Application of Coupled Fixed Point Technique in Solving Integral Equations on Modified Intuitionistic Fuzzy Metric Spaces

Abstract

We establish a common coupled fixed point theorem for weakly compatible mappings on modified intuitionistic fuzzy metric spaces. As an application of our result, we study the existence and uniqueness of the solution to a nonlinear Fredholm integral equation. We also give an example to demonstrate our result.

1. Introduction

The concept of fuzzy metric space has been introduced in several ways. In [1], Kramosil and Michalek introduced the concept of fuzzy metric space. Later on, it is modified by George and Veeramani [2] with the help of continuous t-norms and they defined the Hausdorff topology of fuzzy metric spaces.

Atanassov [3] introduced and studied the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets. Alaca et al. [4] using the idea of intuitionistic fuzzy sets defined the notion of intuitionistic fuzzy metric space with the help of continuous t-norms and continuous t-conorms as a generalization of fuzzy metric space due to Kramosil and Michalek [1]. In [5], Park generalized the notion of fuzzy metric space given by George and Veeramani [2] and introduced the notion of intuitionistic fuzzy metric space.

Gregori et al. [6] pointed out that topologies generated by fuzzy metric and intuitionistic fuzzy metric coincide. In view of this observation, Saadati et al. [7] modified the notion of intuitionistic fuzzy metric and defined the notion of modified intuitionistic fuzzy metric spaces with the help of continuous t-representable.

Bhaskar and Lakshmikantham [8] introduced the notion of coupled fixed point and mixed monotone mappings and gave some coupled fixed point theorems. As an application, they study the existence and uniqueness of solution for periodic boundary value problems. Lakshmikantham and Ciric [9] introduced the concept of coupled coincidence point and proved some common coupled fixed point theorems. Sedghi et al. [10] gave a coupled fixed point theorem for contractions in fuzzy metric space, which was further generalized by Hu [11]. In [12], Hu et al. improved, rectified, and generalized the result obtained in [11].

On the other hand, many scientific and engineering problems can be described by integral equations. Initial and boundary value problems can be transformed into Volterra or Fredholm integral equations. Integral equations can also be created by many mathematical physics models such as diffraction problems, scattering in quantum mechanics, conformal mapping, and water wave. Integral equations or integro-differential equations can be applied in science and engineering. Many areas that are described by integral equations are Volterra’s population growth model, biological species living together, propagation of stocked fish in a new lake, the heat radiation, and so forth.

Very recently, Deshpande et al. [13] proved a common fixed point theorem for mappings under 𝜙-contractive conditions on intuitionistic fuzzy metric spaces. As an application, they study the existence and uniqueness of the solution to a nonlinear Fredholm integral equation.

In this paper, we prove a common coupled fixed point theorem for weakly compatible mappings on modified intuitionistic fuzzy metric spaces. As an application of our result, we study the existence and uniqueness of the solution to a nonlinear Fredholm integral equation, which arise naturally in the theory of signal processing, linear forward modeling, and inverse problems. We also give an example to validate our result. We extend and generalize the results of Hu [11], Hu et al. [12], and Sedghi et al. [10] in the settings of modified intuitionistic fuzzy metric spaces. The result is the genuine generalization of the result of Deshpande et al. [13].

2. Preliminaries

Lemma 1 (Deschrijver and Kerre [14]). Consider the set 𝐿^∗ and operation ≤_𝐿^∗defined by 𝐿^∗=𝑥₁,𝑥₂∶𝑥₁,𝑥₂∈[0,1]²,𝑥₁+𝑥₂≤1, (1)(𝑥₁,𝑥₂)≤_𝐿^∗(𝑦₁,𝑦₂)⇔𝑥₁≤𝑦₁, and 𝑥₂≥𝑦₂ for every (𝑥₁,𝑥₂),(𝑦₁,𝑦₂)∈𝐿^∗. Then (𝐿^∗,≤_𝐿^∗) is a complete lattice.

Definition 2 (Atanassov [3]). An intuitionistic fuzzy set 𝒜_𝜁,𝜂 in a universe 𝑈 is an object 𝒜_𝜁,𝜂={𝜁_𝒜(𝑢),𝜂_𝒜(𝑢)}, where, for all 𝑢∈𝑈,𝜁_𝒜(𝑢)∈[0,1] and 𝜂_𝒜(𝑢)∈[0,1] are called the membership degree and nonmembership degree, respectively, of 𝑢 in 𝒜_𝜁,𝜂 and further they satisfy 𝜁_𝒜(𝑢)+𝜂_𝒜(𝑢)≤1. For every 𝑧_𝑖=(𝑥_𝑖,𝑦_𝑖)∈𝐿^∗, if 𝑐_𝑖∈[0,1] such that ∑^𝑛_𝑗=1𝑐_𝑗=1, then it is easy to see that 𝑐₁𝑥₁,𝑦₁+⋯+𝑐_𝑛𝑥_𝑛,𝑦_𝑛=^𝑛_𝑗=1𝑐_𝑗𝑥_𝑗,𝑦_𝑗=^𝑛_𝑗=1𝑐_𝑗𝑥_𝑗,^𝑛_𝑗=1𝑐_𝑗𝑦_𝑗∈𝐿^∗. (2) We denote its units by 0_𝐿^∗=(0,1) and 1_𝐿^∗=(1,0). Classically, a triangular norm ∗=𝑇 on [0,1] is defined as an increasing, commutative, associative mapping 𝑇∶[0,1]²→[0,1] satisfying 𝑇(1,𝑥)=1∗𝑥=𝑥, for all 𝑥∈[0,1]. A triangular conorm 𝑆=⋄ is defined as an increasing, commutative, associative mapping 𝑆∶[0,1]²→[0,1] satisfying 𝑆(0,𝑥)=0⋄𝑥=𝑥, for all 𝑥∈[0,1]. Using the lattice (𝐿^∗,≤_𝐿^∗), these definitions can be straightforwardly extended.

Definition 3 (Deschrijver et al. [15]). A triangular norm (t-norm) on 𝐿^∗ is a mapping 𝒯∶(𝐿^∗)²→𝐿^∗ satisfying the following conditions:  (∀𝑥∈𝐿^∗)(𝒯(𝑥,1_𝐿^∗)=𝑥) (boundary condition), (∀(𝑥,𝑦)∈(𝐿^∗)²)(𝒯(𝑥,𝑦)=𝒯(𝑦,𝑥)) (commutativity), (∀(𝑥,𝑦,𝑧)∈(𝐿^∗)³)(𝒯(𝑥,𝒯(𝑦,𝑧))=𝒯(𝒯(𝑥,𝑦),𝑧)) (associativity), (∀(𝑥,𝑥^′,𝑦,𝑦^′)∈(𝐿^∗)⁴)(𝑥≤_𝐿^∗𝑥^′and𝑦≤_𝐿^∗𝑦^′⇒𝒯(𝑥,𝑦)≤_𝐿^∗𝒯(𝑥^′,𝑦^′)) (monotonicity).

Definition 4 (Deschrijver and Kerre [14] and Deschrijver et al. [15]). A continuous t-norm 𝒯 on 𝐿^∗ is called continuous t-representable if and only if there exist a continuous t-norm ∗ and a continuous t-conorm ⋄ on [0,1] such that, for all 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂)∈𝐿^∗, 𝒯𝑥,𝑦=𝑥₁∗𝑦₁,𝑥₂⋄𝑦₂. (3) Now define a sequence 𝒯^𝑛 recursively by 𝒯¹=𝒯 and 𝒯^𝑛𝑥⁽¹⁾,…,𝑥^(𝑛+1)=𝒯𝒯^𝑛−1𝑥⁽¹⁾,…,𝑥^(𝑛),𝑥^(𝑛+1), (4) for 𝑛≥2 and 𝑥^(𝑖)∈𝐿^∗.

