Coupled Coincidence Points of Mappings in Ordered Partial Metric Spaces

Abstract

New coupled coincidence point and coupled fixed point results in ordered partial metric spaces under the contractive conditions of Geraghty, Rakotch, and Branciari types are obtained. Examples show that these results are distinct from the known ones.

1. Introduction

In recent years many authors have worked on domain theory in order to equip semantics domain with a notion of distance. In particular, Matthews [1] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. He showed that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. Subsequently, several authors (see, e.g., [2–9]) have proved fixed point theorems in partial metric spaces.

Many generalizations of Banach's contractive condition have been introduced in order to obtain more general fixed point results in metric spaces and their generalizations. We mention here conditions introduced by Geraghty [10], Rakotch [11], and Branciari [12].

The notion of a coupled fixed point was introduced and studied by Bhaskar and Lakshmikantham in [13]. In subsequent papers several authors proved various coupled and common coupled fixed point theorems in partially ordered metric spaces (e.g., [14–17]). These results were applied for investigation of solutions of differential and integral equations. In a recent paper [18], Berinde presented a method of reducing coupled fixed point results in ordered metric spaces to the respective results for mappings with one variable.

In this paper, we further develop the method of Berinde and obtain new coupled coincidence and coupled fixed point results in ordered partial metric spaces, under the contractive conditions of Geraghty, Rakotch, and Branciari types. Examples show that these results are distinct from the known ones. In particular, they show that using the order and/or the partial metric enables conclusions which cannot be obtained in the classical case.

2. Notation and Preliminary Results

2.1. Partial Metric Spaces

The following definitions and details can be seen in [1–9].

Definition 2.1. A partial metric on a nonempty set 𝑋 is a function 𝑝∶𝑋×𝑋→ℝ⁺ such that, for all 𝑥,𝑦,𝑧∈𝑋,(P₁) 𝑥=𝑦⇔𝑝(𝑥,𝑥)=𝑝(𝑥,𝑦)=𝑝(𝑦,𝑦),(P₂)𝑝(𝑥,𝑥)≤𝑝(𝑥,𝑦),(P₃)𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥),(P₄)𝑝(𝑥,𝑦)≤𝑝(𝑥,𝑧)+𝑝(𝑧,𝑦)−𝑝(𝑧,𝑧). The pair (𝑋,𝑝) is called a partial metric space.

It is clear that if 𝑝(𝑥,𝑦)=0, then from (P₁) and (P₂) 𝑥=𝑦. But if 𝑥=𝑦, 𝑝(𝑥,𝑦) may not be 0.

Each partial metric 𝑝 on 𝑋 generates a 𝑇₀ topology 𝜏_𝑝 on 𝑋 which has as a base 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. A sequence {𝑥_𝑛} in (𝑋,𝑝) converges to a point 𝑥∈𝑋, with respect to 𝜏_𝑝, if lim_𝑛→∞𝑝(𝑥,𝑥_𝑛)=𝑝(𝑥,𝑥). This will be denoted as 𝑥_𝑛→𝑥, 𝑛→∞, or lim_𝑛→∞𝑥_𝑛=𝑥.

If 𝑝 is a partial metric on 𝑋, then the function 𝑝^𝑠∶𝑋×𝑋→ℝ⁺ given by𝑝^𝑠𝑥,𝑦=2𝑝𝑥,𝑦−𝑝(𝑥,𝑥)−𝑝𝑦,𝑦 (2.1) is metric on 𝑋. Furthermore, lim_𝑛→∞𝑝^𝑠(𝑥_𝑛,𝑥)=0 if and only if𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝𝑥_𝑛,𝑥=lim_{𝑛,𝑚→∞}𝑝𝑥_𝑛,𝑥_𝑚. (2.2)

A basic example of a partial metric space is the pair (ℝ⁺,𝑝), where 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for all 𝑥,𝑦∈ℝ⁺. The corresponding metric is𝑝^𝑠𝑥,𝑦=2max𝑥,𝑦−𝑥−𝑦=𝑥−𝑦. (2.3) Other examples of partial metric spaces which are interesting from a computational point of view may be found in [1, 19].

Remark 2.2. Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function 𝑝(⋅,⋅) need not be continuous in the sense that 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦 implies 𝑝(𝑥_𝑛,𝑦_𝑛)→𝑝(𝑥,𝑦).

Definition 2.3. Let (𝑋,𝑝) be a partial metric space. Then,(1)a sequence {𝑥_𝑛} in (𝑋,𝑝) is called a Cauchy sequence if lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists (and is finite);(2)the space (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 such that 𝑝(𝑥,𝑥)=lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚).

Lemma 2.4. Let (𝑋,𝑝) be a partial metric space.(a){𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑝) if and only if it is a Cauchy sequence in the metric space (𝑋,𝑝^𝑠).(b)The space (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑝^𝑠) is complete.

Definition 2.5. Let 𝑋 be a nonempty set. Then (𝑋,⪯,𝑝) is called an ordered partial metric space if (i) (𝑋,⪯) is a partially ordered set, and (ii) (𝑋,𝑝) is a partial metric space.
We will say that the space (𝑋,⪯,𝑝) satisfies the ordered-regular condition (abr. (ORC)) if the following holds: if {𝑥_𝑛} is a nondecreasing sequence in 𝑋 with respect to ⪯ such that 𝑥_𝑛→𝑥∈𝑋 as 𝑛→∞, then 𝑥_𝑛⪯𝑥 for all 𝑛∈ℕ.

Definition 2.6 (see [13, 14]). Let (𝑋,⪯) be a partially ordered set, 𝐹∶𝑋×𝑋→𝑋, and 𝑔∶𝑋→𝑋.(1) 𝐹 is said to have 𝑔-mixed monotone property if the following two conditions are satisfied: ∀𝑥₁,𝑥₂,𝑦∈𝑋𝑔𝑥₁⪯𝑔𝑥₂⟹𝐹𝑥₁,𝑦⪯𝐹𝑥₂,𝑦,∀𝑥,𝑦₁,𝑦₂∈𝑋𝑔𝑦₁⪯𝑔𝑦₂⟹𝐹𝑥,𝑦₁⪰𝐹𝑥,𝑦₂. (2.4) If 𝑔=𝑖_𝑋 (the identity map), we say that 𝐹 has the mixed monotone property.(2) A point (𝑥,𝑦)∈𝑋×𝑋 is said to be a coupled coincidence point of 𝐹 and 𝑔 if 𝐹(𝑥,𝑦)=𝑔𝑥 and 𝐹(𝑦,𝑥)=𝑔𝑦 and their common coupled fixed point if 𝐹(𝑥,𝑦)=𝑔𝑥=𝑥 and 𝐹(𝑦,𝑥)=𝑔𝑦=𝑦.

Definition 2.7 (see [20]). Let (𝑋,𝑑) be a metric space, and let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. The pair (𝐹,𝑔) is said to be compatible if lim_𝑛→∞𝑑𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛=0,lim_𝑛→∞𝑑𝑔𝐹y_𝑛,𝑥_𝑛,𝐹𝑔𝑦_𝑛,𝑔𝑥_𝑛=0 (2.5) whenever {𝑥_𝑛} and {𝑦_𝑛} are sequences in 𝑋 such that lim_𝑛→∞𝐹(𝑥_𝑛,𝑦_𝑛)=lim_𝑛→∞𝑔𝑥_𝑛=𝑥 and lim_𝑛→∞𝐹(𝑦_𝑛,𝑥_𝑛)=lim_𝑛→∞𝑔𝑦_𝑛=𝑦 for some 𝑥,𝑦∈𝑋.

