Coupled Fixed Points for Meir-Keeler Contractions in Ordered Partial Metric Spaces

Abstract

In this paper, we prove the existence and uniqueness of a new Meir-Keeler type coupled fixed point theorem for two mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 on a partially ordered partial metric space. We present an application to illustrate our obtained results. Further, we remark that the metric case of our results proved recently in Gordji et al. (2012) have gaps. Therefore, our results revise and generalize some of those presented in Gordji et al. (2012).

1. Introduction and Preliminaries

Fixed point theory is an important tool in the study of nonlinear analysis as it is considered to be the key connection between pure and applied mathematics with wide applications in economics, physical sciences, such as biology, chemistry, physics, differential equations, and almost all engineering fields (see, e.g., [1–13]). From the engineering point of view there are numerous problems in adaptive systems where convergence, optimal performance, and stability are key issues. In this direction many case studies with engineering applications can be described by contraction mappings and their fixed point iterations, such as linear and nonlinear filters, image restoration and image retrieval, and in many other areas where this theory helps to describe and/or understand the phenomenon. Indeed, the relaxation in linear systems, and relaxation and stability in neural networks can be analyzed in this light, where examples for a posteriori and normalized learning algorithms for adaptive filters for monophonic and stereophonic echo cancelation can be presented [14, 15].

Also it is worth mentioning that Matthews introduced the notion of partial metric space, which provides an area with great potential for the development of fixed point theory, as well as tools of conducting studies on denotational semantics of data-flow networks [16].

As a result it is evident that the importance of fixed point theory cannot be ruled out. Banach fixed point theorem [17] is the cornerstone of this topic. The result of Banach has drawn considerable interest of many authors. There are very different approaches in the study of generalization of a Banach fixed point theorem.

One of the interesting generalizations was announced by Matthews [16]. The author introduced the notion of partial metric spaces and proved the analog of Banach fixed point theorem. Roughly speaking, a partial metric space is a generalization of a metric spaces in which self distance of some points may not be zero. Matthews [16] discovered this phenomena when he tried to overcome problems of applying metric space techniques in the subfield of computer science: semantics and domain theory (see, e.g., [18, 19]). After the pioneer result of Mathews, remarkably good results have been reported on partial metric spaces (see, e.g., [20–40]).

On the other hand, considering the existence and uniqueness of a fixed point in partially ordered sets initiated a new trend in fixed point theory. The first result in this direction was given by Turinici [41], where he extended Banach contraction principle in partially ordered sets. Ran and Reurings [42] presented some applications of Turinici's theorem to matrix equations. After this intriguing paper, so many exceptionally good results have been revealed in this direction (see, e.g., [43–50]). Worth mentioning, Gnana Bhaskar and Lakshmikantham [44] introduced the notion of a coupled fixed point in the class of partially ordered metric spaces. Motivated by the above history, we devote this paper to prove the existence and uniqueness of coupled fixed points for a new Meir-Keeler type mappings in ordered partial metric spaces.

First, we recall basic definitions and crucial results. Hereafter, we assume that 𝑋≠∅ and we use the notation 𝑋^𝑘=𝑋×𝑋×⋯×𝑋._𝑘-many (1.1)

Definition 1.1 (see [44]). Let (𝑋,≤) be a partially ordered set and 𝐹∶𝑋×𝑋→𝑋. 𝐹 is said to have the mixed monotone property if 𝐹(𝑥,𝑦) is monotone nondecreasing in 𝑥 and is monotone nonincreasing in 𝑦, that is, for any 𝑥,𝑦∈𝑋, 𝑥₁≤𝑥₂⟹𝐹𝑥₁,𝑦≤𝐹𝑥₂,𝑦,for𝑥₁,𝑥₂𝑦∈𝑋,₁≤𝑦₂⟹𝐹𝑥,𝑦₂≤𝐹𝑥,𝑦₁,for𝑦₁,𝑦₂∈𝑋. (1.2)

Definition 1.2 (see [44]). An element (𝑥,𝑦)∈𝑋×𝑋 is said to be a coupled fixed point of the mapping 𝐹∶𝑋×𝑋→𝑋 if 𝐹𝑥,𝑦=𝑥,𝐹𝑦,𝑥=𝑦. (1.3)

The following two results of Bhaskar and Lakshmikantham in [44] were proved in the context of cone metric spaces in [51].

Theorem 1.3 (see [44]). Let (𝑋,≤) be a partially ordered set and suppose that there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝐹∶𝑋×𝑋→𝑋 be a continuous mapping having the mixed monotone property on 𝑋. Assume that there exists 𝑘∈[0,1) with 𝑑𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑑(𝑥,𝑢)+𝑑𝑦,𝑣,∀𝑢≤𝑥,𝑦≤𝑣. (1.4) If there exists 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑦₀, then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥).

Theorem 1.4 (see [44]). Let (𝑋,≤) be a partially ordered set and suppose that there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping having the mixed monotone property on 𝑋. Suppose that 𝑋 has the following properties: (i)if a nondecreasing sequence {𝑥_𝑛}→𝑥, then 𝑥_𝑛≤𝑥, for all 𝑛, (ii)if a nonincreasing sequence {𝑦_𝑛}→𝑦, then 𝑦≤𝑦_𝑛, for all 𝑛. Assume that there exists a 𝑘∈[0,1) with 𝑑𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑑(𝑥,𝑢)+𝑑𝑦,𝑣,∀𝑢≤𝑥,𝑦≤𝑣. (1.5) If there exists 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀≤𝐹(𝑥₀,𝑦₀) and 𝐹(𝑦₀,𝑥₀)≤𝑦₀, then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥).

Inspired by Definition 1.1, the following concept of a 𝑔-mixed monotone mapping was introduced by Lakshmikantham and Ćirić [47].

Definition 1.5 (see [47]). Let (𝑋,≤) be partially ordered set and 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. 𝐹 is said to have mixed 𝑔-monotone property if 𝐹(𝑥,𝑦) is monotone 𝑔-nondecreasing in 𝑥 and is monotone 𝑔-nonincreasing in 𝑦, that is, for any 𝑥,𝑦∈𝑋, 𝑔𝑥₁≤𝑔𝑥₂⟹𝐹𝑥₁,𝑦≤𝐹𝑥₂,𝑦,for𝑥₁,𝑥₂𝑔𝑦∈𝑋,₁≤𝑔𝑦₂⟹𝐹𝑥,𝑦₂≤𝐹𝑥,𝑦₁,for𝑦₁,𝑦₂∈𝑋. (1.6)

It is clear that Definition 1.5 reduces to Definition 1.1 when 𝑔 is the identity.

