Coupled Fixed-Point Theorems for Contractions in Partially Ordered Metric Spaces and Applications

Abstract

Bhaskar and Lakshmikantham (2006) showed the existence of coupled coincidence points of a mapping 𝐹 from 𝑋×𝑋 into 𝑋 and a mapping 𝑔 from 𝑋 into 𝑋 with some applications. The aim of this paper is to extend the results of Bhaskar and Lakshmikantham and improve the recent fixed-point theorems due to Bessem Samet (2010). Indeed, we introduce the definition of generalized 𝑔-Meir-Keeler type contractions and prove some coupled fixed point theorems under a generalized 𝑔-Meir-Keeler-contractive condition. Also, some applications of the main results in this paper are given.

1. Introduction

The Banach contraction principle [1] is a classical and powerful tool in nonlinear analysis and has been generalized by many authors (see [2–15] and others).

Recently, Bhaskar and Lakshmikantham [16] introduced the notion of a coupled fixed-point of the given two variables mapping. More precisely, let 𝑋 be a nonempty set and 𝐹∶𝑋×𝑋→𝑋 be a given mapping. An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled fixed-point of the mapping 𝐹 if𝐹𝑥,𝑦=𝑥,𝐹𝑦,𝑥=𝑦. (1.1)

They also showed the uniqueness of a coupled fixed-point of the mapping 𝐹 and applied their theorems to the problems of the existence and uniqueness of a solution for a periodic boundary value problem.

Theorem 1.1 (see Zeidler [15]). 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) such that 𝑑𝐹()≤𝑘𝑥,𝑦,𝐹𝑢,𝑣2𝑑(𝑥,𝑢)+𝑑𝑦,𝑣 (1.2) for all 𝑥≥𝑢 and 𝑦≤𝑣. Moreover, if there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀≤𝐹𝑥₀,𝑦₀,𝑦₀≥𝐹𝑦₀,𝑥₀, (1.3) then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥).

Later, in [17], Lakshmikantham and Ćirić investigated some more coupled fixed-point theorems in partially ordered sets, and some others obtained many results on coupled fixed-point theorems in cone metric spaces, intuitionistic fuzzy normed spaces, ordered cone metric spaces and topological spaces (see, e.g., [18–25]).

In [9], Meir and Keeler generalized the well-known Banach fixed-point theorem [1] as follows.

Theorem 1.2 (Meir and Keeler [9]). Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝑋 be a given mapping. Suppose that, for any 𝜖>0, there exists 𝛿(𝜖)>0 such that 𝜖≤𝑑𝑥,𝑦<𝜖+𝛿(𝜖)⟹𝑑𝑇(𝑥),𝑇𝑦<𝜖 (1.4) for all 𝑥,𝑦∈𝑋. Then T admits a unique fixed-point 𝑥₀∈𝑋 and, for all 𝑥∈𝑋, the sequence {𝑇^𝑛(𝑥)} converges to 𝑥₀.

Proposition 1.3 (see [17]). Let (𝑋,𝑑) be a partially ordered metric space and 𝐹∶𝑋×𝑋→𝑋 be a given mapping. If the contraction (1.2) is satisfied, then 𝐹 is a generalized Meir-Keeler type contraction.

Motivated by the results of Bhaskar and Lakshmikantham [16], Lakshmikantham and Ćirić [17], and Samet [26], in this paper, we introduce the definition of 𝑔-Meir-Keeler-contractive mappings and prove some coupled fixed-point theorems under a generalized 𝑔-Meir-Keeler contractive condition.

2. Main Results

Let 𝑋 be a nonempty set. We note that an element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled coincidence point of a mapping 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝐹(𝑥,𝑦)=𝑔(𝑥) and 𝐹(𝑦,𝑥)=𝑔(𝑦) for all 𝑥,𝑦∈𝑋. Also, we say that 𝐹 and 𝑔 are commutative (or commuting) if 𝑔(𝐹(𝑥,𝑦))=𝐹(𝑔(𝑥),𝑔(𝑦)) for all 𝑥,𝑦∈𝑋.

We introduce the following two definitions.

Definition 2.1. Let (𝑋,≤) be a partially ordered set and 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. We say that 𝐹 has the mixed strict 𝑔-monotone property if, for any 𝑥,𝑦∈𝑋, 𝑥₁,𝑥₂∈𝑋,𝑔𝑥₁<𝑔𝑥₂⟹𝐹𝑥₁,𝑦<𝐹𝑥₂,𝑦,𝑦₁,𝑦₂∈𝑋,𝑔𝑦₁<𝑔𝑦₂⟹𝐹𝑥,𝑦₁>𝐹𝑥,𝑦₂. (2.1)

Definition 2.2. Let (𝑋,≤) be a partially ordered set and 𝑑 be a metric on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two given mappings. We say that 𝐹 is a generalized 𝑔-Meir-Keeler type contraction if, for all 𝜖>0, there exists 𝛿(𝜖)>0 such that, for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑔(𝑥)≤𝑔(𝑢) and 𝑔(𝑦)≥𝑔(𝑣), 1𝜖≤2𝑑𝑔(𝑥),𝑔(𝑢)+𝑑𝑔𝑦,𝑔(𝑣)<𝜖+𝛿(𝜖)⟹𝑑𝐹𝑥,𝑦,𝐹(𝑢,𝑣)<𝜖. (2.2)

Lemma 2.3. Let (𝑋,≤) be a partially ordered set and 𝑑 be a metric on 𝑋. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two given mappings. If 𝐹 is a generalized 𝑔-Meir-Keeler type contraction, then we have 𝑑𝐹(<1𝑥,𝑦,𝐹𝑢,𝑣)2𝑑𝑔(𝑥),𝑔(𝑢)+𝑑𝑔𝑦,𝑔(𝑣) (2.3) for all 𝑥,𝑦,𝑢,𝑣 with 𝑔(𝑥)<𝑔(𝑢), 𝑔(𝑦)≥𝑔(𝑣) or 𝑔(𝑥)≤𝑔(𝑢), 𝑔(𝑦)>𝑔(𝑣).

Proof. Let 𝑥,𝑦,𝑢,𝑣∈𝑋 such that 𝑔(𝑥)<𝑔(𝑢) and 𝑔(𝑦)≥𝑔(𝑣) or 𝑔(𝑥)≤𝑔(𝑢) and 𝑔(𝑦)>𝑔(𝑣). Then 𝑑(𝑔(𝑥),𝑔(𝑢))+𝑑(𝑔(𝑦),𝑔(𝑣))>0. Since 𝐹 is a generalized 𝑔-Meir-Keeler type contraction, for 𝜖=(1/2)[𝑑(𝑔(𝑥),𝑔(𝑢))+𝑑(𝑔(𝑦),𝑔(𝑣))], there exists 𝛿(𝜖)>0 such that, for all 𝑥₀,𝑦₀,𝑢₀,𝑣₀∈𝑋 with 𝑔(𝑥₀)≤𝑔(𝑢₀) and 𝑔(𝑦₀)≥𝑔(𝑣₀), 1𝜖≤2𝑑𝑔𝑥₀,𝑔𝑢₀+𝑑𝑔𝑦₀,𝑔𝑣₀<𝜖+𝛿(𝜖)⟹𝑑𝐹𝑥₀,𝑦₀,𝐹𝑢₀,𝑣₀<𝜖. (2.4) Therefore, putting 𝑥₀=𝑥, 𝑦₀=𝑦, 𝑢₀=𝑢 and 𝑣₀=𝑣, we have 𝑑𝐹(<1𝑥,𝑦,𝐹𝑢,𝑣)2𝑑𝑔(𝑥),𝑔(𝑢)+𝑑𝑔𝑦,𝑔(𝑣). (2.5) This completes the proof.

