Common Fixed Point for Self-Mappings Satisfying an Implicit Lipschitz-Type Condition in Kaleva-Seikkala's Type Fuzzy Metric Spaces

Abstract

We first introduce the new real function class ℱ satisfying an implicit Lipschitz-type condition. Then, by using ℱ-type real functions, some common fixed point theorems for a pair of self-mappings satisfying an implicit Lipschitz-type condition in fuzzy metric spaces (in the sense of Kaleva and Seikkala) are established. As applications, we obtain the corresponding common fixed point theorems in metric spaces. Also, some examples are given, which show that there exist mappings which satisfy the conditions in this paper but cannot satisfy the general contractive type conditions.

1. Introduction

In 1984, Kaleva and Seikkala [1] introduced the concept of a fuzzy metric space by setting the distance between two points to be a nonnegative fuzzy real number and studied some of its properties. From then on, some important results for single-valued and multivalued mappings in fuzzy metric spaces, such as coincidence theorems, various fixed point theorems, and so forth, were stated in subsequent work (see [1–11], etc.). Recently, Zhang [12, 13] established some new common fixed point theorems for generalized contractive type mappings in metric spaces and for Lipschitz-type mappings in cone metric spaces. These theorems extended the original contractive type conditions. Moreover, various real function classes satisfying an implicit relation were introduced in [10, 14–23], and some common fixed point theorems for composite mappings satisfying an implicit relation were established in metric spaces and fuzzy metric spaces, respectively.

It is well known that the fuzzy metric space is an important generalization of the ordinary metric space (see [1]). Inspired by [13–23], we establish some common fixed point theorems for new contractive type mappings in fuzzy metric spaces in this paper. In Section 3, we first introduce the new real function class ℱ satisfying an implicit Lipschitz-type condition. Then, in Section 4, by using ℱ-type real functions, some common fixed point theorems for a pair of self-mappings satisfying an implicit Lipschitz-type condition in fuzzy metric spaces are established. In Section 5, as their applications, we obtain the corresponding common fixed point theorems in metric spaces. Also, some examples are given, which show that there exist mappings which satisfy the conditions in this paper but cannot satisfy the general contractive type conditions.

2. Preliminaries and Lemmas

Throughout this paper, let ℤ⁺ be the set of all positive integers, ℝ=(−∞,+∞) and ℝ⁺=[0,+∞). For the details of fuzzy real number, we refer the reader to Kaleva and Seikkala [1], Dubois and Prade [24], and Bag and Samanta [25].

Definition 1 (cf. Dubois and Prade [24]). A mapping 𝜂∶ℝ→[0,1] is called a fuzzy real number or fuzzy interval, whose 𝛼-level set is denoted by [𝜂]_𝛼={𝑡∈ℝ∶𝜂(𝑡)⩾𝛼}, if it satisfies two axioms.(1)There exists 𝑡₀∈ℝ such that 𝜂(𝑡₀)=1.(2)[𝜂]_𝛼=[𝜆_𝛼,𝜌_𝛼] is a closed interval of ℝ for each 𝛼∈(0,1], where −∞<𝜆_𝛼≤𝜌_𝛼<+∞. The set of all such fuzzy real numbers is denoted by 𝐺. If 𝜂∈𝐺 and 𝜂(𝑡)=0 whenever 𝑡<0, then 𝜂 is called a nonnegative fuzzy real number, and by 𝐺⁺ we mean the set of all nonnegative fuzzy real numbers. If 𝜆_𝛼=−∞ and 𝜌_𝛼=+∞ are admissible, then, for the sake of clarity, 𝜂 is called a generalized fuzzy real number. The sets of all generalized fuzzy real numbers or all generalized nonnegative fuzzy real numbers are denoted by 𝐺_∞ and 𝐺⁺_∞, respectively. In that case, if 𝜆_𝛼=−∞, for instance, then [𝜆_𝛼,𝜌_𝛼] means the interval (−∞,𝜌_𝛼].
The notation 0 stands for the fuzzy number satisfying 0(0)=1 and 0(𝑡)=0 if 𝑡≠0. Clearly, 0∈𝐺⁺. ℝ can be embedded in 𝐺: if 𝑎∈ℝ, then 𝑎∈𝐺 satisfies 𝑎(𝑡)=0(𝑡−𝑎).

Lemma 2 (Xiao et al. [8]). Let 𝜂∈𝐺, 𝛼∈(0,1], and [𝜂]_𝛼=[𝜆_𝛼,𝜌_𝛼]. Then(1)lim_{𝑡→−∞}𝜂(𝑡)=0=lim_𝑡→+∞𝜂(𝑡).(2)𝜂(𝑡) is a left continuous and nonincreasing function for 𝑡∈(𝜆₁,+∞).(3)𝜌_𝛼 is a left continuous and nonincreasing function for 𝛼∈(0,1].

Definition 3 (cf. Kaleva and Seikkala [1]). Suppose that 𝑋 is a nonempty set and that 𝑑 is a mapping from 𝑋×𝑋 into 𝐺⁺. Let 𝐿,𝑅∶[0,1]×[0,1]→[0,1] be two symmetric and nondecreasing functions such that 𝐿(0,0)=0 and 𝑅(1,1)=1. For 𝛼∈(0,1] and 𝑥,𝑦∈𝑋, define the mapping 𝑑𝑥,𝑦_𝛼=𝜆_𝛼𝑥,𝑦,𝜌_𝛼𝑥,𝑦. (1) The quadruple (𝑋,𝑑,𝐿,𝑅) is called a fuzzy metric space (briefly, FMS), and 𝑑 is called a fuzzy metric, if  (FM-1) 𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦; (FM-2) 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋; (FM-3) for all 𝑥,𝑦,𝑧∈𝑋∶ (FM-3L) 𝑑(𝑥,𝑦)(𝑠+𝑡)⩾𝐿(𝑑(𝑥,𝑧)(𝑠), 𝑑(𝑧,𝑦)(𝑡)), whenever 𝑠⩽𝜆₁(𝑥,𝑧), 𝑡⩽𝜆₁(𝑧,𝑦) and 𝑠+𝑡⩽𝜆₁(𝑥,𝑦); (FM-3R) 𝑑(𝑥,𝑦)(𝑠+𝑡)⩽𝑅(𝑑(𝑥,𝑧)(𝑠), 𝑑(𝑧,𝑦)(𝑡)), whenever 𝑠⩾𝜆₁(𝑥,𝑧), 𝑡⩾𝜆₁(𝑧,𝑦) and 𝑠+𝑡⩾𝜆₁(𝑥,𝑦).
If 𝑑 is a mapping from 𝑋×𝑋 into 𝐺⁺_∞ and (𝑋,𝑑,𝐿,𝑅) satisfies (FM-1)–(FM-3), then (𝑋,𝑑,𝐿,𝑅) is called a generalized fuzzy metric space (briefly, GFMS).
From Lemma 2 and Definition 3, we obtain the following consequences.

Lemma 4. Let (𝑋,𝑑,𝐿,𝑅) be a FMS, [𝑑(𝑥,𝑦)]_𝛼=[𝜆_𝛼(𝑥,𝑦),𝜌_𝛼(𝑥,𝑦)] for 𝛼∈(0,1], where 𝑥,𝑦∈𝑋 are any two fixed elements. Then (1)lim_{𝑡→−∞}𝑑(𝑥,𝑦)(𝑡)=0=lim_𝑡→+∞𝑑(𝑥,𝑦)(𝑡). (2)𝑑(𝑥,𝑦)(𝑡) is a left continuous and nonincreasing function for  𝑡∈(𝜆₁(𝑥,𝑦),+∞). (3)𝜌_𝛼(𝑥,𝑦) is a left continuous and nonincreasing function for  𝛼∈(0,1].

Lemma 5 (Xiao et al. [8]). Let (𝑋,𝑑,𝐿,𝑅) be a FMS, and suppose that (R-1) 𝑅⩽max; (R-2) lim_𝑎→0⁺𝑅(𝑎,𝑎)=0. Then (R-1) ⇒(R-2).

