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.

The concept of fuzzy metric space has been introduced in several ways. In [

Atanassov [

Gregori et al. [

Bhaskar and Lakshmikantham [

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. [

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 [

Consider the set

An intuitionistic fuzzy set

A triangular norm (

A continuous

A negator on

Let

In this case,

Let

Let

We say that the intuitionistic fuzzy metric space

Throughout this paper,

A sequence

Let

Let

Let

Let

An element

Let

Let

Let

Let

Let

Let

Let

In a modified intuitionistic fuzzy metric space

For convenience, we denote

Let

Let

If not, since

Condition (

Let

It is easy to prove that if

Let

Since

Let

Let

Since

Without loss of generality, we can assume that

Since

Since

Taking

Let

Put

Let

Comparing Theorem

we only use the completeness of

we drop off the continuity of

the concept of compatible mappings has been replaced by weakly compatible mappings.

Next, we give an example to demonstrate Theorem

Let

As an application of the coupled fixed point theorems established in Section

Let

We assume that the functions

(i) Consider

(ii) There exists positive numbers

(iii) Consider

Consider the integral equation (

Consider

Now, by (

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