Coupled Fixed Point Results in Complete Partial Metric Spaces

Abstract

We establish some coupled fixed point theorems for a mapping satisfying some contraction conditions in complete partial metric spaces. Our consequences extend the results of H. Aydi (2011).

1. Introduction and Mathematical Preliminaries

The notion of a partial metric space (PMS) was introduced in 1992 by Matthews [1, 2]. Matthews proved a fixed point theorem on this spaces, analogous to the Banach's fixed point theorem. Recently, many authors have focused on partial metric spaces and their topological properties (see, e.g., [3–9]).

The definition of a partial metric space is given by Matthews (see [1, 2]) as follows:

Definition 1.1. Let 𝑋 be a nonempty set and let 𝑝∶𝑋×𝑋→𝑅⁺ satisfies(P1)𝑥=𝑦⇔𝑝(𝑥,𝑥)=𝑝(𝑦,𝑦)=𝑝(𝑥,𝑦), for all 𝑥,𝑦∈𝑋,(P2)𝑝(𝑥,𝑥)≤𝑝(𝑥,𝑦), for all 𝑥,𝑦∈𝑋,(P3) 𝑝(𝑥,𝑦)=𝑝(𝑦,𝑥), for all 𝑥,𝑦∈𝑋,(P4)𝑝(𝑥,𝑦)≤𝑝(𝑥,𝑧)+𝑝(𝑧,𝑦)−𝑝(𝑧,𝑧), for all 𝑥,𝑦,𝑧∈𝑋.
Then the pair (𝑋,𝑝) is called a partial metric space and 𝑝 is called a partial metric on 𝑋.
The function 𝑑_𝑝∶𝑋×𝑋→𝑅⁺ defined by 𝑑_𝑝𝑥,𝑦=2𝑝𝑥,𝑦−𝑝(𝑥,𝑥)−𝑝𝑦,𝑦 (1.1) satisfies the conditions of a metric on 𝑋; therefore it is a (usual) metric on 𝑋.

Remark 1.2. if 𝑥=𝑦, 𝑝(𝑥,𝑦) may not be 0.(1)A famous example of partial metric spaces is the pair (𝑅⁺,𝑝), where 𝑝(𝑥,𝑦)=max{𝑥,𝑦} for all 𝑥,𝑦∈𝑅⁺. In this case, 𝑑_𝑝 is the Euclidian metric 𝑑_𝑝(𝑥,𝑦)=|𝑥−𝑦|.(2)Each partial metric 𝑝 on 𝑋 generates a 𝑇₀ topology 𝜏_𝑝 on 𝑋 which has a base of open p-balls 𝐵_𝑝(𝑥,𝜀), where 𝑥∈𝑋 and 𝜀>0 (𝐵_𝑝(𝑥,𝜀)={𝑦∈𝑋∶𝑝(𝑥,𝑦)<𝑝(𝑥,𝑥)+𝜀}).

The following concepts has been defined as follows on a partial metric space.

Definition 1.3 (see e.g., [1, 2]). (i) A sequence {𝑥_𝑛} in a PMS (𝑋,𝑝) converges to 𝑥∈𝑋 if and only if 𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝(𝑥,𝑥_𝑛).
(ii) A sequence {𝑥_𝑛} in a PMS (𝑋,𝑝) is called Cauchy if and only if lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚) exists (and is finite).
(iii) A PMS {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).
The concept of coupled fixed point have been introduced in [10] by Bhaskar and Lakshmikantham as follows.

Definition 1.4 (see [10]). An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled fixed point of mapping 𝐹∶𝑋×𝑋→𝑋 if 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥).
Aydi in [11] has obtained some coupled fixed point results for mappings satisfying different contractive conditions on complete partial metric spaces. Some of these results are the following cases.

Theorem 1.5 (see [11, 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: 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝑘𝑝(𝑥,𝑢)+𝑙𝑝𝑦,𝑣, (1.2) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point.

Theorem 1.6 (see [11, 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: 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝑘𝑝𝐹𝑥,𝑦,𝑥+𝑙𝑝(𝐹(𝑢,𝑣),𝑢), (1.3) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point.

