Common Fixed Points of Generalized Cyclic Meir-Keeler-Type Contractions in Partially Ordered Metric Spaces

Abstract

The purpose of this paper is to prove some common point theorems for the generalized cyclic Meir-Keeler-type (𝛼,𝜑,𝐴,𝐵)-contraction in partially ordered metric spaces. Our results generalize many recent common point theorems in the literature.

1. Introduction and Preliminaries

Throughout this paper, by ℝ⁺, we denote the set of all nonnegative real numbers, while ℕ is the set of all natural numbers. Let (𝑋,𝑑) be a metric space, let 𝐷 be a subset of 𝑋, and let 𝑓∶𝐷→𝑋 be a map. We say that 𝑓 is contractive if there exists 𝛼∈[0,1) such that for all 𝑥,𝑦∈𝐷, 𝑑𝑓𝑥,𝑓𝑦≤𝛼⋅𝑑𝑥,𝑦. (1) The well-known Banach fixed point theorem asserts that if 𝐷=𝑋, 𝑓 is contractive, and (𝑋,𝑑) is complete, then 𝑓 has a unique fixed point in 𝑋. It is well known that the Banach contraction principle [1] is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping 𝑓∶𝑋→𝑋 is called a quasicontraction if there exists 𝑘<1 such that 𝑑𝑑,𝑑,𝑓𝑥,𝑓𝑦≤𝑘⋅max𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑓𝑦𝑥,𝑓𝑦,𝑑𝑦,𝑓𝑥 (2) for any 𝑥,𝑦∈𝑋. In 1974, Ćirić [2] introduced these maps and proved an existence and uniqueness fixed point theorem.

The following definitions and results will be needed in the sequel. Let 𝐴 and 𝐵 be two nonempty subsets of a metric space (𝑋,𝑑). A mapping 𝑓∶𝐴∪𝐵→𝐴∪𝐵 is called a cyclic map if 𝑓(𝐴)⊆𝐵 and 𝑓(𝐵)⊆𝐴. In 2003, Kirk et al. [3, 4] proved the following fixed point theorem.

Theorem 1 (see [3, 4]). Let 𝐴 and 𝐵 be two nonempty closed subsets of a complete metric space (𝑋,𝑑), and suppose that 𝑓∶𝐴∪𝐵→𝐴∪𝐵 satisfies(i)𝑓(𝐴)⊂𝐵 and 𝑓(𝐵)⊂𝐴, (ii)𝑑(𝑓𝑥,𝑓𝑦)≤𝑘⋅𝑑(𝑥,𝑦) for all 𝑥∈𝐴, 𝑦∈𝐵, and 𝑘∈(0,1).
Then 𝐴∩𝐵 is nonempty, and 𝑓 has a unique fixed point in 𝐴∩𝐵.

Recently, many authors proved some fixed point theorems for cyclic maps satisfying various contractive conditions (see, [5–20]).

Let 𝑋 be a nonempty set, and let (𝑋,⊑) be a partially ordered set endowed with a metric 𝑑. Then, the triple (𝑋,⊑,𝑑) is called a partially ordered metric space. Two elements 𝑥,𝑦∈𝑋 are said to be comparable if either 𝑥⊑𝑦 or 𝑦⊑𝑥 holds. Altun et al. [21] introduced the notion of weakly increasing mappings and proved some existing theorems.

Definition 2 (see [21]). Let (𝑋,⊑) be a partially ordered set and 𝑓,𝑔∶𝑋→𝑋. Then 𝑓,𝑔 are said to be weakly increasing if 𝑓𝑥⊑𝑔𝑓𝑥 and 𝑔𝑥⊑𝑓𝑔𝑥 for all 𝑥∈𝑋.

And the following definition was introduced in [22].

Definition 3 (see [22]). Let (𝑋,⊑) be a partially ordered set, let 𝐴,𝐵 be closed subsets of 𝑋 with 𝑋=𝐴∪𝐵, and let 𝑓,𝑔∶𝑋→𝑋. Then the pair (𝑓,𝑔) is said to be (𝐴,𝐵)-weakly increasing if 𝑓𝑥⊑𝑔𝑓𝑥 for all 𝑥∈𝐴 and 𝑔𝑥⊑𝑓𝑔𝑥 for all 𝑥∈𝐵.

In this paper, we introduce the new notion of generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction. The purpose of this paper is to prove some common point theorems for the generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction in partially ordered metric spaces. Our results generalize many recent common point theorems in the literature.