Lemma 6. Let (𝑋,𝑑,𝐿,𝑅) be a FMS. Then(1)(R-1) ⇒ for each 𝛼∈(0,1], 𝜌_𝛼(𝑥,𝑦)⩽𝜌_𝛼(𝑥,𝑧)+𝜌_𝛼(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋(cf. [4, 5]).(2)(R-2) ⇒ for each 𝛼∈(0,1] there exists 𝜇=𝜇(𝛼)∈(0,𝛼] such that 𝜌_𝛼(𝑥,𝑦)⩽𝜌_𝜇(𝑥,𝑧)+𝜌_𝜇(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋  (cf. [5, 6, 8]).

Lemma 7 (Kaleva and Seikkala [1]). Let (𝑋,𝑑,𝐿,𝑅) be a FMS with (R-2). Then the family {𝑈(𝜀,𝛼)∶𝜀>0,𝛼∈(0,1]} of sets 𝑈(𝜀,𝛼)={(𝑥,𝑦)∈𝑋×𝑋∶𝜌_𝛼(𝑥,𝑦)<𝜀} forms a basis for a Hausdorff uniformity on 𝑋×𝑋. Moreover, the sets 𝑁_𝑥(𝜀,𝛼)=𝑦∈𝑋∶𝜌_𝛼𝑥,𝑦<𝜀 (2) form a basis for a Hausdorff topology on 𝑋 and this topology is metrizable.

According to Lemma 7, convergence in a FMS (𝑋,𝑑,𝐿,𝑅) can be defined by sequences. A sequence {𝑥_𝑛} in 𝑋 is said to be convergent to 𝑥 (we write 𝑥_𝑛→𝑥 or lim_𝑛→∞𝑥_𝑛=𝑥) if lim_𝑛→∞𝑑(𝑥_𝑛,𝑥)=0; that is, lim_𝑛→∞𝜌_𝛼(𝑥_𝑛,𝑥)=0 for each 𝛼∈(0,1]; {𝑥_𝑛} is called a Cauchy sequence in 𝑋 if lim_{𝑚,𝑛→∞}𝑑(𝑥_𝑚,𝑥_𝑛)=0; equivalently, for any given 𝜀>0 and 𝛼∈(0,1], there exists 𝑁=𝑁(𝜀,𝛼)∈ℤ⁺ such that 𝜌_𝛼(𝑥_𝑚,𝑥_𝑛)<𝜀, whenever 𝑚,𝑛⩾𝑁; (𝑋,𝑑,𝐿,𝑅) is said to be complete, if each Cauchy sequence in 𝑋 is convergent to some point in 𝑋.

Lemma 8 (Kaleva and Seikkala [1]). Let (𝑋,𝑑,𝐿,𝑅) be a FMS with 𝑅⩽max. Then for each 𝛼∈(0,1], 𝜌_𝛼(𝑥,𝑦) is continuous at (𝑥,𝑦)∈𝑋×𝑋.

3. The Real Functions Satisfying an Implicit Lipschitz-Type Condition

Definition 9. A lower semicontinuous function 𝐹∶ℝ⁺₆→ℝ is called a real function satisfying an implicit Lipschitz-type condition, if the following conditions are satisfied. (ℱ-1) 𝐹(𝑡₁,…,𝑡₆) is nonincreasing in 𝑡₅,𝑡₆. (ℱ-2) There exist 𝐴,𝐵>0 with 𝐴𝐵<1 such that for all 𝑢,𝑣⩾0, we have (ℱ-2a) 𝑢⩽𝐴𝑣, whenever 𝐹(𝑢,𝑣,𝑣,𝑢,𝑣+𝑢,0)⩽0, (ℱ-2b) 𝑢⩽𝐵𝑣, whenever 𝐹(𝑢,𝑣,𝑢,𝑣,0,𝑣+𝑢)⩽0. (ℱ-3) For all 𝑢⩾0 with 𝐹(𝑢,𝑢,0,0,𝑢,𝑢)⩽0, we have 𝑢=0. We denote by ℱ the collection of all real functions 𝐹∶ℝ⁺₆→ℝ satisfying an implicit Lipschitz-type condition.

Remark 10. Let 𝐹∈ℱ; if 𝐹(𝑢,0,0,𝑢,𝑢,0)⩽0 or 𝐹(𝑢,0,𝑢,0,0,𝑢)⩽0, then by condition (ℱ-2) of  Definition 9, we have 𝑢=0.

The following examples show that the collection ℱ is a largish class of real functions.

Example 11. Let 𝑎∈(0,1/2). The function 𝐹₁∶ℝ⁺₆→ℝ is defined by 𝐹₁𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑎max𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆; (3) then 𝐹₁∈ℱ.
In fact, it is easy to see that 𝐹₁ is continuous. Also, (ℱ-1) and (ℱ-3) are easy to check. For any 𝑢,𝑣⩾0, if 𝐹₁(𝑢,𝑣,𝑣,𝑢,𝑣+𝑢,0)⩽0 or 𝐹₁(𝑢,𝑣,𝑢,𝑣,0,𝑣+𝑢)⩽0, then we have 𝑢⩽(𝑎/(1−𝑎))𝑣, or 𝑢⩽(𝑎/(1−𝑎))𝑣 respectively. Taking 𝐴=𝐵=𝑎/(1−𝑎), we have 𝐴𝐵=(𝑎/(1−𝑎))²<1; that is, (ℱ-2) holds. Hence 𝐹₁∈ℱ.

Example 12. Let 𝑎,𝑏,𝑐,𝑑>0 with 𝑎+𝑏+𝑐∈(0,1) and 𝑎+𝑑∈(0,1). The function 𝐹₂∶ℝ⁺₆→ℝ is defined by 𝐹₂𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡²₁−𝑡₁𝑎𝑡₂+𝑏𝑡₃+𝑐𝑡₄−𝑑𝑡₅𝑡₆; (4) then 𝐹₂∈ℱ.

Obviously, 𝐹₂ is continuous, and (ℱ-1) and (ℱ-3) are easy to check. For any 𝑢,𝑣⩾0, if 𝐹₂(𝑢,𝑣,𝑣,𝑢,𝑣+𝑢,0)⩽0 or 𝐹₂(𝑢,𝑣,𝑢,𝑣,0,𝑣+𝑢)⩽0, then we have 𝑢²−𝑢(𝑎𝑣+𝑏𝑣+𝑐𝑢)⩽0 or 𝑢²−𝑢(𝑎𝑣+𝑏𝑢+𝑐𝑣)⩽0, respectively, which implies that 𝑢⩽((𝑎+𝑏)/(1−𝑐))𝑣 or 𝑢⩽((𝑎+𝑐)/(1−𝑏))𝑣. Let 𝐴=(𝑎+𝑏)/(1−𝑐) and 𝐵=(𝑎+𝑐)/(1−𝑏). By 𝑎+𝑏+𝑐∈(0,1), we have 𝐴<1 and 𝐵<1; that is, 𝐴𝐵<1. Thus (ℱ-2) holds. Hence 𝐹₂∈ℱ.

Example 13. Let 𝑎∈(0,1/25). The function 𝐹₃∶ℝ⁺₆→ℝ is defined by 𝐹₃𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡³₁−𝑎𝑡³₂+⋯+𝑡³₆; (5) then 𝐹₃∈ℱ.
Obviously, 𝐹₃ is continuous, and (ℱ-1) and (ℱ-3) are easy to check.
For any 𝑢,𝑣⩾0, if 𝐹₃(𝑢,𝑣,𝑣,𝑢,𝑣+𝑢,0)⩽0, then we have 𝑢³−𝑎(2𝑣³+𝑢³+(𝑣+𝑢)³)⩽0, which implies that (1−𝑎)𝑢³⩽2𝑎𝑣³+𝑎(𝑣+𝑢)³⩽3𝑎(𝑣+𝑢)³𝑣⟹𝑢⩽³3𝑎/(1−𝑎)1−³().3𝑎/1−𝑎 (6) Similarly, if 𝐹₃(𝑢,𝑣,𝑢,𝑣,0,𝑣+𝑢)⩽0, then we also have 𝑢⩽𝑣³3𝑎/(1−𝑎)/(1−³3𝑎/(1−𝑎)). Let 𝐴=𝐵=³3𝑎/(1−𝑎)/(1−³3𝑎/(1−𝑎)). Note that 𝑎∈(0,1/25); we have 3𝑎/(1−𝑎)∈(0,1/8)⇒³3𝑎/(1−𝑎)∈(0,1/2); that is, 𝐴=𝐵∈(0,1); thus 𝐴𝐵<1. Therefore (ℱ-2) holds. Hence 𝐹₃∈ℱ.