Theorem 1.7 (see [11, 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: 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝑘𝑝𝐹𝑥,𝑦,𝑢+𝑙𝑝(𝐹(𝑢,𝑣),𝑥), (1.4) for all 𝑥,𝑦,𝑢,𝑣∈𝑋, where 𝑘,𝑙 are nonnegative constants with 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point.

For a survey of fixed point theory, its applications, and related results in partial metric spaces we refer the reader to [4, 5, 12–20] and the references mentioned therein. Also, many researchers have obtained coupled fixed point results for mappings under various contractive conditions in the framework of partial metric spaces (see, e.g., [21, 22]).

In this paper we establish some coupled fixed point results of contractive mappings in the framework of complete partial metric spaces. Our results extend and generalize the results of Aydi [11].

2. Main Results

We recall three easy lemmas which have an essential role in the proof of the main result. These results can be derived easily (see, e.g., [1, 2, 6]).

Lemma 2.1. (1) A sequence {𝑥_𝑛} is a Cauchy sequence in the PMS (𝑋,𝑝) if and only if it is a Cauchy sequence in the metric space (𝑋,𝑑_𝑝).
(2) A PMS (𝑋,𝑝) is complete if and only if the metric space (𝑋,𝑑_𝑝) is complete. Moreover, lim_𝑛→∞𝑑_𝑝𝑥,𝑥_𝑛=0⟺𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝𝑥,𝑥_𝑛=lim_{𝑛,𝑚→∞}𝑝𝑥_𝑛,𝑥_𝑚. (2.1)

Lemma 2.2 (see [3]). Assume that 𝑥_𝑛→𝑧 as 𝑛→∞ in a PMS (𝑋,𝑝) such that 𝑝(𝑧,𝑧)=0. Then, lim_𝑛→∞𝑝(𝑥_𝑛,𝑦)=𝑝(𝑧,𝑦), for every 𝑦∈𝑋.

Lemma 2.3 (see, e.g., [3, 4]). Let (X, p) be a complete PMS. Then,(a)if 𝑝(𝑥,𝑦)=0 then, 𝑥=𝑦,(b)if 𝑥≠𝑦, then 𝑝(𝑥,𝑦)>0.

Throughout this paper, we assume that all of the constants are nonnegative. Our main result is the following. The method of the proof can be found in [11].

