑦_2𝑛,𝑦_2𝑛+2𝑑𝑦,0≤𝜓_2𝑛,𝑦_2𝑛+1,𝑑𝑦_2𝑛,𝑦_2𝑛+1,𝑑𝑦_2𝑛+1,𝑦_2𝑛+2,𝑑𝑦_2𝑛,𝑦_2𝑛+1+𝑑𝑦_2𝑛+1,𝑦_2𝑛+2𝑑𝑦,0≤𝜓_2𝑛+1,𝑦_2𝑛+2,𝑑𝑦_2𝑛+1,𝑦_2𝑛+2,𝑑𝑦_2𝑛+1,𝑦_2𝑛+2,2𝑑𝑦_2𝑛+1,𝑦_2𝑛+2𝑦,0<𝑑_2𝑛+1,𝑦_2𝑛+2, (9) which is a contradiction. Hence 𝑑(𝑦_2𝑛+1,𝑦_2𝑛+2)≤𝑑(𝑦_2𝑛,𝑦_2𝑛+1) for each 𝑛≥1.
Similarly, we can prove that 𝑑(𝑦_2𝑛+1,𝑦_2𝑛)≤𝑑(𝑦_2𝑛,𝑦_2𝑛−1) for each 𝑛≥1.
Therefore, we can conclude that (8) holds.
Let us denote 𝑐_𝑛=𝑑(𝑦_𝑛+1,𝑦_𝑛). Then, from (8), 𝑐_𝑛 is a nonincreasing sequence and bounded below. Thus, it must converge to some 𝑐≥0. If 𝑐>0, then by the above inequalities, we have 𝑐≤𝑐_𝑛+1≤𝜓(𝑐_𝑛,𝑐_𝑛,𝑐_𝑛,2𝑐_𝑛,0). Taking the limit, as 𝑛→∞, we have 𝑐≤𝑐≤𝜓(𝑐,𝑐,𝑐,2𝑐,0)<𝑐, which is a contradiction. Hence, 𝑑𝑦_𝑛+1,𝑦_𝑛⟶0. (10)
Now, we show that {𝑦_𝑛} is a Cauchy sequence.
Suppose that {𝑦_𝑛} is not a Cauchy sequence. Then, there exists 𝜖>0 for which we can find two sequences of natural numbers {𝑚(𝑘)} and {𝑛(𝑘)} with 𝑛(𝑘)>𝑚(𝑘)>𝑘 such that 𝑑𝑦_𝑚(𝑘),𝑦_𝑛(𝑘)≥𝜖,𝑑𝑦_𝑚(𝑘),𝑦_{𝑛(𝑘)−1}<𝜖. (11) From (11), it follows that 𝜖≤𝑑𝑦_𝑚(𝑘),𝑦_𝑛(𝑘)≤𝑑𝑦_𝑚(𝑘),𝑦_{𝑛(𝑘)−1}+𝑑𝑦_{𝑛(𝑘)−1},𝑦_𝑛(𝑘)<𝜖+𝑑𝑦_{𝑛(𝑘)−1},𝑦_𝑛(𝑘). (12) Letting 𝑘→∞ and using (10), we have lim_𝑘→∞𝑑𝑦_𝑚(𝑘),𝑦_𝑛(𝑘)=𝜖. (13)
Again, 𝑑𝑦_{𝑚(𝑘)−1},𝑦_{𝑛(𝑘)−1}≤𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑚(𝑘)+𝑑𝑦_𝑚(𝑘),𝑦_𝑛(𝑘)+𝑑𝑦_𝑛(𝑘),𝑦_{𝑛(𝑘)−1},𝑑𝑦_𝑚(𝑘),𝑦_𝑛(𝑘)≤𝑑𝑦_𝑚(𝑘),𝑦_{𝑚(𝑘)−1}+𝑑𝑦_{𝑚(𝑘)−1},𝑦_{𝑛(𝑘)−1}+𝑑𝑦_{𝑛(𝑘)−1},𝑦_𝑛(𝑘). (14) Letting 𝑘→∞ in the above inequalities and using (10) and (13), we have lim_𝑘→∞𝑑𝑦_{𝑚(𝑘)−1},𝑦_{𝑛(𝑘)−1}=𝜖. (15) Again, 