Definition 1.6 (see [47]). An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled coincidence point of mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝐹𝑥,𝑦=𝑔(𝑥),𝐹𝑦,𝑥=𝑔𝑦, (1.7) and is called a coupled common fixed of 𝐹 and 𝑔, if 𝐹𝑥,𝑦=𝑔(𝑥)=𝑥,𝐹𝑦,𝑥=𝑔𝑦=𝑦. (1.8) The mappings 𝐹 and 𝑔 are said to commute if 𝑔𝐹𝑥,𝑦=𝐹𝑔(𝑥),𝑔𝑦, (1.9) for all 𝑥,𝑦∈𝑋.

Very recently, Gordji et al. [31] replaced mixed  g-monotone  property with a mixed  strict  g-monotone  property and improved the results in [47].

Definition 1.7 (see [31]). Let (𝑋,≤) be a partially ordered set and 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. 𝐹 is said to have the mixed strict 𝑔-monotone property if 𝐹(𝑥,𝑦) is monotone 𝑔-nondecreasing in 𝑥 and is monotone 𝑔-nonincreasing in 𝑦, that is, for any 𝑥,𝑦∈𝑋, 𝑔𝑥₁<𝑔𝑥₂⟹𝐹𝑥₁,𝑦<𝐹𝑥₂,𝑦,for𝑥₁,𝑥₂𝑔𝑦∈𝑋,₁<𝑔𝑦₂⟹𝐹𝑥,𝑦₁>𝐹𝑥,𝑦₂,for𝑦₁,𝑦₂∈𝑋. (1.10)

If we replace 𝑔 with identity map in (1.10), we get the definition of mixed strict monotone property of 𝐹.

A partial metric is a function 𝑝∶𝑋×𝑋→[0,∞) satisfying the following conditions: (P1)If 𝑝(𝑥,𝑥)=𝑝(𝑥,𝑦)=𝑝(𝑦,𝑦), then 𝑥=𝑦, (P2)𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥), (P3)𝑝(𝑥,𝑥)≤𝑝(𝑥,𝑦), (P4)𝑝(𝑥,𝑧)+𝑝(𝑦,𝑦)≤𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧), for all 𝑥,𝑦,𝑧∈𝑋. Then (𝑋,𝑝) is called a partial metric space. If 𝑝 is a partial metric 𝑝 on 𝑋, then the function 𝑑_𝑝∶𝑋×𝑋→[0,∞) given by 𝑑_𝑝𝑥,𝑦=2𝑝𝑥,𝑦−𝑝(𝑥,𝑥)−𝑝𝑦,𝑦 (1.11) is a metric on 𝑋. Each partial metric 𝑝 on 𝑋 generates a 𝑇₀ topology 𝜏_𝑝 on 𝑋 with a base of the family of open 𝑝-balls {𝐵_𝑝(𝑥,𝜀)∶𝑥∈𝑋,𝜀>0}, where 𝐵_𝑝(𝑥,𝜀)={𝑦∈𝑋∶𝑝(𝑥,𝑦)<𝑝(𝑥,𝑥)+𝜀} for all 𝑥∈𝑋 and 𝜀>0. Similarly, closed 𝑝-ball is defined as 𝐵_𝑝[𝑥,𝜀]={𝑦∈𝑋∶𝑝(𝑥,𝑦)≤𝑝(𝑥,𝑥)+𝜀}. For more details see for example [16, 21].

Definition 1.8 (see [16, 21, 33]). Let (𝑋,𝑝) be a partial metric space. (i)A sequence {𝑥_𝑛} in 𝑋 converges to 𝑥∈𝑋 whenever lim_𝑛→∞𝑝(𝑥,𝑥_𝑛)=𝑝(𝑥,𝑥). (ii)A sequence {𝑥_𝑛} in 𝑋 is called Cauchy whenever lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists (and finite). (iii)(𝑋,𝑝) is said to be complete if every Cauchy sequence {𝑥_𝑛} in 𝑋 converges, with respect to 𝜏_𝑝, to a point 𝑥∈𝑋, that is, lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚)=𝑝(𝑥,𝑥). (iv)A mapping 𝑓∶𝑋→𝑋 is said to be continuous at 𝑥₀∈𝑋 if for each 𝜀>0 there exists 𝛿>0 such that 𝑓(𝐵(𝑥₀,𝛿))⊂𝐵(𝑓(𝑥₀),𝜀).

Lemma 1.9 (see [16, 21, 33]). Let (𝑋,𝑝) be a partial metric space. (a)A sequence {𝑥_𝑛} is Cauchy if and only if {𝑥_𝑛} is a Cauchy sequence in the metric space (𝑋,𝑑_𝑝), (b)(𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑑_𝑝) is complete. Moreover, lim_𝑛→∞𝑑_𝑝𝑥,𝑥_𝑛=0⟺lim_𝑛→∞𝑝𝑥,𝑥_𝑛=lim_{𝑛,𝑚→∞}𝑝𝑥_𝑛,𝑥_𝑚=𝑝(𝑥,𝑥). (1.12)

Lemma 1.10 (see [20]). Let (𝑋,𝑝) be a partial metric space. Then (A)If 𝑝(𝑥,𝑦)=0 then 𝑥=𝑦. (B)If 𝑥≠𝑦, then 𝑝(𝑥,𝑦)>0.

Remark 1.11. If 𝑥=𝑦, 𝑝(𝑥,𝑦) may not be 0.

The following two lemmas can be derived from the triangle inequality (P4).

Lemma 1.12 (see [20]). Let 𝑥_𝑛→𝑧 as 𝑛→∞ in a partial metric space (𝑋,𝑝), where 𝑝(𝑧,𝑧)=0. Then lim_𝑛→∞𝑝(𝑥_𝑛,𝑦)=𝑝(𝑧,𝑦) for every 𝑦∈𝑋.

Lemma 1.13 (see [36]). Let lim_𝑛→∞𝑝(𝑥_𝑛,𝑦)=𝑝(𝑦,𝑦) and lim_𝑛→∞𝑝(𝑥_𝑛,𝑧)=𝑝(𝑧,𝑧). If 𝑝(𝑦,𝑦)=𝑝(𝑧,𝑧) then 𝑦=𝑧.

Remark 1.14. Limit of a sequence {𝑥_𝑛} in a partial metric space (𝑋,𝑝) is not unique.

Example 1.15. Consider 𝑋=[0,∞) with 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Then (𝑋,𝑝) is a partial metric space. Clearly, 𝑝 is not a metric. Observe that the sequence {1+1/𝑛²} converges both for example to 𝑥=2 and 𝑦=3, so no uniqueness of the limit.

We give the partial case of a definition given in [31].

Definition 1.16 (see [31]). Let (𝑋,𝑝,≤) be a partially ordered partial metric space. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. The mapping 𝐹 is said to be a 𝑔-Meir-Keeler type contraction if for any 𝜀>0 there exists a 𝛿(𝜀)>0 such that 1𝜀≤2𝑝𝑔(𝑥),𝑔(𝑢)+𝑝𝑔𝑦,𝑔(𝑣)<𝜀+𝛿(𝜀)⟹𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)<𝜀, (1.13) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑔(𝑥)≤𝑔(𝑢), 𝑔(𝑦)≥𝑔(𝑣).