From now on, we suppose that (𝑋,≤) is a partially ordered set, and there exists a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a complete metric space.

Theorem 2.4. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be such that 𝐹(𝑋×𝑋)⊆𝑔(𝑋), 𝑔 is continuous and commutative with 𝐹. Also, suppose that(a)𝐹 has the mixed strict 𝑔-monotone property;(b)𝐹 is a generalized 𝑔-Meir-keeler type contraction;(c)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔(𝑥₀)<𝐹(𝑥₀,𝑦₀) and 𝑔(𝑦₀)>𝐹(𝑦₀,𝑥₀).
Then there exist 𝑥,𝑦∈𝑋 such that 𝑔(𝑥)=𝐹(𝑥,𝑦) and 𝑔(𝑦)=𝐹(𝑦,𝑥); that is, 𝐹 and 𝑔 have a coupled coincidence in 𝑋×𝑋.

Proof. Let 𝑥₀,𝑦₀∈𝑋 be such that 𝑔(𝑥₀)<𝐹(𝑥₀,𝑦₀) and 𝑔(𝑦₀)>𝐹(𝑦₀,𝑥₀). Since 𝐹(𝑋×𝑋)⊆𝑔(𝑋), we can choose 𝑥₁,𝑦₁∈𝑋 such that 𝑔(𝑥₁)=𝐹(𝑥₀,𝑦₀) and 𝑔(𝑦₁)=𝐹(𝑦₀,𝑥₀). Again, from 𝐹(𝑋×𝑋)⊆𝑔(𝑋), we can choose 𝑥₂,𝑦₂∈𝑋 such that 𝑔(𝑥₂)=𝐹(𝑥₁,𝑦₁) and 𝑔(𝑦₂)=𝐹(𝑦₁,𝑥₁).
Continuing this process, we can construct the sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that 𝑔𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛 (2.6) for all 𝑛≥0.
Now, we show that 𝑔𝑥_𝑛<𝑔𝑥_𝑛+1,𝑔𝑦_𝑛>𝑔𝑦_𝑛+1 (2.7) for all 𝑛≥0. For 𝑛=0, we have 𝑔𝑥₀<𝐹𝑥₀,𝑦₀=𝑔𝑥₁,𝑔𝑦₀>𝐹𝑦₀,𝑥₀=𝑔𝑦₁. (2.8) Since 𝐹 has the mixed strict 𝑔-monotone property, then we have 𝑔𝑥₀<𝑔𝑥₁⟹𝐹𝑥₀,𝑦₁<𝐹𝑥₁,𝑦₁,𝑔𝑦₀>𝑔𝑦₁⟹𝐹𝑥₀,𝑦₀<𝐹𝑥₀,𝑦₁. (2.9) It follows that 𝐹(𝑥₀,𝑦₀)<𝐹(𝑥₁,𝑦₁), that is, 𝑔(𝑥₁)<𝑔(𝑥₂).
Similarly, we have 𝑔𝑦₁<𝑔𝑦₀⟹𝐹𝑦₁,𝑥₀<𝐹𝑦₀,𝑥₀,𝑔𝑥₁>𝑔𝑥₀⟹𝐹𝑦₁,𝑥₁<𝐹𝑦₁,𝑥₀. (2.10) Thus it follows that 𝐹(𝑦₁,𝑥₁)<𝐹(𝑦₀,𝑥₀), that is, 𝑔(𝑦₂)<𝑔(𝑦₁).
Again, we have 𝑔𝑥₁<𝑔𝑥₂⟹𝐹𝑥₁,𝑦₂<𝐹𝑥₂,𝑦₂,𝑔𝑦₁>𝑔𝑦₂⟹𝐹𝑥₁,𝑦₁<𝐹𝑥₁,𝑦₂. (2.11) Thus it follows that 𝐹(𝑥₁,𝑦₁)<𝐹(𝑥₂,𝑦₂), that is, 𝑔(𝑥₂)<𝑔(𝑥₃).
Similarly, we have 𝑔𝑦₂<𝑔𝑦₁⟹𝐹𝑦₂,𝑥₁<𝐹𝑦₁,𝑥₁,𝑔𝑥₂>𝑔𝑥₁⟹𝐹𝑦₂,𝑥₂<𝐹𝑦₂,𝑥₁. (2.12) Thus it follows that 𝐹(𝑦₂,𝑥₂)<𝐹(𝑦₁,𝑥₁), that is, 𝑔(𝑦₃)<𝑔(𝑦₂).
Continuing this process for each 𝑛≥1, we get the following: 𝑔𝑥₀<𝑔𝑥₁<𝑔𝑥₂<⋯<𝑔𝑥_𝑛<𝑔𝑥_𝑛+1𝑔𝑦<⋯,₀>𝑔𝑦₁>𝑔𝑦₂>⋯𝑔𝑦_𝑛>𝑔𝑦_𝑛+1>⋯. (2.13) Denote that 𝛿_𝑛∶=𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛+1. (2.14) Since 𝑔(𝑥_𝑛−1)<𝑔(𝑥_𝑛) and 𝑔(𝑦_𝑛−1)>𝑔(𝑦_𝑛), it follows from (2.6) and Lemma 2.3 that 𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1=𝑑𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛<12𝑑𝑔𝑥_𝑛−1,𝑔𝑥_𝑛+𝑑𝑔𝑦_𝑛−1,𝑔𝑦_𝑛. (2.15) Since 𝑔(𝑦_𝑛)<𝑔(𝑦_𝑛−1) and 𝑔(𝑥_𝑛)>𝑔(𝑥_𝑛−1), it follows from (2.6) and Lemma 2.3 that 𝑑𝑔𝑦_𝑛+1,𝑔𝑦_𝑛=𝑑𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑦_𝑛−1,𝑥_𝑛−1<12𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛−1+𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛−1. (2.16) Thus it follows from (2.14)–(2.16) that 𝛿_𝑛<𝛿_𝑛−1. This means that the sequence {𝛿_𝑛/2} is monotone decreasing. Therefore, there exists 𝛿^∗≥0 such that lim_𝑛→∞𝛿_𝑛/2=𝛿^∗, that is, lim_𝑛→∞12𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛+1=𝛿^∗. (2.17)
