Coupled Coincidence Point Theorems for New Types of Mixed Monotone Multivalued Mappings in Partially Ordered Metric Spaces

Abstract

We introduce and study new types of mixed monotone multivalued mappings in partially ordered complete metric spaces. We give relationships between those two types of mappings and prove their coupled fixed point and coupled common fixed point theorems in partially ordered complete metric spaces. Some examples of each type of mappings satisfying the conditions of the main theorems are also given. Our main result includes several recent developments in fixed point theory of mixed monotone multivalued mappings.

1. Introduction and Preliminaries

Let (𝑋,𝑑) be a metric space, and let 𝐶𝐵(𝑋) be the class of all nonempty bounded and closed subsets of 𝑋. For 𝑥∈𝑋 and 𝐴⊆𝑋, we denote 𝑑(𝑥,𝐴)=inf{𝑑(𝑥,𝑦)∶𝑦∈𝐴}. For 𝐴,𝐵∈𝐶𝐵(𝑋), define 𝛿(𝐴,𝐵)=sup𝑑𝑥,𝑦∶𝑥∈𝐴,𝑦∈𝐵. (1) If 𝐴={𝑎}, then we write 𝛿(𝐴,𝐵)=𝛿(𝑎,𝐵). Also in addition, if 𝐵={𝑏}, then 𝛿(𝐴,𝐵)=𝑑(𝑎,𝑏). For all 𝐴,𝐵,𝐶∈𝐶𝐵(𝑋), the definition of 𝛿(𝐴,𝐵) gives the following formulas: (i)𝛿(𝐴,𝐵)=𝛿(𝐵,𝐴);(ii)𝛿(𝐴,𝐵)⩽𝛿(𝐴,𝐶)+𝛿(𝐶,𝐵);(iii)𝛿(𝐴,𝐵)=0 if and only if 𝐴=𝐵={𝑎};(iv)𝛿(𝐴,𝐴)=diam(𝐴).

It is easy to see that the following inequality holds, for 𝑥,𝑦∈𝑋, 𝑑(𝑥,𝐴)−𝑑𝑦,𝐴⩽𝑑𝑥,𝑦, (2) for all 𝐴⊆𝑋. By using the above inequality, the following lemma is obtained.

Lemma 1. Let 𝑋 be a metric space, and let 𝐴 be a nonempty subset of 𝑋. If 𝑔 is defined by 𝑔(𝑥)=𝑑(𝑥,𝐴),∀𝑥∈𝑋, (3) then 𝑔 is continuous.

Let 𝑋 be a nonempty set, 𝐹∶𝑋×𝑋→2^𝑋 (collection of all nonempty subsets of 𝑋), and 𝑔∶𝑋→𝑋. An element (𝑥,𝑦)∈𝑋×𝑋 is called(i)coupled fixed point of 𝐹 if 𝑥∈𝐹(𝑥,𝑦) and 𝑦∈𝐹(𝑦,𝑥),(ii)coupled coincidence point of a hybrid pair {𝐹,𝑔} if 𝑔(𝑥)∈𝐹(𝑥,𝑦) and 𝑔(𝑦)∈𝐹(𝑦,𝑥),(iii)coupled common fixed point of a hybrid pair {𝐹,𝑔} if 𝑥=𝑔(𝑥)∈𝐹(𝑥,𝑦) and 𝑦=𝑔(𝑦)∈𝐹(𝑦,𝑥).

We denote the set of coupled coincidence point of mappings 𝐹 and 𝑔 by 𝐶(𝐹,𝑔). Note that if (𝑥,𝑦)∈𝐶(𝐹,𝑔), then (𝑦,𝑥) is also in 𝐶(𝐹,𝑔).

The hybrid pair {𝐹,𝑔} is called 𝑤-compatible if 𝑔(𝐹(𝑥,𝑦))⊆𝐹(𝑔(𝑥),𝑔(𝑦)) whenever (𝑥,𝑦)∈𝐶(𝐹,𝑔). The mapping 𝑔 is called 𝐹-weakly commuting at some point (𝑥,𝑦)∈𝑋×𝑋 if 𝑔²(𝑥)∈𝐹(𝑔(𝑥),𝑔(𝑦)) and 𝑔²(𝑦)∈𝐹(𝑔(𝑦),𝑔(𝑥)).