Definition 5 (Deschrijver and Kerre [14] and Deschrijver et al. [15]). A negator on 𝐿^∗ is any decreasing mapping 𝒩∶𝐿^∗→𝐿^∗ satisfying 𝒩(0_𝐿^∗)=1_𝐿^∗ and 𝒩(1_𝐿^∗)=0_𝐿^∗. If 𝒩(𝒩(𝑥))=𝑥, for all 𝑥∈𝐿^∗, then 𝒩 is called an involutive negator. A negator on [0,1] is a decreasing mapping 𝑁∶[0,1]→[0,1] satisfying 𝑁(0)=1 and 𝑁(1)=0.𝑁_𝑠 denotes the standard negator on [0,1] defined as, for all 𝑥∈[0,1],𝑁_𝑠(𝑥)=1−𝑥.

Definition 6 (Saadati et al. [7]). Let 𝑀,𝑁 be fuzzy sets from 𝑋²×(0,+∞) to [0,1] such that 𝑀(𝑥,𝑦,𝑡)+𝑁(𝑥,𝑦,𝑡)≤1 for all 𝑥,𝑦∈𝑋 and 𝑡>0. The 3-tuple (𝑋,ℳ_𝑀,𝑁,𝒯) is said to be a modified intuitionistic fuzzy metric space if 𝑋 is an arbitrary (nonempty) set, 𝒯 is a continuous t-representable, and ℳ_𝑀,𝑁 is a mapping 𝑋²×(0,+∞)→𝐿^∗ satisfying the following conditions for every 𝑥,𝑦,𝑧∈𝑋 and 𝑡,𝑠>0:(a)ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)>_𝐿^∗0_𝐿^∗, (b)ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=1_𝐿^∗ if and only if 𝑥=𝑦,(c)ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=ℳ_𝑀,𝑁(𝑦,𝑥,𝑡),(d)ℳ_𝑀,𝑁(𝑥,𝑦,𝑡+𝑠)≥_𝐿^∗𝒯(ℳ_𝑀,𝑁(𝑥,𝑧,𝑡),ℳ_𝑀,𝑁(𝑧,𝑦,𝑠)), (e)ℳ_𝑀,𝑁(𝑥,𝑦,⋅)∶(0,∞)→𝐿^∗ is continuous.
In this case, ℳ_𝑀,𝑁 is called a modified intuitionistic fuzzy metric. Here, ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=(𝑀(𝑥,𝑦,𝑡),𝑁(𝑥,𝑦,𝑡)).

Example 7 (Saadati et al. [7]). Let (𝑋,𝑑) be a metric space. Denote 𝒯(𝑎,𝑏)=(𝑎₁𝑏₁,min{𝑎₂+𝑏₂,1}) for all 𝑎=(𝑎₁,𝑎₂) and 𝑏=(𝑏₁,𝑏₂)∈𝐿^∗ and let 𝑀 and 𝑁 be fuzzy sets on 𝑋²×(0,∞) defined as follows: ℳ_𝑀,𝑁=𝑀=𝑥,𝑦,𝑡𝑥,𝑦,𝑡,𝑁𝑥,𝑦,𝑡ℎ𝑡^𝑛ℎ𝑡^𝑛,+𝑚𝑑𝑥,𝑦𝑚𝑑𝑥,𝑦ℎ𝑡^𝑛,+𝑚𝑑𝑥,𝑦 (5) for all 𝑡,ℎ,𝑚,𝑛∈𝑅⁺. Then (𝑋,ℳ_𝑀,𝑁,𝒯) is a modified intuitionistic fuzzy metric space.

Example 8 (Saadati et al. [7]). Let 𝑋=𝑁. Denote 𝒯(𝑎,𝑏)=(max{0,𝑎₁+𝑏₁−1},𝑎₂+𝑏₂−𝑎₂𝑏₂) for all 𝑎=(𝑎₁,𝑎₂) and 𝑏=(𝑏₁,𝑏₂)∈𝐿^∗ and let 𝑀 and 𝑁 be fuzzy sets on 𝑋²×(0,∞) defined as follows: ℳ_𝑀,𝑁=𝑀=𝑥𝑥,𝑦,𝑡𝑥,𝑦,𝑡,𝑁𝑥,𝑦,𝑡𝑦,𝑦−𝑥𝑦𝑦if𝑥≤𝑦,𝑥,𝑥−𝑦𝑥if𝑦≤𝑥, (6) for all 𝑥,𝑦∈𝑋 and 𝑡>0. Then (𝑋,ℳ_𝑀,𝑁,𝒯) is a modified intuitionistic fuzzy metric space.

Lemma 9 (Fang [16]). We say that the intuitionistic fuzzy metric space (𝑋,ℳ_𝑀,𝑁,𝒯) has property (𝐶), if it satisfies the following condition: ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=𝐶 for all 𝑡>0 implies 𝐶=1_𝐿^∗.

Remark 10. Throughout this paper, (𝑋,ℳ_𝑀,𝑁,𝒯) is a modified intuitionistic fuzzy metric space with property (𝐶).

Definition 11 (Saadati et al. [7]). A sequence {𝑥_𝑛} in a modified intuitionistic fuzzy metric space (𝑋,ℳ_𝑀,𝑁,𝒯) is called a Cauchy sequence if, for each 0<𝜀<1 and 𝑡>0, there exists 𝑛₀∈𝑁 such that ℳ_𝑀,𝑁𝑥_𝑛,𝑥_𝑚,𝑡>_𝐿^∗𝑁_𝑠(𝜀),𝜀, (7) and, for each 𝑛,𝑚≥𝑛₀, here 𝑁_𝑠 is the standard negator. The sequence {𝑥_𝑛} is said to be convergent to 𝑥∈𝑋 in the modified intuitionistic fuzzy metric space (𝑋,ℳ_𝑀,𝑁,𝒯) and denoted by 𝑥−−−−−→_𝑛^ℳ𝑀,𝑁𝑥, if ℳ_𝑀,𝑁(𝑥_𝑛,𝑥,𝑡)→1_𝐿^∗ whenever 𝑛→∞ for every 𝑡>0. A modified intuitionistic fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.

Lemma 12 (Saadati and Park [17]). Let ℳ_𝑀,𝑁 be a modified intuitionistic fuzzy metric. Then, for any 𝑡>0,ℳ_𝑀,𝑁(𝑥,𝑦,𝑡) is nondecreasing with respect to 𝑡, in (𝐿^∗,≤_𝐿^∗) for all 𝑥,𝑦 in 𝑋.

Definition 13 (Saadati et al. [7]). Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space. For 𝑡>0, define the open ball 𝐵(𝑥,𝑟,𝑡) with center 𝑥∈𝑋 and radius 0<𝑟<1, as 𝐵(𝑥,𝑟,𝑡)=𝑦∈𝑋∶ℳ_𝑀,𝑁𝑥,𝑦,𝑡>_𝐿^∗𝑁_𝑠(𝑟),𝑟. (8) A subset 𝐴⊂𝑋 is called open if, for each 𝑥∈𝐴, there exist 𝑡>0 and 0<𝑟<1 such that 𝐵(𝑥,𝑟,𝑡)⊂𝐴. Let 𝜏_𝑀,𝑁 denote the family of all open subsets of 𝑋.𝜏_𝑀,𝑁 is called the topology induced by modified intuitionistic fuzzy metric. This topology is Hausdorff.

