Some Coupled Fixed Point Results on Partial Metric Spaces

Abstract

We give some coupled fixed point results for mappings satisfying different contractive conditions on complete partial metric spaces.

1. Introduction and Preliminaries

For a given partially ordered set 𝑋, Bhaskar and Lakshmikantham in [1] introduced the concept of coupled fixed point of a mapping 𝐹∶𝑋×𝑋→𝑋. Later in [2], Círíc and Lakshmikantham investigated some more coupled fixed point theorems in partially ordered sets. The following is the corresponding definition of a coupled fixed point.

Definition 1.1 (see [3]). An element (𝑥,𝑦)∈𝑋×𝑋 is said to be a coupled fixed point of the mapping 𝐹∶𝑋×𝑋→𝑋 if 𝐹(𝑥,𝑦)=𝑥 and 𝐹(𝑦,𝑥)=𝑦.

Sabetghadam et al. [4] obtained the following.

Theorem 1.2. Let (𝑋,𝑑) be a complete cone metric space. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition for all 𝑥,𝑦,𝑢,𝑣∈𝑋𝑑𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝑘𝑑(𝑥,𝑢)+𝑙𝑑𝑦,𝑣, (1.1) where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point.

In this paper, we give the analogous of this result (and some others in [4]) on partial metric spaces, and we establish some coupled fixed point results.

The concept of partial metric space (𝑋,𝑝) was introduced by Matthews in 1994. In such spaces, the distance of a point in the self may not be zero. First, we start with some preliminaries definitions on the partial metric spaces [3, 5–13].

Definition 1.3 (see ([6–8])). 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).

A partial metric space is a pair (𝑋,𝑝) such that 𝑋 is a nonempty set and 𝑝 is a partial metric on 𝑋.

Remark 1.4. It is clear that if 𝑝(𝑥,𝑦)=0, then from (p1), (p2), and (p3), 𝑥=𝑦. But if 𝑥=𝑦, 𝑝(𝑥,𝑦) may not be 0.

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, (1.2) is a metric on 𝑋.

Definition 1.5 (see ([6–8])). Let (𝑋,𝑝) be a partial metric space. Then, (i)a sequence {𝑥_𝑛} in a partial metric space (𝑋,𝑝) converges to a point 𝑥∈𝑋 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→+∞𝑝(𝑥,𝑥_𝑛);(ii)a sequence {𝑥_𝑛} in a partial metric space (𝑋,𝑝) is called a Cauchy sequence if there exists (and is finite) lim_{𝑛,𝑚→+∞}𝑝(𝑥_𝑛,𝑥_𝑚);(iii)a partial metric space (𝑋,𝑝) is said to be complete if every Cauchy sequence {𝑥_𝑛} in 𝑋 converges to a point 𝑥∈𝑋, that is, 𝑝(𝑥,𝑥)=lim_{𝑛,𝑚→+∞}𝑝(𝑥_𝑛,𝑥_𝑚).

Lemma 1.6 (see ([6, 7, 9])). 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)a partial metric space (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑝^𝑠) is complete; 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. (1.3)

2. Main Results

Our first main result is the following.

Theorem 2.1. Let (𝑋,𝑝) be a complete partial metric space. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition for all 𝑥,𝑦,𝑢,𝑣∈𝑋𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝑘𝑝(𝑥,𝑢)+𝑙𝑝𝑦,𝑣, (2.1) where 𝑘, 𝑙 are nonnegative constants with 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point.