𝑑𝑦_{𝑛(𝑘)−1},𝑦_𝑚(𝑘)≤𝑑𝑦_{𝑚(𝑘)−1},𝑦_{𝑛(𝑘)−1}+𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑚(𝑘),𝑑𝑦_{𝑚(𝑘)−1},𝑦_{𝑛(𝑘)−1}≤𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑚(𝑘)+𝑑𝑦_{𝑛(𝑘)−1},𝑦_𝑚(𝑘). (16) Letting 𝑘→∞ in the above inequalities and using (10) and (15), we have lim_𝑘→∞𝑑𝑦_{𝑛(𝑘)−1},𝑦_𝑚(𝑘)=𝜖. (17) Similarly, we have lim_𝑘→∞𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑛(𝑘)=𝜖. (18) Also, again from (10), (15), and the inequality 𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑛(𝑘)+1−𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑛(𝑘)≤𝑑𝑦_𝑛(𝑘),𝑦_𝑛(𝑘)+1, (19) it follows that lim_𝑘→∞𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑛(𝑘)+1=𝜖. (20) Now, we have 𝑑𝑦_𝑚(𝑘),𝑦_𝑛(𝑘)+1≤𝜓𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑛(𝑘),𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑚(𝑘),𝑑𝑦_𝑛(𝑘),𝑦_𝑛(𝑘)+1,𝑑𝑦_{𝑚(𝑘)−1},𝑦_𝑛(𝑘)+1,𝑑𝑦_𝑛(𝑘),𝑦_𝑚(𝑘). (21) Letting 𝑘→∞, we get 𝜖≤𝜓(𝜖,0,0,𝜖,𝜖)<𝜖, (22) which is a contradiction. Thus {𝑦_𝑛} is a Cauchy sequence. Since 𝑋 is a complete metric space, there exists 𝑧∈𝑋 such that 𝑦_𝑛→𝑧. Therefore, we have lim_𝑛→∞𝑦_2𝑛+1=lim_𝑛→∞𝑇𝑥_2𝑛+1=lim_𝑛→∞𝑓𝑥_2𝑛=𝑧,lim_𝑛→∞𝑦_2𝑛+2=lim_𝑛→∞𝑆𝑥_2𝑛+2=lim_𝑛→∞𝑔𝑥_2𝑛+1=𝑧. (23) Assume that 𝑆 is continuous. Since 𝑓 and 𝑆 are compatible, we have lim_𝑛→∞𝑓𝑆𝑥_2𝑛+2=lim_𝑛→∞𝑆𝑓𝑥_2𝑛+2=𝑆𝑧. (24) Also, 𝑥_2𝑛+1≤𝑔𝑥_2𝑛+1=𝑆𝑥_2𝑛+2. Now, we have 𝑑𝑓𝑆𝑥_2𝑛,𝑔𝑥_2𝑛+1≤𝜓𝑑𝑆²𝑥_2𝑛,𝑇𝑥_2𝑛+1,𝑑𝑆²𝑥_2𝑛,𝑓𝑆𝑥_2𝑛,𝑑𝑇𝑥_2𝑛+1,𝑔𝑥_2𝑛+1,𝑑𝑆²𝑥_2𝑛,𝑔𝑥_2𝑛+1,𝑑𝑇𝑥_2𝑛+1,𝑓𝑆𝑥_2𝑛. (25) Letting 𝑛→∞, we get 𝑑()(𝑑()()((),𝑆𝑧,𝑧≤𝜓𝑆𝑧,𝑧,0,0,𝑑𝑆𝑧,𝑧,𝑑𝑧,𝑆𝑧))<𝑑𝑆𝑧,𝑧 (26) which implies that 𝑆𝑧=𝑧.