Definition 14 (Saadati et al. [7]). Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space. ℳ is said to be continuous on 𝑋×𝑋×(0,∞) if lim_𝑛→∞ℳ_𝑀,𝑁𝑥_𝑛,𝑦_𝑛,𝑡_𝑛=ℳ_𝑀,𝑁𝑥,𝑦,𝑡, (9) whenever a sequence {(𝑥_𝑛,𝑦_𝑛,𝑡_𝑛)} in 𝑋×𝑋×(0,∞) converges to a point (𝑥,𝑦,𝑡)∈𝑋×𝑋×(0,∞), that is; lim_𝑛→∞ℳ_𝑀,𝑁𝑥_𝑛,𝑦,𝑡=ℳ_𝑀,𝑁𝑥,𝑦,𝑡,lim_𝑛→∞ℳ_𝑀,𝑁𝑥,𝑦_𝑛,𝑡=ℳ_𝑀,𝑁𝑥,𝑦,𝑡,lim_𝑛→∞ℳ_𝑀,𝑁𝑥,𝑦,𝑡_𝑛=ℳ_𝑀,𝑁𝑥,𝑦,𝑡. (10)

Lemma 15 (Saadati et al. [7]). Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space. Then ℳ is continuous function on 𝑋×𝑋×(0,∞).

Definition 16 (Bhaskar and Lakshmikantham [8]). An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled fixed point of the mapping 𝐹∶𝑋×𝑋→𝑋 if 𝐹𝑥,𝑦=𝑥,𝐹𝑦,𝑥=𝑦. (11)

Definition 17 (Lakshmikantham and Ciric [9]). Let 𝑋 be a nonempty set. An element (𝑥,𝑦)∈𝑋×𝑋 is called a coupled coincidence point of the mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝐹𝑥,𝑦=𝑔(𝑥),𝐹𝑦,𝑥=𝑔𝑦. (12)

Definition 18 (Lakshmikantham and Ciric [9]). Let 𝑋 be a nonempty set. An element (𝑥,𝑦)∈𝑋×𝑋 is called a common coupled fixed point of the mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑥=𝐹𝑥,𝑦=𝑔(𝑥),𝑦=𝐹𝑦,𝑥=𝑔𝑦. (13)

Definition 19 (Lakshmikantham and Ciric [9]). Let 𝑋 be a nonempty set. An element 𝑥∈𝑋 is called a common fixed point of the mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 if 𝑥=𝐹(𝑥,𝑥)=𝑔(𝑥). (14)

Definition 20 (Lakshmikantham and Ciric [9]). Let 𝑋 be a nonempty set. The mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be commutative if 𝑔𝐹𝑥,𝑦=𝐹𝑔𝑥,𝑔𝑦,∀𝑥,𝑦∈𝑋×𝑋. (15)

Definition 21 (Fang [18]). Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space. The mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are said to be compatible if lim_𝑛→∞ℳ_𝑀,𝑁𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑡=1_𝐿^∗,lim_𝑛→∞ℳ_𝑀,𝑁𝑔𝐹𝑦_𝑛,𝑥_𝑛,𝐹𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑡=1_𝐿^∗, (16) for all 𝑡>0, whenever {𝑥_𝑛} and {𝑦_𝑛} are sequences in 𝑋 such that lim_𝑛→∞𝐹𝑥_𝑛,𝑦_𝑛=lim_𝑛→∞𝑔𝑥_𝑛=𝑥,lim_𝑛→∞𝐹𝑦_𝑛,𝑥_𝑛=lim_𝑛→∞𝑔𝑦_𝑛=𝑦,∀𝑥,𝑦∈𝑋. (17)

Definition 22 (Abbas et al. [19]). Let 𝑋 be a nonempty set. The mappings 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 are called weakly compatible mappings if 𝐹(𝑥,𝑦)=𝑔(𝑥),𝐹(𝑦,𝑥)=𝑔(𝑦) implies that 𝑔𝐹(𝑥,𝑦)=𝐹(𝑔𝑥,𝑔𝑦) and 𝑔𝐹(𝑦,𝑥)=𝐹(𝑔𝑦,𝑔𝑥), for all 𝑥,𝑦∈𝑋.

3. Main Results

Definition 23. Let sup_0<𝑡<1𝒯(𝑡,𝑡)=1_𝐿^∗. A continuous t-representable 𝒯 is said to be continuous t-representable of H-type if the family of functions {𝒯^𝑚(𝑡)}^∞_𝑚=1 is equicontinuous at 𝑡=1_𝐿^∗, where 𝒯¹(𝑡)=𝒯(𝑡,𝑡),𝒯^𝑚+1(𝑡)=𝒯𝑡,𝒯^𝑚(𝑡),𝑚=1,2,…,𝑡∈𝐿^∗. (18) Obviously, 𝒯 is a H-type t-representable if and only if, for any 0<𝜆<1, there exists 0<𝜇<1 such that 𝒯^𝑚(𝑡)>_𝐿^∗𝑁_𝑠(𝜆),𝜆,∀𝑚∈𝑁,when𝑡>_𝐿^∗𝑁_𝑠𝜇,𝜇. (19)

Remark 24. In a modified intuitionistic fuzzy metric space (𝑋,ℳ_𝑀,𝑁,𝒯), whenever ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)>_𝐿^∗(𝑁_𝑠(𝑟),𝑟) for 𝑥,𝑦∈𝑋,𝑡>0 and 0<𝑟<1, we can find a 𝑡₀, 0<𝑡₀<𝑡, such that ℳ_𝑀,𝑁(𝑥,𝑦,𝑡₀)>(𝑁_𝑠(𝑟),𝑟).

Remark 25. For convenience, we denote ℳ_𝑀,𝑁𝑥,𝑦,𝑡^𝑛=𝒯^𝑛−1ℳ_𝑀,𝑁𝑥,𝑦,𝑡, (20) for all 𝑛∈𝑁.

Definition 26. Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space. ℳ_𝑀,𝑁 is said to satisfy the p-property on 𝑋²×(0,∞) if lim_𝑝→∞ℳ_𝑀,𝑁𝑥,𝑦,𝑘^𝑝𝑡^𝑝=1_𝐿^∗, (21) whenever 𝑥,𝑦∈𝑋,𝑘>1, and 𝑝∈𝑁.

Lemma 27. Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space and let ℳ_𝑀,𝑁 satisfy the p-property; then lim_𝑡→∞ℳ_𝑀,𝑁𝑥,𝑦,𝑡=1_𝐿^∗,∀𝑥,𝑦∈𝑋. (22)

Proof. If not, since ℳ_𝑀,𝑁(𝑥,𝑦,⋅) is nondecreasing, there exists 𝑥₀,𝑦₀∈𝑋 such that lim_𝑡→∞ℳ_𝑀,𝑁𝑥₀,𝑦₀,𝑡=𝜆,𝑁_𝑠(𝜆)<1_𝐿^∗, (23) and then, for 𝑘>1, 𝑘^𝑝𝑡→∞ when 𝑝→∞ as 𝑡>0 and we get lim_𝑝→∞ℳ_𝑀,𝑁𝑥₀,𝑦₀,𝑘^𝑝𝑡^𝑝=0_𝐿^∗, (24) which is a contradiction.

Remark 28. Condition (22) cannot guarantee the p-property. See the following example.