2. Main Results

In the sequel, we denote by Ψ the class of functions 𝜓∶ℝ⁺⁵→ℝ⁺ satisfying the following conditions: (𝜓₁)𝜓 is an increasing, continuous function in each coordinate;  (𝜓₂) for all 𝑡∈ℝ⁺, 𝜓(𝑡,𝑡,𝑡,0,2𝑡)≤𝑡, 𝜓(𝑡,𝑡,𝑡,2𝑡,0)≤𝑡, 𝜓(0,0,𝑡,𝑡,0)≤𝑡, and 𝜓(𝑡,0,0,𝑡,𝑡)≤𝑡;  (𝜓₂)𝜓(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅)=0 if and only if 𝑡₁=𝑡₂=𝑡₃=𝑡₄=𝑡₅=0.

We start with the following definition.

Definition 4 (see [23]). Let 𝑓∶𝑋→𝑋 be a self-mapping of a set 𝑋 and 𝛼∶𝑋×𝑋→ℝ⁺. Then 𝑓 is called 𝛼-admissible if 𝑥,𝑦∈𝑋,𝛼𝑥,𝑦≥1⟹𝛼𝑓𝑥,𝑓𝑦≥1. (3)

Definition 5. Let 𝐴,𝐵 be two nonempty subsets of a set 𝑋 with 𝑋=𝐴∪𝐵, let 𝑓∶𝐴→𝐵, 𝑔∶𝐵→𝐴 with 𝑓(𝐴)⊂𝐵 and 𝑔(𝐵)⊂𝐴, and let 𝛼∶𝑋×𝑋→ℝ⁺. Then the pair (𝑓,𝑔) is called 𝛼-admissible if the following conditions hold: (1)𝛼(𝑓𝑥,𝑓𝑥)≥1, ∀𝑥∈𝐴⇒𝛼(𝑔𝑓𝑥,𝑔𝑓𝑥)≥1, (2)𝛼(𝑔𝑦,𝑔𝑦)≥1, ∀𝑦∈𝐵⇒𝛼(𝑓𝑔𝑦,𝑓𝑔𝑦)≥1.

In 1969, Meir and Keeler [24] introduced the following notion of Meir-Keeler-type contraction in a metric space (𝑋,𝑑).

Definition 6. Letting (𝑋,𝑑) be a metric space, 𝑓∶𝑋→𝑋. Then 𝑓 is called a Meir-Keeler-type contraction whenever for each 𝜂>0, there exists 𝛾>0 such that 𝜂≤𝑑𝑥,𝑦<𝜂+𝛾⟹𝑑𝑓𝑥,𝑓𝑦<𝜂. (4)

We now state the new notions of generalized cyclic Meir-Keeler-type (𝜓,𝐴,𝐵)-contractions and generalized Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contractions in partially ordered metric spaces as follows.

Definition 7. Let (𝑋,⊑,𝑑) be a partially ordered metric space, let 𝐴,𝐵 be two nonempty subsets of 𝑋 with 𝑋=𝐴∪𝐵, and let 𝑓∶𝐴→𝐵, 𝑔∶𝐵→𝐴 with 𝑓(𝐴)⊂𝐵 and 𝑔(𝐵)⊂𝐴. Then the pair (𝑓,𝑔) is called a generalized cyclic Meir-Keeler-type (𝜓,𝐴,𝐵)-contraction; if for any comparable elements 𝑥,𝑦∈𝑋 with 𝑥∈𝐴 and 𝑦∈𝐵, we have that for each 𝜂>0 there exists 𝛿>0 such that 𝜂≤𝜓𝑑𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦,𝑑𝑥,𝑔𝑦,𝑑𝑦,𝑓𝑥<𝜂+𝛿⟹𝑑𝑓𝑥,𝑔𝑦<𝜂, (5) where 𝜓∈Ψ.

Definition 8. Let (𝑋,⊑,𝑑) be a partially ordered metric space, let 𝐴,𝐵 be two nonempty subsets of 𝑋 with 𝑋=𝐴∪𝐵, and let 𝑓∶𝐴→𝐵, 𝑔∶𝐵→𝐴 with 𝑓(𝐴)⊂𝐵 and 𝑔(𝐵)⊂𝐴, and 𝛼∶𝑋×𝑋→ℝ⁺. Then (𝑓,𝑔) is called a generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction if the following conditions hold: (1) the pair (𝑓,𝑔) is 𝛼-admissible; (2) for any comparable elements 𝑥,𝑦∈𝑋 with 𝑥∈𝐴 and 𝑦∈𝐵, we have that for each 𝜂>0 there exists 𝛿>0 such that 𝑑𝛼𝑑𝜂≤𝜓𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦,𝑑𝑥,𝑔𝑦,𝑑𝑦,𝑓𝑥<𝜂+𝛿⟹𝛼𝑓𝑥,𝑓𝑥𝑔𝑦,𝑔𝑦𝑓𝑥,𝑔𝑦<𝜂, (6)where 𝜓∈Ψ.