Proof. Choose 𝑥₀,𝑦₀∈𝑋 and set 𝑥₁=𝐹(𝑥₀,𝑦₀) and 𝑦₁=𝐹(𝑦₀,𝑥₀). Repeating this process, set 𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛) and 𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛). Then, by (2.1), we have 𝑝𝑥_𝑛,𝑥_𝑛+1=𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛≤𝑘𝑝𝑥_𝑛−1,𝑥_𝑛+𝑙𝑝𝑦_𝑛−1,𝑦_𝑛, (2.2) and similarly 𝑝𝑦_𝑛,𝑦_𝑛+1=𝑝𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛≤𝑘𝑝𝑦_𝑛−1,𝑦_𝑛+𝑙𝑝𝑥_𝑛−1,𝑥_𝑛. (2.3) Therefore, by letting 𝑑_𝑛=𝑝𝑥_𝑛,𝑥_𝑛+1+𝑝𝑦_𝑛,𝑦_𝑛+1, (2.4) we have 𝑑_𝑛=𝑝𝑥_𝑛,𝑥_𝑛+1+𝑝𝑦_𝑛,𝑦_𝑛+1≤𝑘𝑝𝑥_𝑛−1,𝑥_𝑛+𝑙𝑝𝑦_𝑛−1,𝑦_𝑛+𝑘𝑝𝑦_𝑛−1,𝑦_𝑛+𝑙𝑝𝑥_𝑛−1,𝑥_𝑛=(𝑘+𝑙)𝑝𝑦_𝑛−1,𝑦_𝑛+𝑝𝑥_𝑛−1,𝑥_𝑛=(𝑘+𝑙)𝑑_𝑛−1. (2.5) Consequently, if we set 𝛿=𝑘+𝑙, then, for each 𝑛∈ℕ, we have 𝑑_𝑛≤𝛿𝑑_𝑛−1≤𝛿²𝑑_𝑛−2≤⋯≤𝛿^𝑛𝑑₀. (2.6) If 𝑑₀=0 then 𝑝(𝑥₀,𝑥₁)+𝑝(𝑦₀,𝑦₁)=0. Hence, from Remark 1.4, we get 𝑥₀=𝑥₁=𝐹(𝑥₀,𝑦₀) and 𝑦₀=𝑦₁=𝐹(𝑦₀,𝑥₀), meaning that (𝑥₀,𝑦₀) is a coupled fixed point of 𝐹. Now, let 𝑑₀>0. For each 𝑛≥𝑚, we have, in view of the condition (p4) 𝑝𝑥_𝑛,𝑥_𝑚≤𝑝𝑥_𝑛,𝑥_𝑛−1+𝑝𝑥_𝑛−1,𝑥_𝑛−2−𝑝𝑥_𝑛−1,𝑥_𝑛−1+𝑝𝑥_𝑛−2,𝑥_𝑛−3+𝑝𝑥_𝑛−3,𝑥_𝑛−4−𝑝𝑥_𝑛−3,𝑥_𝑛−3+⋯+𝑝𝑥_𝑚+2,𝑥_𝑚+1+𝑝𝑥_𝑚+1,𝑥_𝑚−𝑝𝑥_𝑚+1,𝑥_𝑚+1≤𝑝𝑥_𝑛,𝑥_𝑛−1+𝑝𝑥_𝑛−1,𝑥_𝑛−2+⋯+𝑝𝑥_𝑚+1,𝑥_𝑚. (2.7) Similarly, we have 𝑝𝑦_𝑛,𝑦_𝑚≤𝑝𝑦_𝑛,𝑦_𝑛−1+𝑝𝑦_𝑛−1,𝑦_𝑛−2+⋯+𝑝𝑦_𝑚+1,𝑦_𝑚. (2.8) Thus, 𝑝𝑥_𝑛,𝑥_𝑚+𝑝𝑦_𝑛,𝑦_𝑚≤𝑑_𝑛−1+𝑑_𝑛−2+⋯+𝑑_𝑚≤𝛿^𝑛−1+𝛿^𝑛−2+⋯+𝛿^𝑚𝑑₀≤𝛿^𝑚𝑑1−𝛿⁰. (2.9) By definition of 𝑝^𝑠, we have 𝑝^𝑠(𝑥,𝑦)≤2𝑝(𝑥,𝑦), so, for any 𝑛≥𝑚𝑝^𝑠𝑥_𝑛,𝑥_𝑚+𝑝^𝑠𝑦_𝑛,𝑦_𝑚≤2𝑝𝑥_𝑛,𝑥_𝑚+2𝑝𝑦_𝑛,𝑦_𝑚𝛿≤2^𝑚𝑑1−𝛿⁰, (2.10) which implies that {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in (𝑋,𝑝^𝑠) because