Now, we show that 𝛿^∗=0. Suppose that 𝛿^∗>0 hold. Let 𝛿^∗=𝜖. Then there exists a positive integer 𝑚 such that 1𝜖≤2𝑑𝑔𝑥_𝑚,𝑔𝑥_𝑚+1+𝑑𝑔𝑦_𝑚,𝑔𝑦_𝑚+1<𝜖+𝛿(𝜖). (2.18) Then, by using (2.7) and the condition (b), we have 𝑑𝐹𝑥_𝑚,𝑦_𝑚,𝐹𝑥_𝑚+1,𝑦_𝑚+1<𝜖, (2.19) and so, by (2.6), we have 𝑑𝑔𝑥_𝑚+1,𝑔𝑥_𝑚+2<𝜖. (2.20)
On the other hand, by (2.15), we have 12𝑑𝑔𝑥_𝑚,𝑔𝑥_𝑚+1+𝑑𝑔𝑦_𝑚,𝑔𝑦_𝑚+1<𝜖, (2.21) which is a contradiction with (2.18). Thus we have 𝜖=𝛿^∗=0, that is, lim_𝑛→∞12𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛+1=0, (2.22) that is, lim_𝑛→∞𝛿_𝑛=0. (2.23)
Now, we prove that {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are Cauchy sequences in 𝑋. Suppose that at least one of {𝑔(𝑥_𝑛)} or {𝑔(𝑦_𝑛)} is not a Cauchy sequence. Then there exist 𝜖>0 and two subsequences {𝑙_𝑘}, {𝑚_𝑘} of integers such that 𝑚_𝑘>𝑙_𝑘≥𝑘 and 𝑑𝑔𝑥_𝑙𝑘,𝑔𝑥_𝑚𝑘≥𝜖2,𝑑𝑔𝑦_𝑙𝑘,𝑔𝑦_𝑚𝑘≥𝜖2 (2.24) for all 𝑘≥1. Thus we have 𝑟_𝑘=𝑑𝑔𝑥_𝑙𝑘,𝑔𝑥_𝑚𝑘+𝑑𝑔𝑦_𝑙𝑘,𝑔𝑦_𝑚𝑘≥𝜖 (2.25) for all 𝑘≥1. Let 𝑚_𝑘 be the smallest number exceeding 𝑙_𝑘 such that (2.25) holds. Then we have 𝑑𝑔𝑥_𝑙𝑘,𝑔𝑥_𝑚𝑘−1+𝑑𝑔𝑦_𝑙𝑘,𝑔𝑦_𝑚𝑘−1<𝜖. (2.26) Thus, from (2.14), (2.25), (2.26) and the triangle inequality, it follows that 𝜖≤𝑟_𝑘≤𝑑𝑔𝑥_l𝑘,𝑔𝑥_𝑚𝑘−1+𝑑𝑔𝑥_𝑚𝑘−1,𝑔𝑥_𝑚𝑘+𝑑𝑔𝑦_𝑙𝑘,𝑔𝑦_𝑚𝑘−1+𝑑𝑔𝑦_𝑚𝑘−1,𝑔𝑦_𝑚𝑘<𝜖+𝛿_𝑚𝑘−1 (2.27) and so 𝜖≤lim_𝑘→∞𝑟_𝑘≤lim_𝑘→∞𝜖+𝛿_𝑚𝑘−1. (2.28) Hence, by (2.23), we have lim_𝑘→∞𝑟_𝑘=𝜖⁺. (2.29) It follows from (2.6), (2.14), and the triangle inequality that 𝑟_𝑘=𝑑𝑔𝑥_𝑙𝑘,𝑔𝑥_𝑚𝑘+𝑑𝑔𝑦_𝑙𝑘,𝑔𝑦_𝑚𝑘≤𝑑𝑔𝑥_𝑙𝑘,𝑔𝑥_𝑙𝑘+1+𝑑𝑔𝑥_𝑙𝑘+1,𝑔𝑥_𝑚𝑘+1+𝑑𝑔𝑥_𝑚𝑘+1,𝑔𝑥_𝑚𝑘+𝑑𝑔𝑦_𝑙𝑘,𝑔𝑦_𝑙𝑘+1+𝑑𝑔𝑦_𝑙𝑘+1,𝑔𝑦_𝑚𝑘+1+𝑑𝑔𝑦_𝑚𝑘+1,𝑔𝑦_𝑚𝑘=𝛿_𝑙𝑘+𝛿_𝑚𝑘+𝑑𝑔𝑥_𝑙𝑘+1,𝑔𝑥_𝑚𝑘+1+𝑑𝑔𝑦_𝑙𝑘+1,𝑔𝑦_𝑚𝑘+1=𝛿_𝑙𝑘+𝛿_𝑚𝑘+𝑑𝐹𝑥_𝑙𝑘,𝑦_𝑙𝑘,𝐹𝑥_𝑚𝑘,𝑦_𝑚𝑘+𝑑𝐹𝑦_𝑙𝑘,𝑥_𝑙𝑘,𝐹𝑦_𝑚𝑘,𝑥_𝑚𝑘. (2.30) Form (2.13) we have 𝑔(𝑥_𝑙𝑘)<𝑔(𝑥_𝑚𝑘) and 𝑔(𝑦_𝑙𝑘)>𝑔(𝑦_𝑚𝑘). Now, it follows from Lemma 2.3 and (2.30) that 𝑟_𝑘<𝛿_𝑙𝑘+𝛿_𝑚𝑘+𝑑𝑔𝑥_𝑙𝑘,𝑔𝑥_𝑚𝑘+𝑑𝑔𝑦_𝑙𝑘,𝑔𝑦_𝑚𝑘, (2.31) that is, 𝑟_𝑘<𝛿_𝑙𝑘+𝛿_𝑚𝑘+𝑟_𝑘. (2.32) This is a contradiction. Therefore, {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are Cauchy sequences. Since 𝑋 is complete, there exist 𝑥,𝑦∈𝑋 such that lim_𝑛→∞𝑔𝑥_𝑛=𝑥,lim_𝑛→∞𝑔𝑦_𝑛=𝑦. (2.33) Since {𝑔(𝑥_𝑛)} is monotone increasing and {𝑔(𝑦_𝑛)} is monotone decreasing, we have 𝑔𝑥_𝑛<𝑥,𝑔𝑦_𝑛>𝑦 (2.34) for all 𝑛≥1. Thus it follows from (2.33) and the continuity of 𝑔 that lim_𝑛→∞𝑔𝑔𝑥_𝑛=𝑔(𝑥),lim_𝑛→∞𝑔𝑔𝑦_𝑛=𝑔𝑦. (2.35) Thus, for all 𝑚≥1, there exists a positive integer 𝑛₀ such that, for all 𝑛≥𝑛₀, 𝑑𝑔𝑔𝑥_𝑛(𝑥<1,𝑔)𝑔𝑔𝑦4𝑚,𝑑_𝑛𝑦<1,𝑔.4𝑚 (2.36) Hence, from (2.6), the commutativity of 𝐹 and 𝑔 and the triangle inequality, we have 𝑑𝐹𝑥,𝑦,𝑔(𝑥)≤𝑑𝐹𝑥,𝑦,𝑔𝑔𝑥_𝑛+𝑑𝑔𝑔𝑥_𝑛(𝑥)𝐹𝐹𝑥,𝑔=𝑑𝑥,𝑦,𝑔_𝑛−1,𝑦_𝑛−1+𝑑𝑔𝑔𝑥_𝑛(𝑥)𝐹𝑔𝑥,𝑔=𝑑𝑥,𝑦,𝐹_𝑛−1,𝑔𝑦_𝑛−1+𝑑𝑔𝑔𝑥_𝑛,𝑔(𝑥). (2.37) Thus, it follows from (2.34), (2.36), and Lemma 2.3 that 𝑑𝐹(𝑥)<1𝑥,𝑦,𝑔2𝑑𝑔𝑔𝑥_𝑛−1,𝑔(𝑥)+𝑑𝑔𝑔𝑦_𝑛−1,𝑔𝑦+𝑑𝑔𝑔𝑥_𝑛(𝑥)<1,𝑔+18𝑚+18𝑚=14𝑚2𝑚⟶0 (2.38) as 𝑚→∞. Therefore, we have 𝐹(𝑥,𝑦)=𝑔(𝑥). Similarly, we can show that 𝐹(𝑦,𝑥)=𝑔(𝑦). This means that 𝐹 and 𝑔 have a coupled coincidence point in 𝑋×𝑋. This completes the proof.

