
Application of Coupled Fixed Point Technique in Solving Integral Equations on Modified Intuitionistic Fuzzy Metric SpacesDOI: 10.1155/2014/348069 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 tnorms 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 tnorms and continuous tconorms 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 trepresentable. 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 integrodifferential equations can be applied in science and engineering. Many areas that are described by