If we replace 𝑔 with the identity in (1.13) and 𝑝 a metric on 𝑋, the 𝐹 is called a Meir-Keeler type contraction.

Definition 1.17. Let (𝑋,𝑝,≤) be a partially ordered partial metric space. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. The mapping 𝐹 is said to be a strict 𝑔-Meir-Keeler type contraction if there exists 0<𝑘<1 such that for any 𝜀>0 there exists a 𝛿(𝜀)>0 such that 𝑘𝜀≤2𝑝𝑔(𝑥),𝑔(𝑢)+𝑝𝑔𝑦,𝑔(𝑣)<𝜀+𝛿(𝜀)⟹𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)<𝜀, (1.14) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑔(𝑥)≤𝑔(𝑢), 𝑔(𝑦)≥𝑔(𝑣).

If we replace 𝑔 with the identity in (1.14) and if 𝑝 a metric on 𝑋, the 𝐹 is called a strict Meir-Keeler type contraction. Further, it can be shown easily that every strict Meir-Keeler (resp., strict 𝑔-Meir-Keeler) type contraction is a Meir-Keeler (resp., 𝑔-Meir-Keeler) type contraction.

Let (𝑋,𝑝) be a partial metric space. Note that the mappings 𝜌_𝑘∶𝑋^𝑘×𝑋^𝑘→[0,+∞) defined by 𝜌_𝑘(𝐱,𝐲)∶=𝑝𝑥₁,𝑦₁+𝑝𝑥₂,𝑦₂+⋯+𝑝𝑥_𝑘,𝑦_𝑘, (1.15) forms a partial metric on 𝑋^𝑘 where 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑘) and 𝐲=(𝑦₁,𝑦₂,…,𝑦_𝑘)∈𝑋^𝑘.

The following fact can be derived easily from Definition 1.16.

Lemma 1.18. Let (𝑋,𝑝,≤) be a partially ordered partial metric space. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. If 𝐹 is a 𝑔-Meir-Keeler type contraction, then one has 𝑝𝐹()<1x,𝑦,𝐹𝑢,𝑣2𝑝𝑔(𝑥),𝑔(𝑢)+𝑝𝑔𝑦,𝑔(𝑣), (1.16) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑔(𝑥)<𝑔(𝑢), 𝑔(𝑦)≥𝑔(𝑣) or 𝑔(𝑥)≤𝑔(𝑢), 𝑔(𝑦)>𝑔(𝑣).

Proof. Without loss of generality, suppose that 𝑔(𝑥)<𝑔(𝑢) and 𝑔(𝑦)≥𝑔(𝑣) where 𝑥,𝑦,𝑢,𝑣∈𝑋. It is clear that 𝑝(𝑔(𝑥),𝑔(𝑢))+𝑝(𝑔(𝑦),𝑔(𝑣))>0. Set 𝜀=(1/2)[𝑝(𝑔(𝑥),𝑔(𝑢))+𝑝(𝑔(𝑦),𝑔(𝑣))]>0. Since 𝐹 is a 𝑔-Meir-Keeler type contraction, then, for this 𝜀, there exits 𝛿=𝛿(𝜀)>0 such that 1𝜀≤2𝑝𝑔𝑥₀,𝑔𝑢₀+𝑝𝑔𝑦₀,𝑔𝑣₀<𝜀+𝛿⟹𝑝𝐹𝑥₀,𝑦₀,𝐹𝑢₀,𝑣₀<𝜀, (1.17) for all 𝑥₀,𝑦₀,𝑢₀,𝑣₀∈𝑋 with 𝑔(𝑥₀)<𝑔(𝑢₀) and 𝑔(𝑦₀)≥𝑔(𝑣₀). The result follows by choosing 𝑥=𝑥₀, 𝑦=𝑦₀, 𝑢=𝑢₀, 𝑧=𝑧₀, that is, 𝑝𝐹()<1𝑥,𝑦,𝐹𝑢,𝑣2𝑝𝑔(𝑥),𝑔(𝑢)+𝑝𝑔𝑦,𝑔(𝑣). (1.18)

Remark 1.19. Let (𝑋,𝑝,≤) be a partially ordered partial metric space. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. If 𝐹 is a strict 𝑔-Meir-Keeler type contraction, then we have 𝑝𝐹()<𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝𝑔(𝑥),𝑔(𝑢)+𝑝𝑔𝑦,𝑔(𝑣), (1.19) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑔(𝑥)<𝑔(𝑢), 𝑔(𝑦)≥𝑔(𝑣) or 𝑔(𝑥)≤𝑔(𝑢), 𝑔(𝑦)>𝑔(𝑣).

Proof. The proof is similar to Lemma 1.18 above.

2. Existence of Coupled Fixed Points

The following theorem is our first main result.

Theorem 2.1. Let (𝑋,𝑝,≤) be a partially ordered partial metric space. Suppose that 𝑋 has the following properties: (a)if {𝑥_𝑛} is a sequence such that 𝑥_𝑛+1>𝑥_𝑛 for each 𝑛=1,2,… and 𝑥_𝑛→𝑥, then 𝑥_𝑛<𝑥 for each 𝑛=1,2,…, (b)if {𝑦_𝑛} is a sequence such that 𝑦_𝑛+1<𝑦_𝑛 for each 𝑛=1,2,… and 𝑦_𝑛→𝑦, then 𝑦_𝑛>𝑦 for each 𝑛=1,2,…. Let 𝑔∶𝑋→𝑋 and 𝐹∶𝑋²→𝑋 be mappings such that 𝐹(𝑋×𝑋)⊂𝑔(𝑋) and 𝑔(𝑋) is a complete subspace of (𝑋,𝑝). Suppose that 𝐹 satisfies the following conditions: (i)𝐹 has the mixed strict 𝑔-monotone property, (ii)𝐹 is a 𝑔-Meir-Keeler type contraction, (iii)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥₀<𝐹𝑥₀,𝑦₀,𝑔𝑦₀≥𝐹𝑦₀,𝑥₀. (2.1)Then 𝐹 and 𝑔 have a coupled coincidence point, that is, there exist 𝑥,𝑦∈𝑋 such that 𝐹𝑥,𝑦=𝑔(𝑥),𝐹𝑦,𝑥=𝑔𝑦. (2.2)