Remark 9. Note that if 𝑓 is a generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction, then we have that for any comparable elements 𝑥,𝑦∈𝑋 with 𝑥∈𝐴 and 𝑦∈𝐵, 𝛼𝛼𝑑𝑑.𝑓𝑥,𝑓𝑥𝑔𝑦,𝑔𝑦𝑓𝑥,𝑔𝑦≤𝜓𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦,𝑑𝑥,𝑔𝑦,𝑑𝑦,𝑓𝑥 (7) Further, if 𝜓𝑑𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦,𝑑𝑥,𝑔𝑦,𝑑𝑦,𝑓𝑥=0, (8) then 𝑑(𝑓𝑥,𝑔𝑦)=0.
On the other hand, if 𝜓𝑑𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦,𝑑𝑥,𝑔𝑦,𝑑𝑦,𝑓𝑥>0, (9) then 𝛼𝛼𝑑𝑑.𝑓𝑥,𝑓𝑥𝑔𝑦,𝑔𝑦𝑓𝑥,𝑔𝑦<𝜓𝑥,𝑦,𝑑𝑥,𝑓𝑥,𝑑𝑦,𝑔𝑦,𝑑𝑥,𝑔𝑦,𝑑𝑦,𝑓𝑥 (10)

We now state our first main result for the generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction as follows.

Theorem 10. Let (𝑋,⊑,𝑑) be a partially ordered complete metric space, let 𝐴,𝐵 be nonempty closed subsets of 𝑋 with 𝑋=𝐴∪𝐵, let 𝛼∶𝑋×𝑋→ℝ⁺, and let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that the pair (𝑓,𝑔) is a generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction and (𝐴,𝐵)-weakly increasing. Suppose that the following conditions hold: (i)𝑓 or 𝑔 is continuous; (ii)there exists 𝑥₀∈𝐴 with 𝛼(𝑓𝑥₀,𝑓𝑥₀)≥1; (iii)if 𝛼(𝑥_𝑛,𝑥_𝑛)≥1 for all 𝑛∈ℕ and lim_𝑛→∞𝑥_𝑛=𝜈, then 𝛼(𝑓𝜈,𝑓𝜈)≥1 and 𝛼(𝑔𝜈,𝑔𝜈)≥1.
Then 𝑓 and 𝑔 have a common fixed point in 𝑋.