Example 14. Let 𝜙₁,…,𝜙₅: ℝ⁺→[0,1) be five continuous functions satisfying the following conditions.(i)𝜙₁(𝑡)+𝜙₄(𝑡)+𝜙₅(𝑡)<1 for all 𝑡∈ℝ⁺,(ii)There exist 𝑎>0, 𝑏>0, 𝐶>0, 𝐷>0 with 𝐶𝐷<𝑎𝑏, such that inf_𝑡⩾01−𝜙₂(𝑡)−𝜙₅(𝑡)=𝑎,inf_𝑡⩾01−𝜙₃(𝑡)−𝜙₄(𝑡)=𝑏,sup_𝑡⩾0𝜙₁(𝑡)+𝜙₂(𝑡)+𝜙₄(𝑡)=𝐶,sup_𝑡⩾0𝜙₁(𝑡)+𝜙₃(𝑡)+𝜙₅(𝑡)=𝐷. (7)We define the function 𝐹₄∶ℝ⁺₆→ℝ as follows: 𝐹₄𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝜙₁𝑡₂𝑡₂+𝜙₂𝑡₂𝑡₃+𝜙₃𝑡₂𝑡₄+𝜙₄𝑡₂𝑡₅+𝜙₅𝑡₂𝑡₆. (8) Then 𝐹₄∈ℱ.

In fact, it is easy to see that 𝐹₄ is continuous, and (ℱ-1) is satisfied.

For any 𝑢,𝑣⩾0, if 𝐹₄(𝑢,𝑣,𝑣,𝑢,𝑣+𝑢,0)⩽0, then we have 𝑢−(𝜙₁(𝑣)𝑣+𝜙₂(𝑣)𝑣+𝜙₃(𝑣)𝑢+𝜙₄(𝑣)(𝑣+𝑢))⩽0, which implies that 𝑢⩽((𝜙₁(𝑣)+𝜙₂(𝑣)+𝜙₄(𝑣))/(1−𝜙₃(𝑣)−𝜙₄(𝑣)))𝑣. From condition (ii), we obtain 𝑢⩽(𝐶/𝑏)𝑣. Similarly, if 𝐹₄(𝑢,𝑣,𝑢,𝑣,0,𝑣+𝑢)⩽0, then we have 𝑢⩽((𝜙₁(𝑣)+𝜙₃(𝑣)+𝜙₅(𝑣))/(1−𝜙₂(𝑣)−𝜙₅(𝑣)))𝑣⩽𝐷/𝑎𝑣. Let 𝐴=𝐶/𝑏, 𝐵=𝐷/𝑎. Note that 𝐶𝐷<𝑎𝑏; we have 𝐴𝐵<1; that is, (ℱ-2) holds.

Furthermore, for any 𝑢⩾0, if 𝐹₄(𝑢,𝑢,0,0,𝑢,𝑢)⩽0, then we have 𝑢−(𝜙₁(𝑢)𝑢+𝜙₄(𝑢)𝑢+𝜙₅(𝑢)𝑢)⩽0, which implies that (1−𝜙₁(𝑢)−𝜙₄(𝑢)−𝜙₅(𝑢))𝑢⩽0. Note that 𝜙₁(𝑡)+𝜙₄(𝑡)+𝜙₅(𝑡)<1 for all 𝑡∈ℝ⁺; we have 𝑢=0. This shows that (ℱ-3) holds. Hence 𝐹₄∈ℱ.

Example 15. Let the function 𝐹₅∶ℝ⁺₆→ℝ be defined by 𝐹₅𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑡₂+120𝑡₂𝑡+21²+𝑡₂+120𝑡₂𝑡+21³+2𝑡₂+15𝑡₂𝑡+6⁴+𝑡₂+140𝑡₂𝑡+42⁵+𝑡₂+12𝑡₂𝑡+3⁶; (9) then 𝐹₅∈ℱ.

In fact, in Example 14, taking 𝜙₁(𝑡)=(𝑡+1)/(20𝑡+21), 𝜙₂(𝑡)=(𝑡+1)/(20𝑡+21), 𝜙₃(𝑡)=(2𝑡+1)/(5𝑡+6),𝜙₄(𝑡)=(𝑡+1)/(40𝑡+42), and 𝜙₅(𝑡)=(𝑡+1)/(2𝑡+3), we obtain five continuous functions 𝜙₁,…,𝜙₅ from ℝ⁺ into [0,1) satisfying the following conditions: 1742⩽𝜙₁(𝑡)+𝜙₄(𝑡)+𝜙₅(𝑡)<2340<1,inf_𝑡⩾01−𝜙₂(𝑡)−𝜙₅(𝑡=9),20inf_𝑡⩾01−𝜙₃(𝑡)−𝜙₄(𝑡)=23,40sup_𝑡⩾0𝜙₁(𝑡)+𝜙₂(𝑡)+𝜙₄(𝑡=1)8,sup_𝑡⩾0𝜙₁(𝑡)+𝜙₃(𝑡)+𝜙₅(𝑡)=19.20 (10)

It is evident that 1/8⋅19/20=19/160<207/800=9/20⋅23/40, and so all conditions of Example 14 are satisfied. Therefore, 𝐹₅∈ℱ.

Example 16. Let 𝑎,𝑏,𝑐,𝑑,𝑒⩾0 with 𝑎+𝑑+𝑒<1,𝑐+𝑑<1 and 𝑏+𝑒<1. There exists 𝛿>0 such that 𝑎+𝑏+𝑐+𝑑+𝑒=1+𝛿 and (𝑐−𝑏)(𝑒−𝑑)>2𝛿. We define the function 𝐹₆∶ℝ⁺₆→ℝ as follows: 𝐹₆𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑎𝑡₂+𝑏𝑡₃+𝑐𝑡₄+𝑑𝑡₅+𝑒𝑡₆; (11) then 𝐹₆∈ℱ.

Obviously, in Example 14, taking 𝜙₁(𝑡)=𝑎,𝜙₂(𝑡)=𝑏, 𝜙₃(𝑡)=𝑐, 𝜙₄(𝑡)=𝑑, and 𝜙₅(𝑡)=𝑒, we obtain five continuous functions 𝜙₁,…,𝜙₅ from ℝ⁺ into [0,1). Moreover, by (𝑐−𝑏)(𝑒−𝑑)>2𝛿, we have 𝑒≠𝑑. Hence, 0<𝑎+𝑑+𝑒<1; that is, condition (i) of Example 14 holds.

Furthermore, 0<1−𝑏−𝑒, 0<1−𝑐−𝑑 is obvious. Note that 𝑎<1 and (𝑐−𝑏)(𝑒−𝑑)>2𝛿; it follows that 𝑎(1+𝛿)+𝑏𝑒+𝑐𝑑<𝑎+𝛿+𝑏𝑒+𝑐𝑑<𝑎−𝛿+𝑏𝑑+𝑐𝑒. (12) By 𝑎+𝑏+𝑐+𝑑+𝑒=1+𝛿, we obtain 𝑎(𝑎+𝑏+𝑐+𝑑+𝑒)+𝑏𝑒+𝑐𝑑+𝑏𝑐+𝑑𝑒<1−𝑏−𝑐−𝑑−𝑒+𝑏𝑑+𝑐𝑒+𝑏𝑐+𝑑𝑒, (13) which implies that (𝑎+𝑏+𝑑)(𝑎+𝑐+𝑒)<(1−𝑐−𝑑)(1−𝑏−𝑒). Hence, condition (ii) of Example 14 is satisfied. Thus, by Example 14, we have 𝐹₆∈ℱ.

Remark 17. The numbers 𝑎,𝑏,𝑐,𝑑,𝑒, and 𝛿 in Example 16 really exist. For example, if we take 𝛿=1/40, 𝑎=1/20, 𝑏=1/20, 𝑐=2/5, 𝑑=1/40, and 𝑒=1/2, then 𝑎+𝑏+𝑐+𝑑+𝑒=1+1/40, 𝑎+𝑑+𝑒=23/40<1, 𝑐+𝑑=17/40<1, 𝑏+𝑒=11/20<1, and (𝑐−𝑏)(𝑒−𝑑)=133/800>1/20; that is, the conditions of Example 16 are satisfied.