2.2. Some Auxiliary Results

Lemma 2.8. (i) Let (𝑋,⪯,𝑝) be an ordered partial metric space. If relation ⊑ is defined on 𝑋² by 𝑌⊑𝑉⟺𝑥⪯𝑢∧𝑦⪰𝑣,𝑌=𝑥,𝑦,𝑉=(𝑢,𝑣)∈𝑋², (2.6) and 𝑃∶𝑋²×𝑋²→ℝ⁺ is given by 𝑃(𝑌,𝑉)=𝑝(𝑥,𝑢)+𝑝𝑦,𝑣,𝑌=𝑥,𝑦,𝑉=(𝑢,𝑣)∈𝑋², (2.7) then (𝑋²,⊑,𝑃) is an ordered partial metric space. The space (𝑋²,⊑,𝑃) is complete iff (𝑋,⪯,𝑝) is complete.
(ii) If 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋, and 𝐹 has the 𝑔-mixed monotone property, then the mapping 𝑇_𝐹∶𝑋²→𝑋² given by 𝑇_𝐹𝑌=𝐹𝑥,𝑦,𝐹𝑦,𝑥,𝑌=𝑥,𝑦∈𝑋² (2.8) is 𝑇_𝑔-nondecreasing with respect to ⊑, that is, 𝑇_𝑔𝑌⊑𝑇_𝑔𝑉⟹𝑇_𝐹𝑌⊑𝑇_𝐹𝑉, (2.9) where 𝑇_𝑔𝑌=𝑇_𝑔(𝑥,𝑦)=(𝑔𝑥,𝑔𝑦).
(iii) If 𝑔 is continuous in (𝑋,𝑝) (i.e., with respect to 𝜏_𝑝), then 𝑇_𝑔 is continuous in (𝑋²,𝑃) (i.e., with respect to 𝜏_𝑃). If 𝐹 is continuous from (𝑋²,𝑃) to (𝑋,𝑝) (i.e., 𝑥_𝑛→𝑥 and 𝑦_𝑛→𝑦 imply 𝐹(𝑥_𝑛,𝑦_𝑛)→𝐹(𝑥,𝑦)), then 𝑇_𝐹 is continuous in (𝑋²,𝑃).

Proof. (i) Relation ⊑ is obviously a partial order on 𝑋². To prove that 𝑃 is a partial metric on 𝑋², only conditions (P₁) and (P₄) are nontrivial.
(P₁) If 𝑌=𝑉∈𝑋², then obviously 𝑝(𝑌,𝑌)=𝑝(𝑌,𝑉)=𝑝(𝑉,𝑉) holds. Conversely, let 𝑝(𝑌,𝑌)=𝑝(𝑌,𝑉)=𝑝(𝑉,𝑉), that is, 𝑝(𝑥,𝑥)+𝑝𝑦,𝑦=𝑝(𝑥,𝑢)+𝑝𝑦,𝑣=𝑝(𝑢,𝑢)+𝑝(𝑣,𝑣). (2.10) We know by (P₂) that 𝑝(𝑥,𝑥)≤𝑝(𝑥,𝑢) and 𝑝(𝑦,𝑦)≤𝑝(𝑦,𝑣). Adding up, we obtain that 𝑝(𝑥,𝑥)+𝑝(𝑦,𝑦)≤𝑝(𝑥,𝑢)+𝑝(𝑦,𝑣), and since in fact equality holds, we conclude that 𝑝(𝑥,𝑥)=𝑝(𝑥,𝑢) and 𝑝(𝑦,𝑦)=𝑝(𝑦,𝑣). Similarly, we get that 𝑝(𝑢,𝑢)=𝑝(𝑥,𝑢) and 𝑝(𝑣,𝑣)=𝑝(𝑦,𝑣). Hence, 𝑝(𝑥,𝑥)=𝑝(𝑥,𝑢)=𝑝(𝑢,𝑢),𝑝𝑦,𝑦=𝑝𝑦,𝑣=𝑝(𝑣,𝑣), (2.11) and applying property (P₁) of partial metric 𝑝, we get that 𝑥=𝑢 and 𝑦=𝑣, that is, 𝑌=𝑉.
(P₄) Let 𝑌=(𝑥,𝑦),𝑉=(𝑢,𝑣),𝑍=(𝑤,𝑧)∈𝑋². Then 𝑃()()≤𝑝()()()+𝑝()()()()()−𝑝()(()()().𝑌,𝑉=𝑝𝑥,𝑢+𝑝𝑦,𝑣𝑥,𝑤+𝑝𝑤,𝑢−𝑝𝑤,𝑤𝑦,𝑧+𝑝𝑧,𝑣−𝑝𝑧,𝑧=𝑝𝑥,𝑤+𝑝𝑦,𝑧+𝑝𝑤,𝑢+𝑝𝑧,𝑣𝑤,𝑤+𝑝𝑧,𝑧)=𝑃𝑌,𝑍+𝑃𝑍,𝑉−𝑃𝑍,𝑍 (2.12)
(ii, iii) The proofs of these assertions are straightforward.

Remark 2.9. Let 𝑝^𝑠 be the metric associated with the partial metric 𝑝 as in (2.1). It is easy to see that, with notation as in the previous lemma, 𝑃^𝑠(𝑌,𝑉)=𝑝^𝑠(𝑥,𝑢)+𝑝^𝑠𝑦,𝑣,𝑌=𝑥,𝑦,𝑉=(𝑢,𝑣)∈𝑋² (2.13) is the associated metric to the partial metric 𝑃 on 𝑋². We note, however, that when we speak about continuity of mappings, we always assume continuity in the sense of the partial metric 𝑝, that is, in the sense of the respective topology 𝜏_𝑝. This should not be confused with the approach given by O'Neill in [3] where both 𝑝- and 𝑝^𝑠-continuity were assumed.

It is easy to see that (using notation as in the previous lemma), the mappings 𝐹 and 𝑔 are 𝑝^𝑠-compatible (in the sense of Definition 2.7) if and only if the mappings 𝑇_𝐹 and 𝑇_𝑔 are 𝑃^𝑠-compatible in the usual sense (i.e., lim_𝑛→∞𝑑(𝑇_𝐹𝑇_𝑔𝑌_𝑛,𝑇_𝑔𝑇_𝐹𝑌_𝑛)=0, whenever {𝑌_𝑛} is a sequence in 𝑋² such that lim_𝑛→∞𝑇_𝐹𝑌_𝑛=lim_𝑛→∞𝑇_𝑔𝑌_𝑛).

Assertions similar to the following lemma (see, e.g., [21]) were used (and proved) in the course of proofs of several fixed point results in various papers.