Proof. By (ii), there exists 𝑥₀∈𝑋 with 𝛼(𝑓𝑥₀,𝑓𝑥₀)≥1. Since 𝑓(𝐴)⊂𝐵 and the pair (𝑓,𝑔) is 𝛼-admissible, there exists 𝑥₁∈𝐵 such that 𝑥₁=𝑓𝑥₀,𝛼𝑔𝑥₁,𝑔𝑥₁=𝛼𝑔𝑓𝑥₀,𝑔𝑓𝑥₀≥1. (11) Since 𝑔(𝐵)⊂𝐴 and the pair (𝑓,𝑔) is 𝛼-admissible, there exists 𝑥₂∈𝐴 such that 𝑥₂=𝑔𝑥₁,𝛼𝑓𝑥₂,𝑓𝑥₂=𝛼𝑓𝑔𝑥₁,𝑓𝑔𝑥₁≥1. (12) Continuing this process, we construct the sequence {𝑥_𝑛} in 𝑋 such that 𝑥_2𝑛+1=𝑓𝑥_2𝑛,𝑥_2𝑛+2=𝑔𝑥_2𝑛+1,𝑥_2𝑛∈𝐴,𝑥_2𝑛+1∈𝐵, (13) and for all 𝑛∈ℕ∪{0}, 𝛼𝑥_2𝑛+1,𝑥_2𝑛+1=𝛼𝑓𝑥_2𝑛,𝑓𝑥_2𝑛𝛼𝑥≥1,_2𝑛+2,𝑥_2𝑛+2=𝛼𝑔𝑥_2𝑛+1,𝑔𝑥_2𝑛+1≥1. (14) Since the pair (𝑓,𝑔) is (𝐴,𝐵)-weakly increasing, we have that 𝑥₁=𝑓𝑥₀⊑𝑔𝑓𝑥₀=𝑔𝑥₁=𝑥₂⊑𝑓𝑔𝑥₁=𝑓𝑥₂=𝑥₃⊑⋯, (15) and so we conclude that for all 𝑛∈ℕ∪{0}, 𝑔𝑓𝑥_2𝑛=𝑔𝑥_2𝑛+1=𝑥_2𝑛+2⊑𝑓𝑔𝑥_2𝑛+1=𝑓𝑥_2𝑛+2=𝑥_2𝑛+3. (16)
Step  1. We will show that {𝑥_𝑛} is a Cauchy sequence in (𝑋,⊑,𝑑).
Case  1. Suppose that 𝑥_2𝑛=𝑥_2𝑛+1 for some 𝑛∈ℕ in the inequality (16). Since 𝑥_2𝑛 and 𝑥_2𝑛+1 are comparable in 𝑋 with 𝑥_2𝑛∈𝐴 and 𝑥_2𝑛+1∈𝐵, by the Remark 9, we have 𝑑𝑥_2𝑛+1,𝑥_2𝑛+2=𝑑𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1≤𝛼𝑓𝑥_2𝑛,𝑓𝑥_2𝑛𝛼𝑔𝑥_2𝑛+1,𝑔𝑥_2𝑛+1𝑑𝑓𝑥_2𝑛,𝑔𝑥_2𝑛+1≤𝜓𝑑𝑥_2𝑛,𝑥_2𝑛+1,𝑑𝑥_2𝑛,𝑓𝑥_2𝑛,𝑑𝑥_2𝑛+1,𝑔𝑥_2𝑛+1,𝑑𝑥_2𝑛,𝑔𝑥_2𝑛+1,𝑑𝑥_2𝑛+1,𝑓𝑥_2𝑛=𝜓𝑑𝑥_2𝑛,𝑥_2𝑛+1,𝑑𝑥_2𝑛,𝑥_2𝑛+1,𝑑𝑥_2𝑛+1,𝑥_2𝑛+2,𝑑𝑥_2𝑛,𝑥_2𝑛+2,𝑑𝑥_2𝑛+1,𝑥_2𝑛+1≤𝜓0,0,𝑑𝑥_2𝑛+1,𝑥_2𝑛+2,𝑑𝑥_2𝑛+1,𝑥_2𝑛+2,0. (17) If 𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)>0, then 𝜓(0,0,𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2), 𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2),0)>0. By Remark 9, we get a contradiction. So we conclude that 𝑑(𝑥_2𝑛+1,𝑥_2𝑛+2)=0; that is, 𝑥_2𝑛+1=𝑥_2𝑛+2. Similarly, we may show that 𝑥_2𝑛+2=𝑥_2𝑛+3. Hence {𝑥_𝑛} is a constant sequence, and so {𝑥_𝑛} is a Cauchy sequence in (𝑋,⊑,𝑑).
Case  2.  Suppose that 𝑥_2𝑛≠𝑥_2𝑛+1 for all 𝑛∈ℕ in the inequality (16).
Substep  1. We show that the sequence {𝑑(𝑥_𝑛,𝑥_𝑛+1)∶𝑛∈ℕ∪{0}} is decreasing.