Now, it follows that 𝑥_2𝑛+1≤𝑔𝑥_2𝑛+1 and 𝑔𝑥_2𝑛+1→𝑧, 𝑥_2𝑛+1≤𝑧. From (5), we have 𝑑𝑓𝑧,𝑔𝑥_2𝑛+1≤𝜓𝑑𝑆𝑧,𝑇𝑥_2𝑛+1,𝑑𝑆𝑧,𝑓𝑧,𝑑𝑇𝑥_2𝑛+1,𝑔𝑥_2𝑛+1,𝑑𝑆𝑧,𝑔𝑥_2𝑛+1,𝑑𝑇𝑥_2𝑛+1,𝑓𝑧. (27) Letting 𝑛→∞, we get 𝑑,𝑓𝑧,𝑧≤𝜓0,𝑑𝑓𝑧,𝑧,0,0,𝑑𝑧,𝑓𝑧<𝑑𝑓𝑧,𝑧 (28) which implies that 𝑓𝑧=𝑧. Since 𝑓(𝑋)⊆𝑇(𝑋), there exists 𝑤∈𝑋 such that 𝑆𝑧=𝑧=𝑓𝑧=𝑇𝑤. Suppose that 𝑔𝑤≠𝑇𝑤. Since 𝑧≤𝑓𝑧=𝑇𝑤≤𝑓𝑇𝑤≤𝑤 implies 𝑧≤𝑤, from (5), we obtain 𝑑𝑑(),𝑑𝑇𝑤,𝑔𝑤=𝑑𝑓𝑧,𝑔𝑤≤𝜓𝑆𝑧,𝑇𝑤,𝑑𝑆𝑧,𝑓𝑧,𝑑𝑇𝑤,𝑔𝑤𝑆𝑧,𝑔𝑤,𝑑𝑇𝑤,𝑓𝑧=𝜓₂,0,0,𝑑𝑇𝑤,𝑔𝑤,𝑑𝑇𝑤,𝑔𝑤,0<𝑑𝑇𝑤,𝑔𝑤 (29) which implies that 𝑇𝑤=𝑔𝑤. Since 𝑔 and 𝑇 are weakly compatible, 𝑔𝑧=𝑔𝑓𝑧=𝑔𝑇𝑤=𝑇𝑔𝑤=𝑇𝑓𝑧=𝑇𝑧. Thus, 𝑧 is a coincidence point of 𝑇 and 𝑔.
Now, 𝑥_2𝑛≤𝑓𝑥_2𝑛 and 𝑓𝑥_2𝑛→𝑧 implies 𝑥_2𝑛≤𝑧. Thus, from (5), we obtain 𝑑𝑓𝑥_2𝑛,𝑔𝑧≤𝜓𝑑𝑆𝑥_2𝑛,𝑇𝑧,𝑑𝑆𝑥_2𝑛,𝑓𝑥_2𝑛,𝑑𝑇𝑧,𝑔𝑧,𝑑𝑆𝑥_2𝑛,𝑔𝑧,𝑑𝑇𝑧,𝑓𝑥_2𝑛. (30) Letting 𝑛→∞, we get 𝑑𝑑,𝑧,𝑔𝑧≤𝜓𝑧,𝑔𝑧,0,0,𝑑𝑧,𝑔𝑧,𝑑𝑧,𝑔𝑧<𝑑𝑧,𝑔𝑧 (31) which implies that 𝑔𝑧=𝑧. Therefore, we have 𝑓𝑧=𝑔𝑧=𝑆𝑧=𝑇𝑧=𝑧.
If 𝑓 is continuous, then, following the similar arguments, also we get the result.
Similarly, the result follows when (b) holds.
Now, suppose that the set of common fixed points of 𝑇, 𝑆, 𝑓, and 𝑔 is well ordered.
We claim that common fixed points of 𝑇, 𝑆, 𝑓, and 𝑔 are unique.
Assume that 𝑇𝑢=𝑆𝑢=𝑓𝑢=𝑔𝑢=𝑢 and 𝑇𝑣=𝑆𝑣=𝑓𝑣=𝑔𝑣=𝑣, but 𝑢≠𝑣. Then, from (5), we have 𝑑()𝑑(),𝑑(𝑑()()(().𝑢,𝑣=𝑑𝑓𝑢,𝑔𝑣≤𝜓𝑆𝑢,𝑇𝑣,𝑑𝑆𝑢,𝑓𝑢,𝑑𝑇𝑣,𝑔𝑣𝑆𝑢,𝑔𝑣,𝑑𝑇𝑣,𝑓𝑢=𝜓𝑢,𝑣,0,0,𝑑𝑢,𝑣,𝑑𝑣,𝑢))<𝑑𝑢,𝑣 (32) This implies that 𝑑(𝑢,𝑣)=0, and hence 𝑢=𝑣.