Example 29. Let (𝑋,𝑑) be an ordinary metric space and let 𝜓 be defined as follows: 𝜓(𝑡)=𝛼√1𝑡,0≤𝑡≤4,1−ln𝑡,𝑡>4, (25) where 𝛼=(1/2)(1−(1/ln4)). Then 𝜓(𝑡) is continuous and increasing in (0,∞),𝜓(𝑡)∈(0,1) and lim_𝑡→∞𝜓(𝑡)=1. Let ℳ_𝑀,𝑁𝑥,𝑦,𝑡=𝜓(𝑡)^{𝑑(𝑥,𝑦)},1−𝜓(𝑡)^{𝑑(𝑥,𝑦)}, (26) for all 𝑥,𝑦∈𝑋 and 𝑡>0, and 𝒯(𝑎,𝑏)=(𝑎₁𝑏₁,𝑎₂+𝑏₂−𝑎₂𝑏₂), for all 𝑎=(𝑎₁,𝑎₂) and 𝑏=(𝑏₁,𝑏₂)∈𝐿^∗. Then (𝑋,ℳ_𝑀,𝑁,𝒯) is a modified intuitionistic fuzzy metric space with lim_𝑡→∞ℳ_𝑀,𝑁𝑥,𝑦,𝑡=lim_𝑡→∞𝜓(𝑡)^{𝑑(𝑥,𝑦)},1−𝜓(𝑡)^{𝑑(𝑥,𝑦)}=1_𝐿^∗,∀𝑥,𝑦∈𝑋. (27) But, for any 𝑥≠𝑦,𝑘>1, and 𝑡>0, lim_𝑝→∞ℳ_𝑀,𝑁𝑥,𝑦,𝑘^𝑝𝑡^𝑝=lim_𝑝→∞𝜓𝑘^𝑝𝑡^{𝑑(𝑥,𝑦)⋅𝑝},lim_𝑝→∞1−𝜓𝑘^𝑝𝑡^{𝑑(𝑥,𝑦)⋅𝑝}=lim_𝑝→∞11−ln𝑘^𝑝𝑡^{𝑑(𝑥,𝑦)⋅𝑝},lim_𝑝→∞11−1−ln(𝑘^𝑝𝑡)^{𝑝⋅𝑑(𝑥,𝑦)}=𝑒^{−𝑑(𝑥,𝑦)/ln𝑘},1−𝑒^{−𝑑(𝑥,𝑦)/ln𝑘}≠1_𝐿^∗. (28) Define Φ={𝜙∶𝑅⁺→𝑅⁺}, where 𝑅⁺=[0,+∞), and each 𝜙∈Φ satisfies the following conditions: (𝜙-1)𝜙 is nondecreasing, (𝜙-2)𝜙 is continuous, (𝜙-3)∑^∞_𝑛=0𝜙^𝑛(𝑡)<+∞ for all 𝑡>0, where 𝜙^𝑛+1(𝑡)=𝜙^𝑛(𝜙(𝑡)), 𝑛∈𝑁.
It is easy to prove that if 𝜙∈Φ, then 𝜙(𝑡)<𝑡 for all 𝑡>0.

Lemma 30. Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space, where 𝒯 is a continuous t-representable of H-type. If there exists 𝜙∈Φ, such that if ℳ_𝑀,𝑁𝑥,𝑦,𝜙(𝑡)≥_𝐿^∗ℳ_𝑀,𝑁𝑥,𝑦,𝑡,∀𝑡>0, (29) then 𝑥=𝑦.

Proof. Since 𝑡>0 and 𝑡>𝜙(𝑡). Using the monotony of ℳ_𝑀,𝑁, we have ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)≥_𝐿^∗ℳ_𝑀,𝑁(𝑥,𝑦,𝜙(𝑡)). Using (29) and the definition of modified intuitionistic fuzzy metric, we have 𝑥=𝑦.

Theorem 31. Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space, where 𝒯 is a continuous t-representable of H-type satisfying (22). Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two weakly compatible mappings and there exists 𝜙∈Φ such that ℳ_𝑀,𝑁𝐹()(𝑡)≥𝑥,𝑦,𝐹𝑢,𝑣,𝜙_𝐿∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑔𝑢,𝑡,ℳ_𝑀,𝑁𝑔𝑦,𝑔𝑣,𝑡, (30) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 and 𝑡>0. Suppose that 𝐹(𝑋×𝑋)⊆𝑔(𝑋) and 𝐹(𝑋×𝑋) or 𝑔(𝑋) is complete. Then there exists a unique 𝑥∈𝑋 such that 𝑥=𝑔(𝑥)=𝐹(𝑥,𝑥).