Lemma 2.10. Let (𝑋,𝑑) be a metric space, and let {𝑥_𝑛} be a sequence in 𝑋 such that {𝑑(𝑥_𝑛+1,𝑥_𝑛)} is decreasing and lim_𝑛→∞𝑑𝑥_𝑛+1,𝑥_𝑛=0. (2.14) If {𝑥_2𝑛} is not a Cauchy sequence, then there exist 𝜀>0 and two sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that the following four sequences tend to 𝜀 when 𝑘→∞: 𝑑𝑥_2𝑚𝑘,𝑥_2𝑛𝑘,𝑑𝑥_2𝑚𝑘,𝑥_2𝑛𝑘+1,𝑑𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘,𝑑𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘+1. (2.15)

As a corollary (applying Lemma 2.10 to the associated metric 𝑝^𝑠 of a partial metric 𝑝, and using Lemma 2.4) we obtain the following.

Lemma 2.11. Let (𝑋,𝑝) be a partial metric space, and let {𝑥_𝑛} be a sequence in 𝑋 such that {𝑝(𝑥_𝑛+1,𝑥_𝑛)} is decreasing and lim_𝑛→∞𝑝𝑥_𝑛+1,𝑥_𝑛=0. (2.16) If {𝑥_2𝑛} is not a Cauchy sequence in (𝑋,𝑝), then there exist 𝜀>0 and two sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that the following four sequences tend to 𝜀 when 𝑘→∞: 𝑝𝑥_2𝑚𝑘,𝑥_2𝑛𝑘,𝑝𝑥_2𝑚𝑘,𝑥_2𝑛𝑘+1,𝑝𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘,𝑝𝑥_{2𝑚𝑘−1},𝑥_2𝑛𝑘+1. (2.17)

3. Coupled Coincidence and Fixed Points under Geraghty-Type Conditions

Let 𝒢 denote the class of real functions 𝛾∶[0,+∞)→[0,1) satisfying the condition 𝛾𝑡_𝑛⟶1implies𝑡_𝑛⟶0. (3.1) An example of a function in 𝒢 may be given by 𝛾(𝑡)=𝑒^−2𝑡 for 𝑡>0 and 𝛾(0)∈[0,1). In an attempt to generalize the Banach contraction principle, Geraghty proved in 1973 the following.

Theorem 3.1 (see [10]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be a self-map. Suppose that there exists 𝛾∈𝒢 such that 𝑑𝑇𝑥,𝑇𝑦≤𝛾𝑑𝑥,𝑦𝑑𝑥,𝑦 (3.2) holds for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point 𝑧∈𝑋 and for each 𝑥∈𝑋 the Picard sequence {𝑇^𝑛𝑥} converges to 𝑧 when 𝑛→∞.

Subsequently, several authors proved such results, including the very recent paper of Ðukić et al. [22].

We begin with the following auxiliary result.

Lemma 3.2. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete. Let 𝑇,𝑆∶𝑋→𝑋 be self-maps such that 𝑆 is continuous, 𝑇𝑋⊂𝑆𝑋, and one of these two subsets of 𝑋 is closed. Suppose that 𝑇 is 𝑆-nondecreasing (with respect to ⪯) and there exists 𝑥₀∈𝑋 with 𝑆𝑥₀⪯𝑇𝑥₀ or 𝑇𝑥₀⪯𝑆𝑥₀. Assume also that there exists 𝛾∈𝒢 such that 𝑝𝑇𝑥,𝑇𝑦≤𝛾𝑝𝑆𝑥,𝑆𝑦𝑝𝑆𝑥,𝑆𝑦 (3.3) holds for all 𝑥,𝑦∈𝑋 such that 𝑆𝑥 and 𝑆𝑦 are comparable. Assume that either 1°: 𝑇 is continuous and the pair (𝑇,𝑆) is 𝑝^𝑠- compatible or 2°: 𝑋 satisfies (ORC). Then, 𝑇 and 𝑆 have a coincidence point in 𝑋.