Let (𝑋,⩽) be a partially ordered set, and suppose that there is a metric 𝑑 on 𝑋 such that (𝑋,𝑑) is a metric space. For 𝑥,𝑦∈𝑋, we write 𝑥<𝑦 if 𝑥⩽𝑦 and 𝑥≠𝑦. On the product space 𝑋×𝑋, we consider the following partial order: ,()()⩽for𝑥,𝑦𝑢,𝑣∈𝑋×𝑋,𝑢,𝑣𝑥,𝑦⟺𝑥⩾𝑢,𝑦⩽𝑣. (4)

The existence of a fixed point for contraction type of mappings in partially ordered metric spaces has been considered recently by Ran and Reurings [1], Gnana Bhaskar and Lakshmikantham [2], Nieto and Rodríguez-López [3, 4], Agarwal et al. [5], Lakshmikantham and Ciric [6], and Harjani and Sadarangani [7].

In [2], Gnana Bhaskar and Lakshmikantham introduced the notions of mixed monotone mappings and a coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings.

In [3], Nieto and Rodríguez-López studied some fixed point theorems for monotone nondecreasing mappings in partially ordered metric spaces. They proved the existence of a fixed point in partially ordered metric spaces and applied the obtained results to study a problem of ordinary differential equations.

In [6], Lakshmikantham and Ciric introduced notions of a mixed 𝑔-monotone mapping and proved coupled coincidence and coupled common fixed point theorems for such nonlinear contractive mappings in partially ordered complete metric spaces. They defined the mixed 𝑔-monotone property as follows.

Let (𝑋,⩽) be a partially ordered set, and let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋. We say that 𝐹 has the mixed 𝑔-monotone property if 𝐹 is monotone 𝑔-nondecreasing in its first argument and is monotone 𝑔-nonincreasing in its second argument; that is, for any 𝑥,𝑦∈𝑋, 𝑥₁,𝑥₂∈𝑋,𝑔𝑥₁⩽𝑔𝑥₂⟹𝐹𝑥₁,𝑦⩽𝐹𝑥₂,𝑦,𝑦₁,𝑦₂∈𝑋,𝑔𝑦₁⩽𝑔𝑦₂⟹𝐹𝑥,𝑦₁⩾𝐹𝑥,𝑦₂. (5)

They proved the following theorem.

Theorem 2 (see [6, Theorem 2.1]). 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. Assume that there is a function 𝜑∶[0,+∞)→[0,+∞) with 𝜑(𝑡)<𝑡 and lim_𝑟→𝑡+𝜑(𝑟)<𝑡 for each 𝑡>0, and also suppose that 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are such that 𝐹 has the mixed 𝑔-monotone property and 𝑑𝐹()𝑑𝑔(𝑥)(𝑢)𝑔𝑦(𝑣)𝑥,𝑦,𝐹𝑢,𝑣⩽𝜑,𝑔+𝑑,𝑔2, (6) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 for which 𝑔(𝑥)⩽𝑔(𝑢) and 𝑔(𝑦)⩾𝑔(𝑣). Suppose 𝐹(𝑋×𝑋)⊆𝑔(𝑋), 𝑔 is continuous and commutes with 𝐹, and also suppose that either (a)𝐹 is continuous or(b)𝑋 has the following properties:(i)if a nondecreasing sequence {𝑥_𝑛}→𝑥 in 𝑋, then 𝑥_𝑛⩽𝑥 for all 𝑛;(ii)if a nonincreasing sequence, {𝑦_𝑛}→𝑦 in 𝑋, then 𝑦⩽𝑦_𝑛 for all 𝑛.If 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 point.

Coupled coincidence and coupled common fixed point theorems for mixed 𝑔-monotone single valued mappings can be applied for solving solutions of some ordinary differential equations, so there are many authors who pay their attention for the existence problem of coupled coincidence and coupled common fixed point theorems; see more detail in [6, 8–11].

In [8], Abbas et al. studied coupled coincidence point and common fixed point theorems for a pair of multivalued and single valued mappings which satisfy a generalized contractive condition in complete metric spaces. In [12], Huang and Fang introduced a class of multivalued mixed increasing operators in Banach spaces and proved some new fixed point and coupled fixed point theorems. In [13], Choudhury and Metiya proved some fixed point theorems for multivalued and single valued mappings in partially ordered metric spaces. In [14], Yin and Guo introduced the new notion of 𝑔-monotone mapping and proved some fixed point theorems for multivalued and single valued 𝑔-increasing mappings in partially ordered metric spaces. The mappings considered in [14] are assumed to satisfy certain metric inequalities which are established by an altering distance function. The main results of [14] extended and improved those of Choudhury and Metiya [13].

Motivated and inspired by these works, we introduce new classes of mixed monotone multivalued mapping in partially ordered metric spaces and prove some existence theorems for coupled fixed point, coupled coincidence point, and coupled common fixed point.