Corollary 2.5. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping satisfying the following conditions:(a)𝐹 has the mixed strict monotone property;(b)𝐹 is a generalized Meir-Keeler type contraction;(c)there exists 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀<𝐹(𝑥₀,𝑦₀) and 𝑦₀>𝐹(𝑦₀,𝑥₀).
Then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥).

Proof. The conclusion follows from Theorem 2.4 by putting 𝑔=𝐼 (: the identity mapping) on 𝑋.

Now, we introduce the product space 𝑋×𝑋 with the following partial order: for all (𝑥,𝑦),(𝑢,𝑣)∈𝑋×𝑋, (𝑢,𝑣)≤𝑥,𝑦⟺𝑢<𝑥,𝑣≥𝑦. (2.39)

Theorem 2.6. Suppose that all the hypotheses of Theorem 2.4 hold and, further, for all (𝑥,𝑦),(𝑥^∗,𝑦^∗)∈𝑋×𝑋, there exists (𝑢,𝑣)∈𝑋×𝑋 such that (𝐹(𝑢,𝑣),𝐹(𝑣,𝑢)) is comparable with (𝐹(𝑥,𝑦),𝐹(𝑦,𝑥)) and (𝐹(𝑥^∗,𝑦^∗),𝐹(𝑦^∗,𝑥^∗)). Then 𝐹 and 𝑔 have a unique coupled common fixed-point, that is, there exists a unique (𝑥,𝑦)∈𝑋×𝑋 such that 𝑥=𝑔(𝑥)=𝐹𝑥,𝑦,𝑦=𝑔𝑦=𝐹𝑦,𝑥. (2.40)

Proof. By Theorem 2.4, the set of coupled coincidences of the mapping 𝐹 and 𝑔 is nonempty.
First, we show that, if (𝑥,𝑦) and (𝑥^∗,𝑦^∗) are coupled coincidence points of 𝐹 and 𝑔, that is, if 𝑔(𝑥)=𝐹𝑥,𝑦,𝑔𝑦=𝐹𝑦,𝑥,𝑔𝑥^∗=𝐹𝑥^∗,𝑦^∗,𝑔𝑦^∗=𝐹𝑦^∗,𝑥^∗, (2.41) then we have 𝑔(𝑥)=𝑔𝑥^∗,𝑔𝑦=𝑔𝑦^∗. (2.42) Put 𝑢₀=𝑢, 𝑣₀=𝑣 and choose 𝑢₁,𝑣₁∈𝑋 such that 𝑔(𝑢₁)=𝐹(𝑢₀,𝑣₀) and 𝑔(𝑣₁)=𝐹(𝑣₀,𝑢₀). Then, similarly as in the proof of Theorem 2.4, we can inductively define the sequences {𝑔(𝑢_𝑛)} and {𝑔(𝑣_𝑛)} such that 𝑔𝑢_𝑛+1=𝐹𝑢_𝑛,𝑣_𝑛,𝑔𝑣_𝑛+1=𝐹𝑣_𝑛,𝑢_𝑛 (2.43) for all 𝑛≥0. Also, if we set 𝑥₀=𝑥, 𝑦₀=𝑦, 𝑥^∗₀=𝑥^∗, and 𝑦^∗₀=𝑦^∗, then we can define the sequences {𝑔(𝑥_𝑛)}, {𝑔(𝑦_𝑛)}, {𝑔(𝑥^∗_𝑛)}, and {𝑔(𝑦^∗_𝑛)} as follows: 𝑔𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,𝑔𝑥^∗_𝑛+1=𝐹𝑥^∗_𝑛,𝑦^∗_𝑛,𝑔𝑦^∗_𝑛+1=𝐹𝑦^∗_𝑛,𝑥^∗_𝑛 (2.44) for all 𝑛≥0. Since 𝐹𝑥,𝑦,𝐹𝑦,𝑥=𝑔𝑥₁,𝑔𝑦₁=𝑔(𝑥)𝑦,(𝐹()(=𝑔𝑢,𝑔𝑢,𝑣,𝐹𝑣,𝑢))₁,𝑔𝑣₁ (2.45) are comparable each other, then 𝑔(𝑥)<𝑔(𝑢₁) and 𝑔(𝑦)≥𝑔(𝑣₁). It is easy to show that (𝑔(𝑥),𝑔(𝑦)), and (𝑔(𝑢_𝑛),𝑔(𝑣_𝑛)) are comparable each other, that is, 𝑔(𝑥)<𝑔(𝑢_𝑛) and 𝑔(𝑦)≥𝑔(𝑣_𝑛) for all 𝑛≥1. Thus it follows from Lemma 2.3 that 𝑑𝑔(𝑥),𝑔𝑢_𝑛+1+𝑑𝑔𝑦,𝑔𝑣_𝑛+1=𝑑𝐹𝑥,𝑦,𝐹𝑢_𝑛,𝑣_𝑛+𝑑𝐹𝑦,𝑥,𝐹𝑣_𝑛,𝑢_𝑛<12𝑑𝑔(𝑥),𝑔𝑢_𝑛+𝑑𝑔𝑦,𝑔𝑣_𝑛+12𝑑𝑔𝑦,𝑔𝑣_𝑛+𝑑𝑔(𝑥),𝑔𝑢_𝑛=𝑑𝑔(𝑥),𝑔𝑢_𝑛+𝑑𝑔𝑦,𝑔𝑣_𝑛 (2.46) and so 12𝑑𝑔(𝑥),𝑔𝑢_𝑛+1+𝑑𝑔𝑦,𝑔𝑣_𝑛+1<12^𝑛𝑑𝑔(𝑥),𝑔𝑢₁+𝑑𝑔𝑦,𝑔𝑣₁⟶0 (2.47) as 𝑛→∞. Therefore, we have lim_𝑛→∞𝑑𝑔(𝑥),𝑔𝑢_𝑛+1=0,lim_𝑛→∞𝑑𝑔𝑦,𝑔𝑣_𝑛+1=0. (2.48)
Similarly, we can prove that lim_𝑛→∞𝑑𝑔𝑥^∗,𝑔𝑢_𝑛+1=0,lim_𝑛→∞𝑑𝑔𝑦^∗,𝑔𝑣_𝑛+1=0. (2.49) Thus, by the triangle inequality, (2.48) and (2.49), we have 𝑑𝑔(𝑥),𝑔𝑥^∗≤𝑑𝑔(𝑥),𝑔𝑢_𝑛+1+𝑑𝑔𝑥^∗,𝑔𝑢_𝑛+1𝑑𝑔𝑦𝑦⟶0,,𝑔^∗≤𝑑𝑔𝑦,𝑔𝑣_𝑛+1+𝑑𝑔𝑦^∗,𝑔𝑣_𝑛+1⟶0 (2.50) as 𝑛→∞, which imply that 𝑔(𝑥)=𝑔(𝑥^∗) and 𝑔(𝑦)=𝑔(𝑦^∗).
Now, we prove that 𝑔(𝑥)=𝑥 and 𝑔(𝑦)=𝑦. Denote that 𝑔(𝑥)=𝑧 and 𝑔(𝑥)=𝑤. Since 𝑔(𝑥)=𝐹(𝑥,𝑦) and 𝑔(𝑦)=𝐹(𝑦,𝑥), by the commutativity of 𝐹 and 𝑔, we have 𝑔(𝑧)𝑔(𝑥)𝐹𝑔(𝑥)𝑦(),𝑔(𝑤)𝑔𝑦𝐹𝑔𝑦(𝑥)().=𝑔=𝑔𝑥,𝑦=𝐹,𝑔=𝐹𝑧,𝑤=𝑔=𝑔𝑦,𝑥=𝐹,𝑔=𝐹𝑤,𝑧 (2.51)(2.52) Thus, (𝑧,w) is a coupled coincidence point of 𝐹 and 𝑔.
Putting 𝑥^∗=𝑧 and 𝑦^∗=𝑤 in (2.52), it follows from (2.42) that 𝑧=𝑔(𝑥)=𝑔𝑥^∗=𝑔(𝑧),𝑤=𝑔𝑦=𝑔𝑦^∗=𝑔(𝑤) (2.53) and so, from (2.51) and (2.52), 𝑧=𝑔(𝑧)=𝐹(𝑧,𝑤),𝑤=𝑔(𝑤)=𝐹(𝑤,𝑧). (2.54) Therefore, (𝑧,𝑤) is a coupled common fixed-point of 𝐹 and 𝑔.
Finally, to prove the uniqueness of the coupled common fixed-point of 𝐹 and 𝑔, assume that (𝑝,𝑞) is another coupled common fixed-point of 𝐹 and 𝑔. Then, by (2.42), we have 𝑝=g(𝑝)=𝑔(𝑧)=𝑧 and 𝑞=𝑔(𝑞)=𝑔(𝑤)=𝑤. This completes the proof.