Proof. The proof follows the lines of proof of [22, Theorems 3.1 and 3.5].
Take 𝑥₀∈𝑋 with, say, 𝑆𝑥₀⪯𝑇𝑥₀, and using that 𝑇 is 𝑆-nondecreasing and that 𝑇𝑋⊂𝑆𝑋 form the sequence {𝑥_𝑛} satisfying 𝑇𝑥_𝑛=𝑆𝑥_𝑛+1, 𝑛=0,1,2,…, and 𝑆𝑥₀⪯𝑇𝑥₀=S𝑥₁⪯𝑇𝑥₁=𝑆𝑥₂⪯⋯⪯𝑇𝑥_𝑛⪯𝑆𝑥_𝑛+1⪯⋯. (3.4) Since 𝑆𝑥_𝑛−1 and 𝑆𝑥_𝑛 are comparable, we can apply the contractive condition to obtain 𝑝𝑆𝑥_𝑛+1,𝑆𝑥_𝑛=𝑝𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝛾𝑝𝑆𝑥_𝑛−1,𝑆𝑥_𝑛𝑝𝑆𝑥_𝑛−1,𝑆𝑥_𝑛≤𝑝𝑆𝑥_𝑛−1,𝑆𝑥_𝑛. (3.5) Consider the following two cases:(1)𝑝(𝑆𝑥_𝑛0+1,𝑆𝑥_𝑛0)=0 for some 𝑛₀∈ℕ;(2)𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)>0 for each 𝑛∈ℕ.Case 1. Under this assumption, we get that 𝑝𝑆𝑥_𝑛0+2,𝑆𝑥_𝑛0+1=𝑝𝑇𝑥_𝑛0+1,𝑇𝑥_𝑛0≤𝛾𝑝𝑆𝑥_𝑛0+1,𝑆𝑥_𝑛0𝑝𝑆𝑥_𝑛0+1,𝑆𝑥_𝑛0=𝛾(0)⋅0=0, (3.6) and it follows that 𝑝(𝑆𝑥_𝑛0+2,𝑆𝑥_𝑛0+1)=0. By induction, we obtain that 𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)=0 for all 𝑛≥𝑛₀ and so 𝑆𝑥_𝑛=𝑆𝑥_𝑛0 for all 𝑛≥𝑛₀. Hence, {𝑆𝑥_𝑛} is a Cauchy sequence, converging to 𝑆𝑥_𝑛0, and 𝑥_𝑛0 is a coincidence point of 𝑆 and 𝑇.Case 2. We will prove first that in this case the sequence {𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)} is strictly decreasing and tends to 0 as 𝑛→∞.
For each 𝑛∈ℕ we have that 0<𝑝𝑆𝑥_𝑛+2,𝑆𝑥_𝑛+1=𝑝𝑇𝑥_𝑛+1,𝑇𝑥_𝑛≤𝛾𝑝𝑆𝑥_𝑛+1,𝑆𝑥_𝑛𝑝𝑆𝑥_𝑛+1,𝑆𝑥_𝑛<𝑝𝑆𝑥_𝑛+1,𝑆𝑥_𝑛. (3.7) Hence, 𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛) is strictly decreasing and bounded from below, thus converging to some 𝑞≥0. Suppose that 𝑞>0. Then, it follows from (3.7) that 𝑝𝑆𝑥_𝑛+2,𝑆𝑥_𝑛+1𝑝𝑆𝑥_𝑛+1,𝑆𝑥_𝑛≤𝛾𝑝𝑆𝑥_𝑛+1,𝑆𝑥_𝑛<1, (3.8) wherefrom, passing to the limit when 𝑛→∞, we get that lim_𝑛→∞𝛾(𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛))=1. Using property (3.1) of the function 𝛾, we conclude that lim_𝑛→∞𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)=0, that is, 𝑞=0, a contradiction. Hence, lim_𝑛→∞𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)=0 is proved.
In order to prove that {𝑆𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑝), suppose the contrary. As was already proved, 𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)→0 as 𝑛→∞, and so, using (P₂), 𝑝(𝑆𝑥_𝑛,𝑆𝑥_𝑛)→0 as 𝑛→∞. Hence, using (2.1), we get that 𝑝^𝑠(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)→0 as 𝑛→∞. Using Lemma 2.11, we obtain that there exist 𝜀>0 and two sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that the following four sequences tend to 𝜀 when 𝑘→∞: 𝑝𝑆𝑥_2𝑚𝑘,𝑆𝑥_2𝑛𝑘,𝑝𝑆𝑥_2𝑚𝑘,𝑆𝑥_2𝑛𝑘+1,𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘,𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘+1. (3.9) Putting in the contractive condition 𝑥=𝑥_{2𝑚𝑘−1} and 𝑦=𝑥_2𝑛𝑘, it follows that 𝑝𝑆𝑥_2𝑚𝑘,𝑆𝑥_2𝑛𝑘+1≤𝛾𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘<𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘. (3.10) Hence, 𝑝𝑆𝑥_2𝑚𝑘,𝑆𝑥_2𝑛𝑘+1𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘≤𝛾𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘<1 (3.11) and lim_𝑘→∞𝛾(𝑝(𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘))=1. Since 𝛾∈𝒢, it follows that lim_𝑘→∞𝑝(𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘)=0, which is in contradiction with 𝜀>0.
Thus, {𝑆𝑥_𝑛} is a Cauchy sequence, both in (𝑋,𝑝) and in (𝑋,𝑝^𝑠). Hence, it converges (in 𝑝 and in 𝑝^𝑠) to a point 𝑆𝑧∈𝑆𝑋 (we suppose 𝑆𝑋 to be closed, that is, complete; the case when 𝑇𝑋 is closed is treated similarly) such that 𝑝(𝑆𝑧,𝑆𝑧)=lim_𝑛→∞𝑝𝑆𝑥_𝑛,𝑆𝑧=lim_{𝑛,𝑚→∞}𝑝𝑆𝑥_𝑛,𝑆𝑥_𝑚. (3.12) Also, it follows easily that lim_𝑛→∞𝑝𝑆𝑥_𝑛,𝑆𝑧=𝑝(𝑆𝑧,𝑆𝑧)=0. (3.13) We will prove that 𝑆 and 𝑇 have a coincidence point.
(i) Suppose that 𝑇∶(𝑋,𝑝)→(𝑋,𝑝) is continuous and that (𝑇,𝑆) is a 𝑝^𝑠-compatible pair. We have that 𝑝^𝑠(𝑇𝑆𝑧,𝑆𝑆𝑧)≤𝑝^𝑠𝑇𝑆𝑧,𝑇𝑆𝑥_𝑛+𝑝^𝑠𝑇𝑆𝑥_𝑛,𝑆𝑇𝑥_𝑛+𝑝^𝑠𝑆𝑇𝑥_𝑛,𝑆𝑆𝑧⟶𝑝^𝑠(𝑇𝑆𝑧,𝑇𝑆𝑧)+0+𝑝^𝑠(𝑆𝑆𝑧,𝑆𝑆𝑧)=0,as𝑛⟶∞. (3.14) It follows that 𝑇(𝑆𝑧)=𝑆(𝑆𝑧) and 𝑆𝑧 is a coincidence point of 𝑇 and 𝑆.
(ii) If (𝑋,𝑝) satisfies (ORC), since {𝑆𝑥_𝑛} is an increasing sequence tending to 𝑆𝑧, we have that 𝑆𝑥_𝑛⪯𝑆𝑧 for each 𝑛∈ℕ. So we can apply (P₄) and the contractive condition to obtain 𝑝(𝑆𝑧,𝑇𝑧)≤𝑝𝑆𝑧,𝑆𝑥_𝑛+1+𝑝𝑇𝑥_𝑛,𝑇𝑧≤𝑝𝑆𝑧,𝑆𝑥_𝑛+1+𝛾𝑝𝑆𝑥_𝑛,𝑆𝑧𝑝𝑆𝑥_𝑛,𝑆𝑧≤𝑝𝑆𝑧,𝑆𝑥_𝑛+1+𝑝𝑆𝑥_𝑛,𝑆𝑧. (3.15) Letting 𝑛→∞ we get 𝑝(𝑆𝑧,𝑇𝑧)=0. Hence, we obtain that 𝑇𝑧=𝑆𝑧 and 𝑧 is a coincidence point.

Now, we are in the position to prove the main result of this section.

Theorem 3.3. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete, and let 𝑔∶𝑋→𝑋 and 𝐹∶𝑋×𝑋→𝑋 be such that 𝐹 has the 𝑔-mixed monotone property. Suppose that(i) there exists 𝛾∈𝒢 such that 𝑝𝐹()𝐹()𝑝𝑝𝑥,𝑦,𝐹𝑢,𝑣+𝑝𝑦,𝑥,𝐹𝑣,𝑢≤𝛾𝑔𝑥,𝑔𝑢+𝑝𝑔𝑦,𝑔𝑣𝑔𝑥,𝑔𝑢+𝑝𝑔𝑦,𝑔𝑣 (3.16) holds for all 𝑥,𝑦,𝑢,𝑣∈𝑋 satisfying (𝑔𝑥⪯𝑔𝑢 and 𝑔𝑦≽𝑔𝑣) or (𝑔𝑥≽𝑔𝑢 and 𝑔𝑦⪯𝑔𝑣);(ii)𝑔 is continuous, 𝐹(𝑋×𝑋)⊂𝑔(𝑋), and one of these two subsets of 𝑋 is closed;(iii)there exist 𝑥₀,𝑦₀∈𝑋 such that (𝑔𝑥₀⪯𝐹(𝑥₀,𝑦₀) and 𝑔𝑦₀≽𝐹(𝑦₀,𝑥₀)) or (𝑔𝑥₀≽𝐹(𝑥₀,𝑦₀) and 𝑔𝑦₀⪯𝐹(𝑦₀,𝑥₀));(iv)𝐹 is continuous and (𝐹,𝑔) is compatible in the sense of Definition 2.7, or(iv^′)(𝑋,⪯,𝑝) satisfies (ORC). Then there exist 𝑥,𝑦∈𝑋 such that 𝑔𝑥=𝐹𝑥,𝑦,𝑔𝑦=𝐹𝑦,𝑥, (3.17) that is, 𝑔 and 𝐹 have a coupled coincidence point.