Proof. Let (𝑥,𝑦)=(𝑥₀,𝑦₀)∈𝑋² be such that 𝑔(𝑥₀)<𝐹(𝑥₀,𝑦₀) and 𝑔(𝑦₀)≥𝐹(𝑦₀,𝑥₀). We construct the sequence {𝑥_𝑛} and {𝑦_𝑛} in the following way. Due to the assumption 𝐹(𝑋×𝑋)⊂𝑔(𝑋), we are able to choose (𝑥₁,𝑦₁)∈𝑋² such that 𝑔(𝑥₁)=𝐹(𝑥₀,𝑦₀) and 𝑔(𝑦₁)=𝐹(𝑦₀,𝑥₀). By repeating the same argument, we can choose (𝑥₂,𝑦₂)∈𝑋² such that 𝑔(𝑥₂)=𝐹(𝑥₁,𝑦₁) and 𝑔(𝑦₂)=𝐹(𝑦₁,𝑥₁). Inductively, we observe that 𝑔𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛∀𝑛=0,1,2,…. (2.3) We claim that, for all 𝑛≥2⋯>𝑔𝑥_𝑛>𝑔𝑥_𝑛−1>⋯>𝑔𝑥₁>𝑔𝑥₀,⋯<𝑔𝑦_𝑛<𝑔𝑦_𝑛−1<⋯<𝑔𝑦₁≤𝑔𝑦₀. (2.4) We will use the mathematical induction to show (2.4). By assumption (iii), we have 𝑔𝑥₀<𝐹𝑥₀,𝑦₀=𝑔𝑥₁,𝑔𝑦₀≥𝐹𝑦₀,𝑥₀=𝑔𝑦₁. (2.5) Assume that the inequalities in (2.4) hold for some 𝑛≥2. Regarding the mixed 𝑔-strict monotone property of 𝐹, we have 𝑔𝑥_𝑛−1<𝑔𝑥_𝑛⟹𝐹𝑥_𝑛−1,𝑦_𝑛−1<𝐹𝑥_n,𝑦_𝑛−1,𝐹𝑦_𝑛−1,𝑥_𝑛−1>𝐹𝑦_𝑛−1,𝑥_𝑛. (2.6) By repeating the same arguments, we observe that 𝑔𝑦_𝑛−1>𝑔𝑦_𝑛⟹𝐹𝑥_𝑛,𝑦_𝑛−1<𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑦_𝑛−1,𝑥_𝑛>𝐹𝑦_𝑛,𝑥_𝑛. (2.7) Combining the above inequalities, together with (2.3), we get 𝑔𝑥_𝑛=𝐹𝑥_𝑛−1,𝑦_𝑛−1<𝐹𝑥_𝑛,𝑦_𝑛=𝑔𝑥_𝑛+1,𝑔𝑦_𝑛=𝐹𝑦_𝑛−1,𝑥_𝑛−1>𝐹𝑦_𝑛,𝑥_𝑛=𝑔𝑦_𝑛+1. (2.8) So, (2.4) holds for all 𝑛≥2. Set Δ_𝑛=𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑛+1. (2.9) Taking Lemma 1.18 and (2.4) into account, we get𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛+1=𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛<12𝑝𝑔𝑥_𝑛−1,𝑔𝑥_𝑛+𝑝𝑔𝑦_𝑛−1,𝑔𝑦_𝑛,𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑛+1=𝑝𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛<12𝑝𝑔𝑥_𝑛−1,𝑔𝑥_𝑛+𝑝𝑔𝑦_𝑛−1,𝑔𝑦_𝑛. (2.10) If we add the previous two inequalities side by side, we obtain that Δ_𝑛<Δ_𝑛−1. Hence, {Δ_𝑛} is monotone decreasing sequence in ℝ. Since the sequence {Δ_𝑛} is bounded below, there exists 𝐿≥0 such that lim_𝑛→∞Δ_𝑛=𝐿.
We prove 𝐿=0. Suppose on the contrary that 𝐿≠0. Thus, there is a positive integer 𝑘 such that for any 𝑛≥𝑘, we have Δ𝜀≤^𝑛2=12𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑛+1<𝜀+𝛿(𝜀), (2.11) where 𝜀=𝐿/2 and 𝛿(𝜀) is chosen by (ii). In particular, for 𝑛=𝑘, we have Δ𝜀≤^𝑘2=12𝑝𝑔𝑥_𝑘,𝑔𝑥_𝑘+1+𝑝𝑔𝑦_𝑘,𝑔𝑦_𝑘+1<𝜀+𝛿(𝜀). (2.12) Regarding the assumption (iii) together with (2.12) and (2.4), we have 𝑝𝐹𝑥_𝑘,𝑦_𝑘,𝐹𝑥_𝑘+1,𝑦_𝑘+1<𝜀, (2.13) which is equivalent to 𝑝𝑔𝑥_𝑘+1,𝑔𝑥_𝑘+2<𝜀. (2.14) Similarly, we have 𝑝𝑔𝑦_𝑘+1,𝑔𝑦_𝑘+2<𝜀. (2.15) Summing the two above inequalities Δ_𝑘+12=12𝑝𝑔𝑥_𝑘+1,𝑔𝑥_𝑘+2+𝑝𝑔𝑦_𝑘+1,𝑔𝑦_𝑘+2<𝜀, (2.16) which contradicts (2.11) for 𝑛=𝑘+1. Thus, 𝜀=𝐿/2=0. That is, lim_𝑛→∞Δ_𝑛=lim_𝑛→∞𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑛+1=0. (2.17) Consequently, we have lim_𝑛→∞𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛+1=0=lim_𝑛→∞𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑛+1. (2.18) By condition (P3), we have 𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛≤𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛+1, (2.19) so letting 𝑛→∞, we get lim_𝑛→∞𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛=0. (2.20) Analogously, we have lim_𝑛→∞𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑛=0. (2.21) We claim that the sequences {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are Cauchy in (𝑔(𝑋),𝑝).