Example 18. Define the function 𝐹₇∶ℝ⁺₆→ℝ as follows: 𝐹₇𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆=𝑡₁−𝑎𝑡₂+𝑏𝑡₃+𝑐𝑡₄+𝑑𝑡₅+𝑒𝑡₆, (14) where 𝑎,𝑏,𝑐,𝑑, and 𝑒 are nonnegative real numbers, with 𝑎+𝑏+𝑐+𝑑+𝑒=1 and either 𝑐>𝑏, 𝑒>𝑑 or 𝑐<𝑏, 𝑒<𝑑. Then 𝐹₇∈ℱ.

In fact, if we take 𝜙₁(𝑡)=𝑎, 𝜙₂(𝑡)=𝑏, 𝜙₃(𝑡)=𝑐, 𝜙₄(𝑡)=𝑑, and 𝜙₅(𝑡)=𝑒, then condition (i) of Example 14 is obviously satisfied. Note that 𝑐>𝑏, 𝑒>𝑑; we have (𝑎+𝑏+𝑑)(𝑎+𝑐+𝑒)=(1−𝑐−𝑒)(1−𝑏−𝑑)<(1−𝑐−𝑑)(1−𝑏−𝑒); that is, condition (ii) of Example 14 is satisfied. Similarly, we can prove the case of 𝑐<𝑏, 𝑒<𝑑. Therefore, by Example 14, 𝐹₇∈ℱ.

4. Main Results

Theorem 19. Let (𝑋,𝑑,𝐿,𝑅) be a complete FMS with 𝑅⩽max and let 𝑆 and 𝑇 be two self-mappings on (𝑋,𝑑,𝐿,𝑅). If there exists 𝐹∈ℱ such that 𝑢−𝐹𝑢,𝑣,𝑤,𝑠,𝑡,𝑔⩾𝜆₁,𝑑𝑑(𝑢)(𝑣)(𝑤),𝑑(𝑠)(𝑡)𝑔,𝑆𝑥,𝑇𝑦𝑆𝑥,𝑇𝑦𝑢−𝐹𝑢,𝑣,𝑤,𝑠,𝑡,𝑔⩽max𝑆𝑥,𝑇𝑦,𝑑𝑥,𝑦,𝑑𝑥,𝑆𝑥)(𝑦,𝑇𝑦,𝑑𝑥,𝑇𝑦,𝑑𝑦,𝑆𝑥 (15) for all 𝑥,𝑦∈𝑋, whenever 𝑢⩾𝜆₁(𝑆𝑥,𝑇𝑦), 𝑣⩾𝜆₁(𝑥,𝑦), 𝑤⩾𝜆₁(𝑥,𝑆𝑥), 𝑠⩾𝜆₁(𝑦,𝑇𝑦), 𝑡⩾𝜆₁(𝑥,𝑇𝑦), and 𝑔⩾𝜆₁(𝑦,𝑆𝑥), then 𝑆 and 𝑇 have a unique common fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_2𝑘+1=𝑆𝑥_2𝑘, 𝑥_2𝑘+2=𝑇𝑥_2𝑘+1, 𝑘=0,1,2,…, converges to the fixed point.