of 0≤𝛿=𝑘+𝑙<1. Since the partial metric space (𝑋,𝑝) is complete, hence thanks to Lemma 1.6, the metric space (𝑋,𝑝^𝑠) is complete, so there exist 𝑢^∗,𝑣^∗∈𝑋 such that lim_𝑛→+∞𝑝^𝑠𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝^𝑠𝑦_𝑛,𝑣^∗=0. (2.11) Again, from Lemma 1.6, we get 𝑝𝑢^∗,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛,𝑝𝑣^∗,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑦_𝑛. (2.12) But, from condition (p2) and (2.6), 𝑝𝑥_𝑛,𝑥_𝑛≤𝑝𝑥_𝑛,𝑥_𝑛+1≤𝑑_𝑛≤𝛿^𝑛𝑑₀, (2.13) so since 𝛿∈[0,1[, hence letting 𝑛→+∞, we get lim_𝑛→+∞𝑝(𝑥_𝑛,𝑥_𝑛)=0. It follows that 𝑝𝑢^∗,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛=0. (2.14) Similarly, we get 𝑝𝑣^∗,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑦_𝑛=0. (2.15) Therefore, we have, using (2.1), 𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗≤𝑝𝐹𝑢^∗,𝑣^∗,𝑥_𝑛+1+𝑝𝑥_𝑛+1,𝑢^∗−𝑝𝑥_𝑛+1,𝑥_𝑛+1𝐹𝑢,by(p4)≤𝑝^∗,𝑣^∗,𝐹𝑥_𝑛,𝑦_𝑛+𝑝𝑥_𝑛+1,𝑢^∗≤𝑘𝑝𝑥_𝑛,𝑢^∗+𝑙𝑝𝑦_𝑛,𝑣^∗+𝑝𝑥_𝑛+1,𝑢^∗, (2.16) and letting 𝑛→+∞, then from (2.14) and (2.15), we obtain 𝑝(𝐹(𝑢^∗,𝑣^∗),𝑢^∗))=0, so 𝐹(𝑢^∗,𝑣^∗)=𝑢^∗. Similarly, we have 𝐹(𝑣^∗,𝑢^∗)=𝑣^∗, meaning that (𝑢^∗,𝑣^∗) is a coupled fixed point of 𝐹.
Now, if (𝑢′,𝑣′) is another coupled fixed point of 𝐹, then 𝑝𝑢^′,𝑢^∗=𝑝𝐹𝑢^′,𝑣^′,𝐹𝑢^∗,𝑣^∗≤𝑘𝑝𝑢^′,𝑢^∗+𝑙𝑝𝑣^′,𝑣^∗,𝑝𝑣^′,𝑣^∗=𝑝𝐹𝑣^′,𝑢^′,𝐹𝑣^∗,𝑢^∗≤𝑘𝑝𝑣^′,𝑣^∗+𝑙𝑝𝑢^′,𝑢^∗. (2.17) It follows that 𝑝𝑢^′,𝑢^∗+𝑝𝑣^′,𝑣^∗≤(𝑘+𝑙)𝑝𝑢^′,𝑢^∗+𝑝𝑣^′,𝑣^∗. (2.18) In view of 𝑘+𝑙<1, this implies that 𝑝(𝑢′,𝑢^∗)+𝑝(𝑣′,𝑣^∗)=0, so 𝑢^∗=𝑢′ and 𝑣^∗=𝑣′. The proof of Theorem 2.1 is completed.