Next, we define two types of relations between the two sets and use these relations to introduce new types of multivalued mappings and show some relationships between these classes of mappings.

Definition 3. Let (𝑋,⩽) be a partially ordered set and let 𝐴, 𝐵 be two nonempty subsets of 𝑋. Then we denote(i)𝐴<^(𝐼)𝐵 if for each 𝑎∈𝐴 there exists 𝑏∈𝐵 such that 𝑎<𝑏;(ii)𝐴<^(𝑆)𝐵 if for each 𝑏∈𝐵 there exists 𝑎∈𝐴 such that 𝑎<𝑏.

Definition 4. Let (𝑋,⩽) be a partially ordered set, and let 𝐹∶𝑋×𝑋→2^𝑋 and 𝑔∶𝑋→𝑋. We say that 𝐹 is mixed 𝑔-monotone of type (𝐴) if for any 𝑥,𝑦∈𝑋𝑥₁,𝑥₂∈𝑋,𝑔𝑥₁<𝑔𝑥₂⟹𝐹𝑥₁,𝑦<^(𝐼)𝐹𝑥₂,𝑦,𝑦₁,𝑦₂∈𝑋,𝑔𝑦₁<𝑔𝑦₂⟹𝐹𝑥,𝑦<₂^(𝐼)𝐹𝑥,𝑦₁. (7)

Definition 5. Let (𝑋,⩽) be a partially ordered set, let 𝐹∶𝑋×𝑋→2^𝑋 be a multivalued mapping, and let 𝑔∶𝑋→𝑋. We say that 𝐹 is mixed 𝑔-monotone of type (𝐵) if, for any 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝑋, 𝑔𝑥₁<𝑔𝑥₂,𝑔𝑦₁>𝑔𝑦₂⟹𝐹𝑥₁,𝑦<₁^(𝐼)𝐹𝑥₂,𝑦₂. (8)
The following proposition gives a relation between multivalued mappings of these two types.

Proposition 6. Let (𝑋,⩽) be a partially ordered set and 𝑑 a metric on 𝑋, 𝐹∶𝑋×𝑋→2^𝑋 and 𝑔∶𝑋→𝑋. If 𝐹 is mixed 𝑔-monotone of type (𝐴), then it is of type (𝐵).

Proof. Let 𝑥₁,𝑥₂,𝑦₁,𝑦₂∈𝑋 such that 𝑔(𝑥₁)<𝑔(𝑥₂) and 𝑔(𝑦₁)>𝑔(𝑦₂). Since 𝐹 is mixed 𝑔-monotone of type (𝐴), we have 𝐹(𝑥₁,𝑦₁)<^(𝐼)𝐹(𝑥₂,𝑦₁) and 𝐹(𝑥₂,𝑦₁)<^(𝐼)𝐹(𝑥₂,𝑦₂). This implies that 𝐹(𝑥₁,𝑦₁)<^(𝐼)𝐹(𝑥₂,𝑦₂).

The converse of the above proposition is not true in general as seen in the following example.

Example 7. Let 𝑋=[1,∞), and let ⩽ be a usual partially ordered on 𝑋; that is, 𝑥⩽𝑦 if and only if 𝑦−𝑥⩾0. Let 𝐹∶𝑋×𝑋→𝐶𝐵(𝑋) and 𝑔∶𝑋→𝑋 be defined by 𝐹=[]=(),𝑥,𝑦21,4,if𝑥,𝑦1,11+31𝑥+𝑦,otherwise, (9) and let 𝑔(𝑥)=𝑥. It is easy to show that 𝐹 is mixed 𝑔-monotone of type (𝐵), but it is not of type (𝐴). Consider 𝑦=1, 𝑥₁=1, and 𝑥₂=2, we have 𝐹(𝑥₁,1)=𝐹(1,1)=[1,4] and 𝐹(𝑥₂,1)=𝐹(2,1)={1+(2/3)(2+1)}={3}. It follows that 𝐹(𝑥₁,1)<^(𝐼)̸𝐹(𝑥₂,1). Hence, 𝐹 is not mixed 𝑔-monotone of type (𝐴).

2. Main Results

In this section, we prove the existence of a coupled fixed points of mixed 𝑔-monotone multivalued mappings of types (𝐴) and (𝐵).