Proof. Firstly, we use (15) to prove that the following inequality: 𝐹𝜌_𝛼𝑆𝑥,𝑇𝑦,𝜌_𝛼𝑥,𝑦,𝜌_𝛼(),𝜌𝑥,𝑆𝑥_𝛼𝑦,𝑇𝑦,𝜌_𝛼𝑥,𝑇𝑦,𝜌_𝛼𝑦,𝑆𝑥⩽0 (16) holds for all 𝑥,𝑦∈𝑋 and 𝛼∈(0,1].
In fact, for each 𝑥,𝑦∈𝑋 and 𝛼∈(0,1], if we set 𝜌_𝛼(𝑆𝑥,𝑇𝑦)=𝑢, 𝜌_𝛼(𝑥,𝑦)=𝑣, 𝜌_𝛼(𝑥,𝑆𝑥)=𝑤, 𝜌_𝛼(𝑦,𝑇𝑦)=𝑠, 𝜌_𝛼(𝑥,𝑇𝑦)=𝑡, 𝜌_𝛼(𝑦,𝑆𝑥)=𝑔, then, for any 𝜀>0, it is obvious that 𝑑(𝑆𝑥,𝑇𝑦)(𝑢+𝜀)<𝛼, 𝑑(𝑥,𝑦)(𝑣+𝜀)<𝛼, 𝑑(𝑥,𝑆𝑥)(𝑤+𝜀)<𝛼, 𝑑(𝑦,𝑇𝑦)(𝑠+𝜀)<𝛼, 𝑑(𝑥,𝑇𝑦)(𝑡+𝜀)<𝛼, 𝑑(𝑦,𝑆𝑥)(𝑔+𝜀)<𝛼, and 𝑢+𝜀⩾𝜆₁(𝑆𝑥,𝑇𝑦), 𝑣+𝜀⩾𝜆₁(𝑥,𝑦), 𝑤+𝜀⩾𝜆₁(𝑥,𝑆𝑥), 𝑠+𝜀⩾𝜆₁(𝑦,𝑇𝑦), 𝑡+𝜀⩾𝜆₁(𝑥,𝑇𝑦), 𝑔+𝜀⩾𝜆₁(𝑦,𝑆𝑥). By (15), we have 𝑢+𝜀−𝐹(𝑢+𝜀,𝑣+𝜀,𝑤+𝜀,𝑠+𝜀,𝑡+𝜀,𝑔+𝜀) ⩾ 𝜆₁(𝑆𝑥,𝑇𝑦) and 𝑑(𝑆𝑥,𝑇𝑦)(𝑢+𝜀−𝐹(𝑢+𝜀,𝑣+𝜀,𝑤+𝜀,𝑠+𝜀,𝑡+𝜀,𝑔+𝜀))<𝛼, which imply that 𝐹𝜌_𝛼𝑆𝑥,𝑇𝑦+𝜀,𝜌_𝛼𝑥,𝑦+𝜀,𝜌_𝛼(𝑥,𝑆𝑥)+𝜀,𝜌_𝛼𝑦,𝑇𝑦+𝜀,𝜌_𝛼𝑥,𝑇𝑦+𝜀,𝜌_𝛼𝑦,𝑆𝑥+𝜀<𝜀; (17) then by the arbitrariness of 𝜀 and the lower semicontinuity of 𝐹, we have 𝐹𝜌_𝛼𝑆𝑥,𝑇𝑦,𝜌_𝛼𝑥,𝑦,𝜌_𝛼(𝑥,𝑆𝑥),𝜌_𝛼,𝜌𝑦,𝑇𝑦_𝛼𝑥,𝑇𝑦,𝜌_𝛼𝑦,𝑆𝑥⩽liminf_𝜀→0𝐹𝜌_𝛼𝑆𝑥,𝑇𝑦+𝜀,𝜌_𝛼𝑥,𝑦+𝜀,𝜌_𝛼()𝜌𝑥,𝑆𝑥+𝜀,_𝛼𝑦,𝑇𝑦+𝜀,𝜌_𝛼𝑥,𝑇𝑦+𝜀,𝜌_𝛼𝑦,𝑆𝑥+𝜀⩽0, (18) for each 𝑥,𝑦∈𝑋 and 𝛼∈(0,1]; that is, the inequality (16) holds for all 𝑥,𝑦∈𝑋 and 𝛼∈(0,1].
For any 𝑥₀∈𝑋, we construct an iterative sequence {𝑥_𝑛} in 𝑋 as follows: 𝑥_2𝑘+1=𝑆𝑥_2𝑘,𝑥_2𝑘+2=𝑇𝑥_2𝑘+1,𝑘=0,1,2,…. (19) For 𝑘=0,1,2,…, applying (16), we obtain for each 𝛼∈(0,1]𝐹𝜌_𝛼𝑆𝑥_2𝑘,𝑇𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘,𝑆𝑥_2𝑘,𝜌_𝛼𝑥_2𝑘+1,𝑇𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘,𝑇𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+1,𝑆𝑥_2𝑘=𝐹𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2,𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2,𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+2,0⩽0. (20) By the known condition 𝑅⩽max and conclusion (1) of Lemma 6, we have 𝜌_𝛼(𝑥_2𝑘,𝑥_2𝑘+2)⩽𝜌_𝛼(𝑥_2𝑘, 𝑥_2𝑘+1)+𝜌_𝛼(𝑥_2𝑘+1,𝑥_2𝑘+2). Note that 𝐹 is nonincreasing in 𝑡₆; it is not difficult to see that 𝐹𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2,𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2,𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1+𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2,0⩽0,foreach𝛼∈(0,1]. (21) Since 𝐹∈ℱ, there exists 𝐴>0 such that 𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2⩽𝐴𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1,foreach𝛼∈(0,1]. (22)
Similarly, for 𝑘=0,1,2,…, applying (16), we obtain for each 𝛼∈(0,1]𝐹𝜌_𝛼𝑆𝑥_2𝑘+2,𝑇𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+2,𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+2,𝑆𝑥_2𝑘+2,𝜌_𝛼𝑥_2𝑘+1,𝑇𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+2,𝑇𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+1,𝑆𝑥_2𝑘+2=𝐹𝜌_𝛼𝑥_2𝑘+3,𝑥_2𝑘+2,𝜌_𝛼𝑥_2𝑘+2,𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+2,𝑥_2𝑘+3,𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2𝜌,0,_𝛼𝑥_2𝑘+1,𝑥_2𝑘+3⩽0. (23) By 𝑅⩽max and (1) of Lemma 6, we have 𝜌_𝛼(𝑥_2𝑘+1,𝑥_2𝑘+3)⩽𝜌_𝛼(𝑥_2𝑘+1,𝑥_2𝑘+2)+𝜌_𝛼(𝑥_2𝑘+2, 𝑥_2𝑘+3). Note that 𝐹 is nonincreasing in 𝑡₆; we obtain 𝐹𝜌_𝛼𝑥_2𝑘+3,𝑥_2𝑘+2,𝜌_𝛼𝑥_2𝑘+2,𝑥_2𝑘+1,𝜌_𝛼𝑥_2𝑘+2,𝑥_2𝑘+3,𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2,0,𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2+𝜌_𝛼𝑥_2𝑘+2,𝑥_2𝑘+3⩽0,foreach𝛼∈(0,1]. (24) Since 𝐹∈ℱ, there exists 𝐵>0 such that 𝜌_𝛼𝑥_2𝑘+3,𝑥_2𝑘+2⩽𝐵𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2,foreach𝛼∈(0,1]. (25) Using inductive method, for 𝑘=0,1,2,…, we can obtain 𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2⩽𝐴𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1⩽𝐴𝐵𝜌_𝛼𝑥_2𝑘−1,𝑥_2𝑘⩽⋯⩽(𝐴𝐵)^𝑘𝐴𝜌_𝛼𝑥₀,𝑥₁,𝜌_𝛼𝑥_2𝑘+2,𝑥_2𝑘+3⩽𝐵𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2⩽𝐵𝐴𝜌_𝛼𝑥_2𝑘,𝑥_2𝑘+1⩽⋯⩽(𝐴𝐵)^𝑘+1𝜌_𝛼𝑥₀,𝑥₁,foreach𝛼∈(0,1]. (26)
Next, we prove that the sequence {𝑥_𝑛} is a Cauchy sequence. For 𝑘<𝑝, by 𝑅⩽max and conclusion (1) of Lemma 6, we have for each 𝛼∈(0,1]𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑝+1⩽𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑘+2+⋯+𝜌_𝛼𝑥_2𝑝,𝑥_2𝑝+1⩽𝐴^𝑝−1_𝑖=𝑘(𝐴𝐵)^𝑖+^𝑝_𝑖=𝑘+1(𝐴𝐵)^𝑖𝜌_𝛼𝑥₀,𝑥₁⩽𝐴(𝐴𝐵)^𝑘+()1−𝐴𝐵𝐴𝐵^𝑘+1𝜌1−𝐴𝐵^𝛼𝑥₀,𝑥₁⩽𝑀(𝐴𝐵)^𝑘𝜌_𝛼𝑥₀,𝑥₁, (27) where 𝑀=(𝐴+𝐴𝐵)/(1−𝐴𝐵). By the similar reasoning process, we have for each 𝛼∈(0,1]𝜌_𝛼𝑥_2𝑘,𝑥_2𝑝+1⩽𝑀(𝐴𝐵)^𝑘𝜌_𝛼𝑥₀,𝑥₁,𝜌_𝛼𝑥_2𝑘,𝑥_2𝑝⩽𝑀(𝐴𝐵)^𝑘𝜌_𝛼𝑥₀,𝑥₁,𝜌_𝛼𝑥_2𝑘+1,𝑥_2𝑝⩽𝑀(𝐴𝐵)^𝑘𝜌_𝛼𝑥₀,𝑥₁. (28) Then there exists 𝑘(𝑛) with (𝑛−1)/2⩽𝑘⩽𝑛/2 for 0<𝑛<𝑚, such that for each 𝛼∈(0,1]𝜌_𝛼𝑥_𝑚,𝑥_𝑛⩽𝑀(𝐴𝐵)^𝑘𝜌_𝛼𝑥₀,𝑥₁. (29) Since 0<𝐴𝐵<1, it is evident that the sequence {𝑥_𝑛} is a Cauchy sequence in 𝑋. By the completeness of 𝑋, we set lim_𝑛→∞𝑥_𝑛=𝑥_∗∈𝑋. Applying (16), we have 𝐹𝜌_𝛼𝑆𝑥_∗,𝑇𝑥_2𝑛+1,𝜌_𝛼𝑥_∗,𝑥_2𝑛+1,𝜌_𝛼𝑥_∗,𝑆𝑥_∗,𝜌_𝛼𝑥_2𝑛+1,𝑇𝑥_2𝑛+1,𝜌_𝛼𝑥_∗,𝑇𝑥_2𝑛+1,𝜌_𝛼𝑥_2𝑘+1,𝑆𝑥_∗=𝐹𝜌_𝛼𝑆𝑥_∗,𝑥_2𝑛+2,𝜌_𝛼𝑥_∗,𝑥_2𝑛+1,𝜌_𝛼𝑥_∗,𝑆𝑥_∗,𝜌_𝛼𝑥_2𝑛+1,𝑥_2𝑛+2,𝜌_𝛼𝑥_∗,𝑥_2𝑛+2,𝜌_𝛼𝑥_2𝑘+1,𝑆𝑥_∗⩽0, (30) for each 𝛼∈(0,1]. Let 𝑛→∞; by the lower semicontinuity of 𝐹 and Lemma 8, we have 𝐹𝜌_𝛼𝑆𝑥_∗,𝑥_∗,0,𝜌_𝛼𝑥_∗,𝑆𝑥_∗,0,0,𝜌_𝛼𝑥_∗,𝑆𝑥_∗⩽liminf_𝑛→∞𝐹𝜌_𝛼𝑆𝑥_∗,𝑥_2𝑛+2,𝜌_𝛼𝑥_∗,𝑥_2𝑛+1,𝜌_𝛼𝑥_∗,𝑆𝑥_∗,𝜌_𝛼𝑥_2𝑛+1,𝑥_2𝑛+2,𝜌_𝛼𝑥_∗,𝑥_2𝑛+2,𝜌_𝛼𝑥_2𝑘+1,𝑆𝑥_∗⩽0, (31) for each 𝛼∈(0,1]. By Remark 10, 𝜌_𝛼(𝑆𝑥_∗,𝑥_∗)=0 for each 𝛼∈(0,1], which implies that 𝑥_∗ is a fixed point of 𝑆.
Similarly, for each 𝛼∈(0,1], we have 𝐹𝜌_𝛼𝑆𝑥_2𝑛,𝑇𝑥_∗,𝜌_𝛼𝑥_2𝑛,𝑥_∗,𝜌_𝛼𝑥_2𝑛,𝑆𝑥_2𝑛,𝜌_𝛼𝑥_∗,𝑇𝑥_∗,𝜌_𝛼𝑥_2𝑛,𝑇𝑥_∗,𝜌_𝛼𝑥_∗,𝑆𝑥_2𝑛=𝐹𝜌_𝛼𝑥_2𝑛+1,𝑇𝑥_∗,𝜌_𝛼𝑥_2𝑛,𝑥_∗,𝜌_𝛼𝑥_2𝑛,𝑥_2𝑛+1,𝜌_𝛼𝑥_∗,𝑇𝑥_∗,𝜌_𝛼𝑥_2𝑛,𝑇𝑥_∗,𝜌_𝛼𝑥_∗,𝑥_2𝑛+1⩽0. (32) Let 𝑛→∞; by the lower semicontinuity of 𝐹 and Lemma 8, we have for each 𝛼∈(0,1]𝐹𝜌_𝛼𝑥_∗,𝑇𝑥_∗,0,0,𝜌_𝛼𝑥_∗,𝑇𝑥_∗,𝜌_𝛼𝑥_∗,𝑇𝑥_∗,0⩽0. (33) By Remark 10, 𝜌_𝛼(𝑥_∗,𝑇𝑥_∗)=0 for each 𝛼∈(0,1], which implies that 𝑥_∗ is also a fixed point of 𝑇. Thus 𝑥_∗ is a common fixed point of 𝑆,𝑇.
Lastly, we prove the uniqueness of the common fixed point. If 𝑥^∗ is another common fixed point of 𝑆,𝑇, then by (16), we have for each 𝛼∈(0,1]𝐹𝜌_𝛼𝑆𝑥_∗,𝑇𝑥^∗,𝜌_𝛼𝑥_∗,𝑥^∗,𝜌_𝛼𝑥_∗,𝑆𝑥_∗,𝜌_𝛼𝑥^∗,𝑇𝑥^∗,𝜌_𝛼𝑥_∗,𝑇𝑥^∗,𝜌_𝛼𝑥^∗,𝑆𝑥_∗=𝐹𝜌_𝛼𝑥_∗,𝑥^∗,𝜌_𝛼𝑥_∗,𝑥^∗𝜌,0,0,_𝛼𝑥_∗,𝑥^∗,𝜌_𝛼𝑥^∗,𝑥_∗⩽0. (34) Note that 𝐹∈ℱ, and by (ℱ-3) of Definition 9, we obtain 𝜌_𝛼(𝑥^∗,𝑥_∗)=0 for each 𝛼∈(0,1]; hence 𝑥^∗=𝑥_∗. The uniqueness is proved and we complete the proof of the theorem.