It is worth noting that when the constants in Theorem 2.1 are equal, we have the following corollary

Corollary 2.2. Let (𝑋,𝑝) be a complete partial metric space. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition for all 𝑥, 𝑦,𝑢,𝑣∈𝑋𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝(𝑥,𝑢)+𝑝𝑦,𝑣, (2.19) where 0≤𝑘<1. Then, 𝐹 has a unique coupled fixed point.

Example 2.3. Let 𝑋=[0,+∞[ endowed with the usual partial metric 𝑝 defined by 𝑝∶𝑋×𝑋→[0,+∞[ with 𝑝(𝑥,𝑦)=max{𝑥,𝑦}. The partial metric space (𝑋,𝑝) is complete because (𝑋,𝑝^𝑠) is complete. Indeed, for any 𝑥,𝑦∈𝑋, 𝑝^𝑠𝑥,𝑦=2𝑝𝑥,𝑦−𝑝(𝑥,𝑥)−𝑝𝑦,𝑦=2max𝑥,𝑦−𝑥+𝑦=𝑥−𝑦, (2.20) Thus, (𝑋,𝑝^𝑠) is the Euclidean metric space which is complete. Consider the mapping 𝐹∶𝑋×𝑋→𝑋 defined by 𝐹(𝑥,𝑦)=(𝑥+𝑦)/6. For any 𝑥,𝑦,𝑢,𝑣∈𝑋, we have 𝑝𝐹(=1𝑥,𝑦,𝐹𝑢,𝑣)6≤1max𝑥+𝑦,𝑢+𝑣6{}=1max𝑥,𝑢+max𝑦,𝑣6𝑝(𝑥,𝑢)+𝑝𝑦,𝑣. (2.21) which is the contractive condition (2.19) for 𝑘=1/3. Therefore, by Corollary 2.2, 𝐹 has a unique coupled fixed point, which is (0,0). Note that if the mapping 𝐹∶𝑋×𝑋→𝑋 is given by 𝐹(𝑥,𝑦)=(𝑥+𝑦)/2, then 𝐹 satisfies the contractive condition (2.19) for 𝑘=1, that is, 𝑝𝐹(=1𝑥,𝑦,𝐹𝑢,𝑣)2≤1max𝑥+𝑦,𝑢+𝑣2{}=1max𝑥,𝑢+max𝑦,𝑣2𝑝(𝑥,𝑢)+𝑝𝑦,𝑣. (2.22) In this case, (0,0) and (1,1) are both coupled fixed points of 𝐹, and, hence, the coupled fixed point of 𝐹 is not unique. This shows that the condition 𝑘<1 in Corollary 2.2, and hence 𝑘+𝑙<1 in Theorem 2.1 cannot be omitted in the statement of the aforesaid results.

Theorem 2.4. Let (𝑋,𝑝) be a complete partial metric space. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition for all 𝑥,𝑦,𝑢,𝑣∈𝑋𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝑘𝑝𝐹𝑥,𝑦,𝑥+𝑙𝑝(𝐹(𝑢,𝑣),𝑢), (2.23) where 𝑘, 𝑙 are nonnegative constants with 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point.