Proof. Let relation ⊑, partial metric 𝑃, and mappings 𝑇_𝐹, 𝑇_𝑔 on 𝑋² be defined as in Lemma 2.8. Then (𝑋²,⊑,𝑃) is an ordered partial metric space which is complete and 𝑇_𝐹 is a 𝑇_𝑔-nondecreasing self-map on 𝑋². Moreover,(i)there exists 𝛾∈𝒢 such that 𝑃𝑇_𝐹𝑌,𝑇_𝐹𝑉≤𝛾𝑃𝑇_𝑔𝑌,𝑇_𝑔𝑉𝑃𝑇_𝑔𝑌,𝑇_𝑔𝑉 (3.18) holds for all ⊑-comparable 𝑌,𝑉∈𝑋²;(ii)𝑇_𝑔 is continuous, 𝑇_𝐹(𝑋²)⊂𝑇_𝑔(𝑋²), and one of these two subsets of 𝑋² is closed;(iii)there exists 𝑌₀∈𝑋² such that 𝑇_𝑔𝑌₀ and 𝑇_𝐹𝑌₀ are comparable;(iv)𝑇_𝐹 is continuous and the pair (𝑇_𝐹,𝑇_𝑔) is 𝑃^𝑠-compatible, or(iv^′)(𝑋²,⊑,𝑃) satisfies (ORC).Thus, all the conditions of Lemma 3.2 are satisfied (with 𝑇=𝑇_𝐹 and 𝑆=𝑇_𝑔). Hence, there exists 𝑌∈𝑋² such that 𝑇_𝑔𝑌=𝑇_𝐹𝑌, that is, there exist 𝑥,𝑦∈𝑋 such that 𝑔𝑥=𝐹(𝑥,𝑦) and 𝑔𝑦=𝐹(𝑦,𝑥). Therefore, 𝑔 and 𝐹 have a coupled coincidence point.

Putting 𝑔=𝑖_𝑋 (the identity map) in Theorem 3.3, we obtain the following.

Corollary 3.4. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete, and let 𝐹∶𝑋×𝑋→𝑋 have the mixed monotone property. Suppose that(i)there exists 𝛾∈𝒢 such that 𝑝𝐹()𝐹()𝑝()𝑝()𝑥,𝑦,𝐹𝑢,𝑣+𝑝𝑦,𝑥,𝐹𝑣,𝑢≤𝛾𝑥,𝑢+𝑝𝑦,𝑣𝑥,𝑢+𝑝𝑦,𝑣 (3.19) holds for all 𝑥,𝑦,𝑢,𝑣∈𝑋 satisfying (𝑥⪯𝑢 and 𝑦≽𝑣) or (𝑥≽𝑢 and 𝑦⪯𝑣);(ii)there exist 𝑥₀,𝑦₀∈𝑋 such that (𝑥₀⪯𝐹(𝑥₀,𝑦₀) and 𝑦₀≽𝐹(𝑦₀,𝑥₀)) or (𝑥₀≽𝐹(𝑥₀,𝑦₀) and 𝑦₀⪯𝐹(𝑦₀,𝑥₀));(iii)𝐹 is continuous, or(iii^′)(𝑋,⪯,𝑝) satisfies (ORC).Then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹𝑥,𝑦,𝑦=𝐹𝑦,𝑥, (3.20) that is, 𝐹 has a coupled fixed point.

If 𝑝=𝑑 is a standard metric, this reduces to [15, Corollary 2.3]. The following example shows how Corollary 3.4 can be used.

Example 3.5. Let 𝑋=[0,1/2] be ordered by the standard relation ≤. Consider the partial metric 𝑝 on 𝑋 given by 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Let 𝐹∶𝑋×𝑋→𝑋 be given as 𝐹=1𝑥,𝑦6𝑥−𝑦,𝑥≥𝑦,0,𝑥<𝑦. (3.21) Finally, take 𝛾∈𝒢 given as 𝛾(𝑡)=𝑒^−𝑡/(𝑡+1) for 𝑡>0 and 𝛾(0)∈[0,1). We will show that conditions of Corollary 3.4 hold true.

Take arbitrary 𝑥,𝑦,𝑢,𝑣∈𝑋 satisfying 𝑥≥𝑢 and 𝑦≤𝑣 (the other possible case is treated symmetrically), and denote 𝐿=𝑝(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))+𝑝(𝐹(𝑦,𝑥),𝐹(𝑣,𝑢)) and 𝑅=𝛾(𝑝(𝑥,𝑢)+𝑝(𝑦,𝑣))(𝑝(𝑥,𝑢)+𝑝(𝑦,𝑣)). Consider the six possible cases: 1°: 0≤𝑦≤𝑢≤𝑣≤𝑥≤1/2, 2°: 0≤𝑦≤𝑣≤𝑢≤𝑥≤1/2, 3°: 0≤𝑢≤𝑦≤𝑥≤𝑣≤1/2, 4°: 0≤𝑢≤𝑥≤𝑦≤𝑣≤1/2, 5°: 0≤𝑦≤𝑢≤𝑥≤𝑣≤1/2, and 6°: 0≤𝑢≤𝑦≤𝑣≤𝑥≤1/2. It is easy to check that in all these cases 𝐿≤(1/6)(𝑥+𝑣) and that 𝑅=(𝑒^{−(𝑥+𝑣)}/(𝑥+𝑣+1))(𝑥+𝑣). Since16𝑒𝑡≤^−𝑡𝑡+1⋅𝑡 (3.22) holds for each 𝑡∈[0,1] (as far as it holds for 𝑡=1), we obtain that 𝐿≤𝑅. The conditions of Corollary 3.4 are satisfied and 𝐹 has a coupled fixed point (which is (0,0)).

4. Coupled Coincidence and Fixed Points under Rakotch-Type Conditions

Let ℛ denote the class of real functions 𝜌∶[0,+∞)→[0,1) satisfying the conditionlimsup_𝑠→𝑡𝜌(𝑠)<1foreach𝑡>0. (4.1) Rakotch proved in 1962 the following.

Theorem 4.1 (see [11]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be a self-map. Suppose that there exists 𝜌∈ℛ such that 𝑑𝑇𝑥,𝑇𝑦≤𝜌𝑑𝑥,𝑦𝑑𝑥,𝑦 (4.2) holds for all 𝑥,𝑦∈𝑋. Then 𝑇 has a unique fixed point 𝑧∈𝑋.

We will prove the respective result for the existence of a coupled coincidence point in the frame of partial metric spaces. We begin with the following auxiliary result, which may be of interest on its own.

Lemma 4.2. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete. Let 𝑇,𝑆∶𝑋→𝑋 be self-maps such that 𝑆 is continuous, 𝑇𝑋⊂𝑆𝑋, and one of these two subsets of 𝑋 is closed. Suppose that 𝑇 is 𝑆-nondecreasing (with respect to ⪯) and there exists 𝑥₀∈𝑋 with 𝑆𝑥₀⪯𝑇𝑥₀ or 𝑇𝑥₀⪯𝑆𝑥₀. Assume also that there exists 𝜌∈ℛ such that 𝑝𝑇𝑥,𝑇𝑦≤𝜌𝑝𝑆𝑥,𝑆𝑦𝑝𝑆x,𝑆𝑦 (4.3) holds for all 𝑥,𝑦∈𝑋 such that 𝑆𝑥 and 𝑆𝑦 are comparable. Assume that either 1°: 𝑇 is continuous and the pair (𝑇,𝑆) is 𝑝^𝑠-compatible or 2°: 𝑋 satisfies (ORC). Then, 𝑇 and 𝑆 have a coincidence point in 𝑋.