Theorem 2.4. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝛼₁𝑝(𝑥,𝑢)+𝛼₂𝑝𝑦,𝑣+𝛼₃𝑝𝐹𝑥,𝑦,𝑥+𝛼₄𝑝𝐹𝑦,𝑥,𝑦+𝛼₅𝑝𝐹𝑥,𝑦,𝑢+𝛼₆𝑝𝐹𝑦,𝑥,𝑣+𝛼₇𝑝(𝐹(𝑢,𝑣),𝑥)+𝛼₈𝑝𝐹(𝑣,𝑢),𝑦+𝛼₉𝑝(𝐹(𝑢,𝑣),𝑢)+𝛼₁₀𝑝(𝐹(𝑣,𝑢),𝑣), (2.2) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where ∑¹⁰_𝑖=1𝛼_𝑖<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Proof. Let 𝑥₀,𝑦₀∈𝑋 be arbitrary. Define 𝑥₁,𝑦₁∈𝑋 such that 𝑥₁=𝐹(𝑥₀,𝑦₀) and 𝑦₁=𝐹(𝑦₀,𝑥₀) and in this way, we construct the sequences {𝑥_𝑛} and {𝑦_𝑛} as 𝑥_𝑛=𝐹(𝑥_𝑛−1,𝑦_𝑛−1) and 𝑦_𝑛=𝐹(𝑦_𝑛−1,𝑥_𝑛−1), for all 𝑛≥0.
We will complete the proof in three steps.
Step I. Let 𝛿_𝑛=𝑝(𝑥_𝑛−1,𝑥_𝑛)+𝑝(𝑦_𝑛−1,𝑦_𝑛). We will show that lim_𝑛→∞𝛿_𝑛=0.
Using (2.2) we obtain that 𝑝𝑥_𝑛,𝑥_𝑛+1=𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛≤𝛼₁𝑝𝑥_𝑛−1,𝑥_𝑛+𝛼₂𝑝𝑦_𝑛−1,𝑦_𝑛+𝛼₃𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝑥_𝑛−1+𝛼₄𝑝𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝑦_𝑛−1+𝛼₅𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝑥_𝑛+𝛼₆𝑝𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝑦_𝑛+𝛼₇𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛−1+𝛼₈𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛−1+𝛼₉𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛+𝛼₁₀𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛=𝛼₁𝑝𝑥_𝑛−1,𝑥_𝑛+𝛼₂𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₃𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₄𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₅𝑝𝑥_𝑛,𝑥_𝑛+𝛼₆𝑝𝑦_𝑛,𝑦_𝑛+𝛼₇𝑝𝑥_𝑛+1,𝑥_𝑛−1+𝛼₈𝑝𝑦_𝑛+1,𝑦_𝑛−1+𝛼₉𝑝𝑥_𝑛+1,𝑥_𝑛+𝛼₁₀𝑝𝑦_𝑛+1,𝑦_𝑛≤𝛼₁𝑝𝑥_𝑛−1,𝑥_𝑛+𝛼₂𝑝𝑦_𝑛−1,𝑦_𝑛+𝛼₃𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₄𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₅−𝛼₇𝑝𝑥_𝑛,𝑥_𝑛+𝛼₆−𝛼₈𝑝𝑦_𝑛,𝑦_𝑛+𝛼₇𝑝𝑥_𝑛+1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₈𝑝𝑦_𝑛+1,𝑦_𝑛+𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₉𝑝𝑥_𝑛+1,𝑥_𝑛+𝛼₁₀𝑝𝑦_𝑛+1,𝑦_𝑛. (2.3)
Analogously, starting from 𝑝(𝑥_𝑛+1,𝑥_𝑛)=𝑝(𝐹(𝑥_𝑛,𝑦_𝑛),𝐹(𝑥_𝑛−1,𝑦_𝑛−1)), we have 𝑝𝑥_𝑛+1,𝑥_𝑛=𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥_𝑛−1,𝑦_𝑛−1≤𝛼₁𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₂𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₃𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛+𝛼₄𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛+𝛼₅𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛−1+𝛼₆𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛−1+𝛼₇𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝑥_𝑛+𝛼₈𝑝𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝑦_𝑛+𝛼₉𝑝𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝑥_𝑛−1+𝛼₁₀𝑝𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝑦_𝑛−1=𝛼₁𝑝𝑥_𝑛−1,𝑥_𝑛+𝛼₂𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₃𝑝𝑥_𝑛+1,𝑥_𝑛+𝛼₄𝑝𝑦_𝑛+1,𝑦_𝑛+𝛼₅𝑝𝑥_𝑛+1,𝑥_𝑛−1+𝛼₆𝑝𝑦_𝑛+1,𝑦_𝑛−1+𝛼₇𝑝𝑥_𝑛,𝑥_𝑛+𝛼₈𝑝𝑦_𝑛,𝑦_𝑛+𝛼₉𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₁₀𝑝𝑦_𝑛,𝑦_𝑛−1≤𝛼₁𝑝𝑥_𝑛−1,𝑥_𝑛+𝛼₂𝑝𝑦_𝑛−1,𝑦_𝑛+𝛼₃𝑝𝑥_𝑛+1,𝑥_𝑛+𝛼₄𝑝𝑦_𝑛+1,𝑦_𝑛+𝛼₅𝑝𝑥_𝑛+1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₆𝑝𝑦_𝑛+1,𝑦_𝑛+𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₇−𝛼₅𝑝𝑥_𝑛,𝑥_𝑛+𝛼₈−𝛼₆𝑝𝑦_𝑛,𝑦_𝑛+𝛼₉𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₁₀𝑝𝑦_𝑛,𝑦_𝑛−1. (2.4)
In a similar way, we have 𝑝𝑦_𝑛,𝑦_𝑛+1=𝑝𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛≤𝛼₁𝑝𝑦_𝑛−1,𝑦_𝑛+𝛼₂𝑝𝑥_𝑛−1,𝑥_𝑛+𝛼₃𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₄𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₅−𝛼₇𝑝𝑦_𝑛,𝑦_𝑛+𝛼₆−𝛼₈𝑝𝑥_𝑛,𝑥_𝑛+𝛼₇𝑝𝑦_𝑛+1,𝑦_𝑛+𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₈𝑝𝑥_𝑛+1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₉𝑝𝑦_𝑛+1,𝑦_𝑛+𝛼₁₀𝑝𝑥_𝑛+1,𝑥_𝑛. (2.5)