Proof. We take the same sequences {𝑥_𝑛} and {𝑦_𝑛} given in the proof of Theorem 2.1 by 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛forany𝑛∈ℕ. (2.24) Applying (2.23), we get 𝑝𝑥_𝑛,𝑥_𝑛+1≤𝛿𝑝𝑥_𝑛−1,𝑥_𝑛𝑝𝑦_𝑛,𝑦_𝑛+1≤𝛿𝑝𝑦_𝑛−1,𝑦_𝑛 (2.25)(2.26) where 𝛿=𝑘/(1−𝑙). By definition of 𝑝^𝑠, we have 𝑝^𝑠𝑥_𝑛,𝑥_𝑛+1≤2𝑝𝑥_𝑛,𝑥_𝑛+1≤2𝛿^𝑛𝑝𝑥₁,𝑥₀,𝑝^𝑠𝑦_𝑛,𝑦_𝑛+1≤2𝑝𝑦_𝑛,𝑦_𝑛+1≤2𝛿^𝑛𝑝𝑦₁,𝑦₀. (2.27)(2.28) Since 𝑘+𝑙<1, hence 𝛿<1, so the sequences {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in the metric space (𝑋,𝑝^𝑠). The partial metric space (𝑋,𝑝) is complete, hence from Lemma 1.6, (𝑋,𝑝^𝑠) is complete, so there exist 𝑢^∗,𝑣^∗∈𝑋 such that lim_𝑛→+∞𝑝^𝑠𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝^𝑠𝑦_𝑛,𝑣^∗=0. (2.29) From Lemma 1.6, we get 𝑝𝑢^∗,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛,𝑝𝑣^∗,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑦_𝑛. (2.30) By the condition (p2) and (2.25), we have 𝑝𝑥_𝑛,𝑥_𝑛≤𝑝𝑥_𝑛,𝑥_𝑛+1≤𝛿^𝑛𝑝𝑥₁,𝑥₀, (2.31) so lim_𝑛→+∞𝑝(𝑥_𝑛,𝑥_𝑛)=0. It follows that 𝑝𝑢^∗,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛=0. (2.32) Similarly, we find 𝑝𝑣^∗,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑦_𝑛=0. (2.33) Therefore, by (2.23), 𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗≤𝑝𝐹𝑢^∗,𝑣^∗,𝑥_𝑛+1+𝑝𝑥_𝑛+1,𝑢^∗=𝑝𝐹𝑢^∗,𝑣^∗,𝐹𝑥_𝑛,𝑦_𝑛+𝑝𝑥_𝑛+1,𝑢^∗≤𝑘𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗+𝑙𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛+𝑝𝑥_𝑛+1,𝑢^∗=𝑘𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗+𝑙𝑝𝑥_𝑛+1,𝑥_𝑛+𝑝𝑥_𝑛+1,𝑢^∗, (2.34) and letting 𝑛→+∞, then from (2.27)–(2.32), we obtain 𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗≤𝑘𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗. (2.35) From the preceding inequality, we can deduce a contradiction if we assume that 𝑝(𝐹(𝑢^∗,𝑣^∗),𝑢^∗)≠0, because in that case we conclude that 1≤𝑘 and now this inequality is, in fact, a contradiction, so 𝑝(𝐹(𝑢^∗,𝑣^∗),𝑢^∗)=0, that is, 𝐹(𝑢^∗,𝑣^∗)=𝑢^∗. Similarly, we have 𝐹(𝑣^∗,𝑢^∗)=𝑣^∗, meaning that (𝑢^∗,𝑣^∗) is a coupled fixed point of 𝐹. Now, if (𝑢′,𝑣′) is another coupled fixed point of 𝐹, then, in view of (2.23), 𝑝𝑢^′,𝑢^∗=𝑝𝐹𝑢^′,𝑣^′,𝐹𝑢^∗,𝑣^∗≤𝑘𝑝𝐹𝑢^′,𝑣^′,𝑢^′+𝑙𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗=𝑘𝑝𝑢^′,𝑢^′+𝑙𝑝𝑢^∗,𝑢^∗≤𝑘𝑝𝑢^′,𝑢^∗+𝑙𝑝𝑢^′,𝑢^∗=(𝑘+𝑙)𝑝𝑢^′,𝑢^∗,using(p2), (2.36) that is, 𝑝(𝑢′,𝑢^∗)=0 since (𝑘+𝑙)<1. It follows that 𝑢^∗=𝑢′. Similarly, we can have 𝑣^∗=𝑣′, and the proof of Theorem 2.4 is completed.

Theorem 2.5. Let (𝑋,𝑝) be a complete partial metric space. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition for all 𝑥, 𝑦,𝑢,𝑣∈𝑋𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝑘𝑝𝐹𝑥,𝑦,𝑢+𝑙𝑝(𝐹(𝑢,𝑣),𝑥), (2.37) where 𝑘,𝑙 are nonnegative constants with 𝑘+2𝑙<1. Then, 𝐹 has a unique coupled fixed point.