Subcase  1. If 𝑛 is even, then we let 𝑛=2𝑚 for some 𝑚∈ℕ. Since 𝑥_2𝑚∈𝐴, 𝑥_2𝑚+1∈𝐵, and 𝑥_2𝑚, 𝑥_2𝑚+1 are comparable in 𝑋, we have 𝑑𝑥_𝑛+1,𝑥_𝑛+2=𝑑𝑥_2𝑚+1,𝑥_2𝑚+2=𝑑𝑓𝑥_2𝑚,𝑔𝑥_2𝑚+1≤𝛼𝑓𝑥_2𝑚,𝑓𝑥_2𝑚𝛼𝑔𝑥_2𝑚+1,𝑔𝑥_2𝑚+1𝑑𝑓𝑥_2𝑚,𝑔𝑥_2𝑚+1<𝜓𝑑𝑥_2𝑚,𝑥_2𝑚+1,𝑑𝑥_2𝑚,𝑓𝑥_2𝑚,𝑑𝑥_2𝑚+1,𝑔𝑥_2𝑚+1,𝑑𝑥_2𝑚,𝑔𝑥_2𝑚+1,𝑑𝑥_2𝑚+1,𝑓𝑥_2𝑚=𝜓𝑑𝑥_2𝑚,𝑥_2𝑚+1,𝑑𝑥_2𝑚,𝑥_2𝑚+1,𝑑𝑥_2𝑚+1,𝑥_2𝑚+2,𝑑𝑥_2𝑚,𝑥_2𝑚+2,𝑑𝑥_2𝑚+1,𝑥_2𝑚+1≤𝜓𝑑𝑥_2𝑚,𝑥_2𝑚+1,𝑑𝑥_2𝑚,𝑥_2𝑚+1,𝑑𝑥_2𝑚+1,𝑥_2𝑚+2,𝑑𝑥_2𝑚,𝑥_2𝑚+1+𝑑𝑥_2𝑚+1,𝑥_2𝑚+2𝑑𝑥,0=𝜓_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2,0. (18) If 𝑑(𝑥_2𝑚,𝑥_2𝑚+1)<𝑑(𝑥_2𝑚+1,𝑥_2𝑚+2), then the above inequality becomes 𝑑𝑥_2𝑚+1,𝑥_2𝑚+2<𝜓𝑑𝑥_2𝑚+1,𝑥_2𝑚+2,𝑑𝑥_2𝑚+1,𝑥_2𝑚+2,𝑑𝑥_2𝑚+1,𝑥_2𝑚+2,2𝑑𝑥_2𝑚+1,𝑥_2𝑚+2𝑥,0≤𝑑_2𝑚+1,𝑥_2𝑚+2, (19) which is a contradiction. So we have that 𝑑𝑥_2𝑚+1,𝑥_2𝑚+2≤𝑑𝑥_2𝑚,𝑥_2𝑚+1. (20)
Subcase  2. If 𝑛 is odd, then we let 𝑛=2𝑚+1 for some 𝑚∈ℕ. Since 𝑥_2𝑚+2∈𝐴, 𝑥_2𝑚+3∈𝐵 and 𝑥_2𝑚+2, 𝑥_2𝑚+3 are comparable in 𝑋, we have 𝑑𝑥_𝑛+2,𝑥_𝑛+1=𝑑𝑥_2𝑚+3,𝑥_2𝑚+2=𝑑𝑓𝑥_2𝑚+2,𝑔𝑥_2𝑚+1≤𝛼𝑓𝑥_2𝑚+2,𝑓𝑥_2𝑚+2𝛼𝑔𝑥_2𝑚+1,𝑔𝑥_2𝑚+1×𝑑𝑓𝑥_2𝑚+2,𝑔𝑥_2𝑚+1<𝜓𝑑𝑥_2𝑚+2,𝑥_2𝑚+1,𝑑𝑥_2𝑚+2,𝑓𝑥_2𝑚+2,𝑑𝑥_2𝑚+1,𝑔𝑥_2𝑚+1,𝑑𝑥_2𝑚+2,𝑔𝑥_2𝑚+1,𝑑𝑥_2𝑚+1,𝑓𝑥_2𝑚+2=𝜓𝑑𝑥_2𝑚+2,𝑥_2𝑚+1,𝑑𝑥_2𝑚+2,𝑥_2𝑚+3,𝑑𝑥_2𝑚+1,𝑥_2𝑚+2,𝑑𝑥_2𝑚+2,𝑥_2𝑚+2,𝑑𝑥_2𝑚+1,𝑥_2𝑚+3≤𝜓𝑑𝑥_2𝑚+2,𝑥_2𝑚+1,𝑑𝑥_2𝑚+2,𝑥_2𝑚+3,𝑑𝑥_2𝑚+1,𝑥_2𝑚+2,0,𝑑𝑥_2𝑚+1,𝑥_2𝑚+2+𝑑𝑥_2𝑚+2,𝑥_2𝑚+3=𝜓𝑑𝑥_𝑛+1,𝑥_𝑛,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+1,0,𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2. (21) If 𝑑(𝑥_2𝑚+1,𝑥_2𝑚+2)<𝑑(𝑥_2𝑚+2,𝑥_2𝑚+3), then the above inequality becomes 𝑑𝑥_2𝑚+2,𝑥_2𝑚+3<𝜓𝑑𝑥_2𝑚+2,𝑥_2𝑚+3,𝑑𝑥_2𝑚+2,𝑥_2𝑚+3,𝑑𝑥_2𝑚+3,𝑥_2𝑚+3,0,2𝑑𝑥_2𝑚+2,𝑥_2𝑚+3≤𝑑𝑥_2𝑚+2,𝑥_2𝑚+3, (22) which is a contradiction. So we have that 𝑑𝑥_2𝑚+2,𝑥_2𝑚+3<𝑑𝑥_2𝑚+1,𝑥_2𝑚+2. (23) From (20) and (23), we conclude that 𝑑𝑥₊₁,𝑥_𝑛+2<𝑑𝑥_𝑛+1,𝑥_𝑛+1. (24) From the above argument, we have that the sequence {𝑑(𝑥_𝑛,𝑥_𝑛+1)∶𝑛∈ℕ∪{0}} is decreasing, and it must converge to some 𝜂≥0; that is, lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=𝜂. (25)
Substep  2. We next claim that lim_𝑛→∞𝑑𝑥_𝑛,𝑓𝑥_𝑛+1=0. (26) Notice that 𝜂=inf{𝑑(𝑓𝑥_𝑛,𝑓𝑥_𝑛+1)∶𝑛∈ℕ∪{0}}. We claim that 𝜂=0. Suppose, to the contrary, that 𝜂>0.