Proof. The proof is similar to the proof of Lemma 3.2, so we note only the basic step.
With 𝛾 replaced by 𝜌, (3.7) shows that 𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛) is strictly decreasing, thus converging to some 𝑞≥0. Suppose that 𝑞>0. Then, it follows that 𝑞≤limsup_𝑛→∞𝜌𝑝𝑆𝑥_𝑛+1,𝑆𝑥_𝑛<𝑞, (4.4) a contradiction. Hence, lim_𝑛→∞𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)=0 is proved. The rest of the proof is identical to that of Lemma 3.2.

The following example shows that the existence of order may be crucial.

Example 4.3. Let 𝑋=[0,2] be endowed with the partial metric defined by 𝑝=[],(](].𝑥,𝑦𝑥−𝑦,𝑥,𝑦∈0,1max𝑥,𝑦,𝑥∈1,2∨𝑦∈1,2 (4.5) The order ⪯ is given by 𝑥⪯𝑦⟺𝑥=𝑦∨𝑥,𝑦∈(1,2]∧𝑥≤𝑦. (4.6) Then it is easy to check that (𝑋,⪯,𝑝) is an ordered partial metric space which is complete and satisfies (ORC). Consider the mappings 𝑆,𝑇∶𝑋→𝑋 given as [],𝑆𝑥=𝑥,𝑇𝑥=𝑥1−𝑥,𝑥∈0,12,𝑥∈(1,2]. (4.7) Take the function 𝜌∈ℛ given by 𝜌(𝑡)=1/2. It is easy to check that the contractive condition (4.3) holds. Indeed, if 𝑆𝑥=𝑥 and 𝑆𝑦=𝑦∈𝑋 are comparable, say 𝑥≽𝑦, then either 𝑥=𝑦∈[0,1] or (𝑥,𝑦∈(1,2] and 𝑥≥𝑦). In the first case, 𝑝(𝑆𝑥,𝑆𝑦)=𝑝(𝑥,𝑥)=0 and 𝑝(𝑇𝑥,𝑇𝑦)=|(1−𝑥)−(1−𝑥)|=0; hence, 𝑝(𝑇𝑥,𝑇𝑦)≤𝜌(0)⋅0 trivially holds (whichever function 𝜌∈ℛ is chosen). In the second case, 𝑝(𝑆𝑥,𝑆𝑦)=𝑝(𝑥,𝑦)=𝑥 and 𝑝𝑥𝑇𝑥,𝑇𝑦=𝑝2,𝑦2=𝑥−𝑦2<𝑥2 (4.8) and 𝑝(𝑇𝑥,𝑇𝑦)≤(1/2)⋅𝑥=𝜌(𝑝(𝑆𝑥,𝑆𝑦))𝑝(𝑆𝑥,𝑆𝑦). All the conditions of Lemma 4.2 are fulfilled and mappings 𝑇 and 𝑆 have a coincidence point (𝑥=1/2).

On the other hand, consider the same problem, but without order. Then the contractive condition does not hold and the conclusion about the coincidence point cannot be obtained in this way. Indeed, take any 𝑥,𝑦∈[0,1] with 𝑥≠𝑦. Then, 𝑝(𝑆𝑥,𝑆𝑦)=𝑝(𝑥,𝑦)=|𝑥−𝑦| and 𝑝(𝑇𝑥,𝑇𝑦)=|(1−𝑥)−(1−𝑦)|=|𝑥−𝑦|. Hence, if 𝑝(𝑇𝑥,𝑇𝑦)≤𝜌(𝑝(𝑆𝑥,𝑆𝑦))𝑝(𝑆𝑥,𝑆𝑦), then 𝜌(𝑝(𝑆𝑥,𝑆𝑦))≥1 and 𝜌∉ℛ.

Now, we are in the position to give

Theorem 4.4. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete, and let 𝑔∶𝑋→𝑋 and 𝐹∶𝑋×𝑋→𝑋 be such that 𝐹 has the 𝑔-mixed monotone property. Suppose that(i)there exists 𝜌∈ℛ such that 𝑝𝐹()𝐹()𝑝𝑝𝑥,𝑦,𝐹𝑢,𝑣+𝑝𝑦,𝑥,𝐹𝑣,𝑢≤𝜌𝑔𝑥,𝑔𝑢+𝑝𝑔𝑦,𝑔𝑣𝑔𝑥,𝑔𝑢+𝑝𝑔𝑦,𝑔𝑣 (4.9) holds for all 𝑥,𝑦,𝑢,𝑣∈𝑋 satisfying (𝑔𝑥⪯𝑔𝑢 and 𝑔𝑦≽𝑔𝑣) or (𝑔𝑥≽𝑔𝑢 and 𝑔𝑦⪯𝑔𝑣) and conditions (ii), (iii), (iv), respectively, (iv′) of Theorem 3.3 hold true. Then there exist 𝑥,𝑦∈𝑋 such that 𝑔𝑥=𝐹𝑥,𝑦,𝑔𝑦=𝐹𝑦,𝑥, (4.10) that is, 𝑔 and 𝐹 have a coupled coincidence point.

Proof. The proof is analogous to the proof of Theorem 3.3, and so is omitted.

Putting 𝑔=𝑖_𝑋 (the identity map) in Theorem 4.4, we obtain the following.

Corollary 4.5. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete, and let 𝐹∶𝑋×𝑋→𝑋 have the mixed monotone property. Suppose that(i)there exists 𝜌∈ℛ such that 𝑝𝐹()𝐹()𝑝()𝑝()𝑥,𝑦,𝐹𝑢,𝑣+𝑝𝑦,𝑥,𝐹𝑣,𝑢≤𝜌𝑥,𝑢+𝑝𝑦,𝑣𝑥,𝑢+𝑝𝑦,𝑣 (4.11) holds for all 𝑥,𝑦,𝑢,𝑣∈𝑋 satisfying (𝑥⪯𝑢 and 𝑦≽𝑣) or (𝑥≽𝑢 and 𝑦⪯𝑣) and conditions (ii), (iii), respectively, (iii′) of Corollary 3.4 hold true. Then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹𝑥,𝑦,𝑦=𝐹𝑦,𝑥, (4.12) that is, 𝐹 has a coupled fixed point.

5. Coupled Coincidence and Fixed Points under Integral Conditions

Denote by Φ the set of functions 𝜑∶ℝ⁺→ℝ⁺ that are Lebesgue integrable and summable (having finite integral) on each compact subset of ℝ⁺, and satisfying condition ∫^𝜀₀𝜑(𝑡)𝑑𝑡>0 for each 𝜀>0. Branciari [12] was the first to use integral-type contractive conditions in order to obtain fixed point results. He proved the following.

Theorem 5.1 (see [12]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be a self-map satisfying ₀^{𝑑(𝑇𝑥,𝑇𝑦)}𝜑(𝑡)𝑑𝑡≤𝑐₀^{𝑑(𝑥,𝑦)}𝜑(𝑡)𝑑𝑡 (5.1) for some 𝑐∈(0,1) and some 𝜑∈Φ. Then 𝑇 has a unique fixed point 𝑧∈𝑋 and for each 𝑥∈𝑋 the Picard sequence {𝑇^𝑛𝑥} converges to 𝑧 when 𝑛→∞.