Proof. Since, 𝑘+2𝑙<1, hence 𝑘+𝑙<1, and as a consequence the proof of the uniqueness in this theorem is as trivial as in the other results. To prove the existence of the fixed point, choose the sequences {𝑥_𝑛} and {𝑦_𝑛} like in the proof of Theorem 2.1, that is 𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,forany𝑛∈ℕ. (2.38) Applying again (2.37), we have 𝑝𝑥_𝑛,𝑥_𝑛+1=𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛≤𝑘𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝑥_𝑛+𝑙𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛−1=𝑘𝑝𝑥_𝑛,𝑥_𝑛+𝑙𝑝𝑥_𝑛+1,𝑥_𝑛−1≤𝑘𝑝𝑥_𝑛+1,𝑥_𝑛+𝑙𝑝𝑥_𝑛+1,𝑥_𝑛−1𝑥,by(p2)≤𝑘𝑝_𝑛+1,𝑥_𝑛+𝑙𝑝𝑥_𝑛+1,𝑥_𝑛+𝑙𝑝𝑥_𝑛,𝑥_𝑛−1−𝑙𝑝𝑥_𝑛,𝑥_𝑛≤()𝑝𝑥,using(p4)𝑘+𝑙_𝑛,𝑥_𝑛+1+𝑙𝑝𝑥_𝑛−1,𝑥_𝑛. (2.39) It follows that for any 𝑛∈ℕ^∗𝑝𝑥_𝑛,𝑥_𝑛+1≤𝑙𝑝𝑥1−𝑙−𝑘^𝑛−1,𝑥_𝑛. (2.40) Let us take 𝛿=𝑙/(1−𝑙−𝑘). Hence, we deduce 𝑝^𝑠𝑥_𝑛,𝑥_𝑛+1≤2𝑝𝑥_𝑛,𝑥_𝑛+1≤2𝛿^𝑛𝑝𝑥₀,𝑥₁. (2.41) Under the condition 0≤𝑘+2𝑙<1, we get 0≤𝛿<1. From this fact, we immediately obtain that {𝑥_𝑛} is Cauchy in the complete metric space (𝑋,𝑝^𝑠). Of course, similar arguments apply to the case of the sequence {𝑦_𝑛} in order to prove that 𝑝^𝑠𝑦_𝑛,𝑦_𝑛+1≤2𝑝𝑦_𝑛,𝑦_𝑛+1≤2𝛿^𝑛𝑝𝑦₀,𝑦₁, (2.42) and, thus, that the sequence {𝑦_𝑛} is Cauchy in (𝑋,𝑝^𝑠). Therefore, there exist 𝑢^∗,𝑣^∗∈𝑋 such that lim_𝑛→+∞𝑝^𝑠𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝^𝑠𝑦_𝑛,𝑣^∗=0. (2.43) Thanks to Lemma 1.6, we have lim_𝑛→+∞𝑝𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛=𝑝𝑢^∗,𝑢^∗,lim_𝑛→+∞𝑝𝑦_𝑛,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑦_𝑛=𝑝𝑣^∗,𝑣^∗. (2.44) The condition (p2) together with (2.41) yield that 𝑝𝑥_𝑛,𝑥_𝑛≤𝑝𝑥_𝑛,𝑥_𝑛+1≤𝛿^𝑛𝑝𝑥₀,𝑥₁, (2.45) hence letting 𝑛→+∞, we get lim_𝑛→+∞𝑝(𝑥_𝑛,𝑥_𝑛)=0. It follows that 𝑝𝑢^∗,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑢^∗=lim_𝑛→+∞𝑝𝑥_𝑛,𝑥_𝑛=0. (2.46) Similarly, we have 𝑝𝑣^∗,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑣^∗=lim_𝑛→+∞𝑝𝑦_𝑛,𝑦_𝑛=0. (2.47) Therefore, we have, using (2.37), 𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗≤𝑝𝐹𝑢^∗,𝑣^∗,𝑥_𝑛+1+𝑝𝑥_𝑛+1,𝑢^∗=𝑝𝐹𝑢^∗,𝑣^∗,𝐹𝑥_𝑛,𝑦_𝑛+𝑝𝑥_𝑛+1,𝑢^∗≤𝑘𝑝𝐹𝑢^∗,𝑣^∗,𝑥_𝑛+𝑙𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑢^∗+𝑝𝑥_𝑛+1,𝑢^∗=𝑘𝑝𝐹𝑢^∗,𝑣^∗,𝑥_𝑛+𝑙𝑝𝑥_𝑛+1,𝑢^∗+𝑝𝑥_𝑛+1,𝑢^∗≤𝑘𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗+𝑘𝑝𝑢^∗,𝑥_𝑛+𝑙𝑝𝑥_𝑛+1,𝑢^∗+𝑝𝑥_𝑛+1,𝑢^∗,using(p4). (2.48) Letting 𝑛→+∞ yields, using (2.46), 𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗≤𝑘𝑝𝐹𝑢^∗,𝑣^∗,𝑢^∗, (2.49) and since 𝑘<1, we have 𝑝(𝐹(𝑢^∗,𝑣^∗),𝑢^∗)=0, that is, 𝐹(𝑢^∗,𝑣^∗)=𝑢^∗. Similarly, thanks to (2.47), we get 𝐹(𝑣^∗,𝑢^∗)=𝑣^∗, and hence (𝑢^∗,𝑣^∗) is a coupled fixed point of 𝐹.