Proof. Let 𝑥₀,𝑦₀∈𝑋 be two arbitrary points in 𝑋. Since 𝐹(𝑋×𝑋)⊆𝑔(𝑋), we can choose 𝑥₁,𝑦₁∈𝑋 such that 𝑔𝑥₁=𝐹𝑥₀,𝑦₀,𝑔𝑦₁=𝐹𝑦₀,𝑥₀. (31) Continuing in this way, we can construct two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 such that 𝑔𝑥_𝑛+1=𝐹𝑥_𝑛,𝑦_𝑛,𝑔𝑦_𝑛+1=𝐹𝑦_𝑛,𝑥_𝑛,∀𝑛≥0. (32) The proof is divided into 4 steps.
Step 1. Prove that {𝑔𝑥_𝑛} and {𝑔𝑦_𝑛} are Cauchy sequences.
Since 𝒯 is a continuous t-representable of H-type, therefore, for any 𝜆>0, there exists a 𝜇>0 such that 𝒯^𝑚−1𝑁_𝑠𝜇,𝜇≥_𝐿^∗𝑁_𝑠(𝜆),𝜆,∀𝑚∈𝑁. (33) Since lim_𝑡→∞ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=1_𝐿^∗ for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀≥_𝐿^∗𝑁_𝑠𝜇,ℳ,𝜇_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀≥_𝐿^∗𝑁_𝑠𝜇,𝜇. (34) On the other hand, since 𝜙∈Φ, by condition (𝜙-3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then, for any 𝑡>0, there exists 𝑛₀∈𝑁 such that 𝑡>^∞_𝑘=𝑛0𝜙^𝑘𝑡₀. (35) From condition (30), we have ℳ_𝑀,𝑁𝑔𝑥₁,𝑔𝑥₂,𝜙𝑡₀=ℳ_𝑀,𝑁𝐹𝑥₀,𝑦₀,𝐹𝑥₁,𝑦₁,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀,ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀,ℳ_𝑀,𝑁𝑔𝑦₁,𝑔𝑦₂,𝜙𝑡₀=ℳ_𝑀,𝑁𝐹𝑦₀,𝑥₀,𝐹𝑦₁,𝑥₁,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀,ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀. (36) Similarly, we can also get ℳ_𝑀,𝑁𝑔𝑥₂,𝑔𝑥₃,𝜙²𝑡₀=ℳ_𝑀,𝑁𝐹𝑥₁,𝑦₁,𝐹𝑥₂,𝑦₂,𝜙²𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥₁,𝑔𝑥₂,𝜙𝑡₀,ℳ_𝑀,𝑁𝑔𝑦₁,𝑔𝑦₂,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀²,ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀²,ℳ_𝑀,𝑁𝑔𝑦₂,𝑔𝑦₃,𝜙²𝑡₀=ℳ_𝑀,𝑁𝐹𝑦₁,𝑥₁,𝐹𝑦₂,𝑥₂,𝜙²𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑦₁,𝑔𝑦₂,𝜙𝑡₀,ℳ_𝑀,𝑁𝑔𝑥₁,𝑔𝑥₂,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀²,ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀². (37) Continuing in the same way, we can get ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑥_𝑛+1,𝜙^𝑛𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀^2𝑛−1,ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀^2𝑛−1,ℳ_𝑀,𝑁𝑔𝑦_𝑛,𝑔𝑦_𝑛+1,𝜙^𝑛𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀^2𝑛−1,ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀^2𝑛−1. (38) Now, from (33), (34), and (35), for 𝑚>𝑛≥𝑛₀, we have ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑥_𝑚≥,𝑡_𝐿∗ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑥_𝑚,^∞_𝑘=𝑛0𝜙^𝑘𝑡₀≥_𝐿^∗ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑥_𝑚,^𝑚−1_𝑘=𝑛𝜙^𝑘𝑡₀≥_𝐿^∗𝒯^{𝑚−𝑛−1}ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑥_𝑛+1,𝜙^𝑛𝑡₀,ℳ_𝑀,𝑁𝑔𝑥_𝑛+1,𝑔𝑥_𝑛+2,𝜙^𝑛+1𝑡₀ℳ,…,_𝑀,𝑁𝑔𝑥_𝑚−1,𝑔𝑥_𝑚,𝜙^𝑚−1𝑡₀≥_𝐿^∗𝒯^{𝑚−𝑛−1}𝒯ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀^2𝑛−1,ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀^2𝑛−1,𝒯ℳ_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀^2𝑛,ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀^2𝑛𝒯ℳ,…,_𝑀,𝑁𝑔𝑥₀,𝑔𝑥₁,𝑡₀^2𝑚−2,ℳ_𝑀,𝑁𝑔𝑦₀,𝑔𝑦₁,𝑡₀^2𝑚−2≥_𝐿^∗𝒯ℳ_𝑀,𝑁(𝑔𝑥₀,𝑔𝑥₁,𝑡₀)^{2𝑚−1−2𝑛−1},ℳ_𝑀,𝑁(𝑔𝑦₀,𝑔𝑦₁,𝑡₀)^{2𝑚−1−2𝑛−1}≥_𝐿^∗𝒯^{2𝑚−2𝑛−1}𝑁_𝑠𝜇≥,𝜇_𝐿∗𝑁_𝑠(𝜆),𝜆, (39) which implies that ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑥_𝑚,𝑡>_𝐿^∗𝑁_𝑠(𝜆),𝜆, (40) for all 𝑚,𝑛∈𝑁 with 𝑚>𝑛≥𝑛₀ and 𝑡>0. So {𝑔𝑥_𝑛} is a Cauchy sequence. Similarly, we can get that {𝑔𝑦_𝑛} is also a Cauchy sequence.
Step 2. Prove that 𝑔 and 𝐹 have a coupled coincidence point.
Without loss of generality, we can assume that 𝑔(𝑋) is complete; then there exist 𝑥,𝑦∈𝑔(𝑋) and 𝑎,𝑏∈𝑋 such that lim_𝑛→∞𝑔𝑥_𝑛=lim_𝑛→∞𝐹𝑥_𝑛,𝑦_𝑛=𝑔𝑎=𝑥,lim_𝑛→∞𝑔𝑦_𝑛=lim_𝑛→∞𝐹𝑦_𝑛,𝑥_𝑛=𝑔𝑏=𝑦. (41) From (30), we get ℳ_𝑀,𝑁𝐹𝑥_𝑛,𝑦_𝑛()(𝑡)≥,𝐹𝑎,𝑏,𝜙_𝐿∗𝒯ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑎,𝑡,ℳ_𝑀,𝑁𝑔𝑦_𝑛,𝑔𝑏,𝑡. (42) Since ℳ_𝑀,𝑁 is continuous, taking limit as 𝑛→∞, we have ℳ_𝑀,𝑁𝑔𝑎,𝐹(𝑎,𝑏),𝜙(𝑡)=1_𝐿^∗, (43) which implies that 𝐹(𝑎,𝑏)=𝑔𝑎=𝑥. (44) Similarly, we can show that 𝐹(𝑏,𝑎)=𝑔𝑏=𝑦. (45) Since 𝐹 and 𝑔 are weakly compatible, we get that 𝑔𝐹(𝑎,𝑏)=𝐹𝑔𝑎,𝑔𝑏,𝑔𝐹(𝑏,𝑎)=𝐹𝑔𝑏,𝑔𝑎, (46) which implies that 𝑔𝑥=𝐹𝑥,𝑦,𝑔𝑦=𝐹𝑦,𝑥. (47)
Step 3. Prove that 𝑔𝑥=𝑦 and 𝑔𝑦=𝑥.
Since 𝒯 is a continuous t-representable of H-type. Therefore, for any 𝜆>0, there exists a 𝜇>0 such that 𝒯^𝑚−1𝑁_𝑠𝜇,𝜇≥_𝐿^∗𝑁_𝑠(𝜆),𝜆,∀𝑚∈𝑁. (48) Since lim_𝑡→∞ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=1_𝐿^∗ for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝑡₀≥_𝐿^∗𝑁_𝑠𝜇,ℳ,𝜇_𝑀,𝑁𝑔𝑦,𝑥,𝑡₀≥_𝐿^∗𝑁_𝑠𝜇,𝜇. (49) On the other hand, since 𝜙∈Φ, by condition (𝜙-3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then, for any 𝑡>0, there exists 𝑛₀∈𝑁 such that 𝑡>^∞_𝑘=𝑛0𝜙^𝑘𝑡₀. (50) Since ℳ_𝑀,𝑁𝑔𝑥,𝑔𝑦_𝑛+1,𝜙𝑡₀=ℳ_𝑀,𝑁𝐹𝑥,𝑦,𝐹𝑦_𝑛,𝑥_𝑛,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑔𝑦_𝑛,𝑡₀∗ℳ_𝑀,𝑁𝑔𝑦,𝑔𝑥_𝑛,𝑡₀, (51) thus ℳ_𝑀,𝑁𝑔𝑥,𝑔𝑦_𝑛+1,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑔𝑦_𝑛,𝑡₀∗ℳ_𝑀,𝑁𝑔𝑦,𝑔𝑥_𝑛,𝑡₀. (52) Letting 𝑛→∞ in (52), by using (41), we get ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝑡₀,ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝑡₀. (53) Similarly, we can get ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝑡₀,ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝑡₀. (54) From (53) and (54), we have 𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝜙𝑡₀,ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁(𝑔𝑥,𝑦,𝑡₀)²,ℳ_𝑀,𝑁(𝑔𝑦,𝑥,𝑡₀)². (55) By this way, we can get, for all 𝑛∈𝑁, 𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝜙^𝑛𝑡₀,ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝜙^𝑛𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝜙^𝑛−1𝑡₀²,ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝜙^𝑛−1𝑡₀²≥_𝐿^∗𝒯ℳ_𝑀,𝑁(𝑔𝑥,𝑦,𝑡₀)^2𝑛,ℳ_𝑀,𝑁(𝑔𝑦,𝑥,𝑡₀)^2𝑛. (56) Thus, 𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝜙^𝑛𝑡₀,ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝜙^𝑛𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁(𝑔𝑥,𝑦,𝑡₀)^2𝑛,ℳ_𝑀,𝑁(𝑔𝑦,𝑥,𝑡₀)^2𝑛. (57) Then, by (48), (49), (50), and (57), we have 𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝑡,ℳ_𝑀,𝑁≥𝑔𝑦,𝑥,𝑡_𝐿∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,^∞_𝑘=𝑛0𝜙^𝑘𝑡₀,ℳ_𝑀,𝑁𝑔𝑦,𝑥,^∞_𝑘=𝑛0𝜙^𝑘𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝜙^𝑛0𝑡₀,ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝜙^𝑛0𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁(𝑔𝑥,𝑦,𝑡₀)^2𝑛0,ℳ_𝑀,𝑁(𝑔𝑦,𝑥,𝑡₀)^2𝑛0≥_𝐿^∗𝒯^2𝑛0+1−1𝑁_𝑠𝜇≥,𝜇_𝐿∗𝑁_𝑠(𝜆),𝜆. (58) So for any 𝜆>0 we have 𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑦,𝑡,ℳ_𝑀,𝑁𝑔𝑦,𝑥,𝑡≥_𝐿^∗𝑁_𝑠(𝜆),𝜆,∀𝑡>0. (59) We can get that 𝑔𝑥=𝑦 and 𝑔𝑦=𝑥.
Step 4. Prove that 𝑥=𝑦.
Since 𝒯 is a continuous t-representable of H-type. Therefore, for any 𝜆>0, there exists a 𝜇>0 such that 𝒯^𝑚−1𝑁_𝑠𝜇,𝜇≥_𝐿^∗𝑁_𝑠(𝜆),𝜆,∀𝑚∈𝑁. (60) Since lim_𝑡→∞ℳ_𝑀,𝑁(𝑥,𝑦,𝑡)=1_𝐿^∗ for all 𝑥,𝑦∈𝑋, there exists 𝑡₀>0 such that ℳ_𝑀,𝑁𝑥,𝑦,𝑡₀≥_𝐿^∗𝑁_𝑠𝜇,𝜇. (61) On the other hand, since 𝜙∈Φ, by condition (𝜙-3), we have ∑^∞_𝑛=1𝜙^𝑛(𝑡₀)<∞. Then, for any 𝑡>0, there exists 𝑛₀∈𝑁 such that 𝑡>^∞_𝑘=𝑛0𝜙^𝑘𝑡₀. (62) Since, for 𝑡₀>0, ℳ_𝑀,𝑁𝑔𝑥_𝑛+1,𝑔𝑦_𝑛+1,𝜙𝑡₀=ℳ_𝑀,𝑁𝐹𝑥_𝑛,𝑦_𝑛,𝐹𝑦_𝑛,𝑥_𝑛,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑡₀,ℳ_𝑀,𝑁𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑡₀, (63) thus ℳ_𝑀,𝑁𝑔𝑥_𝑛+1,𝑔𝑦_𝑛+1,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑡₀,ℳ_𝑀,𝑁𝑔𝑦_𝑛,𝑔𝑥_𝑛,𝑡₀. (64) Letting 𝑛→∞ in the above inequality, we get ℳ_𝑀,𝑁𝑥,𝑦,𝜙𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁𝑥,𝑦,𝑡₀,ℳ_𝑀,𝑁𝑦,𝑥,𝑡₀. (65) Thus, by (60), (61), (62), and (65), we have ℳ_𝑀,𝑁≥𝑥,𝑦,𝑡_𝐿∗ℳ_𝑀,𝑁𝑥,𝑦,^∞_𝑘=𝑛0𝜙^𝑘𝑡₀≥_𝐿^∗ℳ_𝑀,𝑁𝑥,𝑦,𝜙^𝑛0𝑡₀≥_𝐿^∗𝒯ℳ_𝑀,𝑁(𝑥,𝑦,𝑡₀)^2𝑛0−1,ℳ_𝑀,𝑁(𝑦,𝑥,𝑡₀)^2𝑛0−1≥_𝐿^∗𝒯^2𝑛0+1−3𝑁_𝑠𝜇≥,𝜇_𝐿∗𝑁_𝑠(𝜆),𝜆, (66) which implies that 𝑥=𝑦. Thus, we have proved that 𝐹 and 𝑔 have a unique common fixed point in 𝑋. This completes the proof of Theorem 31.