Analogously, starting from 𝑝(𝑦_𝑛+1,𝑦_𝑛)=𝑝(𝐹(𝑦_𝑛,𝑥_𝑛),𝐹(𝑦_𝑛−1,𝑥_𝑛−1)), we have 𝑝𝑦_𝑛+1,𝑦_𝑛=𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦_𝑛−1,𝑥_𝑛−1≤𝛼₁𝑝𝑦_𝑛−1,𝑦_𝑛+𝛼₂𝑝𝑥_𝑛−1,𝑥_𝑛+𝛼₃𝑝𝑦_𝑛+1,𝑦_𝑛+𝛼₄𝑝𝑥_𝑛+1,𝑥_𝑛+𝛼₅𝑝𝑦_𝑛+1,𝑦_𝑛+𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₆𝑝𝑥_𝑛+1,𝑥_𝑛+𝑝𝑥_𝑛,𝑥_𝑛−1+𝛼₇−𝛼₅𝑝𝑦_𝑛,𝑦_𝑛+𝛼₈−𝛼₆𝑝𝑥_𝑛,𝑥_𝑛+𝛼₉𝑝𝑦_𝑛,𝑦_𝑛−1+𝛼₁₀𝑝𝑥_𝑛,𝑥_𝑛−1. (2.6)
Adding (2.3), (2.4), (2.5), and (2.6) we obtain that 2𝛿_𝑛+1≤2𝛼₁+2𝛼₂+𝛼₃+𝛼₄+𝛼₅+𝛼₆+𝛼₇+𝛼₈+𝛼₉+𝛼₁₀𝛿_𝑛+𝛼₃+𝛼₄+𝛼₅+𝛼₆+𝛼₇+𝛼₈+𝛼₉+𝛼₁₀𝛿_𝑛+1, (2.7) or, equivalently, 𝛿_𝑛+1≤𝜆𝛿_𝑛, (2.8) where, 𝜆=[2𝛼₁+2𝛼₂+𝛼₃+𝛼₄+𝛼₅+𝛼₆+𝛼₇+𝛼₈+𝛼₉+𝛼₁₀]/(2−[𝛼₃+𝛼₄+𝛼₅+𝛼₆+𝛼₇+𝛼₈+𝛼₉+𝛼₁₀]).
Repeating the above mentioned process, we have 𝛿_𝑛+1≤𝜆𝛿_𝑛≤𝜆²𝛿_𝑛−1≤⋯≤𝜆^𝑛+1𝛿₀, (2.9) where, from our assumption about coefficients 𝛼_𝑖, 𝜆∈[0,1); hence, lim_𝑛→∞𝛿_𝑛=0. (2.10)
Step II. {𝑥_𝑛} and {𝑦_𝑛} are Cauchy.
If 𝛿₀=0 then, 𝑝(𝑥₀,𝑥₁)+𝑝(𝑦₀,𝑦₁)=0. Hence, we get 𝑥₀=𝑥₁=𝐹(𝑥₀,𝑦₀) and 𝑦₀=𝑦₁=𝐹(𝑦₀,𝑥₀); that is, (𝑥₀,𝑦₀) is a coupled fixed point of 𝐹. Now, let 𝛿₀>0. For each 𝑚≥𝑛, we have 𝑝𝑥_𝑚,𝑥_𝑛+𝑝𝑦_𝑚,𝑦_𝑛≤𝑝𝑥_𝑚,𝑥_𝑚−1+𝑝𝑦_𝑚,𝑦_𝑚−1+𝑝𝑥_𝑚−1,𝑥_𝑚−2+𝑝𝑦_𝑚−1,𝑦_𝑚−2𝑥+⋯+𝑝_𝑛+1,𝑥_𝑛+𝑝𝑦_𝑛+1,𝑦_𝑛=𝛿_𝑚+𝛿_𝑚−1+⋯+𝛿_𝑛+1≤𝜆^𝑚+𝜆^𝑚−1+⋯+𝜆^𝑛+1𝛿₀≤𝜆^𝑛+1𝛿1−𝜆⁰. (2.11)
So, we have lim_{𝑛,𝑚→∞}𝑝(𝑥_𝑛,𝑥_𝑚)+𝑝(𝑦_𝑛,𝑦_𝑚)=0. This proves that {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in (𝑋,𝑝) and hence {𝑥_𝑛} and {𝑦_𝑛} are Cauchy sequences in the metric space (𝑋,𝑑_𝑝). From Lemma 2.1, (𝑋,𝑑_𝑝) is complete, so {𝑥_𝑛} and {𝑦_𝑛} converge to some 𝑥,𝑦∈𝑋, respectively; that is, lim_𝑛→∞𝑑_𝑝(𝑥_𝑛,𝑥)=0 and lim_𝑛→∞𝑑_𝑝(𝑦_𝑛,𝑦)=0. Therefore, from Lemma 2.1 and (2.10), we have 𝑝(𝑥,𝑥)=lim_𝑛→∞𝑝𝑥_𝑛,𝑥=lim_{𝑛,𝑚→∞}𝑝𝑥_𝑛,𝑥_𝑚𝑝=0,𝑦,𝑦=lim_𝑛→∞𝑝𝑦_𝑛,𝑦=lim_{𝑛,𝑚→∞}𝑝𝑦_𝑛,𝑦_𝑚=0. (2.12)(2.13)
Step III. We will show that 𝐹 has a unique coupled fixed point.
From the above step, lim_𝑛→∞𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥=lim_𝑛→∞𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝑦=0. (2.14)
Next, we will prove that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥).
We have 𝑝𝑥,𝐹𝑥,𝑦≤𝑝𝑥,𝑥_𝑛+1+𝑝𝑥_𝑛+1,𝐹𝑥,𝑦−𝑝𝑥_𝑛+1,𝑥_𝑛+1. (2.15)
Taking the limit as 𝑛→∞ in the above inequality, as 𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑦_𝑛) and using triangle inequality and (2.12), we have 𝑝𝑥,𝐹𝑥,𝑦≤lim_𝑛→∞𝑝𝑥,𝑥_𝑛+1+lim_𝑛→∞𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥,𝑦=lim_𝑛→∞𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥,𝑦. (2.16)
But, for all 𝑛≥0, from (2.2), 𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥,𝑦≤𝛼₁𝑝𝑥_𝑛,𝑥+𝛼₂𝑝𝑦_𝑛,𝑦+𝛼₃𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥_𝑛+𝛼₄𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝑦_𝑛+𝛼₅𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝑥+𝛼₆𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝑦+𝛼₇𝑝𝐹𝑥,𝑦,𝑥_𝑛+𝛼₈𝑝𝐹𝑦,𝑥,𝑦_𝑛+𝛼₉𝑝𝐹𝑥,𝑦,𝑥+𝛼₁₀𝑝𝐹𝑦,𝑥,𝑦=𝛼₁𝑝𝑥_𝑛,𝑥+𝛼₂𝑝𝑦_𝑛,𝑦+𝛼₃𝑝𝑥_𝑛+1,𝑥+𝛼₄𝑝𝑦_𝑛+1,𝑦+𝛼₅𝑝𝑥_𝑛+1,𝑥+𝛼₆𝑝𝑦_𝑛+1,𝑦+𝛼₇𝑝𝐹𝑥,𝑦,𝑥_𝑛+𝛼₈𝑝𝐹𝑦,𝑥,𝑦_𝑛+𝛼₉𝑝𝐹𝑥,𝑦,𝑥+𝛼₁₀𝑝𝐹𝑦,𝑥,𝑦. (2.17)
In the above inequality, if 𝑛→∞, using (2.12) and Lemma 2.2 we have lim_𝑛→∞𝑝𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑥,𝑦≤𝛼₇+𝛼₉𝑝𝐹𝑥,𝑦,𝑥+𝛼₈+𝛼₁₀𝑝𝐹𝑦,𝑥,𝑦. (2.18)
Analogously, 𝑝𝑦,𝐹𝑦,𝑥≤𝑝𝑦,𝑦_𝑛+1+𝑝𝑦_𝑛+1,𝐹𝑦,𝑥−𝑝𝑦_𝑛+1,𝑦_𝑛+1. (2.19)
Taking the limit as 𝑛→∞ in the above inequality, since 𝑦_𝑛+1=𝐹(𝑦_𝑛,𝑥_𝑛) and using triangle inequality and (2.13), we have 𝑝𝑦,𝐹𝑦,𝑥≤lim_𝑛→∞𝑝𝑦,𝑦_𝑛+1+lim_𝑛→∞𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦,𝑥=lim_𝑛→∞𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦,𝑥. (2.20)
Similar to (2.17), we have lim_𝑛→∞𝑝𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦,𝑥≤𝛼₇+𝛼₉𝑝𝐹𝑦,𝑥,𝑦+𝛼₈+𝛼₁₀𝑝𝐹𝑥,𝑦,𝑥. (2.21)
Adding (2.18) and (2.21) and using (2.15) and (2.19), we obtain that 𝑝≤𝛼𝑥,𝐹𝑥,𝑦+𝑝𝑦,𝐹𝑦,𝑥₇+𝛼₈+𝛼₉+𝛼₁₀𝑝𝑥,𝐹𝑥,𝑦+𝑝𝑦,𝐹𝑦,𝑥. (2.22)
Therefore, 𝑝(𝑥,𝐹(𝑥,𝑦))+𝑝(𝑦,𝐹(𝑦,𝑥))=0; that is, 𝐹(𝑥,𝑦)=𝑥 and 𝐹(𝑦,𝑥)=𝑦.