Take an arbitrary 𝜀>0. It follows from (2.17) that there exists 𝑘∈ℕ such that 12𝑝𝑔𝑥_𝑘,𝑔𝑥_𝑘+1+𝑝𝑔𝑦_𝑘,𝑔𝑦_𝑘+1<𝛿(𝜀). (2.22) Without loss of the generality, assume that 𝛿(𝜀)≤𝜀 and define the following set Π∶=𝑥,𝑦∈𝑋²∶𝑝𝑥,𝑔𝑥_𝑘+𝑝𝑦,𝑔𝑦_𝑘<2(𝜀+𝛿(𝜀))and𝑥>𝑔𝑥_𝑘,𝑦≤𝑔𝑦_𝑘. (2.23) Take ∧=(𝑔(𝑋),𝑔(𝑋))∩Π. We claim that 𝐹𝑝,𝑞,𝐹𝑞,𝑝∈∧∀𝑥,𝑦=𝑔𝑝,𝑔𝑞∈∧where𝑝,𝑞∈𝑋. (2.24) Take (𝑥,𝑦)=(𝑔(𝑝),𝑔(𝑞))∈Π. Then, by (2.22) and the triangle inequality (which still holds for partial metrics) we have 12𝑝𝑔𝑥_𝑘,𝐹𝑝,𝑞+𝑝𝑔𝑦_𝑘≤1,𝐹𝑞,𝑝2𝑝𝑔𝑥_𝑘,𝑔𝑥_𝑘+1+𝑝𝑔𝑥_𝑘+1+1,𝐹𝑝,𝑞2𝑝𝑔𝑦_𝑘,𝑔𝑦_𝑘+1+𝑝𝑔𝑦_𝑘+1=1,𝐹𝑞,𝑝2𝑝𝑔𝑥_𝑘,𝑔𝑥_𝑘+1+𝑝𝑔𝑦_𝑘,𝑔𝑦_𝑘+1+12𝑝𝐹𝑝,𝑞,𝐹𝑥_𝑘,𝑦_𝑘+12𝑝𝐹𝑦_𝑘,𝑥_𝑘(𝜀)+1,𝐹𝑞,𝑝<𝛿2𝑝𝐹𝑝,𝑞,𝐹𝑥_𝑘,𝑦_𝑘+12𝑝𝐹𝑦_𝑘,𝑥_𝑘,𝐹𝑞,𝑝. (2.25) We distinguish two cases.
First Case. (1/2)[𝑝(𝑥,𝑔(𝑥_𝑘))+𝑝(𝑦,𝑔(𝑦_𝑘))]=(1/2)[𝑝(𝑔(𝑝),𝑔(𝑥_𝑘))+𝑝(𝑔(𝑞),𝑔(𝑦_𝑘))]≤𝜀.
By Lemma 1.18 and the definition of Π, the inequality (2.25) turns into 12𝑝𝑔𝑥_𝑘,𝐹𝑝,𝑞+𝑝𝑔𝑦_𝑘(𝜀)+1,𝐹𝑞,𝑝<𝛿2𝑑𝐹𝑝,𝑞,𝐹𝑥_𝑘,𝑦_𝑘+12𝑑𝐹𝑦_𝑘,𝑥_𝑘(𝜀)+1,𝐹𝑞,𝑝<𝛿212𝑝𝑔𝑝,𝑔𝑥_𝑘+𝑝𝑔𝑞,𝑔𝑦_𝑘+12𝑝𝑔𝑦_𝑘,𝑔𝑞+𝑝𝑔𝑥_𝑘𝑝(𝜀)+1,𝑔=𝛿2𝑝𝑔𝑝,𝑔𝑥_𝑘+𝑝𝑔𝑞,𝑔𝑦_𝑘≤𝛿(𝜀)+𝜀. (2.26)
Second Case. 𝜀<(1/2)[𝑝(𝑥,𝑔(𝑥_𝑘))+𝑝(𝑦,𝑔(𝑦_𝑘))]=(1/2)[𝑝(𝑔(𝑝),𝑔(𝑥_𝑘))+𝑝(𝑔(𝑞),𝑔(𝑦_𝑘))]<𝜀+𝛿(𝜀).
In this case, we have 1𝜀<2𝑝𝑔𝑝,𝑔𝑥_𝑘+𝑝𝑔𝑞,𝑔𝑦_𝑘<𝜀+𝛿(𝜀). (2.27) Since 𝑥=𝑔𝑝>𝑔𝑥_𝑘 and 𝑦=𝑔𝑞≤𝑔𝑦_𝑘, by (ii), we get 𝑝𝐹𝑝,𝑞,𝐹𝑥_𝑘,𝑦_𝑘<𝜀. (2.28) Also, we have 1𝜀<2𝑝𝑔𝑦_𝑘,𝑔𝑞+𝑝𝑔𝑥_𝑘,𝑔𝑝<𝜀+𝛿(𝜀). (2.29) Since 𝑥=𝑔(𝑝)>𝑔(𝑥_𝑘) and 𝑦=𝑔(𝑞)≤𝑔(𝑦_k), by (ii), we get 𝑝𝐹𝑦_𝑘,𝑥_𝑘,𝐹𝑞,𝑝<𝜀. (2.30) Thus, combining (2.25), (2.28) and (2.30), we obtain 12𝑝𝑔𝑥_𝑘,𝐹𝑝,𝑞+𝑝𝑔𝑦_𝑘,𝐹𝑞,𝑝<𝜀+𝛿(𝜀). (2.31) On the other hand, using (i), it is obvious that 𝐹𝑝,𝑞>𝑔𝑥_𝑘,𝐹𝑞,𝑝≤𝑔𝑦_𝑘. (2.32) We conclude that (𝐹(𝑝,𝑞),𝐹(𝑞,𝑝))∈Π. Since 𝐹(𝑋×𝑋)⊂𝑔(𝑋), so 𝐹𝑝,𝑞,𝐹𝑞,𝑝∈∧, (2.33) that is, (2.24) holds. By (2.22), we have (𝑔(𝑥_𝑘+1),𝑔(𝑦_𝑘+1))∈∧. This implies with (2.24) that 𝑔𝑥_𝑘+1,𝑔𝑦_𝑘+1∈∧⟹𝐹𝑥_𝑘+1,𝑦_𝑘+1,𝐹𝑦_𝑘+1,𝑥_𝑘+1=𝑔𝑥_𝑘+2,𝑔𝑦_𝑘+2⟹𝐹𝑥∈∧_𝑘+2,𝑦_𝑘+2,𝐹𝑦_𝑘+2,𝑥_𝑘+2=𝑔𝑥_𝑘+3,𝑔𝑦_𝑘+3𝑔𝑥∈∧⟹⋯⟹_𝑛,𝑔𝑦_𝑛∈∧⟹⋯. (2.34) Then, for all 𝑛>𝑘, we have (𝑔𝑥_𝑛,𝑔𝑦_𝑛)∈∧. This implies that for all 𝑛,𝑚>𝑘, we have 𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑚+𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑚≤𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑘+𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑘+𝑝𝑔𝑥_𝑘,𝑔𝑥_𝑚+𝑝𝑔𝑦_𝑘,𝑔𝑦_𝑚<4(𝜀+𝛿(𝜀))≤8𝜀. (2.35) Thus, the sequences {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are Cauchy in (𝑔(𝑋),𝑝). By Lemma 1.9, {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are also Cauchy in (𝑔(𝑋),𝑑_𝑝). Again by Lemma 1.9, (𝑔(𝑋),𝑑_𝑝)) is complete. Thus, there exist 𝑥,𝑦∈𝑋 such that by using (2.20) and (2.21), we arrive at lim_𝑛→∞𝑑_𝑝𝑔(𝑥),𝑔𝑥_𝑛=0⟺𝑝𝑔(𝑥),𝑔(𝑥)=lim_𝑛→∞𝑝𝑔(𝑥),𝑔𝑥_𝑛=lim_𝑛→∞𝑝𝑔𝑥_𝑛,𝑔𝑥_𝑛=0,lim_𝑛→∞𝑑_𝑝𝑔𝑦,𝑔𝑦_𝑛=0⟺𝑝𝑔𝑦,𝑔𝑦=lim_𝑛→∞𝑝𝑔𝑦,𝑔𝑦_𝑛=lim_𝑛→∞𝑝𝑔𝑦_𝑛,𝑔𝑦_𝑛=0. (2.36) Since the sequences {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are monotone increasing and monotone decreasing, respectively, by properties (a) and (b), we conclude that 𝑔𝑥_𝑛<𝑔(𝑥),𝑔𝑦_𝑛>𝑔𝑦, (2.37) for each 𝑛≥0. Therefore, having in mind that 𝐹 is a 𝑔-Meir-Keeler type contraction, by (2.37) and Lemma 1.18, we get 𝑝𝑔𝑥_𝑛+1,𝐹𝑥,𝑦=𝑝𝐹𝑥_𝑛,𝑦_𝑛<1,𝐹𝑥,𝑦2𝑝𝑔𝑥_𝑛,𝑔(𝑥)+𝑝𝑔𝑦_𝑛,𝑔𝑦. (2.38) From (2.36), by Lemma 1.12, we obtain 𝑝𝑔(𝑥),𝐹𝑥,𝑦=lim_𝑛→∞𝑝𝑔𝑥_𝑛+1≤1,𝐹𝑥,𝑦2lim_𝑛→∞𝑝𝑔𝑥_𝑛,𝑔(𝑥)+𝑝𝑔𝑦_𝑛,𝑔𝑦=0, (2.39) so 𝐹(𝑥,𝑦)=𝑔(𝑥). Analogously we get 𝐹(𝑦,𝑥)=𝑔(𝑦).