If 𝑛 is even, by the argument of Subcase 1 and the inequality (25), we have lim_𝑛→∞𝜓𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛,𝑥_𝑛+1,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2,0=𝜂. (27) Since (𝑓,𝑔) is a generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction, corresponding to 𝜂 use and taking into account the above (27), there exist 𝛿>0 and a natural number 𝑘 such that 𝜂≤𝜓𝑑𝑥_𝑘,𝑥_𝑘+1,𝑑𝑥_𝑘,𝑥_𝑘+1,𝑑𝑥_𝑘+1,𝑥_𝑘+2,𝑑𝑥_𝑘,𝑥_𝑘+1+𝑑𝑥_𝑘+1,𝑥_𝑘+2,0<𝜂+𝛿⟹𝛼𝑓𝑥_𝑘,𝑓𝑥_𝑘𝛼𝑔𝑥_𝑘+1,𝑔𝑥_𝑘+1𝑑𝑓𝑥_𝑘,𝑔𝑥_𝑘+1<𝜂, (28) which implies 𝑑𝑓𝑥_𝑘,𝑔𝑥_𝑘+1≤𝛼𝑓𝑥_𝑘,𝑓𝑥_𝑘𝛼𝑔𝑥_𝑘+1,𝑔𝑥_𝑘+1𝑑𝑓𝑥_𝑘,𝑔𝑥_𝑘+1<𝜂. (29) So we get a contradiction, since 𝜂=inf{𝑑(𝑥_𝑛,𝑥_𝑛+1)∶𝑛∈ℕ∪{0}. Thus we have that lim_𝑛→∞𝑑𝑥_𝑛,𝑓𝑥_𝑛+1=0. (30)
If 𝑛 is odd, by the argument of Subcase 2 and the inequality (25), we have lim_𝑛→∞𝜓𝑑𝑥_𝑛+1,𝑥_𝑛,𝑑𝑥_𝑛+1,𝑥_𝑛+2,𝑑𝑥_𝑛,𝑥_𝑛+1,0,𝑑𝑥_𝑛,𝑥_𝑛+1+𝑑𝑥_𝑛+1,𝑥_𝑛+2=𝜂. (31) Similarly, we can prove that lim_𝑛→∞𝑑𝑥_𝑛,𝑓𝑥_𝑛+1=0. (32)
Substep  3. We show that {𝑥_𝑛} is a Cauchy sequence in (𝑋,⊑,𝑑). It is sufficient to show that {𝑥_2𝑛} is a Cauchy sequence in (𝑋,⊑,𝑑).
Suppose, to the contrary, that {𝑥_2𝑛} is not a Cauchy sequence in (𝑋,⊑,𝑑). Then there exist 𝜖>0 and two subsequences {𝑥_2𝑚(𝑘)} and {𝑥_2𝑛(𝑘)} of {𝑥_2𝑛} such that 𝑛(𝑘) is the smallest integer for which 𝑛(𝑘)>𝑚(𝑘)>𝑘, 𝑑𝑥_2𝑚(𝑘),𝑥_2𝑛(𝑘)≥𝜖,𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑛(𝑘)−2}<𝜖, (33) and we get 𝜖≤𝑑𝑥_2𝑚(𝑘),𝑥_2𝑛(𝑘)≤𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑛(𝑘)−2}+𝑑𝑥_{2𝑛(𝑘)−2},𝑥_{2𝑛(𝑘)−1}+𝑑𝑥_{2𝑛(𝑘)−1},𝑥_2𝑛(𝑘)<𝜖+𝑑𝑥_{2𝑛(𝑘)−2},𝑥_{2𝑛(𝑘)−1}+𝑑𝑥_{2𝑛(𝑘)−1},𝑥_2𝑛(𝑘). (34) Letting 𝑘→∞ in the above inequality, we get lim_𝑛→∞𝑑𝑥_2𝑚(𝑘),𝑥_2𝑛(𝑘)=𝜖. (35) On the other hand, we also obtain that 𝜖≤𝑑𝑥_2𝑚(𝑘),𝑥_2𝑛(𝑘)≤𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑛(𝑘)−1}+𝑑𝑥_{2𝑛(𝑘)−1},𝑥_2𝑛(𝑘)≤𝑑𝑥_{2𝑚(𝑘)+1},𝑥_{2𝑛(𝑘)−1}+𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑚(𝑘)+1}+𝑑𝑥_{2𝑛(𝑘)−1},𝑥_2𝑛(𝑘)≤𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑛(𝑘)−1}+2𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑚(𝑘)+1}+𝑑𝑥_{2𝑛(𝑘)−1},𝑥_2𝑛(𝑘)≤𝑑𝑥_2𝑚(𝑘),𝑥_2𝑛(𝑘)+2𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑚(𝑘)+1}+2𝑑𝑥_{2𝑛(𝑘)−1},𝑥_2𝑛(𝑘). (36) Letting 𝑘→∞ in the above inequality, we get