Theorem 8. 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. Suppose 𝐹∶𝑋×𝑋→𝐶𝐵(𝑋) and 𝑔∶𝑋→𝑋 are such that 𝐹 is mixed 𝑔-monotone of type (𝐵). Assume that there is a function 𝜑∶[0,∞)→[0,∞) with 𝜑(𝑡)<𝑡 and lim_𝑟→𝑡+𝜑(𝑟)<𝑡 for each 𝑡>0 and 𝛿𝐹()𝑑𝑔(𝑥)(𝑢)𝑔𝑦(𝑣)𝑥,𝑦,𝐹𝑢,𝑣⩽𝜑,𝑔+𝑑,𝑔2, (10) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 for which 𝑔(𝑥)>𝑔(𝑢) and 𝑔(𝑦)<𝑔(𝑣). Suppose that 𝐹(𝑋×𝑋)⊆𝑔(𝑋), 𝑔(𝑋) is a complete subset of 𝑋, and suppose that(a)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥<₀^(𝐼)𝐹𝑥₀,𝑦₀,𝑔𝑦>₀^(𝑆)𝐹𝑦₀,𝑥₀, (11)(b)𝑋 has the following properties:(i)if an increasing sequence {𝑥_𝑛}→𝑥, then 𝑥_𝑛<𝑥 for all 𝑛;(ii)if a decreasing sequence {𝑦_𝑛}→𝑦, then 𝑦<𝑦_𝑛 for all 𝑛.Then there exist 𝑥,𝑦∈𝑋 such that 𝑔(𝑥)∈𝐹𝑥,𝑦,𝑔𝑦∈𝐹𝑦,𝑥. (12)