Conversely, if 𝑇, 𝑆, 𝑓, and 𝑔 have only one common fixed point, then the set of common fixed point of 𝑓, 𝑔, 𝑆, and 𝑇 being singleton is well ordered. This completes the proof.

Example 8. Consider 𝑋=[0,1]∪{2,3,4,…} with usual ordering and 𝑑=[][],𝑥,𝑦𝑥−𝑦if𝑥,𝑦∈0,1,𝑥≠𝑦;𝑥+𝑦ifatleastoneof𝑥or𝑦∉0,1𝑥≠𝑦;0if𝑥=𝑦. (33) Then (𝑋,≤,𝑑) is a complete partially ordered metric space.
Let 𝑓,𝑔,𝑆, and 𝑇 be self-mappings on 𝑋 defined as 𝑓(𝑥)=10if𝑥=0;21if𝑥∈0,2;11if𝑥∈2;{};𝑔(𝑥)=1,1𝑥if𝑥∈2,3,4,…0if𝑥=0;21if𝑥∈0,2;1𝑥if𝑥∈2∪{};𝑇(𝑥)=1,12,3,4,…0if𝑥≤2;121if𝑥∈2;{};𝑆(𝑥)=1,1𝑥−1if𝑥∈2,3,4,…0if𝑥≤2;12𝑥−1if𝑥∈2;{}.,1𝑥if𝑥∈2,3,4,… (34) Define function 𝜓∶[0,∞)⁵→[0,∞) by the formula 𝜓𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅=67max𝑡₁,𝑡₂,𝑡₃,𝑡₄+𝑡₅2. (35) Note that 𝑓,𝑔,𝑆, and 𝑇 satisfy all the conditions given in Theorem 7. Moreover, 0 is a common fixed point of 𝑓,𝑔,𝑆, and 𝑇.

If 𝑓=𝑔, then we have the following result.

Corollary 9. Let (𝑋,≤) be a partially ordered set, and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose that 𝑇, 𝑓, and 𝑆 are self-mappings on 𝑋, the pairs (𝑇,𝑓) and (𝑆,𝑓) are partially weakly increasing with 𝑓(𝑋)⊆𝑇(𝑋) and 𝑓(𝑋)⊆𝑆(𝑋), and the dominating mapping 𝑓 is a weak annihilator of 𝑇 and 𝑆. Further, suppose that there exists the function 𝜓∈Ψ such that, for any two comparable elements 𝑥,𝑦∈𝑋, 𝑑𝑑,𝑑𝑓𝑥,𝑓𝑦≤𝜓𝑆𝑥,𝑇𝑦,𝑑𝑆𝑥,𝑓𝑥,𝑑𝑇𝑦,𝑓𝑦𝑆𝑥,𝑓𝑦,𝑑𝑇𝑦,𝑓𝑥 (36) holds. If, for a nondecreasing sequence {𝑥_𝑛} with 𝑥_𝑛≤𝑦_𝑛 for all 𝑛≥1, 𝑦_𝑛→𝑢 implies that 𝑥_𝑛≤𝑢 and either(a)𝑓,𝑆 are compatible, 𝑓 or 𝑆 is continuous, and 𝑓,𝑇 are weakly compatible or(b)𝑓,𝑇 are compatible, 𝑓 or 𝑇 is continuous, and 𝑓,𝑆 are weakly compatible, then 𝑓,𝑆, and 𝑇 have a common fixed point in 𝑋. Moreover, the set of common fixed points of 𝑓,𝑆, and 𝑇 is well ordered if and only if 𝑓,𝑆, and 𝑇 have one and only one common fixed point in 𝑋.