According to the proof of Theorem 19, we can easily obtain the following corollary.

Corollary 20. Let (𝑋,𝑑,𝐿,𝑅) be a complete FMS with 𝑅⩽max and let 𝑆 and 𝑇 be two self-mappings on (𝑋,𝑑,𝐿,𝑅). If there exists 𝐹∈ℱ such that (16) holds for all 𝑥,𝑦∈𝑋 and 𝛼∈(0,1], then 𝑆 and 𝑇 have a unique common fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_2𝑘+1=𝑆𝑥_2𝑘, 𝑥_2𝑘+2=𝑇𝑥_2𝑘+1, 𝑘=0,1,2,…, converges to the fixed point.

Theorem 21. Let (𝑋,𝑑,𝐿,𝑅) be a complete FMS with 𝑅⩽max. Let 𝜙₁,…,𝜙₅∶ℝ⁺→[0,1) be five continuous functions which satisfy the following conditions:(i)𝜙₁(𝑡)+𝜙₄(𝑡)+𝜙₅(𝑡)<1 for all 𝑡∈ℝ⁺,(ii)0<inf_𝑡⩾0{1−𝜙₂(𝑡)−𝜙₅(𝑡)}=𝑎, 0<inf_𝑡⩾0{1−𝜙₃(𝑡)−𝜙₄(𝑡)}=𝑏, 0<sup_𝑡⩾0{𝜙₁(𝑡)+𝜙₂(𝑡)+𝜙₄(𝑡)}=𝐶, 0<sup_𝑡⩾0{𝜙₁(𝑡)+𝜙₃(𝑡)+𝜙₅(𝑡)}=𝐷 and 𝐶𝐷<𝑎𝑏.Let 𝑆 and 𝑇 be two self-mappings on (𝑋,𝑑,𝐿,𝑅) such that 𝜌_𝛼𝑆𝑥,𝑇𝑦⩽𝜙₁𝜌_𝛼𝑥,𝑦𝜌_𝛼𝑥,𝑦+𝜙₂𝜌_𝛼𝑥,𝑦×𝜌_𝛼(𝑥,𝑆𝑥)+𝜙₃𝜌_𝛼𝑥,𝑦𝜌_𝛼𝑦,𝑇𝑦+𝜙₄𝜌_𝛼𝑥,𝑦𝜌_𝛼𝑥,𝑇𝑦+𝜙₅𝜌_𝛼𝑥,𝑦𝜌_𝛼𝑦,𝑆𝑥, (35) for all 𝑥,𝑦∈𝑋 and 𝛼∈(0,1]. Then 𝑆 and 𝑇 have a unique common fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_2𝑘+1=𝑆𝑥_2𝑘,𝑥_2𝑘+2=𝑇𝑥_2𝑘+1,𝑘=0,1,2,…, converges to the fixed point.

Proof. Taking 𝐹(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)=𝑡₁−[𝜙₁(𝑡₂)𝑡₂+𝜙₂(𝑡₂)𝑡₃+𝜙₃(𝑡₂)𝑡₄+𝜙₄(𝑡₂)𝑡₅+𝜙₅(𝑡₂)𝑡₆], from the known conditions (i) and (ii) and Example 14, we obtain 𝐹=𝐹₄∈ℱ. Furthermore, from (35) we can easily derive the inequality (16). Then by Corollary 20, the theorem is proved.

Corollary 22. Let (𝑋,𝑑,𝐿,𝑅) be a complete FMS with 𝑅⩽max and let 𝑆 and 𝑇 be two self-mappings on (𝑋,𝑑,𝐿,𝑅). If there exist 𝑎,𝑏,𝑐,𝑑,𝑒⩾0 and 𝛿>0 with 𝑎+𝑏+𝑐+𝑑+𝑒=1+𝛿, 𝑎+𝑑+𝑒<1, 𝑐+𝑑<1, 𝑏+𝑒<1 and (𝑐−𝑏)(𝑒−𝑑)>2𝛿, such that 𝜌_𝛼𝑆𝑥,𝑇𝑦⩽𝑎𝜌_𝛼𝑥,𝑦+𝑏𝜌_𝛼(𝑥,𝑆𝑥)+𝑐𝜌_𝛼𝑦,𝑇𝑦+𝑑𝜌_𝛼𝑥,𝑇𝑦+𝑒𝜌_𝛼𝑦,𝑆𝑥, (36) for all 𝑥,𝑦∈𝑋 and 𝛼∈(0,1], then 𝑆 and 𝑇 have a unique common fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_2𝑘+1=𝑆𝑥_2𝑘, 𝑥_2𝑘+2=𝑇𝑥_2𝑘+1, 𝑘=0,1,2,…, converges to the fixed point.

Proof. Taking 𝐹(𝑡₁,𝑡₂,𝑡₃,𝑡₄,𝑡₅,𝑡₆)=𝑡₁−(𝑎𝑡₂+𝑏𝑡₃+𝑐𝑡₄+𝑑𝑡₅+𝑒𝑡₆), note that the known conditions and Example 16, we have 𝐹=𝐹₆∈ℱ. Furthermore, from (36) we can easily derive the inequality (16). Then by Corollary 20, the corollary is proved.

5. Applications to the Ordinary Metric Spaces and Examples

In this section, we first establish some common fixed point theorems for a pair of self-mappings satisfying an implicit Lipschitz-type condition in complete metric spaces. After that, we give two examples, by which we can claim that our conclusions are really generalizations of the early results.

Let (𝑋,𝜌) be an ordinary metric space and 𝑑𝑡)=,,𝑥,𝑦(1,𝑡=𝜌𝑥,𝑦0,𝑡≠𝜌𝑥,𝑦∀𝑥,𝑦∈𝑋,𝑡∈ℝ. (37) Then (𝑋,𝑑,min,max) is a FMS (cf. [1, 9]). It is easy to see that (𝑋,𝜌) and (𝑋,𝑑,min,max) are homeomorphic and 𝜌(𝑥,𝑦)=𝜌_𝛼(𝑥,𝑦) for all 𝛼∈(0,1].