Corollary 2.7. Suppose that all the hypotheses of Corollary 2.5 hold and, further, for all (𝑥,𝑦) and (𝑥^∗,𝑦^∗)∈𝑋×𝑋, there exists (𝑢,𝑣)∈𝑋×𝑋 that is comparable with (𝑥,𝑦) and (𝑥^∗,𝑦^∗). Then there exists a unique 𝑥∈𝑋 such that 𝑥=𝐹(𝑥,𝑥).

3. Applications

Now, we give some applications of the main results in Section 2.

Theorem 3.1. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two given mappings. Assume that there exists a function 𝜑∶[0,+∞)→[0,+∞) satisfying the following conditions:(a)𝜑(0)=0 and 𝜑(𝑡)>0 for any 𝑡>0;(b)𝜑 is nondecreasing and right continuous;(c)for any 𝜖>0, there exists 𝛿(𝜖)>0 such that, for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑔(𝑥)≤𝑔(𝑢) and 𝑔(𝑦)≥𝑔(𝑣),1𝜖≤𝜑2𝑑𝑔(𝑥),𝑔(𝑢)+𝑑𝑔𝑦,𝑔(𝑣)<𝜖+𝛿(𝜖)⟹𝜑𝑑𝐹𝑥,𝑦,𝐹(𝑢,𝑣)<𝜖. (3.1) Then 𝐹 is a generalized 𝑔-Meir-Keeler type contraction.

Proof. For any 𝜖>0, it follows from (a) that 𝜑(𝜖)>0 and so there exists 𝛼>0 such that, for all 𝑢,𝑣,𝑢^∗,𝑣^⋆∈𝑋 with 𝑔(𝑢)≤𝑔(𝑢^∗) and 𝑔(𝑣)≥𝑔(𝑣^∗), 𝜑(𝜖)1≤𝜑2𝑑𝑔(𝑢),𝑔𝑢^∗+𝑑𝑔(𝑣),𝑔𝑣^∗(𝜖)𝑑𝐹()𝑢<𝜑+𝛼⟹𝜑𝑢,𝑣,𝐹^∗,𝑣^∗<𝜑(𝜖). (3.2) From the right continuity of 𝜑, there exists 𝛿>0 such that 𝜑(𝜖+𝛿)<𝜑(𝜖)+𝛼. For any 𝑥,𝑦,𝑢,𝑣∈𝑋 such that 𝑔(𝑥)≤𝑔(𝑢), 𝑔(𝑦)≥𝑔(𝑣) and 1𝜖≤2𝑑𝑔(𝑥),𝑔(𝑢)+𝑑𝑔𝑦,𝑔(𝑣)<𝜖+𝛿, (3.3) since 𝜑 is nondecreasing function, we get the following: 𝜑(𝜖)1≤𝜑2𝑑𝑔(𝑥),𝑔(𝑢)+𝑑𝑔𝑦,𝑔(𝑣)<𝜑(𝜖+𝛼)<𝜑(𝜖)+𝛼. (3.4) By (3.2), we have 𝜑[𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))]<𝜑(𝜖) and so 𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))<𝜖. Therefore, it follows that 𝐹 is a generalized 𝑔-Meir-Keeler type contraction. This completes the proof.

Corollary 3.2 (see [26, Theorem  3.1]). Let 𝐹∶𝑋×𝑋→𝑋 be a given mapping. Assume that there exists a function 𝜑∶[0,+∞)→[0,+∞) satisfying the following conditions:(a)𝜑(0)=0 and 𝜑(𝑡)>0 for any 𝑡>0;(b)𝜑 is nondecreasing and right continuous;(c)for any 𝜖>0, there exists 𝛿(𝜖)>0 such that 𝑥≤𝑢, 𝑦≥𝑣 and 1𝜖≤𝜑2𝑑(𝑥,𝑢)+𝑑𝑦,𝑣<𝜖+𝛿(𝜖)⟹𝜑𝑑𝐹𝑥,𝑦,𝐹(𝑢,𝑣)<𝜖. (3.5)Then 𝐹 is a generalized Meir-Keeler type contraction.

The following result is an immediate consequence of Theorems 2.4 and 3.1.

Corollary 3.3. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two given mappings such that 𝐹(𝑋×𝑋)⊆𝑔(𝑋), 𝑔 is continuous and commutative with 𝐹. Also, suppose that(a)𝐹 has the mixed strict 𝑔-monotone property;(b)for any 𝜖>0, there exists 𝛿(𝜖)>0 such that, for all 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑔(𝑥)≤𝑔(𝑢) and 𝑔(𝑦)≥𝑔(𝑣), 𝜖≤₀^{(1/2)[𝑑(𝑔(𝑥),𝑔(𝑢))+𝑑(𝑔(𝑦),𝑔(𝑣))]}𝜑(𝑡)𝑑𝑡<𝜖+𝛿(𝜖)⟹₀^{𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))}𝜑(𝑡)𝑑𝑡<𝜖, (3.6) where 𝜑 is a locally integrable function from [0,+∞) into itself satisfying the following condition: ^𝑠₀𝜑(𝑡)𝑑𝑡>0 (3.7) for all 𝑠>0;(c)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔(𝑥₀)<𝐹(𝑥₀,𝑦₀) and 𝑔(𝑦₀)>𝐹(𝑦₀,𝑥₀).Then there exists (𝑥,𝑦)∈𝑋×𝑋 such that 𝑔(𝑥)=𝐹(𝑥,𝑦) and 𝑔(𝑦)=𝐹(𝑦,𝑥). Moreover, if 𝑔(𝑥₀) and 𝑔(𝑦₀) are comparable to each other, then 𝐹 and 𝑔 have a unique coupled common fixed-point in 𝑋×𝑋.