Corollary 10. Let (𝑋,≤) be a partially ordered set, and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose that 𝑇, 𝑓, and 𝑔 are self-mappings on 𝑋, the pairs (𝑇,𝑓) and (𝑇,𝑔) are partially weakly increasing with 𝑓(𝑋)⊆𝑇(𝑋) and 𝑔(𝑋)⊆𝑇(𝑋), and the dominating mappings 𝑓 and 𝑔 are weak annihilators of 𝑇. Further, suppose that there exists the function 𝜓∈Ψ such that, for any two comparable elements 𝑥,𝑦∈𝑋, 𝑑𝑑,𝑑𝑓𝑥,𝑔𝑦≤𝜓𝑇𝑥,𝑇𝑦,𝑑𝑇𝑥,𝑓𝑥,𝑑𝑇𝑦,𝑔𝑦𝑇𝑥,𝑔𝑦,𝑑𝑇𝑦,𝑓𝑥 (37) holds. If, for a nondecreasing sequence {𝑥_𝑛} with 𝑥_𝑛≤𝑦_𝑛 for all 𝑛≥1, 𝑦_𝑛→𝑢 implies that 𝑥_𝑛≤𝑢 and either(a)𝑓,𝑇 are compatible, 𝑓 or 𝑇 is continuous, and 𝑔,𝑇 are weakly compatible or(b)𝑔,𝑇 are compatible, 𝑔 or 𝑇 is continuous, and 𝑓,𝑇 are weakly compatible, then 𝑓,𝑔, and 𝑇 have a common fixed point in 𝑋. Moreover, the set of common fixed points of 𝑓,𝑔, and 𝑇 is well ordered if and only if 𝑓,𝑔, and 𝑇 have one and only one common fixed point in 𝑋.

Corollary 11. Let (𝑋,≤) be a partially ordered set, and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose that 𝑇 and 𝑓 are self-mappings on 𝑋, the pair (𝑇,𝑓) is partially weakly increasing with 𝑓(𝑋)⊆𝑇(𝑋), and the dominating mapping 𝑓 is a weak annihilator of 𝑇. Further, suppose that there exists the function 𝜓∈Ψ such that, for any two comparable elements 𝑥,𝑦∈𝑋, 𝑑𝑑,𝑑𝑓𝑥,𝑓𝑦≤𝜓𝑇𝑥,𝑇𝑦,𝑑𝑇𝑥,𝑓𝑥,𝑑𝑇𝑦,𝑓𝑦𝑇𝑥,𝑓𝑦,𝑑𝑇𝑦,𝑓𝑥 (38) holds. If, for a nondecreasing sequence {𝑥_𝑛} with 𝑥_𝑛≤𝑦_𝑛 for all 𝑛≥1, 𝑦_𝑛→𝑢 implies that 𝑥_𝑛≤𝑢 and, further, 𝑓,𝑇 are compatible, 𝑓 or 𝑇 is continuous, and 𝑓,𝑇 are weakly compatible, then 𝑓 and 𝑇 have a common fixed point in 𝑋. Moreover, the set of common fixed points of 𝑓 and 𝑇 is well ordered if and only if 𝑓 and 𝑇 have one and only one common fixed point in 𝑋.

3. Applications

The aim of the section is to apply our new results to mappings involving contractions of integral type. For this purpose, denote by Λ the set of functions 𝜇∶[0,∞)→[0,∞) satisfying the following hypotheses:(h1)𝜇 is a Lebesgue-integrable mapping on each compact of [0,∞);(h2)for any 𝜖>0, we have ∫^𝜖₀𝜇(𝑡)>0.

Corollary 12. Let (𝑋,≤) be a partially ordered set, and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose that 𝑇, 𝑓, 𝑔, and 𝑆 are self-mappings on 𝑋, the pairs (𝑇,𝑓) and (𝑆,𝑔) are partially weakly increasing with 𝑓(𝑋)⊆𝑇(𝑋) and 𝑔(𝑋)⊆𝑆(𝑋), and the dominating mappings 𝑓 and 𝑔 are weak annihilators of 𝑇 and 𝑆, respectively. Further, suppose that there exists the function 𝜓∈Ψ such that, for any two comparable elements 𝑥,𝑦∈𝑋, ₀^{𝑑(𝑓𝑥,𝑔𝑦)}𝛼(𝑠)≤𝑑𝑠_{0𝜓(𝑑(𝑆𝑥,𝑇𝑦),𝑑(𝑆𝑥,𝑓𝑥),𝑑(𝑇𝑦,𝑔𝑦),𝑑(𝑆𝑥,𝑔𝑦),𝑑(𝑇𝑦,𝑓𝑥))}𝛼(𝑠)𝑑𝑠 (39) holds, where 𝛼∈Λ. If, for a nondecreasing sequence {𝑥_𝑛} with 𝑥_𝑛≤𝑦_𝑛 for all 𝑛≥1, 𝑦_𝑛→𝑢 implies that 𝑥_𝑛≤𝑢 and either(a)𝑓,𝑆 are compatible, 𝑓 or 𝑆 is continuous, and 𝑔,𝑇 are weakly compatible or(b)𝑔,𝑇 are compatible, 𝑔 or 𝑇 is continuous, and 𝑓,𝑆 are weakly compatible, then 𝑓,𝑔,𝑆, and 𝑇 have a common fixed point in 𝑋. Moreover, the set of common fixed points of 𝑓,𝑔,𝑆, and 𝑇 is well ordered if and only if 𝑓,𝑔,𝑆, and 𝑇 have one and only one common fixed point in 𝑋.