Subsequently, several authors proved such results, including the very recent paper of Liu et al. [23].

We begin with the following.

Lemma 5.2. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete. Let 𝑇,𝑆∶𝑋→𝑋 be self-maps such that 𝑆 is continuous, 𝑇𝑋⊂𝑆𝑋, and one of these two subsets of 𝑋 is closed. Suppose that 𝑇 is 𝑆-nondecreasing (with respect to ⪯) and there exists 𝑥₀∈𝑋 with 𝑆𝑥₀⪯𝑇𝑥₀ or 𝑇𝑥₀⪯𝑆𝑥₀. Assume also that there exists 𝜌∈ℛ and 𝜑∈Φ such that ₀^{𝑝(𝑇𝑥,𝑇𝑦)}𝜑(𝑡)𝑑𝑡≤𝜌𝑝𝑆𝑥,𝑆𝑦₀^{𝑝(𝑆𝑥,𝑆𝑦)}𝜑(𝑡)𝑑𝑡 (5.2) holds for all 𝑥,𝑦∈𝑋 such that 𝑆𝑥 and 𝑆𝑦 are comparable. Assume that either 1°:𝑇 is continuous and the pair (𝑇,𝑆) is 𝑝^𝑠-compatible or 2^∘∶𝑋 satisfies (ORC). Then, 𝑇 and 𝑆 have a coincidence point in 𝑋.

Proof. Take 𝑥₀∈𝑋 with, say, 𝑆𝑥₀⪯𝑇𝑥₀, and using that 𝑇 is 𝑆-nondecreasing and that 𝑇𝑋⊂𝑆𝑋 form the sequence {𝑥_𝑛} satisfying 𝑇𝑥_𝑛=𝑆𝑥_𝑛+1, 𝑛=0,1,2,…, and 𝑆𝑥₀⪯𝑇𝑥₀=𝑆𝑥₁⪯𝑇𝑥₁=S𝑥₂⪯⋯⪯𝑇𝑥_𝑛⪯𝑆𝑥_𝑛+1⪯⋯. (5.3) Since 𝑆𝑥_𝑛−1 and 𝑆𝑥_𝑛 are comparable, we can apply the contractive condition (5.2) to obtain ^{𝑝(𝑆𝑥𝑛+1,𝑆𝑥𝑛)}₀𝜑(𝑡)𝑑𝑡=^{𝑝(𝑇𝑥𝑛,𝑇𝑥𝑛−1)}₀𝜑(𝑡)𝑝𝑑𝑡≤𝜌𝑆𝑥_𝑛,𝑆𝑥_𝑛−1^{𝑝(𝑆𝑥𝑛,𝑆𝑥𝑛−1)}₀𝜑(𝑡)𝑑𝑡≤^{𝑝(𝑆𝑥𝑛,𝑆𝑥𝑛−1)}₀𝜑(𝑡)𝑑𝑡. (5.4) If 𝑝(𝑆𝑥_𝑛0,𝑆𝑥_𝑛0−1)=0 for some 𝑛₀∈ℕ, then it easily follows that 𝑆𝑥_𝑛0−1=𝑆𝑥_𝑛0=𝑆𝑥_𝑛0+1=⋯ and 𝑥_𝑛0 is a point of coincidence of 𝑆 and 𝑇. Suppose that 𝑝(𝑆𝑥_𝑛,𝑆𝑥_𝑛+1)>0 for each 𝑛∈ℕ (and, hence, the strict inequality holds in (5.4)). We will prove that {𝑝(𝑆𝑥_𝑛,𝑆𝑥_𝑛−1)} is a nonincreasing sequence. Suppose, to the contrary, that 𝑝(𝑆𝑥_𝑛0+1,𝑆𝑥_𝑛0)>𝑝(𝑆𝑥_𝑛0,𝑆𝑥_𝑛0−1) for some 𝑛₀∈ℕ. Then, since 𝜑∈Φ, we get that 0<^{𝑝(𝑆𝑥𝑛0,𝑆𝑥𝑛0−1)}₀𝜑(𝑡)𝑑𝑡≤^{𝑝(𝑆𝑥𝑛0+1,𝑆𝑥𝑛0)}₀𝜑(𝑡)𝑑𝑡=^{𝑝(𝑇𝑥𝑛0,𝑇𝑥𝑛0−1)}₀𝜑(𝑡)𝑝𝑑𝑡≤𝜌𝑆𝑥_𝑛0,𝑆𝑥_𝑛0−1^{𝑝(𝑆𝑥𝑛0,𝑆𝑥𝑛0−1)}₀𝜑(𝑡)𝑑𝑡<^{𝑝(𝑆𝑥𝑛0,𝑆𝑥𝑛0−1)}₀𝜑(𝑡)𝑑𝑡, (5.5) a contradiction.
Denote by 𝑞≥0 the limit of the nonincreasing sequence {𝑝(𝑆𝑥_𝑛,𝑆𝑥_𝑛−1)} of positive numbers. Suppose that 𝑞>0. Then, using Lemmas 2.1 and 2.2 from [23], the contractive condition (5.2), and property (4.1) of function 𝜌, we get that 0<^𝑞₀𝜑(𝑡)𝑑𝑡=limsup_𝑛→∞^{𝑝(𝑇𝑥𝑛,𝑇𝑥𝑛−1)}₀𝜑(𝑡)𝑑𝑡≤limsup_𝑛→∞𝜌𝑝𝑆𝑥_𝑛,𝑆𝑥_𝑛−1^{𝑝(𝑆𝑥𝑛,𝑆𝑥𝑛−1)}₀𝜑(𝑡)𝑑𝑡≤limsup_𝑛→∞𝜌𝑝𝑆𝑥_𝑛,𝑆𝑥_𝑛−1limsup_𝑛→∞^{𝑝(𝑆𝑥𝑛,𝑆𝑥𝑛−1)}₀𝜑(𝑡)≤𝑑𝑡limsup_𝑠→𝑞𝜌(𝑠)^𝑞₀𝜑(𝑡)𝑑𝑡<^𝑞₀𝜑(𝑡)𝑑𝑡, (5.6) a contradiction. We conclude that lim_𝑛→∞𝑝(𝑆𝑥_𝑛+1,𝑆𝑥_𝑛)=0.
Now we prove that {𝑆𝑥_𝑛} is a Cauchy sequence in (𝑋,𝑝) (and in (𝑋,𝑝^𝑠)). If this were not the case, as in the proof of Lemma 3.2, using Lemma 2.11, we would get that there exist 𝜀>0 and two sequences {𝑚_𝑘} and {𝑛_𝑘} of positive integers such that the following four sequences tend to 𝜀 when 𝑘→∞: 𝑝𝑆𝑥_2𝑚𝑘,𝑆𝑥_2𝑛𝑘,𝑝𝑆𝑥_2𝑚𝑘,𝑆𝑥_2𝑛𝑘+1,𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘,𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘+1. (5.7) Putting in the contractive condition 𝑥=𝑥_{2𝑚𝑘−1} and 𝑦=𝑥_2𝑛𝑘, it follows that ^{𝑝(𝑇𝑥𝑘2𝑚−1,𝑇𝑥𝑘2𝑛)}₀𝜑(𝑡)𝑑𝑡≤𝜌𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘^{𝑝(𝑆𝑥𝑘2𝑚−1,𝑆𝑥𝑘2𝑛)}₀. (5.8) Passing to the upper limit as 𝑘→∞ and using properties of functions 𝜑∈Φ, 𝜌∈ℛ as well as [23, Lemma 2.1], we get a contradiction 0<^𝜀₀𝜑(𝑡)𝑑𝑡≤limsup_𝑛→∞𝜌𝑝𝑆𝑥_{2𝑚𝑘−1},𝑆𝑥_2𝑛𝑘^𝜀₀𝜑(𝑡)𝑑𝑡<^𝜀₀𝜑(𝑡)𝑑𝑡. (5.9) Thus, {𝑆𝑥_𝑛} is a Cauchy sequence, converging to some 𝑆𝑧∈𝑆𝑋 (which we suppose to be closed in 𝑋) such that 𝑝(𝑆𝑧,𝑆𝑧)=0. We will prove that 𝑇 and 𝑆 have a coincidence point.
(i) Suppose that 𝑇∶(𝑋,𝑝)→(𝑋,𝑝) is continuous and (𝑇,𝑆) is 𝑝^𝑠-compatible. As in the proof of Lemma 3.2, we get that 𝑇(𝑆𝑧)=𝑆(𝑆𝑧) and 𝑆𝑧 is a coincidence point of 𝑇 and 𝑆.
(ii) If (𝑋,𝑝) satisfies (ORC), since {𝑆𝑥_𝑛} is an increasing sequence tending to 𝑆𝑧, we have that 𝑆𝑥_𝑛⪯𝑆𝑧 for each 𝑛∈ℕ. So we can apply (P₄) and the contractive condition to obtain 𝑝(𝑆𝑧,𝑇𝑧)≤𝑝𝑆𝑧,𝑆𝑥_𝑛+1+𝑝𝑇𝑥_𝑛,𝑇𝑧≤𝑝𝑆𝑧,𝑆𝑥_𝑛+1+𝜌𝑝𝑆𝑥_𝑛,𝑆𝑧^{𝑝(𝑆𝑥𝑛}₀^,𝑆𝑧)𝜑(𝑡)𝑑𝑡. (5.10) Passing to the upper limit as 𝑛→∞, using properties of functions 𝜌∈ℛ, 𝜑∈Φ, as well as [23, Lemma 2.1], we get that 𝑝(𝑆𝑧,𝑇𝑧)=0. Hence, we obtain that 𝑇𝑧=𝑆𝑧.