Taking 𝑔=𝐼 (the identity mapping) in Theorem 31, we get the following consequence.

Corollary 32. Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a complete modified intuitionistic fuzzy metric space, where 𝒯 is continuous t-representable of H-type satisfying (22). Let 𝐹∶𝑋×𝑋→𝑋 and there exist 𝜙∈Φ such that ℳ_𝑀,𝑁𝐹()(𝑡)≥𝑥,𝑦,𝐹𝑢,𝑣,𝜙_𝐿∗𝒯ℳ_𝑀,𝑁(𝑥,𝑢,𝑡),ℳ_𝑀,𝑁𝑦,𝑣,𝑡, (67) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 and 𝑡>0. Then there exists 𝑥∈𝑋 such that 𝑥=𝐹(𝑥,𝑥).

Put 𝜙(𝑡)=𝑘𝑡, where 0<𝑘<1, in Theorem 31; we get the following.

Corollary 33. Let (𝑋,ℳ_𝑀,𝑁,𝒯) be a modified intuitionistic fuzzy metric space such that ℳ_𝑀,𝑁 has p-property. Let 𝐹∶𝑋×𝑋→𝑋 and 𝑔∶𝑋→𝑋 be two functions such that ℳ_𝑀,𝑁𝐹()≥𝑥,𝑦,𝐹𝑢,𝑣,𝑘𝑡_𝐿∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑔𝑢,𝑡,ℳ_𝑀,𝑁𝑔𝑦,𝑔𝑣,𝑡, (68) for all 𝑥,𝑦,𝑢,𝑣∈𝑋 and 𝑡>0, where 0<𝑘<1. Suppose that 𝐹(𝑋×𝑋)⊆𝑔(𝑋) and 𝐹(𝑋×𝑋) or 𝑔(𝑋) is complete. Then there exists a unique 𝑥∈𝑋 such that 𝑥=𝑔(𝑥)=𝐹(𝑥,𝑥).

Remark 34. Comparing Theorem 31 in the present paper with Theorem  3.1 in [13], we can see that Theorem 31 is a genuine generalization of Theorem 3.1 in the sense that(1)we only use the completeness of 𝑔(𝑋) or 𝐹(𝑋×𝑋);(2)we drop off the continuity of 𝑔;(3)the concept of compatible mappings has been replaced by weakly compatible mappings.

Next, we give an example to demonstrate Theorem 31.