Proof. Let 𝑥₀,𝑦₀∈𝑋 be such that {𝑔(𝑥₀)}<^(𝐼)𝐹(𝑥₀,𝑦₀) and {𝑔(𝑦₀)}>^(𝑆)𝐹(𝑦₀,𝑥₀). Since 𝐹(𝑋×𝑋)⊆𝑔(𝑋), it follows that there exist 𝑥₁,𝑦₁∈𝑋 such that 𝑔(𝑥₁)∈𝐹(𝑥₀,𝑦₀), 𝑔(𝑦₁)∈𝐹(𝑦₀,𝑥₀) and 𝑔(𝑥₀)<𝑔(𝑥₁), 𝑔(𝑦₀)>𝑔(𝑦₁). Since 𝐹 is mixed 𝑔-monotone of type (𝐵), we obtain 𝐹𝑥₀,𝑦<₀^(𝐼)𝐹𝑥₁,𝑦₁,𝐹𝑦₁,𝑥<₁^(𝐼)𝐹𝑦₀,𝑥₀. (13) Again from 𝐹(𝑋×𝑋)⊆𝑔(𝑋), we can choose 𝑥₂,𝑦₂∈𝑋 such that 𝑔(𝑥₂)∈𝐹(𝑥₁,𝑦₁), 𝑔(𝑦₂)∈𝐹(𝑦₁,𝑥₁) and 𝑔(𝑥₁)<𝑔(𝑥₂), 𝑔(𝑦₁)>𝑔(𝑦₂). By continuing this process, we obtain sequences {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} in 𝑋 such that 𝑔𝑥_𝑛+1∈𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1∈𝐹𝑦_𝑛,𝑥_𝑛𝑔𝑥,∀𝑛⩾0,₀<𝑔𝑥₁<⋯<𝑔𝑥_𝑛𝑔𝑦<⋯,₀>𝑔𝑦₁>⋯>𝑔𝑦_𝑛>⋯. (14)(15)(16)
Put 𝛼_𝑛=𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛+1,∀𝑛⩾0. (17) We show that 𝛼_𝑛𝛼⩽2𝜑^𝑛−12. (18) Since 𝑔(𝑥_𝑛−1)<𝑔(𝑥_𝑛) and 𝑔(𝑦_𝑛−1)>𝑔(𝑦_𝑛), from (14) and (10) we have 𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1⩽𝛿𝐹𝑥_𝑛−1,𝑦_𝑛−1,𝐹𝑥_𝑛,𝑦_𝑛𝑑𝑔𝑥⩽𝜑^𝑛−1,𝑔𝑥_𝑛+𝑑𝑔𝑦_𝑛−1,𝑔𝑦_𝑛2𝛼=𝜑^𝑛−12. (19) Similarly, we have 𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛+1⩽𝛿𝐹𝑦_𝑛−1,𝑥_𝑛−1,𝐹𝑦_𝑛,𝑥_𝑛𝑑𝑔𝑦⩽𝜑^𝑛−1,𝑔𝑦_𝑛+𝑑𝑔𝑥_𝑛−1,𝑔𝑥_𝑛2𝛼=𝜑^𝑛−12. (20) Adding (19) and (20), we obtain (18).
Since 𝜑(𝑡)<𝑡 for 𝑡>0, it follows From (18) that a sequence {𝛼_𝑛} is monotone decreasing. Therefore, there is some 𝛼⩾0 such that lim_𝑛→∞𝛼_𝑛=𝛼. (21) We show that 𝛼=0. Suppose, to the contrary, that 𝛼>0. Then, by taking the limit as 𝑛→∞ on both sides of (18), we have 𝛼=lim_𝑛→∞𝛼_𝑛⩽2lim_𝑛→∞𝜑𝛼_𝑛−12=2lim_{𝛼𝑛→𝛼+}𝜑𝛼_𝑛−12𝛼<22=𝛼, (22) which is a contradiction. Thus, 𝛼=0; that is, lim_𝑛→∞𝑑𝑔𝑥_𝑛,𝑔𝑥_𝑛+1+𝑑𝑔𝑦_𝑛,𝑔𝑦_𝑛−1=0. (23)
Now we prove that {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are Cauchy sequences. Suppose, to the contrary, that at least one of {𝑔(𝑥_𝑛)} or {𝑔(𝑦_𝑛)} is not a Cauchy sequence. Then there exists an 𝜀>0 and two subsequences of integers {𝑙(𝑘)}, {𝑚(𝑘)}, 𝑚(𝑘)>𝑙(𝑘)⩾𝑘 with 𝑟_𝑘=𝑑𝑔𝑥_𝑙(𝑘),𝑔𝑥_𝑚(𝑘)+𝑑𝑔𝑦_𝑙(𝑘),𝑔𝑦_𝑚(𝑘)⩾𝜀for𝑘∈ℕ. (24) We may also assume that 𝑑𝑔𝑥_𝑙(𝑘),𝑔𝑥_{𝑚(𝑘)−1}+𝑑𝑔𝑦_𝑙(𝑘),𝑔𝑦_{𝑚(𝑘)−1}<𝜀 (25) by choosing 𝑚(𝑘) to be the smallest number exceeding 𝑙(𝑘) for which (24) holds. From (24) and (25) and by the triangle inequality, we have 𝜀⩽𝑟_𝑘⩽𝑑𝑔𝑥_𝑙(𝑘),𝑔𝑥_{𝑚(𝑘)−1}+𝑑𝑔𝑥_{𝑚(𝑘)−1},𝑔𝑥_𝑚(𝑘)+𝑑𝑔𝑦_𝑙(𝑘),𝑔𝑦_{𝑚(𝑘)−1}+𝑑𝑔𝑦_{𝑚(𝑘)−1},𝑔𝑦_𝑚(𝑘)=𝑑𝑔𝑥_𝑙(𝑘),𝑔𝑥_{𝑚(𝑘)−1}+𝑑𝑔𝑦_𝑙(𝑘),𝑔𝑦_{𝑚(𝑘)−1}+𝛼_{𝑚(𝑘)−1}<𝜀+𝛼_{𝑚(𝑘)−1}. (26) By taking the limit as 𝑘→∞, we get lim_𝑘→∞𝑟_𝑘=𝜀. (27) By the triangle inequality, we have 𝑟_𝑘=𝑑𝑔𝑥_𝑙(𝑘),𝑔𝑥_𝑚(𝑘)+𝑑𝑔𝑦_𝑙(𝑘),𝑔𝑦_𝑚(𝑘)⩽𝑑𝑔𝑥_𝑙(𝑘),𝑔𝑥_𝑙(𝑘)+1+𝑑𝑔𝑥_𝑙(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝑑𝑔𝑥_𝑚(𝑘)+1,𝑔𝑥_𝑚(𝑘)+𝑑𝑔𝑦_𝑙(𝑘),𝑔𝑦_𝑙(𝑘)+1+𝑑𝑔𝑦_𝑙(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1+𝑑𝑔𝑦_𝑚(𝑘)+1,𝑔𝑦_𝑚(𝑘)=𝑑𝑔𝑥_𝑙(𝑘),𝑔𝑥_𝑙(𝑘)+1+𝑑𝑔𝑦_𝑙(𝑘),𝑔𝑦_𝑙(𝑘)+1+𝑑𝑔𝑥_𝑚(𝑘),𝑔𝑥_𝑚(𝑘)+1+𝑑𝑔𝑦_𝑚(𝑘),𝑔𝑦_𝑚(𝑘)+1+𝑑𝑔𝑥_𝑙(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝑑𝑔𝑦_𝑙(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1. (28) Hence, 𝑟_𝑘⩽𝛼_𝑙(𝑘)+𝛼_𝑚(𝑘)+𝑑𝑔𝑥_𝑙(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1+𝑑𝑔𝑦_𝑙(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1. (29) From (15) and (16), we have 𝑔(𝑥_𝑙(𝑘))<𝑔(𝑥_𝑚(𝑘)) and 𝑔(𝑦_𝑙(𝑘))>𝑔(𝑦_𝑚(𝑘)). By (14) and (iii), we have 𝑑𝑔𝑥_𝑙(𝑘)+1,𝑔𝑥_𝑚(𝑘)+1⩽𝛿𝐹𝑥_𝑙(𝑘),𝑦_𝑙(𝑘),𝐹𝑥_𝑚(𝑘),𝑦_𝑚(𝑘)𝑑𝑔𝑥⩽𝜑^𝑙(𝑘),𝑔𝑥_𝑚(𝑘)+𝑑𝑔𝑦_𝑙(𝑘),𝑔𝑦_𝑚(𝑘)2𝑟=𝜑^𝑘2. (30) Similarly, we have 𝑑𝑔𝑦_𝑙(𝑘)+1,𝑔𝑦_𝑚(𝑘)+1⩽𝛿𝐹𝑦_𝑙(𝑘),𝑥_𝑙(𝑘),𝐹𝑦_𝑚(𝑘),𝑥_𝑚(𝑘)𝑑𝑔𝑦⩽𝜑^𝑙(𝑘),𝑔𝑦_𝑚(𝑘)+𝑑𝑔𝑥_𝑙(𝑘),𝑔𝑥_𝑚(𝑘)2𝑟=𝜑^𝑘2. (31) It follows from (30), (31), and (29) that 𝑟_𝑘⩽𝛼_𝑙(𝑘)+𝛼_𝑚(𝑘)𝑟+2𝜑^𝑘2. (32) By using (23) and (27) together with the property of 𝜑, we have 𝜀⩽2lim_𝑘→∞𝜑𝑟_𝑘2=2lim_{𝑟𝑘→𝜀+}𝜑𝑟_𝑘2𝜀<22=𝜀, (33) which is a contradiction. Thus, {𝑔(𝑥_𝑛)} and {𝑔(𝑦_𝑛)} are Cauchy sequences. Since 𝑔(𝑋) is complete, there exist 𝑥,𝑦∈𝑋 such that lim_𝑛→∞𝑔𝑥_𝑛=𝑔(𝑥),lim_𝑛→∞𝑔𝑦_𝑛=𝑔𝑦. (34) Finally, we claim 𝑔(𝑥)∈𝐹(𝑥,𝑦) and 𝑔(𝑦)∈𝐹(𝑦,𝑥).
From (34) and (ii), we obtain 𝑔(𝑥_𝑛)<𝑔(𝑥), and 𝑔(𝑦_𝑛)>𝑔(𝑦), for all 𝑛⩾0. By (iii), we have 𝑑𝐹𝑥,𝑦,𝑔𝑥_𝑛+1⩽𝛿𝐹𝑥,𝑦,𝐹𝑥_𝑛,𝑦_𝑛𝑑𝑔𝑥⩽𝜑^𝑛,𝑔(𝑥)+𝑑𝑔𝑦_𝑛,𝑔𝑦2. (35) This implies by Lemma 1 and the property of 𝜑 that 𝑑(𝐹(𝑥,𝑦),𝑔(𝑥))=0. Hence, 𝑔(𝑥)∈𝐹(𝑥,𝑦). Similarly, we can show that 𝑔(𝑦)∈𝐹(𝑦,𝑥). Thus, (𝑥,𝑦) is a coupled coincidence point of 𝐹 and 𝑔.