Theorem 23. Let (𝑋,𝜌) be a complete metric space and let 𝑆 and 𝑇 be two self-mappings on (𝑋,𝜌). If there exists 𝐹∈ℱ such that 𝐹𝜌(),𝜌𝑆𝑥,𝑇𝑦,𝜌𝑥,𝑦,𝜌𝑥,𝑆𝑥𝑦,𝑇𝑦,𝜌𝑥,𝑇𝑦,𝜌𝑦,𝑆𝑥⩽0, (38) for all 𝑥,𝑦∈𝑋. Then 𝑆 and 𝑇 have a unique common fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_2𝑘+1=𝑆𝑥_2𝑘, 𝑥_2𝑘+2=𝑇𝑥_2𝑘+1, 𝑘=0,1,2,…, converges to the fixed point.

Proof. Note that the topology and completeness of (𝑋,𝜌) and the induced FMS (𝑋,𝑑,min,max) are coincident, as well as 𝜌(𝑥,𝑦)=𝜌_𝛼(𝑥,𝑦) for all 𝛼∈(0,1]; it is not difficult to see that the inequality (16) holds as a result of (38). Moreover, the other conditions of Corollary 20 are satisfied; thus by Corollary 20, the theorem is proved.

Applying the same method, we can obtain the following theorem and corollary by virtue of Theorem 21 and Corollary 22, respectively.

Theorem 24. Let (𝑋,𝜌) be a complete metric space. Let 𝜙₁,…,𝜙₅∶ℝ⁺→[0,1) be five continuous functions which satisfy the following conditions:(i)𝜙₁(𝑡)+𝜙₄(𝑡)+𝜙₅(𝑡)<1 for all 𝑡∈ℝ⁺,(ii)0<inf_𝑡⩾0{1−𝜙₂(𝑡)−𝜙₅(𝑡)}=𝑎, 0<inf_𝑡⩾0{1−𝜙₃(𝑡)−𝜙₄(𝑡)}=𝑏, 0<sup_𝑡⩾0{𝜙₁(𝑡)+𝜙₂(𝑡)+𝜙₄(𝑡)}=𝐶, 0<sup_𝑡⩾0{𝜙₁(𝑡)+𝜙₃(𝑡)+𝜙₅(𝑡)}=𝐷 and 𝐶𝐷<𝑎𝑏.Let 𝑆 and 𝑇 be two self-mappings on (𝑋,𝜌) such that 𝜌𝑆𝑥,𝑇𝑦⩽𝜙₁𝜌𝑥,𝑦𝜌𝑥,𝑦+𝜙₂𝜌𝑥,𝑦𝜌(𝑥,𝑆𝑥)+𝜙₃𝜌𝑥,𝑦𝜌𝑦,𝑇𝑦+𝜙₄𝜌𝑥,𝑦𝜌𝑥,𝑇𝑦+𝜙₅𝜌𝑥,𝑦𝜌𝑦,𝑆𝑥, (39) for all 𝑥,𝑦∈𝑋. Then 𝑆 and 𝑇 have a unique common fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_2𝑘+1=𝑆𝑥_2𝑘, 𝑥_2𝑘+2=𝑇𝑥_2𝑘+1, 𝑘=0,1,2,…, converges to the fixed point.

Corollary 25. Let (𝑋,𝜌) be a complete metric space and let 𝑆 and 𝑇 be two self-mappings on (𝑋,𝜌). If there exist 𝑎,𝑏,𝑐,𝑑,𝑒⩾0, and 𝛿>0 with 𝑎+𝑏+𝑐+𝑑+𝑒=1+𝛿, 𝑎+𝑑+𝑒<1, 𝑐+𝑑<1, 𝑏+𝑒<1, and (𝑐−𝑏)(𝑒−𝑑)>2𝛿, such that 𝜌(),𝑆𝑥,𝑇𝑦⩽𝑎𝜌𝑥,𝑦+𝑏𝜌𝑥,𝑆𝑥+𝑐𝜌𝑦,𝑇𝑦+𝑑𝜌𝑥,𝑇𝑦+𝑒𝜌𝑦,𝑆𝑥 (40) for all 𝑥,𝑦∈𝑋, then 𝑆 and 𝑇 have a unique common fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_2𝑘+1=𝑆𝑥_2𝑘, 𝑥_2𝑘+2=𝑇𝑥_2𝑘+1, 𝑘=0,1,2,…, converges to the fixed point.

By Theorem 23 and Example 18, we can also obtain the following corollary.

Corollary 26. Let (𝑋,𝜌) be a complete metric space and let 𝑆 and 𝑇 be two self-mappings on (𝑋,𝜌). If there exist 𝑎,𝑏,𝑐,𝑑,𝑒⩾0 with 𝑎+𝑏+𝑐+𝑑+𝑒=1 and either 𝑐>𝑏, 𝑒>𝑑 or 𝑐<𝑏, 𝑒<𝑑, such that 𝜌(),𝑆𝑥,𝑇𝑦⩽𝑎𝜌𝑥,𝑦+𝑏𝜌𝑥,𝑆𝑥+𝑐𝜌𝑦,𝑇𝑦+𝑑𝜌𝑥,𝑇𝑦+𝑒𝜌𝑦,𝑆𝑥 (41) for all 𝑥,𝑦∈𝑋, then 𝑆 and 𝑇 have a unique common fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_2𝑘+1=𝑆𝑥_2𝑘, 𝑥_2𝑘+2=𝑇𝑥_2𝑘+1, 𝑘=0,1,2,…, converges to the fixed point.

Remark 27. In the conditions of Theorem 1 of [13], if we suppose the functions 𝐴(𝑥,𝑦), 𝐵(𝑥,𝑦), 𝐶(𝑥,𝑦), 𝐷(𝑥,𝑦), and 𝐸(𝑥,𝑦) to be continuous, then the theorem is just Theorem 24 in this paper. Therefore, in some sense, Theorem 23 is the improvement and generalizations of Theorem  1 in [13]. Meanwhile, Corollaries 25 and 26 are just Corollaries 1 and  2 in [13], respectively.
If we assume 𝑇=𝑆, then, by Corollary 26, we can obtain the following corollary.

Corollary 28. Let (𝑋,𝜌) be a complete metric space and let 𝑆 be a self-mapping on (𝑋,𝜌). If there exist 𝑎,𝑏,𝑐,𝑑,𝑒⩾0 with 𝑎+𝑏+𝑐+𝑑+𝑒=1 and either 𝑐>𝑏, 𝑒>𝑑 or 𝑐<𝑏, 𝑒<𝑑, such that 𝜌(),𝑆𝑥,𝑆𝑦⩽𝑎𝜌𝑥,𝑦+𝑏𝜌𝑥,𝑆𝑥+𝑐𝜌𝑦,𝑆𝑦+𝑑𝜌𝑥,𝑆𝑦+𝑒𝜌𝑦,𝑆𝑥 (42) for all 𝑥,𝑦∈𝑋, then 𝑆 has a unique fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_𝑛+1=𝑆𝑥_𝑛, 𝑛=0,1,2,…, converges to the fixed point.

Corollary 29. Let (𝑋,𝜌) be a complete metric space and let 𝑆 be a self-mapping on (𝑋,𝜌). If there exist 𝐴,𝐵,𝐶∈[0,1) with 𝐴+𝐵+𝐶<1, such that 𝜌𝑆𝑥,𝑆𝑦⩽𝐴𝜌𝑥,𝑦+𝐵𝜌(𝑥,𝑆𝑥)+𝐶𝜌𝑦,𝑆𝑦, (43) for all 𝑥,𝑦∈𝑋, then 𝑆 has a unique fixed point in 𝑋. Moreover, for any 𝑥₀∈𝑋, the iterative process 𝑥_𝑛+1=𝑆𝑥_𝑛,𝑛=0,1,2,…, converges to the fixed point.