Corollary 3.4. Let 𝐹∶𝑋×𝑋→𝑋 be a mapping satisfying the following conditions:(a)𝐹 has the mixed strict monotone property;(b) for any 𝜖>0, there exists 𝛿(𝜖)>0 such that 𝑥≤𝑢, 𝑦≥𝑣 and 𝜖≤₀^{(1/2)[𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)]}𝜑(𝑡)𝑑𝑡<𝜖+𝛿(𝜖)⟹₀^{[𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))]}𝜑(𝑡)𝑑𝑡<𝜖, (3.8) where 𝜑 is a locally integrable function from [0,+∞) into itself satisfying ^𝑠₀𝜑(𝑡)𝑑𝑡>0 (3.9) for all 𝑠>0;(c) there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀<𝐹(𝑥₀,𝑦₀) and 𝑦₀>𝐹(𝑦₀,𝑥₀).Then there exists (𝑥,𝑦)∈𝑋×𝑋 such that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥). Moreover, if 𝑥₀ and 𝑦₀ are comparable to each other, then 𝐹 has a unique coupled common fixed-point in 𝑋×𝑋.

Corollary 3.5. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two given mappings such that 𝐹(𝑋×𝑋)⊆𝑔(𝑋), 𝑔 is continuous and commutes with 𝐹. Also, suppose that(a)𝐹 has the mixed strict 𝑔-monotone property;(b)for any 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑔(𝑥)≤𝑔(u) and 𝑔(𝑦)≥𝑔(𝑣), ₀^{[𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))]}𝜑(𝑡)𝑑𝑡≤𝑘₀^{(1/2)[𝑑(𝑔(𝑥),𝑔(𝑢))+𝑑(𝑔(𝑦),𝑔(𝑣))]}𝜑(𝑡)𝑑𝑡, (3.10) where 𝑘∈(0,1) and 𝜑 is a locally integrable function from [0,+∞) into itself satisfying ^𝑠₀𝜑(𝑡)𝑑𝑡>0 (3.11) for all 𝑠>0;(c)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔(𝑥₀)<𝐹(𝑥₀,𝑦₀) and 𝑔(𝑦₀)>𝐹(𝑦₀,𝑥₀).Then there exists (𝑥,𝑦)∈𝑋×𝑋 such that 𝑔(𝑥)=𝐹(𝑥,𝑦) and 𝑔(𝑦)=𝐹(𝑦,𝑥). Moreover, if 𝑔(𝑥₀) and 𝑔(𝑦₀) are comparable to each other, then 𝐹 and 𝑔 have a unique coupled common fixed-point in 𝑋×𝑋.

Proof. For any 𝜖>0, if we take 𝛿(𝜖)=(1/𝑘−1)𝜖 and apply Corollary 3.3, then we can get the conclusion.

Corollary 3.6. Let F∶𝑋×𝑋→𝑋 be a mapping satisfying the following conditions:(a)𝐹 has the mixed strict monotone property,(b)for any 𝑥,𝑦,𝑢,𝑣∈𝑋 with 𝑥≤𝑢 and 𝑦≥𝑣, ₀^{𝑑(𝐹(𝑥,𝑦),𝐹(𝑢,𝑣))}𝜑(𝑡)𝑑𝑡≤𝑘₀^{(1/2)[𝑑(𝑥,𝑢)+𝑑(𝑦,𝑣)]}𝜑(𝑡)𝑑𝑡, (3.12) where 𝑘∈(0,1) and 𝜑 is a locally integrable function from [0,+∞) into itself satisfying ^𝑠₀𝜑(𝑡)𝑑𝑡>0 (3.13) for all 𝑠>0;(c)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑥₀<𝐹(𝑥₀,𝑦₀) and 𝑦₀>𝐹(𝑦₀,𝑥₀).Then there exist 𝑥,𝑦∈𝑋 such that 𝑥=𝐹(𝑥,𝑦) and 𝑦=𝐹(𝑦,𝑥). Moreover, if 𝑥₀ and 𝑦₀ are comparable to each other, then 𝐹 has a unique coupled common fixed-point in 𝑋×𝑋.

Finally, by using the above results, we show the existence of solutions for the following integral equation: 𝑥(𝑡),𝑦(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))+𝜆𝑥(𝑠)−𝑓𝑠,𝑦(𝑠)+𝜆𝑦(𝑠)𝑑𝑠,_𝑇0𝐺(𝑡,𝑠)𝑓𝑠,𝑦(𝑠)+𝜆𝑦(𝑠)−𝑓(𝑠,𝑥(𝑠))+𝜆𝑥(𝑠)𝑑𝑠, (3.14) where 𝑥,𝑦∈𝐶(𝐼,ℝ) (: the set of continuous functions from 𝐼 into ℝ), 𝑇>0, 𝑓∶𝐼×ℝ→ℝ is a continuous function and 𝐺()=𝑒𝑡,𝑠^{𝜆(𝑇+𝑠−𝑡)}𝑒^𝜆𝑇𝑒−1,if0≤𝑠<𝑡≤𝑇;_{𝜆(𝑠−𝑡)}𝑒^𝜆𝑇−1,if0≤𝑡<𝑠≤𝑇. (3.15)

Definition 3.7. A lower solution for the integral equation (3.14) is an element (𝛼,𝛽)∈𝐶¹(𝐼,ℝ)×𝐶¹(𝐼,ℝ) such that 𝛼^′(𝑡)(𝑡)((𝑡(𝑡)(0)(𝑇),𝛽+𝜆𝛽≤𝑓𝑡,𝛼))−𝑓𝑡,𝛽,𝛼<𝛼_′(𝑡)+𝜆𝛼(𝑡)≥𝑓𝑡,𝛽(𝑡)−𝑓(𝑡,𝛼(𝑡)),𝛽(0)≥𝛽(𝑇), (3.16) where 𝐶¹(𝐼,ℝ) denotes the set of differentiable functions from 𝐼 into ℝ.

Now, we prove the existence of solutions for the integral equation (3.14) by using the existence of a lower solution for the integral equation (3.14).

Theorem 3.8. Let 𝒜 be the class of the functions 𝜑∶[0,∞)→[0,∞) satisfying the following conditions:(a)𝜑 is increasing;(b)for any 𝑥≥0, there exists 𝑘∈[0,1) such that 𝜑(𝑥)<(𝑘/2)𝑥.In the integral equation (3.14), suppose that there exists 𝜆>0 such that, for all 𝑥,𝑦∈ℝ with 𝑦>𝑥, 0<𝑓𝑡,𝑦+𝜆𝑦−𝑓(𝑡,𝑥)+𝜆𝑥≤𝜆𝜑𝑦−𝑥, (3.17) where 𝜑∈𝒜. If a lower solution of the integral equation (3.14) exists, then a solution of the integral equation (3.14) exists.