Remark 2.5. (1) If in the above theorem, we assume that 𝛼_𝑖=0, for all 3≤𝑖≤10, then we obtain the result of Aydi in [11] which is noted here in Theorem 1.5.
(2) If in the above theorem, 𝛼_𝑖=0, for all 1≤𝑖≤10, unless 𝑖=3,9, then we obtain the result of Aydi in [11] which is mentioned here in Theorem 1.6.
(3) If in the above theorem, we assume that 𝛼_𝑖=0, for all 3≤𝑖≤10, except that 𝑖≠5,7, then we obtain the result of Aydi in [11] (Theorem 1.7).
Many results can be deduced from the above theorem as follows.

Corollary 2.6. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝛼₁𝑝𝐹𝑥,𝑦,𝑥+𝛼₂𝑝𝐹𝑦,𝑥,𝑦+𝛼₃𝑝(𝐹(𝑢,𝑣),𝑢)+𝛼₄𝑝(𝐹(𝑣,𝑢),𝑣), (2.23) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where ∑⁴_𝑖=1𝛼_𝑖<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Corollary 2.7. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝛼₁𝑝𝐹𝑥,𝑦,𝑢+𝛼₂𝑝𝐹𝑦,𝑥,𝑣+𝛼₃𝑝(𝐹(𝑢,𝑣),𝑥)+𝛼₄𝑝𝐹(𝑣,𝑢),𝑦, (2.24) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where ∑⁴_𝑖=1𝛼_𝑖<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Corollary 2.8. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝛼₁𝑝𝐹𝑥,𝑦,𝑥+𝛼₂𝑝𝐹𝑦,𝑥,𝑦+𝛼₃𝑝(𝐹(𝑢,𝑣),𝑥)+𝛼₄𝑝𝐹(𝑣,𝑢),𝑦, (2.25) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where 𝛼₁+𝛼₂+𝛼₃+𝛼₄<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Corollary 2.9. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹𝑥,𝑦,𝐹(𝑢,𝑣)≤𝛼₁𝑝𝐹𝑥,𝑦,𝑢+𝛼₂𝑝𝐹𝑦,𝑥,𝑣+𝛼₃𝑝(𝐹(𝑢,𝑣),𝑢)+𝛼₄𝑝(𝐹(𝑣,𝑢),𝑣), (2.26) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where ∑⁴_𝑖=1𝛼_𝑖<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Also, we have the following results, when the constants in the above corollaries are equal.