When the constants in Theorems 2.4 and 2.5 are equal, we get the following corollaries.

Corollary 2.6. Let (𝑋,𝑝) be a complete partial metric space. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition for all 𝑥,𝑦,𝑢,𝑣∈𝑋𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝𝐹𝑥,𝑦,𝑥+𝑝(𝐹(𝑢,𝑣),𝑢), (2.50) where 0≤𝑘<1. Then, 𝐹 has a unique coupled fixed point.

Corollary 2.7. Let (𝑋,𝑝) be a complete partial metric space. Suppose that the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the following contractive condition for all 𝑥,𝑦,𝑢,𝑣∈𝑋𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝𝐹𝑥,𝑦,𝑢+𝑝(𝐹(𝑢,𝑣),𝑥), (2.51) where 0≤𝑘<2/3. Then, 𝐹 has a unique coupled fixed point.

Proof. The condition 0≤𝑘<2/3 follows from the hypothesis on 𝑘 and 𝑙 given in Theorem 2.5.

Remark 2.8. (i) Theorem 2.1 extends the Theorem  2.2 of [4] on the class of partial metric spaces.
(ii) Theorem 2.4 extends the Theorem  2.5 of [4] on the class of partial metric spaces.

Remark 2.9. Note that in Theorem 2.4, if the mapping 𝐹∶𝑋×𝑋→𝑋 satisfies the contractive condition (2.23) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, then 𝐹 also satisfies the following contractive condition: 𝑝𝐹()𝐹()(𝐹())𝐹.𝑥,𝑦,𝐹𝑢,𝑣=𝑝𝑢,𝑣,𝐹𝑥,𝑦≤𝑘𝑝𝑢,𝑣,𝑢+𝑙𝑝𝑥,𝑦,𝑥 (2.52) Consequently, by adding (2.23) and (2.52), 𝐹 also satisfies the following: 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝑘+𝑙2𝑝(𝐹(𝑢,𝑣),𝑢)+𝑘+𝑙2𝑝𝐹𝑥,𝑦,𝑥, (2.53) which is a contractive condition of the type (2.50) in Corollary 2.6 with equal constants. Therefore, one can also reduce the proof of general case (2.23) in Theorem 2.4 to the special case of equal constants. A similar argument is valid for the contractive conditions (2.37) in Theorem 2.5 and (2.51) in Corollary 2.7.

Acknowledgment

The author thanks the editor and the referees for their kind comments and suggestions to improve this paper.