Remark 2.2. We remak that Theorem 2.1 has been proved recently in [31] in the category of partially ordered metric spaces. However, they proceed the proof without using the assumptions (a) and (b) stated in our Theorem. They claimed that 𝑔(𝑥_𝑛)<𝑥 and 𝑔(𝑦_𝑛)>𝑦 by using the fact that the sequences {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are increasing and decreasing, respectively. In our belief, this step is not true and cannot be achieved without using the assumptions (a) and (b). Actually, this may not be true if the partial ordering, for example, is obtained via nonstrongly minihedral cones.

Corollary 2.3. Let (𝑋,𝑝,≤) be a complete ordered partial metric space. Suppose that 𝑋 has the following properties: (a)if {𝑥_𝑛} is a sequence such that 𝑥_𝑛+1>𝑥_𝑛 for each 𝑛=1,2,… and 𝑥_𝑛→𝑥, then 𝑥_𝑛<𝑥 for each 𝑛=1,2,…, (b)if {𝑦_𝑛} is a sequence such that 𝑦_𝑛+1<𝑦_𝑛 for each 𝑛=1,2,… and 𝑦_𝑛→𝑦, then 𝑦_𝑛>𝑦 for each 𝑛=1,2,….Let 𝐹∶𝑋²→𝑋 be a given mapping. Suppose that 𝐹 satisfies the following conditions: (i)𝐹 has the mixed strict monotone property, (ii)𝐹 is a Meir-Keeler type contraction, (iii)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀<𝐹𝑥₀,𝑦₀,𝑦₀≥𝐹𝑦₀,𝑥₀. (2.40)Then, 𝐹 has a coupled fixed point, that is, there exist 𝑥,𝑦∈𝑋 such that 𝐹𝑥,𝑦=𝑥,𝐹𝑦,𝑥=𝑦. (2.41)

Proof. It follows by taking 𝑔=𝐼_𝑋, the identity mapping on 𝑋, in Theorem 2.1.

3. Uniqueness of Coupled Fixed Points

Let (𝑋,≤) be a partially ordered set. We endow 𝑋×𝑋 by the following order (denoted ≤_𝑔) (𝑢,𝑣)≤_𝑔𝑥,𝑦⟺𝑔(𝑢)<𝑔(𝑥),𝑔𝑦≤𝑔(𝑣),∀𝑥,𝑦,(𝑢,𝑣)∈𝑋×𝑋. (3.1) Moreover, (𝑢,𝑣) and (𝑥,𝑦) are called 𝑔-comparable if either (𝑢,𝑣)≤_𝑔(𝑥,𝑦) or (𝑢,𝑣)≤_𝑔(𝑥,𝑦). In case 𝑔=𝐼_𝑋, we shortly say that (𝑢,𝑣) and (𝑥,𝑦) are comparable and denote by (𝑢,𝑣)≤(𝑥,𝑦). In this section, we will prove the uniqueness of the coupled fixed point.

Theorem 3.1. In addition to the hypotheses of Theorem 2.1, assume that for all non 𝑔-comparable points (𝑥,𝑦), (𝑥^∗,𝑦^∗)∈𝑋², there exists (𝑎,𝑏)∈𝑋² such that (𝐹(𝑎,𝑏),𝐹(𝑏,𝑎)) is comparable to both (𝑔(𝑥),𝑔(𝑦)) and (𝑔(𝑥^∗),𝑔(𝑦^∗)). Further, assume that 𝐹 and 𝑔 commute and 𝐹 is a strict 𝑔-Meir-Keeler type contraction. Then, 𝐹 and 𝑔 have a unique coupled common fixed point, that is, there exists (𝑢,𝑣)∈𝑋² such that 𝑢=𝑔(𝑢)=𝐹(𝑢,𝑣),𝑣=𝑔(𝑣)=𝐹(𝑣,𝑢). (3.2)