Example 9. Let 𝑋=[1,∞) and let ⩽ be a usual partially ordered on 𝑋; that is, 𝑥⩽𝑦 if and only if 𝑦−𝑥⩾0. Let 𝐹∶𝑋×𝑋→𝐶𝐵(𝑋) and 𝑔∶𝑋→𝑋 be defined by 𝐹=4𝑥,𝑦1,3=(),2,if𝑥,𝑦1,11+91𝑥+𝑦,otherwise, (36) and let 𝑔(𝑥)=𝑥. It is easy to check that 𝐹 is a mixed 𝑔-monotone of type (𝐵). Now we prove that (10) holds. Let 𝑥,𝑦,𝑢,𝑣∈𝑋 such that 𝑔(𝑥)>𝑔(𝑢) and 𝑔(𝑦)<𝑔(𝑣). Then 𝑥>𝑢 and 𝑦<𝑣. So (𝑥,𝑦)≠(1,1) and (𝑢,𝑣)≠(1,1). And we have 𝛿𝐹()2𝑥,𝑦,𝐹𝑢,𝑣=𝛿1+91𝑥+𝑦,21+91𝑢+𝑣=21+91𝑥+𝑦2−1−91𝑢+𝑣=29)+1(𝑥−𝑢𝑦−1𝑣⩽29||+1𝑥−𝑢𝑦−1𝑣<29||+⩽4𝑥−𝑢𝑦−𝑣9⋅12||+𝑑()𝑥−𝑢𝑦−𝑣⩽𝜑𝑥,𝑢+𝑑𝑦,𝑣2𝑑𝑔(𝑥)(𝑢)𝑔𝑦(𝑣)=𝜑,𝑔+𝑑,𝑔2, (37) where 𝜑(𝑡)=(4/9)𝑡. By Theorem 8, we obtain that 𝐹 has a coupled fixed point. Note that (1,1) is a coupled fixed point of 𝐹.