Corollary 13. Let (𝑋,≤) be a partially ordered set, and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose that 𝑇, 𝑓, and 𝑔 are self-mappings on 𝑋, the pairs (𝑇,𝑓) and (𝑇,𝑔) are partially weakly increasing with 𝑓(𝑋)⊆𝑇(𝑋) and 𝑔(𝑋)⊆𝑇(𝑋), and the dominating mappings 𝑓 and 𝑔 are weak annihilators of 𝑇. Further, suppose that there exists the function 𝜓∈Ψ such that, for any two comparable elements 𝑥,𝑦∈𝑋, ₀^{𝑑(𝑓𝑥,𝑔𝑦)}𝛼(𝑠)≤𝑑𝑠_{0𝜓(𝑑(𝑇𝑥,𝑇𝑦),𝑑(𝑇𝑥,𝑓𝑥),𝑑(𝑇𝑦,𝑔𝑦),𝑑(𝑇𝑥,𝑔𝑦),𝑑(𝑇𝑦,𝑓𝑥))}𝛼(𝑠)𝑑𝑠 (40) holds, where 𝛼∈Λ. If, for a nondecreasing sequence {𝑥_𝑛} with 𝑥_𝑛≤𝑦_𝑛 for all 𝑛≥1, 𝑦_𝑛→𝑢 implies that 𝑥_𝑛≤𝑢 and either(a)𝑓,𝑇 are compatible, 𝑓 or 𝑇 is continuous, and 𝑔,𝑇 are weakly compatible or(b)𝑔,𝑇 are compatible, 𝑔 or 𝑇 is continuous, and 𝑓,𝑇 are weakly compatible, then 𝑓,𝑔, and 𝑇 have a common fixed point in 𝑋. Moreover, the set of common fixed points of 𝑓,𝑔, and 𝑇 is well ordered if and only if 𝑓,𝑔, and 𝑇 have one and only one common fixed point in 𝑋.

Corollary 14. Let (𝑋,≤) be a partially ordered set, and suppose that there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Suppose that 𝑇 and 𝑓 are self-mappings on 𝑋, the pair (𝑇,𝑓) is a partially weakly increasing with 𝑓(𝑋)⊆𝑇(𝑋), and the dominating mapping 𝑓 is a weak annihilator of 𝑇. Further, suppose that there exists the function 𝜓∈Ψ such that, for any two comparable elements 𝑥,𝑦∈𝑋, ₀^{𝑑(𝑓𝑥,𝑓𝑦)}𝛼(𝑠)≤𝑑𝑠_{0𝜓(𝑑(𝑇𝑥,𝑇𝑦),𝑑(𝑇𝑥,𝑓𝑥),𝑑(𝑇𝑦,𝑓𝑦),𝑑(𝑇𝑥,𝑓𝑦),𝑑(𝑇𝑦,𝑓𝑥))}𝛼(𝑠)𝑑𝑠 (41) holds, where 𝛼∈Λ. If, for a nondecreasing sequence {𝑥_𝑛} with 𝑥_𝑛≤𝑦_𝑛 for all 𝑛≥1, 𝑦_𝑛→𝑢 implies that 𝑥_𝑛≤𝑢 and, further, 𝑓,𝑇 are compatible, 𝑓 or 𝑇 is continuous, and 𝑓,𝑇 are weakly compatible, then 𝑓 and 𝑇 have a common fixed point in 𝑋. Moreover, the set of common fixed points of 𝑓 and 𝑇 is well ordered if and only if 𝑓 and 𝑇 have one and only one common fixed point in 𝑋.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.