lim_𝑛→∞𝑑𝑥_2𝑚(𝑘),𝑥_2𝑛(𝑘)=lim_𝑛→∞𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑛(𝑘)−1}=lim_𝑛→∞𝑑𝑥_{2𝑚(𝑘)+1},𝑥_{2𝑛(𝑘)−1}=𝜖. (37) Since 𝑑𝑥_{2𝑚(𝑘)+1},𝑥_{2𝑛(𝑘)−1}≤𝑑𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘)+𝑑𝑥_2𝑛(𝑘),𝑥_{2𝑛(𝑘)−1}≤𝑑𝑥_{2𝑚(𝑘)+1},𝑥_{2𝑛(𝑘)−1}+2𝑑𝑥_2𝑛(𝑘),𝑥_{2𝑛(𝑘)−1}, (38) letting 𝑘→∞ in the above inequality, we have lim_𝑛→∞𝑑𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘)=𝜖. (39) Since 𝑥_2𝑚(𝑘)∈𝐴, 𝑥_{2𝑛(𝑘)−1}∈𝐵, and 𝑥_2𝑚(𝑘), 𝑥_{2𝑛(𝑘)−1} are comparable in 𝑋, we have 𝑑𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘)=𝑑𝑓𝑥_2𝑚(𝑘),𝑔𝑥_{2𝑛(𝑘)−1}≤𝛼𝑓𝑥_2𝑚(𝑘),𝑓𝑥_2𝑚(𝑘)𝛼𝑔𝑥_{2𝑛(𝑘)−1},𝑔𝑥_{2𝑛(𝑘)−1}×𝑑𝑓𝑥_2𝑚(𝑘),𝑔𝑥_{2𝑛(𝑘)−1}<𝜓𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑛(𝑘)−1},𝑑𝑥_2𝑚(𝑘),𝑓𝑥_2𝑚(𝑘),𝑑𝑥_{2𝑛(𝑘)−1},𝑔𝑥_{2𝑛(𝑘)−1},𝑑𝑥_2𝑚(𝑘),𝑔𝑥_{2𝑛(𝑘)−1},𝑑𝑥_{2𝑛(𝑘)−1},𝑓𝑥_2𝑚(𝑘)=𝜓𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑛(𝑘)−1},𝑑𝑥_2𝑚(𝑘),𝑥_{2𝑚(𝑘)+1},𝑑𝑥_{2𝑛(𝑘)−1},𝑥_2𝑛(𝑘),𝑑𝑥_2𝑚(𝑘),𝑥_2𝑛(𝑘),𝑑𝑥_{2𝑛(𝑘)−1},𝑥_{2𝑚(𝑘)+1}. (40) Letting 𝑘→∞ in the above inequality and using (37) and (39), we get 𝜖=lim_𝑛→∞𝑑𝑥_{2𝑚(𝑘)+1},𝑥_2𝑛(𝑘)<𝜓(𝜖,0,0,𝜖,𝜖)≤𝜖, (41) which implies a contradiction. So we get that {𝑥_𝑛} is a Cauchy sequence in (𝑋,⊑,𝑑).
Step  2. Finally, we prove the existence of common fixed point of 𝑓 and 𝑔.
Since (𝑋,⊑,𝑑) is complete and {𝑥_𝑛} is a Cauchy sequence in (𝑋,⊑,𝑑), there exists 𝜈∈𝑋 such that lim_𝑛→∞𝑥_𝑛=lim_𝑛→∞𝑥_2𝑛=lim_𝑛→∞𝑥_2𝑛−1=𝜈. (42) From (42) and since 𝛼(𝑥_𝑛,𝑥_𝑛)≥1 for all 𝑛∈ℕ, we have 𝛼(𝑓𝜈,𝑓𝜈)≥1 and 𝛼(𝑔𝜈,𝑔𝜈)≥1.
Since {𝑥_2𝑛} is a sequence in 𝐴 and 𝐴 is closed, by (42), we have that 𝜈∈𝐴. Similarly, since {𝑥_2𝑛+1} is a sequence in 𝐵 and 𝐵 is closed, by (42), we have that 𝜈∈𝐵. We now claim that 𝜈 is a common fixed point of 𝑓 and 𝑔. Without loss of generality, we assume that 𝑓 is continuous, and by (42), we have 𝑥_2𝑛+1=𝑓𝑥_2𝑛⟶𝜈,as𝑛→∞. (43) By the uniqueness of the limit, we have that 𝜈=𝑓𝜈.
Since 𝜈⊑𝜈 with 𝜈∈𝐴 and 𝜈∈𝐵, we have 𝑑𝛼𝑑𝑑(),𝑑.𝜈,𝑔𝜈=𝑑𝑓𝜈,𝑔𝜈≤𝛼𝑓𝜈,𝑓𝜈𝑔𝜈,𝑔𝜈𝑓𝜈,𝑔𝜈<𝜓𝜈,𝜈,𝑑𝜈,𝑓𝜈,𝑑𝜈,𝑔𝜈𝜈,𝑔𝜈,𝑑𝜈,𝑓𝜈=𝜓0,0,𝑑𝜈,𝑔𝜈,𝑑𝜈,𝑔𝜈,0≤𝑑𝜈,𝑔𝜈 (44) This implies that 𝜈=𝑔𝜈. So we complete the proof.