The following results are obtained directly from Theorem 8 and Proposition 6.

Corollary 10. 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. Suppose that 𝐹∶𝑋×𝑋→𝐶𝐵(𝑋) and 𝑔∶𝑋→𝑋 are such that 𝐹 is a mixed 𝑔-monotone of type (𝐴). Assume that there is a function 𝜑∶[0,∞)→[0,∞) with 𝜑(𝑡)<𝑡 and lim_𝑟→𝑡+𝜑(𝑟)<𝑡 for each 𝑡>0 and 𝛿𝐹()𝑑𝑔(𝑥)(𝑢)𝑔𝑦(𝑣)𝑥,𝑦,𝐹𝑢,𝑣⩽𝜑,𝑔+𝑑,𝑔2, (38) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 for which 𝑔(𝑥)>𝑔(𝑢) and 𝑔(𝑦)<𝑔(𝑣). Suppose that 𝐹(𝑋×𝑋)⊆𝑔(𝑋), that 𝑔(𝑋) is a complete subset of 𝑋; suppose that(a)there exist 𝑥₀,𝑦₀∈𝑋 such that 𝑔𝑥<₀^(𝐼)𝐹𝑥₀,𝑦₀,𝑔𝑦>₀^(𝑆)𝐹𝑦₀,𝑥₀, (39)(b)𝑋 has the following properties:(i)if an increasing sequence {𝑥_𝑛}→𝑥, then 𝑥_𝑛<𝑥 for all 𝑛;(ii)if a decreasing sequence {𝑦_𝑛}→𝑦, then 𝑦<𝑦_𝑛 for all 𝑛.Then there exist 𝑥,𝑦∈𝑋 such that 𝑔(𝑥)∈𝐹𝑥,𝑦,𝑔𝑦∈𝐹𝑦,𝑥. (40)

Note that the contractive condition (38) is weaker than the condition (5) in [6].

Theorem 11. In addition to the hypotheses of Theorem 8 and Corollary 10, if one of the following conditions holds: (a)𝐹 and 𝑔 are 𝑤-compatible, {𝑔^𝑛(𝑥)} is an increasing sequence which converges to 𝑢, {𝑔^𝑛(𝑦)} is a decreasing sequence which converges to 𝑣 for some (𝑥,𝑦)∈𝐶(𝐹,𝑔), 𝑢,𝑣∈𝑋, and 𝑔 is continuous at 𝑢 and 𝑣,(b)𝑔 is 𝐹-weakly commuting for some (𝑥,𝑦)∈𝐶(𝐹,𝑔), 𝑔²(𝑥)=𝑔(𝑥), and 𝑔²(𝑦)=𝑔(𝑦),(c)𝑔 is continuous at 𝑥 and 𝑦 for some (𝑥,𝑦)∈𝐶(𝐹,𝑔); and lim_𝑛→∞𝑔^𝑛(𝑢)=𝑥, lim_𝑛→∞𝑔^𝑛(𝑣)=𝑦 for some 𝑢,𝑣∈𝑋,(d){(𝑔(𝑥),𝑔(𝑦))∶(𝑥,𝑦)∈𝐶(𝐹,𝑔)} is singleton subset of 𝐶(𝐹,𝑔).Then 𝐹 and 𝑔 have a coupled common fixed point.