Proof. The set of coupled coincidence points of 𝐹 and 𝑔 is not empty due to Theorem 2.1. If (𝑥,𝑦) is the only coupled coincidence point of 𝐹 and 𝑔, then commutativity of 𝐹 and 𝑔 implies that 𝑔𝑔(𝑥)=𝑔𝐹𝑥,𝑦=𝐹𝑔(𝑥),𝑔𝑦,𝑔𝑔𝑦=𝑔𝐹𝑦,𝑥=𝐹𝑔𝑦,𝑔(𝑥). (3.3) Hence, (𝑢,𝑣)=(𝑔(𝑥),𝑔(𝑦)) is a coupled coincidence point of 𝐹 and 𝑔 and by uniqueness we conclude that 𝐹𝑥,𝑦=𝑔(𝑥)=𝑥,𝐹𝑦,𝑥=𝑔𝑦=𝑦. (3.4)
Now suppose that (𝑥,𝑦),(𝑥^∗,𝑦^∗)∈𝑋² are two coupled coincidence points of 𝐹 and 𝑔. We show that 𝑔(𝑥)=𝑔(𝑥^∗) and 𝑔(𝑦)=𝑔(𝑦^∗). To this end we distinguish the following two cases.
First Case. (𝑥,𝑦) is 𝑔-comparable to (𝑥^∗,𝑦^∗) with respect to the ordering in 𝑋², where 𝐹𝑥,𝑦=𝑔(𝑥),𝐹𝑦,𝑥=𝑔𝑦,𝐹𝑥^∗,𝑦^∗=𝑔𝑥^∗,𝐹𝑦^∗,𝑥^∗=𝑔𝑦^∗. (3.5) Without loss of the generality, we may assume that 𝑔(𝑥)=𝐹𝑥,𝑦<𝐹𝑥^∗,𝑦^∗=𝑔𝑥^∗,𝑔𝑦=𝐹𝑦,𝑥≥𝐹𝑦^∗,𝑥^∗=𝑔𝑦^∗. (3.6) By definition of 𝜌₂ and Lemma 1.18 we have 0<𝜌₂𝑔(𝑥),𝑔𝑦,𝑔𝑥^∗,𝑔𝑦^∗=𝑝𝑔(𝑥),𝑔𝑥^∗+𝑝𝑔𝑦^∗𝑦𝐹𝑥,𝑔=𝑝𝑥,𝑦,𝐹^∗,𝑦^∗+𝑝𝐹𝑦^∗,𝑥^∗,𝑔(𝑥)𝑥,𝐹𝑦,𝑥<𝑝,𝑔^∗+𝑝𝑔𝑦^∗,𝑔𝑦=𝜌₂𝑔(𝑥),𝑔𝑦,𝑔𝑥^∗,𝑔𝑦^∗, (3.7) which is a contradiction. Therefore, we have (𝑔(𝑥),𝑔(𝑦))=(𝑔(𝑥^∗),𝑔(𝑦^∗)). Hence 𝑔(𝑥)=𝑔𝑥^∗,𝑔𝑦=𝑔𝑦^∗. (3.8)
Second Case. (𝑥,𝑦) is not 𝑔-comparable to (𝑥^∗,𝑦^∗).
By assumption, there exists (𝑎,𝑏)∈𝑋² such that (𝐹(𝑎,𝑏),𝐹(𝑏,𝑎)) is comparable to both (𝑔(𝑥),𝑔(𝑦)) and (𝑔(𝑥^∗),𝑔(𝑦^∗)). Then, we have 𝑔(𝑥)=𝐹𝑥,𝑦<𝐹(𝑎,𝑏),𝐹𝑥^∗,𝑦^∗=𝑔𝑥^∗(),𝑔𝑦()𝑦<𝐹𝑎,𝑏=𝐹𝑦,𝑥≥𝐹𝑏,𝑎,𝐹^∗,𝑥^∗=𝑔𝑦^∗≥𝐹(𝑏,𝑎), (3.9) Setting 𝑥=𝑥₀, 𝑦=𝑦₀, 𝑎=𝑎₀, 𝑏=𝑏₀, and 𝑥^∗=𝑥^∗₀, 𝑦^∗=𝑦^∗₀ as in the proof of Theorem 2.1, we get 𝑔𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1=F𝑦_𝑛,𝑥_𝑛𝑔𝑎∀𝑛=0,1,2,…,_𝑛+1=𝐹𝑎_𝑛,𝑏_𝑛,𝑔𝑏_𝑛+1=𝐹𝑏_𝑛,𝑎_𝑛𝑔𝑥∀𝑛=0,1,2,…,_∗𝑛+1=𝐹𝑥^∗_𝑛,𝑦^∗_𝑛,𝑔𝑦^∗_𝑛+1=𝐹𝑦^∗_𝑛,𝑥^∗_𝑛∀𝑛=0,1,2,…. (3.10) Since (𝐹(𝑥,𝑦),𝐹(𝑦,𝑥))=(𝑔(𝑥),𝑔(𝑦))=(𝑔(𝑥₁),𝑔(𝑦₁)) is comparable with (𝐹(𝑎,𝑏),𝐹(𝑏,𝑎))=(𝑔(𝑎₁),𝑔(𝑏₁)), we have 𝑔(𝑥)<𝑔(𝑎₁) and 𝑔(𝑏₁)≤𝑔(𝑦). By using that 𝐹 has the mixed 𝑔-strict monotone property, we observe that 𝑔(𝑥)<𝑔(𝑎_𝑛) and 𝑔(𝑏_𝑛)≤𝑔(𝑦) for all 𝑛≥1. Thus, by Remark 1.19, we get that 𝑝𝑔(𝑥),𝑔𝑎_𝑛+1+𝑝𝑔𝑦,𝑔𝑏_𝑛+1=𝑝𝐹𝑥,𝑦,𝐹𝑎_𝑛,𝑏_𝑛+𝑝𝐹𝑏_𝑛,𝑎_𝑛<𝑘,𝐹𝑦,𝑥2𝑝𝑔(𝑥),𝑔𝑎_𝑛+𝑝𝑔𝑦,𝑔𝑏_𝑛+𝑘2𝑝𝑔𝑦,𝑔𝑏_𝑛+𝑝𝑔(𝑥),𝑔𝑎_𝑛=𝑘𝑝g(𝑥),𝑔𝑎_𝑛+𝑝𝑔𝑦,𝑔𝑏_𝑛. (3.11)
Inductively, we derive that 𝑝𝑔(𝑥),𝑔𝑎_𝑛+1+𝑝𝑔𝑦,𝑔𝑏_𝑛+1<𝑘^𝑛𝑝𝑔(𝑥),𝑔𝑎₁+𝑝𝑔𝑦,𝑔𝑏₁. (3.12) The right hand side of above inequality tends to zero as 𝑛→∞. Hence, lim_𝑛→∞𝑝𝑔(𝑥),𝑔𝑎_𝑛+1+𝑝𝑔𝑦,𝑔𝑏_𝑛+1=0. (3.13) Analogously, we get that lim_𝑛→∞𝑝𝑔𝑥^∗,𝑔𝑎_𝑛+1+𝑝𝑔𝑦^∗,𝑔𝑏_𝑛+1=0. (3.14) By the triangle inequality, we have 𝑝𝑔(𝑥),𝑔𝑥^∗≤𝑝𝑔(𝑥),𝑔𝑎_𝑛+1+𝑝𝑔𝑥^∗,𝑔𝑎_𝑛+1−𝑝𝑔𝑎_𝑛+1,𝑔𝑎_𝑛+1≤𝑝𝑔(𝑥),𝑔𝑎_𝑛+1+𝑝𝑔𝑥^∗,𝑔𝑎_𝑛+1𝑝𝑔𝑦𝑦⟶0as𝑛⟶∞,,𝑔^∗≤𝑝𝑔𝑦,𝑔𝑏_𝑛+1+𝑝𝑔𝑦^∗,𝑔𝑏_𝑛+1−𝑝𝑔𝑏_𝑛+1,𝑔𝑏_𝑛+1≤𝑝𝑔𝑦,𝑔𝑏_𝑛+1+𝑝𝑔𝑦^∗,𝑔𝑏_𝑛+1⟶0as𝑛⟶∞. (3.15) Combining all observation above, we get that 𝑝(𝑔(𝑥^∗),𝑔(𝑥))=0 and 𝑝(𝑔(𝑦^∗),𝑔(𝑦))=0. Therefore, 𝑔(𝑥)=𝑔𝑥^∗,𝑔𝑦=𝑔𝑦^∗. (3.16) In both cases above, we have shown that (3.16) holds. Now, let 𝑔(𝑥)=𝑢 and 𝑔(𝑦)=𝑣. By the commutativity of 𝐹 and 𝑔 with the fact that 𝑔(𝑥)=𝐹(𝑥,𝑦) and 𝐹(𝑦,𝑥)=𝑔(𝑦), we have 𝑔(𝑢)𝑔(𝑥)𝐹𝑔(𝑥)𝑦(),𝑔(𝑣)𝑔𝑦𝐹𝑔𝑦(𝑥)=().=𝑔=𝑔𝑥,𝑦=𝐹,𝑔=𝐹𝑢,𝑣=𝑔=𝑔𝑦,𝑥=𝐹,𝑔𝐹𝑣,𝑢 (3.17) Thus, (𝑢,𝑣) is a coupled coincidence point of 𝐹 and 𝑔. Setting 𝑢=𝑥^∗ and 𝑣=𝑦^∗ in (3.17). Then, by (3.16) we have 𝑢=𝑔(𝑥)=𝑔𝑥^∗=𝑔(𝑢),𝑣=𝑔𝑦=𝑔𝑦^∗=𝑔(𝑣). (3.18) From (3.17) we get that 𝑢=𝑔(𝑢)=𝐹(𝑢,𝑣),𝑣=𝑔(𝑣)=𝐹(𝑣,𝑢). (3.19) Hence, the pair (𝑢,𝑣) is the coupled common fixed point of 𝐹 and 𝑔.
Finally, we prove the uniqueness of the coupled common fixed point of 𝐹 and 𝑔. Actually, if (𝑧,𝑤) is another coupled common fixed point of 𝐹 and 𝑔, then 𝑢=𝑔(𝑢)=𝑔(𝑧)=𝑧,𝑣=𝑔(𝑣)=𝑔(𝑤)=𝑤 (3.20) follows from (3.16).