Proof. Define a mapping 𝐹∶𝐶(𝐼,ℝ)×𝐶(𝐼,ℝ)→𝐶(𝐼,ℝ) by 𝐹𝑥(𝑡),𝑦(𝑡)=^𝑇₀𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠))+𝜆𝑥(𝑠)−𝑓𝑠,𝑦(𝑠)+𝜆𝑦(𝑠)𝑑𝑠. (3.18)
Note that, if (𝑥(𝑡),𝑦(𝑡))∈𝐶(𝐼,ℝ)×𝐶(𝐼,ℝ) is a coupled fixed-point of 𝐹, then (𝑥(𝑡),𝑦(𝑡)) is a solution of the integral equation (3.14).
Now, we check the hypotheses in Corollary 2.5 as follows:(1)𝑋×𝑋=𝐶(𝐼,ℝ)×𝐶(𝐼,ℝ) is a partially ordered set if we define the order relation in 𝑋×𝑋 as follows: (𝑢(𝑡),𝑣(𝑡))≤𝑥(𝑡),𝑦(𝑡)iff𝑢(𝑡)<𝑥(𝑡),𝑣(𝑡)≥𝑦(𝑡) (3.19) for all (𝑥(𝑡),𝑦(𝑡)),(𝑢(𝑡),𝑣(𝑡))∈𝑋×𝑋 and 𝑡∈𝐼.(2)(𝑋,𝑑) is a complete metric space if we define a metric 𝑑 as follows: 𝑑𝑥(𝑡),𝑦(𝑡)=sup_𝑡∈𝐼𝑥(𝑡)−𝑦(𝑡)∶𝑥(𝑡),𝑦(𝑡)∈𝑋. (3.20)(3)The mapping 𝐹 has the mixed strict monotone property. In fact, by hypothesis, if 𝑥₂>𝑥₁, then we have 𝑓𝑡,𝑥₂+𝜆𝑥₂>𝑓𝑡,𝑥₁+𝜆𝑥₁, (3.21)which implies that, for any 𝑡∈𝐼, ^𝑇₀𝑓𝑠,𝑥₂(𝑠)+𝜆𝑥₂(𝑠)(𝑠)(𝑠𝐺()>−𝑓𝑠,𝑦−𝜆𝑦)𝑡,𝑠𝑑𝑠_𝑇0𝑓𝑠,𝑥₁(𝑠)+𝜆𝑥₁(𝑠)−𝑓𝑠,𝑦(𝑠)−𝜆𝑦(𝑠)𝐺(𝑡,𝑠)𝑑𝑠, (3.22)that is, 𝐹𝑥₂(𝑡),𝑦(𝑡)>𝐹𝑥₁(𝑡),𝑦(𝑡). (3.23)
Similarly, if 𝑦₁<𝑦₂, then we have 𝑓𝑡,𝑦₂+𝜆𝑦₂>𝑓𝑡,𝑦₁+𝜆𝑦₁, (3.24) which implies that, for any 𝑡∈𝐼, ^𝑇₀𝑓(𝑠,𝑥(𝑠))+𝜆𝑥(𝑠)−𝑓𝑠,𝑦₂(𝑠)−𝜆𝑦₂(𝑠𝐺()<)𝑡,𝑠𝑑𝑠_𝑇0𝑓(𝑠,𝑥(𝑠))+𝜆𝑥(𝑠)−𝑓𝑠,𝑦₁(𝑠)−𝜆𝑦₁(𝑠)𝐺(𝑡,𝑠)𝑑𝑠, (3.25) that is, 𝐹𝑥(𝑡),𝑦₂(𝑡)<𝐹𝑥(𝑡),𝑦₁(𝑡). (3.26)
Now, we show that 𝐹 satisfies (1.2). In fact, let (x,𝑦)≤(𝑢,𝑣) and 𝑡∈𝐼. Then we have 𝑑𝐹𝑥(𝑡)(𝑡)(𝑢(𝑡)(𝑡𝐹𝑥(𝑡)(𝑡)(𝑢(𝑡)(𝑡,𝑦,𝐹,𝑣))=sup,𝑦−𝐹,𝑣))∶𝑡∈𝐼=sup_𝑡∈𝐼^𝑇₀𝐺(𝑓((𝑠(𝑠)(𝑠)(𝑠−𝑡,𝑠)𝑠,𝑥))+𝜆𝑥−𝑓𝑠,𝑦−𝜆𝑦)𝑑𝑠_𝑇0𝐺(𝑡,𝑠)𝑓(𝑠,𝑢(𝑠))+𝜆𝑢(𝑠)−𝑓(𝑠,𝑣(𝑠))−𝜆𝑣(𝑠)𝑑𝑠≤sup_𝑡∈𝐼^𝑇₀𝐺(𝑓((𝑠(𝑠)((𝑠(𝑠)((𝑠(𝑠)(𝑠)(𝑠𝑡,𝑠)𝑠,𝑥))+𝜆𝑥−𝑓𝑠,𝑢))−𝜆𝑢+𝑓𝑠,𝑣))+𝜆𝑣−𝑓𝑠,𝑦−𝜆𝑦)𝑑𝑠. (3.27) Since the function 𝜑(𝑥) is increasing and (𝑥,𝑦)≤(𝑢,𝑣), we have 𝜑(𝑥(𝑠)−𝑢(𝑠))≤𝜑(𝑑(𝑥(𝑠),𝑢(𝑠))),𝜑𝑣(𝑠)−𝑦(𝑠)≤𝜑𝑑𝑣(𝑠),𝑦(𝑠), (3.28) we obtain the following: 𝑑𝐹𝑥(𝑡),𝑦(𝑡),𝐹(𝑢(𝑡),𝑣(𝑡))≤sup_𝑡∈𝐼^𝑇₀𝐺(𝑡,𝑠)𝜆𝜑(𝑥(𝑠)−𝑢(𝑠))+𝜆𝜑𝑣(𝑠)−𝑦(𝑠)𝑑𝑠≤𝜆sup_𝑡∈𝐼^𝑇₀𝐺(𝜑(𝑑(𝑥(𝑠)(𝑠𝑑𝑣(𝑠)(𝑠)𝜑(𝑑(𝑥(𝑠)(𝑠𝑑𝑣(𝑠)(𝑠)𝑡,𝑠),𝑢)))+𝜑,𝑦𝑑𝑠=𝜆,𝑢)))+𝜑,𝑦⋅sup_𝑡∈𝐼^𝑇₀𝐺()𝜑(𝑑(𝑥(𝑠)(𝑠𝑑𝑣(𝑠)(𝑠)𝑡,𝑠𝑑𝑠=𝜆,𝑢)))+𝜑,𝑦⋅sup_𝑡∈𝐼1𝑒^𝜆𝑇1−1𝜆𝑒^{𝜆(𝑇+𝑠−𝑡)𝑡}₀+1𝜆𝑒^{𝜆(𝑠−𝑡)𝑇}_𝑡𝜑(𝑑(𝑥(𝑠)(𝑠𝑑𝑣(𝑠)(𝑠)⋅1=𝜆,𝑢)))+𝜑,𝑦𝜆𝑒^𝜆𝑇𝑒−1^𝜆𝑇(𝑑(𝑥(𝑠)(𝑠𝑑𝑣(𝑠)(𝑠)<𝑘−1=𝜑,𝑢)))+𝜑,𝑦2𝑑(𝑥(𝑠)(𝑠𝑣(𝑠)(𝑠)≤𝑘,𝑢))+𝑑,𝑦2𝑥(𝑡)(𝑡}+𝑘sup{|−𝑢)|∶𝑡∈𝐼2𝑣(𝑡)(𝑡=𝑘sup−𝑦)∶𝑡∈𝐼2𝑑(𝑥(𝑡),𝑢(𝑡))+𝑑𝑦(𝑡),𝑣(𝑡). (3.29) Then, by Proposition 1.3, 𝐹 is a generalized Meir-Keeler type contraction.
Finally, let (𝛼(𝑡),𝛽(𝑡))∈𝐶¹(𝐼,ℝ)×𝐶¹(𝐼,ℝ) be a lower solution for the integral equation (3.14). Then we show that 𝛼<𝐹𝛼,𝛽,𝛽≥𝐹𝛽,𝛼. (3.30) Indeed, we have 𝛼^′(𝑡)+𝜆𝛽(𝑡)≤𝑓(𝑡,𝛼(𝑡))−𝑓(𝑡,𝛽(𝑡)) for any 𝑡∈𝐼 and so 𝛼^′(𝑡)+𝜆𝛼(𝑡)≤𝑓(𝑡,𝛼(𝑡))−𝑓𝑡,𝛽(𝑡)+𝜆𝛼(𝑡)−𝜆𝛽(𝑡) (3.31) for any 𝑡∈𝐼. Multiplying by 𝑒^𝜆𝑡 in (3.31), we get the following: 𝛼(𝑡)𝑒^𝜆𝑡′≤𝑓(𝑡,𝛼(𝑡))+𝜆𝛼(𝑡)−𝑓𝑡,𝛽(𝑡)+𝜆𝛽(𝑡)𝑒^𝜆𝑡 (3.32) for any 𝑡∈𝐼, which implies that 𝛼(𝑡)𝑒^𝜆𝑡≤𝛼(0)+^𝑡₀𝑓(𝑠,𝛼(𝑠))+𝜆𝛼(𝑠)−𝑓𝑠,𝛽(𝑠)−𝜆𝛽(𝑠)𝑒^𝜆𝑠𝑑𝑠 (3.33) for any 𝑡∈𝐼. This implies that 𝛼(0)𝑒^𝜆𝑡<𝛼(𝑇)𝑒^𝜆𝑇≤𝛼(0)+^𝑇₀𝑓(𝑠,𝛼(𝑠))+𝜆𝛼(𝑠)−𝑓𝑠,𝛽(𝑠)−𝜆𝛽(𝑠)𝑒^𝜆𝑠𝑑𝑠 (3.34) and so 𝛼(0)<^𝑇₀𝑒^𝜆𝑠𝑒^𝜆𝑇𝑓((𝑠(𝑠)(𝑠(𝑠−1𝑠,𝛼))+𝜆𝛼−𝑓𝑠,𝛽)−𝜆𝛽)𝑑𝑠. (3.35) Thus it follows from (3.35) and (3.33) that 𝛼(𝑡)𝑒^𝜆𝑡<^𝑇_𝑡𝑒^𝜆𝑠𝑒^𝜆𝑇𝑓((𝑠(𝑠)(𝑠(𝑠+−1𝑠,𝛼))+𝜆𝛼−𝑓𝑠,𝛽)−𝜆𝛽)𝑑𝑠_𝑡0𝑒^{𝜆(𝑇+𝑠)}𝑒^𝜆𝑇𝑓((𝑠(𝑠)(𝑠)(𝑠−1𝑠,𝛼))+𝜆𝛼−𝑓𝑠,𝛽−𝜆𝛽)𝑑𝑠, (3.36) and so 𝛼(𝑡)<^𝑡₀𝑒^{𝜆(𝑇+𝑠−𝑡)}𝑒^𝜆𝑇𝑓((𝑠(𝑠)(𝑠(𝑠+−1𝑠,𝛼))+𝜆𝛼−𝑓𝑠,𝛽)−𝜆𝛽)𝑑𝑠_𝑇𝑡𝑒^{𝜆(𝑠−𝑡)}𝑒^𝜆𝑇𝑓((𝑠(𝑠)(𝑠)(𝑠−1𝑠,𝛼))+𝜆𝛼−𝑓𝑠,𝛽−𝜆𝛽)𝑑𝑠. (3.37) Hence we have 𝛼(𝑡)<^𝑇₀𝐺(𝑡,𝑠)𝑓(𝑠,𝛼(𝑠))+𝜆𝛼(𝑠)−𝑓𝑠,𝛽(𝑠)−𝜆𝛽(𝑠)𝑑𝑠=𝐹𝛼(𝑡),𝛽(𝑡) (3.38) for any 𝑡∈𝐼.
Similarly, we have 𝛽(𝑡)≥𝐹(𝛽(𝑡),𝛼(𝑡)). Therefore, by Corollary 2.5, 𝐹 has a coupled fixed-point.