Example 35. Let 𝑋={0,1,1/2,1/3,…,1/𝑛,…} and 𝒯(𝑎,𝑏)=(min{𝑎₁,𝑏₁},max{𝑎₂,𝑏₂}), for all 𝑎=(𝑎₁,𝑎₂) and 𝑏=(𝑏₁,𝑏₂)∈𝐿^∗. Define ℳ_𝑀,𝑁=𝑡𝑥,𝑦,𝑡,𝑡+𝑥−𝑦𝑥−𝑦,𝑡+𝑥−𝑦∀𝑥,𝑦∈𝑋,𝑡>0. (69) Then (𝑋,ℳ_𝑀,𝑁,𝒯) is a modified intuitionistic fuzzy metric space. Let 𝜙(𝑡)=𝑡/2. Let 𝑔∶𝑋→𝑋 and 𝐹∶𝑋×𝑋→𝑋 be defined as 𝑔(𝑥)=10,𝑥=0,1,𝑥=,12𝑛+112𝑛+1,𝑥=,𝐹=12𝑛𝑥,𝑦(2𝑛+1)⁴,=1𝑥,𝑦,12𝑛,2𝑛0,otherwise. (70) Let 𝑥_𝑛=𝑦_𝑛=1/2𝑛. We have 𝑔𝑥_𝑛=1𝐹𝑥2𝑛+1⟶0,_𝑛,𝑦_𝑛=1(2𝑛+1)⁴⟶0as𝑛⟶∞, (71) but ℳ_𝑀,𝑁𝐹𝑔𝑥_𝑛,𝑔𝑦_𝑛,𝑔𝐹𝑥_𝑛,𝑦_𝑛,𝑡=ℳ_𝑀,𝑁(0,1,𝑡)↛1_𝐿^∗, (72) so 𝑔 and 𝐹 are not compatible. From 𝐹(𝑥,𝑦)=𝑔(𝑥) and 𝐹(𝑦,𝑥)=𝑔(𝑦), we can get (𝑥,𝑦)=(0,0) and we have 𝑔𝐹(0,0)=𝐹(𝑔0,𝑔0), which implies that 𝐹 and 𝑔 are weakly compatible. The following result is easy to be verified: 𝑡𝑡𝑋+𝑡≥min,𝑡𝑌+𝑡,𝑋𝑍+𝑡𝑌𝑋+𝑡≤max,𝑍𝑌+𝑡,{}𝑍+𝑡⟺𝑋≤max𝑌,𝑍,∀𝑋,𝑌,𝑍≥0,𝑡>0. (73) By the definition of ℳ_𝑀,𝑁,𝜙 and the result above, we can get inequality (30): ℳ_𝑀,𝑁𝐹()(𝑡)≥𝑥,𝑦,𝐹𝑢,𝑣,𝜙_𝐿∗𝒯ℳ_𝑀,𝑁𝑔𝑥,𝑔𝑢,𝑡,ℳ_𝑀,𝑁𝑔𝑦,𝑔𝑣,𝑡, (74) which is equivalent to the following: 2𝐹𝑥,𝑦−𝐹(𝑢,𝑣)≤max𝑔𝑥−𝑔𝑢,𝑔𝑦−𝑔𝑣. (75) Now, we verify inequality (75). Let 𝐴={1/2𝑛,𝑛∈𝑁},𝐵=𝑋−𝐴. By the symmetry and without loss of generality, (𝑥,𝑦),(𝑢,𝑣) have six possibilities.
Case 1. (𝑥,𝑦)∈𝐵×𝐵,(𝑢,𝑣)∈𝐵×𝐵. It is obvious that (75) holds.
Case 2. (𝑥,𝑦)∈𝐵×𝐵,(𝑢,𝑣)∈𝐵×𝐴. It is obvious that (75) holds.
Case 3. (𝑥,𝑦)∈𝐵×𝐵,(𝑢,𝑣)∈𝐴×𝐴. If 𝑢≠𝑣, (75) holds. If 𝑢=𝑣, let 𝑢=𝑣=1/2𝑛, and then 2𝐹(=2𝑥,𝑦−𝐹𝑢,𝑣)(2𝑛+1)⁴,max𝑔𝑥−𝑔𝑢,𝑔𝑦−𝑔𝑣=2𝑛,2𝑛+1 (76) which implies that (75) holds.
Case 4. (𝑥,𝑦)∈𝐵×𝐴,(𝑢,𝑣)∈𝐵×𝐴. It is obvious that (75) holds.
Case 5. (𝑥,𝑦)∈𝐵×𝐴,(𝑢,𝑣)∈𝐴×𝐴. If 𝑢≠𝑣, (75) holds. If 𝑢=𝑣, let 𝑥∈𝐵,𝑦=1/2𝑗,𝑢=𝑣=1/2𝑛, and then 2𝐹(=2𝑥,𝑦−𝐹𝑢,𝑣)(2𝑛+1)⁴,,1max𝑔𝑥−𝑔𝑢𝑔𝑦−𝑔𝑣=max,12𝑛+1−12𝑗+1,2𝑛+1 (77) or max𝑔𝑥−𝑔𝑢,𝑔𝑦−𝑔𝑣=max2𝑛,12𝑛+1−12𝑗+1.2𝑛+1 (78) So, (75) holds.
Case 6. (𝑥,𝑦)∈𝐴×𝐴 and (𝑢,𝑣)∈𝐴×𝐴. If 𝑥≠𝑦 and 𝑢≠𝑣, (75) holds. If 𝑥≠𝑦,𝑢=𝑣, let 𝑥=1/2𝑖, 𝑦=1/2𝑗,𝑖≠𝑗, and 𝑢=𝑣=1/2𝑛. Then 2𝐹(=2𝑥,𝑦−𝐹𝑢,𝑣)(2𝑛+1)⁴,,1max𝑔𝑥−𝑔𝑢𝑔𝑦−𝑔𝑣=max−12𝑖+1,12𝑛+1−12𝑗+1.2𝑛+1 (79) So, (75) holds. If 𝑥=𝑦 and 𝑢=𝑣, let 𝑥=𝑦=1/2𝑖, 𝑢=𝑣=1/2𝑛. Then 2𝐹(1𝑥,𝑦−𝐹𝑢,𝑣)=2(2𝑖+1)⁴−1(2𝑛+1)⁴,,=1max𝑔𝑥−𝑔𝑢𝑔𝑦−𝑔𝑣−12𝑖+1.2𝑛+1 (80) So, (75) holds. Then all the conditions in Theorem 31 are satisfied and 0 is the unique common fixed point of 𝑔 and 𝐹.

4. Application to Integral Equations

As an application of the coupled fixed point theorems established in Section 3 of our paper, we study the existence and uniqueness of the solution to a Fredholm nonlinear integral equation. We will consider the following integral equation: 𝑥𝑝=^𝑏_𝑎𝐾₁𝑝,𝑞+𝐾₂×𝑓𝑞𝑞𝑝,𝑝,𝑞𝑞,𝑥+𝑔𝑞,𝑥𝑑𝑞+ℎ (81) for all 𝑝∈𝐼=[𝑎,𝑏].

Let Θ denote the set of all functions 𝜃∶[0,+∞)→[0,+∞) satisfying the following: (i_𝜃)𝜃 is nondecreasing; (ii_𝜃)𝜃(𝑝)≤𝑝.

We assume that the functions 𝐾₁,𝐾₂,𝑓,𝑔 fulfill the following conditions.

Assumption 36. (i) Consider 𝐾₁𝑝,𝑞≥0,𝐾₂𝑝,𝑞≤0,∀𝑝,𝑞∈𝐼. (82)
(ii) There exists positive numbers 𝜆,𝜇, and 𝜃∈Θ such that, for all 𝑥,𝑦∈𝑅 with 𝑥≥𝑦, the following conditions hold: 0≤𝑓𝑞,𝑥−𝑓𝑞,𝑦≤𝜆𝜃𝑥−𝑦,−𝜇𝜃𝑥−𝑦≤𝑔𝑞,𝑥−𝑔𝑞,𝑦≤0. (83)
(iii) Consider max𝜆,𝜇sup_𝑝∈𝐼^𝑏_𝑎𝐾₁𝑝,𝑞−𝐾₂1𝑝,𝑞𝑑𝑞≤8. (84)

Theorem 37. Consider the integral equation (81) with 𝐾₁,𝐾₂∈𝐶(𝐼×𝐼,ℝ), 𝑓,𝑔∈𝐶(𝐼×ℝ,ℝ), and ℎ∈𝐶(𝐼,𝑅). Suppose that Assumption 36 is satisfied. Then the integral equation (81) has a unique solution in 𝐶(𝐼,𝑅).

