#### 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 -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 and operation defined by
**, and for every . Then is a complete lattice.*

*Definition 2 (Atanassov [3]). *An intuitionistic fuzzy set in a universe is an object , where, for all and are called the membership degree and nonmembership degree, respectively, of in and further they satisfy . For every , if such that , then it is easy to see that
We denote its units by and . Classically, a triangular norm on is defined as an increasing, commutative, associative mapping satisfying , for all . A triangular conorm is defined as an increasing, commutative, associative mapping satisfying , for all . Using the lattice , these definitions can be straightforwardly extended.

*Definition 3 (Deschrijver et al. [15]). *A triangular norm (*t*-norm) on is a mapping satisfying the following conditions: (boundary condition), (commutativity), (associativity), (monotonicity).

*Definition 4 (Deschrijver and Kerre [14] and Deschrijver et al. [15]). *A continuous* t*-norm on is called continuous* t*-representable if and only if there exist a continuous* t*-norm and a continuous* t*-conorm on such that, for all ,
Now define a sequence recursively by and
for and .

*Definition 5 (Deschrijver and Kerre [14] and Deschrijver et al. [15]). *A negator on is any decreasing mapping satisfying and . If , for all , then is called an involutive negator. A negator on is a decreasing mapping satisfying and denotes the standard negator on defined as, for all .

*Definition 6 (Saadati et al. [7]). *Let be fuzzy sets from to such that for all and . The 3-tuple is said to be a modified intuitionistic fuzzy metric space if is an arbitrary (nonempty) set, is a continuous* t*-representable, and is a mapping satisfying the following conditions for every and :(a),
(b) if and only if ,(c),(d),
(e) is continuous.

In this case, is called a modified intuitionistic fuzzy metric. Here, .

*Example 7 (Saadati et al. [7]). *Let be a metric space. Denote for all and and let and be fuzzy sets on defined as follows:
for all . Then is a modified intuitionistic fuzzy metric space.

*Example 8 (Saadati et al. [7]). *Let . Denote for all and and let and be fuzzy sets on defined as follows:
for all and . Then is a modified intuitionistic fuzzy metric space.

Lemma 9 (Fang [16]). *We say that the intuitionistic fuzzy metric space has property , if it satisfies the following condition: for all implies .*

*Remark 10. *Throughout this paper, is a modified intuitionistic fuzzy metric space with property .

*Definition 11 (Saadati et al. [7]). *A sequence in a modified intuitionistic fuzzy metric space is called a Cauchy sequence if, for each and , there exists such that
and, for each , here is the standard negator. The sequence is said to be convergent to in the modified intuitionistic fuzzy metric space and denoted by , if whenever for every . 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 be a modified intuitionistic fuzzy metric. Then, for any is nondecreasing with respect to , in for all in .*

*Definition 13 (Saadati et al. [7]). *Let be a modified intuitionistic fuzzy metric space. For , define the open ball with center and radius , as
A subset is called open if, for each , there exist and such that . Let denote the family of all open subsets of is called the topology induced by modified intuitionistic fuzzy metric. This topology is Hausdorff.

*Definition 14 (Saadati et al. [7]). *Let be a modified intuitionistic fuzzy metric space. is said to be continuous on if
whenever a sequence in converges to a point , that is;

Lemma 15 (Saadati et al. [7]). *Let be a modified intuitionistic fuzzy metric space. Then is continuous function on .*

*Definition 16 (Bhaskar and Lakshmikantham [8]). *An element is called a coupled fixed point of the mapping if

*Definition 17 (Lakshmikantham and Ciric [9]). *Let be a nonempty set. An element is called a coupled coincidence point of the mappings and if

*Definition 18 (Lakshmikantham and Ciric [9]). *Let be a nonempty set. An element is called a common coupled fixed point of the mappings and if

*Definition 19 (Lakshmikantham and Ciric [9]). *Let be a nonempty set. An element is called a common fixed point of the mappings and if

*Definition 20 (Lakshmikantham and Ciric [9]). *Let be a nonempty set. The mappings and are said to be commutative if

*Definition 21 (Fang [18]). *Let be a modified intuitionistic fuzzy metric space. The mappings and are said to be compatible if
for all , whenever and are sequences in such that

*Definition 22 (Abbas et al. [19]). *Let be a nonempty set. The mappings and are called weakly compatible mappings if implies that and , for all .

#### 3. Main Results

*Definition 23. *Let . A continuous* t*-representable is said to be continuous* t*-representable of H-type if the family of functions is equicontinuous at , where
Obviously, is a H-type* t*-representable if and only if, for any , there exists such that

*Remark 24. *In a modified intuitionistic fuzzy metric space , whenever for and , we can find a , , such that .

*Remark 25. *For convenience, we denote
for all .

*Definition 26. *Let be a modified intuitionistic fuzzy metric space. is said to satisfy the* p*-property on if
whenever , and .