Corollary 2.10. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝()+𝑙𝑥,𝑢+𝑝𝑦,𝑣2𝑝𝐹𝐹+𝑟𝑥,𝑦,𝑥+𝑝𝑦,𝑥,𝑦2𝑝𝐹𝐹+𝑠𝑥,𝑦,𝑢+𝑝𝑦,𝑥,𝑣2𝑝(𝐹())𝐹()+𝑡𝑢,𝑣,𝑥+𝑝𝑣,𝑢,𝑦2𝑝(𝐹(𝑢,𝑣),𝑢)+𝑝(𝐹(𝑣,𝑢),𝑣), (2.27) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where 𝑘+𝑙+𝑟+𝑠+𝑡<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Corollary 2.11. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝𝐹𝐹+𝑙𝑥,𝑦,𝑥+𝑝𝑦,𝑥,𝑦2𝑝(𝐹(𝑢,𝑣),𝑢)+𝑝(𝐹(𝑣,𝑢),𝑣), (2.28) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Corollary 2.12. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝𝐹𝐹+𝑙𝑥,𝑦,𝑢+𝑝𝑦,𝑥,𝑣2𝑝(𝐹(𝑢,𝑣),𝑥)+𝑝𝐹(𝑣,𝑢),𝑦, (2.29) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Corollary 2.13. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝𝐹𝐹+𝑙𝑥,𝑦,𝑥+𝑝𝑦,𝑥,𝑦2𝑝(𝐹(𝑢,𝑣),𝑥)+𝑝𝐹(𝑣,𝑢),𝑦, (2.30) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where 𝑘+𝑙<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Corollary 2.14. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝()+𝑙𝑥,𝑢+𝑝𝑦,𝑣4𝑝𝐹𝐹(𝐹())(𝐹()+𝑟𝑥,𝑦,𝑥+𝑝𝑦,𝑥,𝑦+𝑝𝑢,𝑣,𝑢+𝑝𝑣,𝑢,𝑣)4𝑝𝐹𝑥,𝑦,𝑢+𝑝𝐹𝑦,𝑥,𝑣+𝑝(𝐹(𝑢,𝑣),𝑥)+𝑝𝐹(𝑣,𝑢),𝑦, (2.31) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where 𝑘+𝑙+𝑟<1. Then, 𝐹 has a unique coupled fixed point in 𝑋.