Proof. Consider 𝑋=𝐶(𝐼,𝑅). It is easy to check that (𝑋,ℳ_𝑀,𝑁,𝒯) is a complete modified intuitionistic fuzzy metric space with respect to the modified intuitionistic fuzzy metric: ℳ_𝑀,𝑁=𝑡𝑥,𝑦,𝑡,𝑡+𝑥−𝑦𝑥−𝑦,𝑡+𝑥−𝑦∀𝑥,𝑦∈𝑋,𝑡>0, (85) for all 𝑥,𝑦∈𝑋 and 𝑡>0 with 𝒯(𝑎,𝑏)=(min{𝑎₁,𝑏₁},max{𝑎₂,𝑏₂}), for all 𝑎=(𝑎₁,𝑎₂) and 𝑏=(𝑏₁,𝑏₂)∈𝐿^∗. Define now the mapping 𝐹∶𝑋×𝑋→𝑋 by 𝐹𝑥,𝑦𝑝=^𝑏_𝑎𝐾₁𝑓𝑞𝑞+𝑝,𝑞𝑞,𝑥+𝑔𝑞,𝑦𝑑𝑞_𝑏𝑎𝐾₂𝑓𝑞𝑞𝑝,𝑝,𝑞𝑞,𝑦+𝑔𝑞,𝑥𝑑𝑞+ℎ (86) for all 𝑝∈𝐼 and 𝜙(𝑡)=𝑡/2 for all 𝑡>0. Now, for all 𝑥,𝑦,𝑢,𝑣∈𝑋, using (83), we have 𝐹𝑥,𝑦𝑝−𝐹(𝑢,𝑣)𝑝=^𝑏_𝑎𝐾₁𝑓𝑞𝑞+𝑝,𝑞𝑞,𝑥+𝑔𝑞,𝑦𝑑𝑞_𝑏𝑎𝐾₂𝑓𝑞𝑞−𝑝,𝑞𝑞,𝑦+𝑔𝑞,𝑥𝑑𝑞_𝑏𝑎𝐾₁𝑓𝑞𝑞−𝑝,𝑞𝑞,𝑢+𝑔𝑞,𝑣𝑑𝑞_𝑏𝑎𝐾₂𝑓𝑞𝑞=𝑝,𝑞𝑞,𝑣+𝑔𝑞,𝑢𝑑𝑞_𝑏𝑎𝐾₁𝑓𝑞𝑞𝑞𝑞+𝑝,𝑞𝑞,𝑥−𝑓𝑞,𝑢+𝑔𝑞,𝑦−𝑔𝑞,𝑣𝑑𝑞_𝑏𝑎𝐾₂𝑓𝑞𝑞𝑞𝑞=𝑝,𝑞𝑞,𝑦−𝑓𝑞,𝑣+𝑔𝑞,𝑥−𝑔𝑞,𝑢𝑑𝑞_𝑏𝑎𝐾₁𝑓𝑞𝑞−𝑔𝑞𝑞−𝑝,𝑞𝑞,𝑥−𝑓𝑞,𝑢𝑞,𝑣−𝑔𝑞,𝑦𝑑𝑞_𝑏𝑎𝐾₂𝑓𝑞𝑞−𝑔𝑞𝑞≤𝑝,𝑞𝑞,𝑣−𝑓𝑞,𝑦𝑞,𝑥−𝑔𝑞,𝑢𝑑𝑞_𝑏𝑎𝐾₁𝑥𝑞𝑞𝑣𝑞𝑞−𝑝,𝑞𝜆𝜃−𝑢+𝜇𝜃−𝑦𝑑𝑞_𝑏𝑎𝐾₂𝑣𝑞𝑞𝑥𝑞𝑞𝑝,𝑞𝜆𝜃−𝑦+𝜇𝜃−𝑢𝑑𝑞. (87) Thus 𝐹𝑥,𝑦𝑝−𝐹(𝑢,𝑣)𝑝≤^𝑏_𝑎𝐾₁𝑥𝑞𝑞𝑣𝑞𝑞−𝑝,𝑞𝜆𝜃−𝑢+𝜇𝜃−𝑦𝑑𝑞_𝑏𝑎𝐾₂𝑣𝑞𝑞𝑥𝑞𝑞𝑝,𝑞𝜆𝜃−𝑦+𝜇𝜃−𝑢𝑑𝑞. (88) Since the function 𝜃 is nondecreasing, so we have 𝜃𝑥𝑞𝑞𝑥𝑞𝑞,𝜃𝑣𝑞𝑞𝑣𝑞𝑞.−𝑢≤𝜃−𝑢−𝑦≤𝜃−𝑦 (89) Hence, by (88), in view of the fact that 𝐾₂(𝑝,𝑞)≤0, we get 𝐹𝑥,𝑦𝑝−𝐹(𝑢,𝑣)𝑝≤^𝑏_𝑎𝐾₁𝑥𝑞𝑞𝑣𝑞𝑞−𝑝,𝑞𝜆𝜃−𝑢+𝜇𝜃−𝑦𝑑𝑞_𝑏𝑎𝐾₂𝑣𝑞𝑞𝑥𝑞𝑞𝑝,𝑞𝜆𝜃−𝑦+𝜇𝜃−𝑢𝑑𝑞≤^𝑏_𝑎𝐾₁𝜃𝑥𝑞𝑞𝜃𝑣𝑞𝑞−𝑝,𝑞max𝜆,𝜇−𝑢+max𝜆,𝜇−𝑦𝑑𝑞_𝑏𝑎𝐾₂𝜃𝑣𝑞𝑞𝜃𝑥𝑞𝑞𝑝,𝑞max𝜆,𝜇−𝑦+max𝜆,𝜇−𝑢𝑑𝑞, (90) as all the quantities on the right hand side of (88) are nonnegative. Now, by using (84), we get 𝐹𝑥,𝑦−𝐹(𝑢,𝑣)≤max𝜆,𝜇^𝑏_𝑎𝐾₁𝑝,𝑞−𝐾₂⋅𝜃𝑥𝑞𝑞𝑣𝑞𝑞𝑝,𝑞𝑑𝑞−𝑢+𝜃−𝑦≤max𝜆,𝜇sup_𝑝∈𝐼^𝑏_𝑎𝐾₁𝑝,𝑞−𝐾₂⋅𝜃𝑥𝑞𝑞𝑣𝑞𝑞≤𝜃𝑝,𝑞𝑑𝑞−𝑢+𝜃−𝑦(|𝑥−𝑢|)+𝜃𝑣−𝑦8. (91) Thus 2𝐹(≤𝜃𝑥,𝑦−𝐹𝑢,𝑣)(|𝑥−𝑢|)+𝜃𝑣−𝑦4. (92) Now, since 𝜃 is nondecreasing, we have 𝜃||+,𝜃|+,(|𝑥−𝑢|)≤𝜃𝑥−𝑢𝑦−𝑣𝑦−𝑣≤𝜃|𝑥−𝑢𝑦−𝑣 (93) which, by (ii_𝜃), implies 𝜃(|𝑥−𝑢|)+𝜃𝑦−𝑣2||+≤||+|,,≤𝜃𝑥−𝑢𝑦−𝑣𝑥−𝑢𝑦−𝑣≤2max|𝑥−𝑢𝑦−𝑣 (94) and so 𝜃(|𝑥−𝑢|)+𝜃𝑦−𝑣4≤max|𝑥−𝑢|,𝑦−𝑣. (95) Thus, by (92) and (95), we get 2𝐹𝑥,𝑦−𝐹(𝑢,𝑣)≤max|𝑥−𝑢|,𝑦−𝑣. (96)
Now, by (96) and (73), it follows that ℳ_𝑀,𝑁𝐹𝑥,𝑦,𝐹(𝑢,𝑣),𝜙(𝑡)=ℳ_𝑀,𝑁𝐹(),𝑡𝑥,𝑦,𝐹𝑢,𝑣2=𝑡/2𝐹(,𝐹(𝑡/2+𝑥,𝑦−𝐹𝑢,𝑣)𝑥,𝑦−𝐹𝑢,𝑣)𝐹(=𝑡𝑡/2+𝑥,𝑦−𝐹𝑢,𝑣)𝐹(,2𝐹(𝑡+2𝑥,𝑦−𝐹𝑢,𝑣)𝑥,𝑦−𝐹𝑢,𝑣)𝐹(≥𝑡+2𝑥,𝑦−𝐹𝑢,𝑣)_𝐿∗𝑡|,,|,𝑡+max|𝑥−𝑢𝑦−𝑣max|𝑥−𝑢𝑦−𝑣|,≥𝑡+max|𝑥−𝑢𝑦−𝑣_𝐿∗𝑡min||,𝑡𝑡+𝑥−𝑢,||𝑡+𝑦−𝑣max𝑥−𝑢||,𝑡+𝑥−𝑢𝑦−𝑣≥𝑡+𝑦−𝑣_𝐿∗𝒯ℳ_𝑀,𝑁(𝑥,𝑢,𝑡),ℳ_𝑀,𝑁𝑦,𝑣,𝑡, (97) which is the condition in (67), shows that all hypotheses of Corollary 32 are satisfied. This proves that 𝐹 has a unique fixed point 𝑥∈𝑋; that is, 𝑥=𝐹(𝑥,𝑥), and therefore 𝑥∈𝐶(𝐼,𝑅) is the unique solution of the integral equation (81).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.