Proof. By Theorem 8, 𝐹 and 𝑔 have a coupled coincidence point (𝑥,𝑦)∈𝑋×𝑋. Suppose that (a) holds. Then {𝑔^𝑛(𝑥)} is an increasing sequence which converges to 𝑢, and {𝑔^𝑛(𝑦)} is a decreasing sequence which converges to 𝑣 for some (𝑥,𝑦)∈𝐶(𝐹,𝑔), 𝑢,𝑣∈𝑋. Hence, 𝑔^𝑛(𝑥)<𝑢 and 𝑔^𝑛(𝑦)>𝑣 for all 𝑛⩾1. Since 𝑔 is continuous at 𝑢 and 𝑣, we have that 𝑢 and 𝑣 are fixed points of 𝑔; that is, 𝑢=𝑔(𝑢) and 𝑣=𝑔(𝑣). As 𝐹 and 𝑔 are 𝑤-compatible, (𝑔^𝑛(𝑥),𝑔^𝑛(𝑦))∈𝐶(𝐹,𝑔) for all 𝑛⩾1 and 𝑔^𝑛(𝑥)∈𝐹(𝑔^𝑛−1(𝑥),𝑔^𝑛−1(𝑦)), 𝑔^𝑛(𝑦)∈𝐹(𝑔^𝑛−1(𝑦),𝑔^𝑛−1(𝑥)). Using (10), we obtain 𝑑𝑔(𝑢)()𝑔(𝑢),𝐹𝑢,𝑣⩽𝑑,𝑔^𝑛(𝑥)+𝑑𝑔^𝑛(𝑥)()𝑔(𝑢),𝐹𝑢,𝑣⩽𝑑,𝑔^𝑛(𝑥)+𝛿𝐹𝑔^𝑛−1(𝑥),𝑔^𝑛−1𝑦()𝑔(𝑢),𝐹𝑢,𝑣⩽𝑑,𝑔^𝑛(𝑥)𝑑𝑔+𝜑^𝑛−1(𝑥),𝑢+𝑑𝑔^𝑛−1𝑦,𝑣2. (41) By taking limit as 𝑛→∞ and using the property of 𝜑, we obtain 𝑑(𝑔(𝑢),𝐹(𝑢,𝑣))=0, and hence 𝑔(𝑢)∈𝐹(𝑢,𝑣). Similarly, we can show that 𝑔(𝑣)∈𝐹(𝑣,𝑢). Consequently, 𝑢=𝑔(𝑢)∈𝐹(𝑢,𝑣) and 𝑣=𝑔(𝑣)∈𝐹(𝑣,𝑢). Hence, (𝑢,𝑣) is a coupled common fixed point of 𝐹 and 𝑔. Suppose that (b) holds. That is 𝑔 is 𝐹-weakly commuting, 𝑔²(𝑥)=𝑔(𝑥), and 𝑔²(𝑦)=𝑔(𝑦) for some (𝑥,𝑦)∈𝐶(𝐹,𝑔). Then 𝑔(𝑥)=𝑔²(𝑥)∈𝐹(𝑔(𝑥),𝑔(𝑦)), and 𝑔(𝑦)=𝑔²(𝑦)∈𝐹(𝑔(𝑦),𝑔(𝑥)). Hence, (𝑔(𝑥),𝑔(𝑦)) is a coupled common fixed point of 𝐹 and 𝑔. Suppose now that (c) holds. Then there exists (𝑥,𝑦)∈𝐶(𝐹,𝑔) such that 𝑔 is continuous at 𝑥 and 𝑦 and lim_𝑛→∞𝑔^𝑛(𝑢)=𝑥, lim_𝑛→∞𝑔^𝑛(𝑣)=𝑦 for some 𝑢,𝑣∈𝑋. By the continuity of 𝑔 at 𝑥 and 𝑦, we get 𝑥=𝑔(𝑥)∈𝐹(𝑥,𝑦) and 𝑦=𝑔(𝑦)∈𝐹(𝑦,𝑥). Hence, (𝑥,𝑦) is coupled common fixed point of 𝐹 and 𝑔. Finally, suppose that (d) holds. Let {(𝑔(𝑥),𝑔(𝑦))∶(𝑥,𝑦)∈𝐶(𝐹,𝑔)}={(𝑔(𝑥₀),𝑔(𝑦₀))}⊆𝐶(𝐹,𝑔). Then (𝑔(𝑥₀),𝑔(𝑦₀))∈𝐶(𝐹,𝑔). Hence, (𝑔(𝑥₀),𝑔(𝑦₀)) is a coupled common fixed point of 𝐹 and 𝑔.

Acknowledgment

This work was supported by Chiang Mai University, Chiang Mai, Thailand.