Putting 𝑆=𝑖_𝑋 and 𝜌(𝑡)=𝑐∈(0,1) in Lemma 5.2, we obtain an ordered partial metric extension of Branciari's Theorem 5.1. The following example shows that this extension is proper.

Example 5.3. Let 𝑋=[0,1] be endowed with the standard metric and order, and consider 𝑇,𝑆∶𝑋→𝑋 defined by 𝑇𝑥=𝑥/(1+𝑥), and 𝑆𝑥=𝑥, 𝑥∈𝑋. Take 𝜑∈Φ given by 𝜑(𝑡)=1. Then condition (5.1) does not hold. Indeed, ₀^{𝑑(𝑇𝑥,𝑇𝑦)}1⋅𝑑𝑡≤𝑐₀^{𝑑(𝑥,𝑦)}1⋅𝑑𝑡 (5.11) would imply that 𝑥−𝑦(1+𝑥)1+𝑦≤𝑐𝑥−𝑦 (5.12) for all 𝑥,𝑦∈𝑋. But taking 𝑥≠𝑦 and 𝑥,𝑦→0+ would give that 1≤𝑐, a contradiction.

On the other hand, consider the same problem in an (ordered) partial metric space, with the partial metric given by 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. Then, condition (5.2) reduces to₀^{𝑝(𝑇𝑥,𝑇𝑦)}1⋅𝑑𝑡≤𝑐₀^{𝑝(𝑥,𝑦)}1⋅𝑑𝑡, (5.13) which is, for 𝑥≥𝑦, equivalent to 𝑥/(1+𝑥)≤𝑐𝑥 and holds true (for each 𝑥∈𝑋) if 𝑐∈[1/2,1). Obviously, 𝑇 has a (unique) fixed point 𝑥=0.

The following theorem is obtained from Lemma 5.2 in a similar way as Theorems 3.3 and 4.4 from respective lemmas.

Theorem 5.4. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete, and let 𝑔∶𝑋→𝑋 and 𝐹∶𝑋×𝑋→𝑋 be such that 𝐹 has the 𝑔-mixed monotone property. Suppose that(i)there exists 𝜌∈ℛ and 𝜑∈Φ such that ₀^{𝑝(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))+𝑝(𝐹(𝑦,𝑥),𝐹(𝑣,𝑢))}𝜑(𝑡)𝑝𝑑𝑡≤𝜌𝑔𝑥,𝑔𝑢+𝑝𝑔𝑦,𝑔𝑣₀^{𝑝(𝑔𝑥,𝑔𝑢)+𝑝(𝑔𝑦,𝑔𝑣)}𝜑(𝑡)𝑑𝑡 (5.14) holds for all 𝑥,𝑦,𝑢,𝑣∈𝑋 satisfying (𝑔𝑥⪯𝑔𝑢 and 𝑔𝑦≽𝑔𝑣) or (𝑔𝑥≽𝑔𝑢 and 𝑔𝑦⪯𝑔𝑣); and conditions (ii), (iii), (iv), respectively, (iv′) of Theorem 3.3 hold true. Then there exist 𝑥,𝑦∈𝑋 such that 𝑔𝑥=𝐹𝑥,𝑦,𝑔𝑦=𝐹𝑦,𝑥, (5.15) that is, 𝑔 and 𝐹 have a coupled coincidence point.

Putting 𝑔=𝑖_𝑋 in Theorem 5.4, we obtain the following.

Corollary 5.5. Let (𝑋,⪯,𝑝) be an ordered partial metric space which is complete, and let 𝐹∶𝑋×𝑋→𝑋 have the mixed monotone property. Suppose that(i)there exists 𝜌∈ℛ and 𝜑∈Φ such that ₀^{𝑝(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))+𝑝(𝐹(𝑦,𝑥),𝐹(𝑣,𝑢))}𝜑(𝑡)𝑝()𝑑𝑡≤𝜌𝑥,𝑢+𝑝𝑦,𝑣₀^{𝑝(𝑥,𝑢)+𝑝(𝑦,𝑣)}𝜑(𝑡)𝑑𝑡 (5.16) holds for all 𝑥,𝑦,𝑢,𝑣∈𝑋 satisfying (𝑥⪯𝑢 and 𝑦≽𝑣) or (𝑥≽𝑢 and 𝑦⪯𝑣) and conditions (ii), (iii), respectively, (iii′) of Corollary 3.4 hold true. Then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹𝑥,𝑦,𝑦=𝐹𝑦,𝑥, (5.17) that is, 𝐹 has a coupled fixed point.

Acknowledgments

The authors are thankful to the referees for very useful suggestions that helped to improve the paper. They are also thankful to the Ministry of Science and Technological Development of Serbia.