Example 3.9. In the integral equation (3.14), we put 𝜆=1.5, 𝑓(𝑢,𝑣)=𝑢−𝑣 for all (𝑢,𝑣)∈𝐼×ℝ and 𝑇=0.5. Then 𝑓 is a continuous function, and we have 𝑥(𝑡),𝑦(𝑡)=₀^0.5𝐺(𝑡,𝑠)0.5𝑥(𝑠)−0.5𝑦(𝑠)𝑑𝑠,₀^0.5𝐺(𝑡,𝑠)0.5𝑦(𝑠)−0.5𝑥(𝑠)𝑑𝑠, (3.39) where 𝑥,𝑦∈𝐶(𝐼,ℝ), and 𝐺()=𝑒𝑡,𝑠^{1.5(0.5+𝑠−𝑡)}𝑒^0.75𝑒−1,if0≤𝑠<𝑡≤0.5,_{1.5(𝑠−𝑡)}𝑒^0.75−1,if0≤𝑡<𝑠≤0.5. (3.40) Also, (𝛼(𝑡),𝛽(𝑡))=(−2𝑒^−0.5𝑡,3𝑒^−0.5𝑡) is a lower solution of (3.39). Moreover, if we define 𝜑(𝑥)=𝑥/3 for all 𝑥∈[0,∞), then 𝜑 is increasing and, for any 𝑥>0, there exists 𝑘=1/1.1∈[0,1) such that 𝜑(𝑥)=𝑥/3<(𝑘/2)𝑥=𝑥/2.2. For all 𝑥,𝑦∈ℝ with 𝑦>𝑥, we have 0<𝑓𝑡,𝑦+𝜆𝑦−𝑓(𝑡,𝑥)+𝜆𝑥=0.5𝑦−𝑥≤𝜆𝜑𝑦−𝑥=1.5𝑦−𝑥3=0.5𝑦−𝑥. (3.41) Therefore, all the conditions of Theorem 3.8 hold, and a solution of (3.39) exists.

Acknowledgment

This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011.0021821).