Applying Theorem 10 and if we let 𝛼(𝑥,𝑦)=1, then we immediately get the following theorem.

Theorem 11. Let (𝑋,⊑,𝑑) be a partially ordered complete metric space, let 𝐴,𝐵 be nonempty closed subsets of 𝑋 with 𝑋=𝐴∪𝐵, and let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that the pair (𝑓,𝑔) is a generalized cyclic Meir-Keeler-type (𝜓,𝐴,𝐵)-contraction and (𝐴,𝐵)-weakly increasing. If 𝑓 or 𝑔 is continuous, then 𝑓 and 𝑔 have a common fixed point in 𝑋.

We next state our second main result for the generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction as follows.

Theorem 12. Let (𝑋,⊑,𝑑) be a partially ordered complete metric space, let 𝐴,𝐵 be nonempty closed subsets of 𝑋 with 𝑋=𝐴∪𝐵, let 𝛼∶𝑋×𝑋→ℝ⁺, and let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that the pair (𝑓,𝑔) is a generalized cyclic Meir-Keeler-type (𝛼,𝜓,𝐴,𝐵)-contraction and (𝐴,𝐵)-weakly increasing. Suppose that the following conditions hold: (i)if {𝑥_𝑛} is a nondecreasing sequence in 𝑋 and lim_𝑛→∞𝑥_𝑛=𝜈, then 𝑥_𝑛⊑𝜈; (ii)there exists 𝑥₀∈𝐴 with 𝛼(𝑓𝑥₀,𝑓𝑥₀)≥1; (iii)if 𝛼(𝑥_𝑛,𝑥_𝑛)≥1 for all 𝑛∈ℕ and lim_𝑛→∞𝑥_𝑛=𝜈, then 𝛼(𝑓𝜈,𝑓𝜈)≥1 and 𝛼(𝑔𝜈,𝑔𝜈)≥1.
Then 𝑓 and 𝑔 have a common fixed point in 𝑋.

Proof. From the same proof’s process of Theorem 10, we can construct a nondecreasing sequence {𝑥_𝑛} in 𝑋 with 𝑥_2𝑛∈𝐴, 𝑥_2𝑛+1∈𝐵, and 𝑥_𝑛→𝜈 for some 𝜈∈𝑋. Since 𝑥_𝑛→𝜈 and 𝐴, 𝐵 are nonempty closed subsets of 𝑋, we have 𝑥_2𝑛→𝜈, 𝑥_2𝑛+1→𝜈, and 𝜈∈𝐴∩𝐵. By the condition (i), we get 𝑥_𝑛⊑𝜈 for all 𝑛∈ℕ.
Since 𝑥_2𝑛∈𝐴 and 𝜈∈𝐵, we have 𝑑𝑥_2𝑛+1,𝑔𝜈=𝑑𝑓𝑥_2𝑛,𝑔𝜈≤𝛼𝑓𝑥_2𝑛,𝑓𝑥_2𝑛𝛼𝑔𝜈,𝑔𝜈𝑑𝑓𝑥_2𝑛𝑑𝑥,𝑔𝜈<𝜓_2𝑛,𝜈,𝑑𝑥_2𝑛,𝑓𝑥_2𝑛,𝑑𝑥,𝑑𝜈,𝑔𝜈_2𝑛,𝑔𝜈,𝑑𝜈,𝑓𝑥_2𝑛=𝜓𝑑𝑥_2𝑛,𝜈,𝑑𝑥_2𝑛,𝑥_2𝑛+1,𝑑𝑥,𝑑𝜈,𝑔𝜈_2𝑛,𝑔𝜈,𝑑𝜈,𝑥_2𝑛+1. (45) Letting 𝑛→∞ in the above inequality, we get 𝑑𝜈,𝑔𝜈<𝜓0,0,𝑑𝜈,𝑔𝜈,𝑑𝜈,𝑔𝜈,0≤𝑑𝜈,𝑔𝜈. (46) This implies that 𝑑(𝜈,𝑔𝜈)=0; that is, 𝜈=𝑔𝜈. Similarly, we may show that 𝜈=𝑓𝜈. So 𝜈 is a common fixed point of 𝑓 and 𝑔.

Applying Theorem 12, it is easy to get the following theorem.

Theorem 13. Let (𝑋,⊑,𝑑) be a partially ordered complete metric space, let 𝐴,𝐵 be nonempty closed subsets of 𝑋 with 𝑋=𝐴∪𝐵, and let 𝑓,𝑔∶𝑋→𝑋 be two mappings such that the pair (𝑓,𝑔) is a generalized cyclic Meir-Keeler-type (𝜓,𝐴,𝐵)-contraction and (𝐴,𝐵)-weakly increasing. Suppose that the following condition holds: if {𝑥_𝑛} is a nondecreasing sequence in 𝑋 and lim_𝑛→∞𝑥_𝑛=𝜈, then 𝑥_𝑛⊑𝜈.
Then 𝑓 and 𝑔 have a common fixed point in 𝑋.