Narrow Results

In this paper, we adopt a fixed point approach towards a problem of finding fuzzy distance between two subsets of a fuzzy metric space. This is accomplished simultaneously by determining two different pairs of points which are obtained by finding an optimal approximate solution of a coupled fixed point equation for an appropriate coupled contractive operator. By its nature, the problem is a global optimization problem. The coupled contraction defined here has certain properties suitable for applications to the optimization problem considered in this paper. The main result is illustrated with an example. By applications of the main theorem, a new coupled fixed point result is derived in fuzzy metric space and a new coupled proximity point result is obtained in metric spaces. This work is a continuation of a newly emerged line of research on global optimization in the context of fuzzy metric spaces.

We introduce $\alpha$-coupled fixed point for $𝜃$-contractions in the space of bounded functions on a set $S$. As an application of our result, we prove the existence and uniqueness of solutions for a class of systems of functional equations arising in dynamic programming.

In this paper, we discuss the relation theoretic fixed point results for rational type contractions by using $C$-class functions. Also, we prove that the existence and uniqueness of coupled fixed point theorem under mixed monotone property and $C$-class function. Moreover, we give some examples in support of our proved result. Finally, we give an application to find a solution to a nonlinear integral equation.