Proof. If 𝐵⩾𝐶, by 𝐴+𝐵+𝐶<1, we can take 4𝛿=1−(𝐴+𝐵+𝐶), and let 𝑎=𝐴, 𝑏=𝐵+𝛿, 𝑐=𝐶, 𝑑=2𝛿, and 𝑒=𝛿; then we have 𝑎+𝑏+𝑐+𝑑+𝑒=1, 𝑐<𝑏, 𝑒<𝑑, and for 𝑥,𝑦∈𝑋, 𝜌()(),𝑆𝑥,𝑆𝑦⩽𝐴𝜌𝑥,𝑦+𝐵𝜌𝑥,𝑆𝑥+𝐶𝜌𝑦,𝑆𝑦⩽𝑎𝜌𝑥,𝑦+𝑏𝜌𝑥,𝑆𝑥+𝑐𝜌𝑦,𝑆𝑦+𝑑𝜌𝑥,𝑆𝑦+𝑒𝜌𝑦,𝑆𝑥 (44) which imply that the conditions of Corollary 28 are satisfied. Similarly, we can prove the case of 𝐵⩽𝐶. Therefore, by Corollary 28, the corollary is proved.

Remark 30. Corollary 29 is just the fixed point theorem for Reich-type contraction mappings in [26, 27]. Meanwhile, we can easily see that the conditions of Corollary 28 generalize the Reich-type contractive conditions in Corollary 29.

Remark 31. If we take 𝐹=𝐹₁ in Theorem 23, then Theorem 23 is the generalizations of the fixed point theorem for mappings satisfying Cirić-type [28] contractive conditions to the common fixed point theorem for a pair of self-mappings with 𝑎∈(0,1/2).

Now we give two examples, by which we can claim that the theorems in this paper are really the generalizations of the general contractive type fixed point theorems.

Example 32. Suppose 𝑋={0,1,2}, 𝜌 is an ordinary metric; then (𝑋,𝜌) is a complete metric space.
We define two mappings 𝑆,𝑇∶𝑋→𝑋 as follows: 𝑆(0)=𝑆(1)=𝑆(2)=0,𝑇(0)=𝑇(1)=0,𝑇(2)=1; then 𝜌(𝑆(0)(0(𝑆(0)(1(𝑆(1)(0(𝑆(1)(1(𝑆(2)(0(𝑆(2)(1𝜌(𝑆(0)(2(𝑆(1)(2(𝑆(2)(2,𝑇))=𝜌,𝑇))=𝜌,𝑇))=𝜌,𝑇))=𝜌,𝑇))=𝜌,𝑇))=0,,𝑇))=𝜌,𝑇))=𝜌,𝑇))=1. (45) We take 𝛿=1/40, 𝑎=1/40, 𝑏=1/40, 𝑐=9/20, 𝑑=1/40, and 𝑒=1/2; then 𝑎+𝑏+𝑐+𝑑+𝑒=1+𝛿, 𝑎+𝑑+𝑒<1, 𝑐+𝑑<1,𝑏+𝑒<1, and (𝑐−𝑏)(𝑒−𝑑)>2𝛿; that is, these numbers satisfy the conditions of Corollary 25. Moreover, 𝜌(𝑆(0)(2⩽1,𝑇))𝜌()+1400,2𝜌((0+9400,𝑆))𝜌((2+1202,𝑇))𝜌((2+1400,𝑇))2𝜌(2,𝑆(0))=61,𝜌(𝑆(1)(2⩽140,𝑇))𝜌()+1401,2𝜌((1+9401,𝑆))𝜌((2+1202,𝑇))𝜌((2+1401,𝑇))2𝜌((1=32,𝑆))2,𝜌(𝑆(2)(2⩽1,𝑇))𝜌()+1402,2𝜌((2+9402,𝑆))𝜌((2+1202,𝑇))𝜌((2+1402,𝑇))2𝜌(2,𝑆(2))=61.40 (46) Hence 𝑆,𝑇 satisfy the nonexpansive condition, and they have a unique common fixed point. But for any nonnegative real numbers 𝛼,𝛽,𝛾 with 0<𝛼+2𝛽+2𝛾<1, we have ()𝜌((1((2𝜌((2((1(𝑆(1)(2;𝛼𝜌1,2+𝛽1,𝑆))+𝜌2,𝑇))+𝛾1,𝑇))+𝜌2,𝑆))=𝛼+2𝛽+2𝛾<1=𝜌,𝑇)) (47) thus 𝑆,𝑇 cannot satisfy the general contractive type condition 0<𝛼+2𝛽+2𝛾<1.

Remark 33. For the examples satisfying the conditions 𝑎+𝑏+𝑐+𝑑+𝑒=1 and either 𝑐>𝑏, 𝑒>𝑑 or 𝑐<𝑏, 𝑒<𝑑 in Corollary 26, we refer the reader to Example  2 in [13].

Example 34. Suppose 𝑋={0,1,2};𝜌 is a discrete metric, then (𝑋,𝜌) is a complete metric space.
Likewise, we define the mappings 𝑆,𝑇∶𝑋→𝑋 as follows: 𝑆(0)=𝑆(1)=𝑆(2)=0,𝑇(0)=𝑇(1)=0,𝑇(2)=1; then 𝜌(𝑆(0)(0(𝑆(0)(1(𝑆(1)(0(𝑆(1)(1(𝑆(2)(0(𝑆(2)(1𝜌(𝑆(0)(2(𝑆(1)(2(𝑆(2)(2,𝑇))=𝜌,𝑇))=𝜌,𝑇))=𝜌,𝑇))=𝜌,𝑇))=𝜌,𝑇))=0,,𝑇))=𝜌,𝑇))=𝜌,𝑇))=1. (48) We take 𝛿=1/40, 𝑎=1/40, 𝑏=1/40, 𝑐=9/20, 𝑑=1/40, 𝑒=1/2, then 𝑎+𝑏+𝑐+𝑑+𝑒=1+𝛿, 𝑎+𝑑+𝑒<1, 𝑐+𝑑<1, 𝑏+𝑒<1, and (𝑐−𝑏)(𝑒−𝑑)>2𝛿; that is, these numbers satisfy the conditions of Corollary 25. Moreover, 𝜌(𝑆(0)(2=1,𝑇))𝜌()+1400,2𝜌((0+9400,𝑆))𝜌((2+1202,𝑇))𝜌((2+1400,𝑇))2𝜌((0𝜌(𝑆(1)(2=12,𝑆))=1,,𝑇))𝜌()+1401,2𝜌((1+9401,𝑆))𝜌((2+1202,𝑇))𝜌((2+1401,𝑇))2𝜌((1𝜌(𝑆(2)(2=12,𝑆))=1,,𝑇))𝜌()+1402,2𝜌((2+9402,𝑆))𝜌((2+1202,𝑇))𝜌((2+1402,𝑇))2𝜌(2,𝑆(2))=1. (49) Hence 𝑆,𝑇 satisfy the nonexpansive condition, and they have a unique common fixed point. While for any nonnegative real numbers 𝛼,𝛽,𝛾,𝜂,𝜇<1 with 𝛼+𝛽+𝛾+𝜂+𝜇<1, we have ()((1((2((2((1(𝑆(1)(2;𝛼𝜌1,2+𝛽𝜌1,𝑆))+𝛾𝜌2,𝑇))+𝜂𝜌1,𝑇))+𝜇𝜌2,𝑆))=𝛼+𝛽+𝛾+𝜇<1=𝜌,𝑇)) (50) thus 𝑆,𝑇 cannot satisfy the general contractive type condition 𝛼+𝛽+𝛾+𝜂+𝜇<1.

6. Conclusion

In this paper, we studied the existence and uniqueness of fixed points for nonlinear contractions in fuzzy metric spaces in the sense of Kaleva-Seikkala. By virtue of a level-cut method, we established relationships between fuzzy metric and a family of quasi-metrics. Using them, we obtained some common fixed point theorems for a pair of self-mappings satisfying an implicit Lipschitz-type conditions in fuzzy metric spaces. Obviously, the present investigation enriches our knowledge of fixed points in fuzzy metric spaces. But, the work in this paper with respect to fixed point in fuzzy setting is based on 𝑅⩽max. The question whether the discussion can be carried out under the weaker condition (R-2) deserves our further investigation.

Acknowledgments

This work was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 13KJB110004) and the Qing Lan Project of Jiangsu Province of China.