Corollary 2.15. Let (𝑋,𝑝) be a complete partial metric space and 𝐹∶𝑋×𝑋→𝑋 be a mapping such that 𝑝𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑝𝐹𝐹+𝑙𝑥,𝑦,𝑢+𝑝𝑦,𝑥,𝑣2𝑝(𝐹(𝑢,𝑣),𝑢)+𝑝(𝐹(𝑣,𝑢),𝑣), (2.32) for every pairs (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, where 𝑘+𝑙<1. Then 𝐹 has a unique coupled fixed point in 𝑋.

Example 2.16. Let 𝑋=[0,∞) and 𝑝 on 𝑋 be given as 𝑝(𝑎,𝑏)=max{𝑎,𝑏}. Obviously, the partial metric space (𝑋,𝑝) is complete (see, e.g., Example  2.3 of [11]).
Define 𝐹∶𝑋×𝑋→𝑋 as 𝐹(𝑥,𝑦)=(𝑥+𝑦)/30, for all 𝑥,𝑦∈𝑋.
Now, we have 𝑝𝐹(=1𝑥,𝑦,𝐹𝑢,𝑣)≤130max𝑥+𝑦,𝑢+𝑣{}≤127max𝑥,𝑢+max𝑦,𝑣{}+127max𝑥,𝑢+max𝑦,𝑣27max𝑥+𝑦30,𝑥+max𝑦+𝑥+130,𝑦27max𝑥+𝑦30,𝑢+max𝑦+𝑥+130,𝑣27max𝑢+𝑣30,𝑥+max𝑣+𝑢+130,𝑦27max𝑢+𝑣30,𝑢+max𝑣+𝑢30,𝑣=𝛼₁𝑝(𝑥,𝑢)+𝛼₂𝑝𝑦,𝑣+𝛼₃𝑝𝐹𝑥,𝑦,𝑥+𝛼₄𝑝𝐹𝑦,𝑥,𝑦+𝛼₅𝑝𝐹𝑥,𝑦,𝑢+𝛼₆𝑝𝐹𝑦,𝑥,𝑣+𝛼₇𝑝(𝐹(𝑢,𝑣),𝑥)+𝛼₈𝑝𝐹(𝑣,𝑢),𝑦+𝛼₉𝑝(𝐹(𝑢,𝑣),𝑢)+𝛼₁₀𝑝(𝐹(𝑣,𝑢),𝑣). (2.33)
Thus, (2.2) is satisfied with 𝛼_𝑖=1/27. Obviously, all the conditions of Theorem 2.4 are satisfied. Moreover, (0,0) is the unique coupled fixed point of 𝐹.