Remark 3.2. We remark that Theorem 3.1 above has been recently proved in [31] without assuming that the mapping 𝐹 is a strict 𝑔-Meir-Keeler contraction. This leads to a gap in the proof of Theorem  2.6 there.

Corollary 3.3. Suppose that all the hypotheses of Corollary 2.3 hold, and further, for all (𝑥,𝑦), (𝑥^∗,𝑦^∗)∈𝑋², there exists (𝑎,𝑏)∈𝑋² that is comparable to (𝑥,𝑦) and (𝑥^∗,𝑦^∗). Further, assume that 𝐹 is a strict Meir-Keeler type contraction. Then, 𝐹 has a unique coupled fixed point.

4. Applications

Motivated by Suzuki [52] and on the same lines of Theorem  3.1 of [53], one can prove the following result.

Theorem 4.1. Let (𝑋,𝑝,≤) be a partially ordered partial metric space. Let 𝐹∶𝑋²→𝑋 and 𝑔∶𝑋→𝑋 be given mappings such that 𝐹(𝑋²)⊂𝑔(𝑋). Assume that there exists a function 𝜃 from [0,+∞) into itself satisfying the following:(I)𝜃(0)=0 and 𝜃(𝑡)>0 for every 𝑡>0,(II)𝜃 is nondecreasing and right continuous,(III)for every 𝜀>0, there exists 𝛿(𝜀)>0 such that 1𝜀≤𝜃2𝑝𝑔𝑥,𝑔𝑢+𝑝𝑔𝑦,𝑔𝑣<𝜀+𝛿(𝜀)⟹𝜃𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)<𝜀, (4.1)for all 𝑔𝑥≤𝑔𝑢 and 𝑔𝑦≥𝑔𝑣.
Then, 𝐹 is a 𝑔-Meir-Keeler type function.

The following result is an immediate consequence of Theorems 2.1 and 4.1.

Corollary 4.2. Let (𝑋,𝑝,≤) be a partially ordered complete partial metric space. Given 𝐹∶𝑋²→𝑋 and 𝑔∶𝑋→𝑋 such that 𝐹(𝑋²)⊂𝑔(𝑋), 𝑔(𝑋) is a complete subspace and the following hypotheses hold:(i)𝐹 has the mixed 𝑔-strict monotone property,(ii)for every 𝜀>0, there exists 𝛿(𝜀)>0 such that 𝜀≤₀^{(1/2)[𝑝(𝑔𝑥,𝑔𝑢)+𝑝(𝑔𝑦,𝑔𝑣)]}𝜙(𝑡)𝑑𝑡<𝜀+𝛿(𝜀)⟹₀^{𝑝(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))}𝜙(𝑡)𝑑𝑡<𝜀, (4.2)for all 𝑔𝑥≤𝑔𝑢 and 𝑔𝑦≥𝑔𝑣, where 𝜙∶[0,+∞)→[0,+∞) is a locally integrable function satisfying ∫^𝑠₀𝜙(𝑡)𝑑𝑡>0 for all 𝑠>0,(iii)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥₀<𝐹𝑥₀,𝑦₀,𝑔𝑦₀≥𝐹𝑦₀,𝑥₀. (4.3)Assume that the hypotheses (a) and (b) given in Theorem 2.1 hold. Then, 𝐹 and 𝑔 have a coupled coincidence point.

To end this paper, we give the following corollary.

Corollary 4.3. Let (𝑋,𝑝,≤) be a partially ordered partial metric space. Given 𝐹∶𝑋²→𝑋 and 𝑔∶𝑋→𝑋 such that 𝐹(𝑋²)⊂𝑔(𝑋), 𝑔(𝑋) is a complete subspace and the following hypotheses hold:(i)F has the mixed 𝑔-strict monotone property,(ii)for all 𝑔𝑥≤𝑔𝑢 and 𝑔𝑦≥𝑔𝑣, ₀^{𝑝(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))}𝜙(𝑡)𝑑𝑡≤𝑘₀^{(1/2)[𝑝(𝑔𝑥,𝑔𝑢)+𝑝(𝑔𝑦,𝑔𝑣)]}𝜙(𝑡)𝑑𝑡, (4.4)where 𝑘∈(0,1) and 𝜙 is a locally integrable function from [0,+∞) into itself satisfying ∫^𝑠₀𝜙(𝑡)𝑑𝑡>0 for all 𝑠>0,(iii)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥₀<𝐹𝑥₀,𝑦₀,𝑔𝑦₀≥𝐹𝑦₀,𝑥₀. (4.5)Assume that the hypotheses (a) and (b) of Theorem 2.1 hold. Then, 𝐹 and 𝑔 have a coupled coincidence point.

Proof. For all 𝜀>0, we take 𝛿(𝜀)=(1/𝑘−1)𝜀 and we apply Corollary 4.2.