Lemma 27. *Let be a modified intuitionistic fuzzy metric space and let satisfy the p-property; then
*

*Proof. *If not, since is nondecreasing, there exists such that
and then, for , when as and we get
which is a contradiction.

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

*Example 29. *Let be an ordinary metric space and let be defined as follows:
where . Then is continuous and increasing in and . Let
for all and , and , for all and . Then is a modified intuitionistic fuzzy metric space with
But, for any , and ,
Define , where , and each satisfies the following conditions: is nondecreasing, is continuous, for all , where , .

It is easy to prove that if , then for all .

Lemma 30. *Let be a modified intuitionistic fuzzy metric space, where is a continuous t-representable of H-type. If there exists , such that if
*

*then .*

*Proof. *Since and . Using the monotony of , we have . Using (29) and the definition of modified intuitionistic fuzzy metric, we have .

Theorem 31. *Let be a modified intuitionistic fuzzy metric space, where is a continuous t-representable of H-type satisfying (22). Let and be two weakly compatible mappings and there exists such that
*

*for all and . Suppose that and or is complete. Then there exists a unique such that .*

*Proof. *Let be two arbitrary points in . Since , we can choose such that
Continuing in this way, we can construct two sequences and in such that
The proof is divided into 4 steps. *Step **1.* Prove that and are Cauchy sequences.

Since is a continuous* t*-representable of H-type, therefore, for any , there exists a such that
Since for all , there exists such that
On the other hand, since , by condition , we have . Then, for any , there exists such that
From condition (30), we have
Similarly, we can also get
Continuing in the same way, we can get
Now, from (33), (34), and (35), for , we have
which implies that
for all with and . So is a Cauchy sequence. Similarly, we can get that is also a Cauchy sequence. *Step **2.* Prove that and have a coupled coincidence point.

Without loss of generality, we can assume that is complete; then there exist and such that
From (30), we get
Since is continuous, taking limit as , we have
which implies that
Similarly, we can show that
Since and are weakly compatible, we get that
which implies that
*Step **3.* Prove that and .

Since is a continuous* t*-representable of H-type. Therefore, for any , there exists a such that
Since for all , there exists such that
On the other hand, since , by condition , we have . Then, for any , there exists such that
Since
thus
Letting in (52), by using (41), we get
Similarly, we can get
From (53) and (54), we have
By this way, we can get, for all ,
Thus,
Then, by (48), (49), (50), and (57), we have
So for any we have
We can get that and .*Step **4.* Prove that .

Since is a continuous* t*-representable of H-type. Therefore, for any , there exists a such that
Since for all , there exists such that
On the other hand, since , by condition , we have . Then, for any , there exists such that
Since, for ,
thus
Letting in the above inequality, we get
Thus, by (60), (61), (62), and (65), we have
which implies that . Thus, we have proved that and have a unique common fixed point in . This completes the proof of Theorem 31.

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

Corollary 32. *Let be a complete modified intuitionistic fuzzy metric space, where is continuous t-representable of H-type satisfying (22). Let and there exist such that
*

*for all and . Then there exists such that .*

Put , where , in Theorem 31; we get the following.

Corollary 33. *Let be a modified intuitionistic fuzzy metric space such that has p-property. Let and be two functions such that
*

*for all and , where . Suppose that and or is complete. Then there exists a unique such that .*

*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 or ;(2)we drop off the continuity of ;(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 and , for all and . Define
Then is a modified intuitionistic fuzzy metric space. Let . Let and be defined as
Let . We have
but
so and are not compatible. From and , we can get and we have , which implies that and are weakly compatible. The following result is easy to be verified:
By the definition of and the result above, we can get inequality (30):
which is equivalent to the following:
Now, we verify inequality (75). Let . By the symmetry and without loss of generality, have six possibilities. *Case **1. *. It is obvious that (75) holds.*Case **2. *. It is obvious that (75) holds. *Case **3. *. If , (75) holds. If , let , and then
which implies that (75) holds. *Case **4. *. It is obvious that (75) holds. *Case **5. *. If , (75) holds. If , let , and then
or
So, (75) holds. *Case **6. * and . If and , (75) holds. If , let , , and . Then
So, (75) holds. If and , let , . Then
So, (75) holds. Then all the conditions in Theorem 31 are satisfied and is the unique common fixed point of and .

#### 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: for all .

Let denote the set of all functions satisfying the following: is nondecreasing; .

We assume that the functions fulfill the following conditions.

*Assumption 36. *(i) Consider

(ii) There exists positive numbers , and such that, for all with , the following conditions hold:

(iii) Consider

Theorem 37. *Consider the integral equation (81) with , , and . Suppose that Assumption 36 is satisfied. Then the integral equation (81) has a unique solution in .*

*Proof. *Consider . It is easy to check that is a complete modified intuitionistic fuzzy metric space with respect to the modified intuitionistic fuzzy metric:
for all and with , for all and . Define now the mapping by
for all and for all . Now, for all , using (83), we have