Narrow Results

This paper is about the generalization of the concept of fuzzy metric. Here we introduce g-fuzzy metric space of order $n$. In particular case, it reduces to George- and Veeramani-type fuzzy metric and Q-fuzzy metric. Some non-trivial examples of g-fuzzy metric space are given and its topological structure, the notion of convergent sequence, Cauchy sequence, etc. are also studied.

In this paper, we construct an Iterated Function System (IFS) on a fuzzy metric space for generating fractal sets in that fuzzy metric space. A new cyclic contraction mapping is defined and utilized in the construction of the IFS. It is shown that the cyclic contraction has a unique fixed point when the space is complete. These contractions automatically do not possess continuity property. A fuzzy variant of the Hausdorff metric is used on the set of compact subsets of the fuzzy metric space. Two illustrative examples are discussed. Further, the fixed point theorem obtained here effectively generalizes a recent result in the literature.

In this paper, we investigate the fractals generated by the iterated function system of fuzzy contractions in the fuzzy metric spaces by generalizing the Hutchinson-Barnsley theory. We prove some existence and uniqueness theorems of fractals in the standard fuzzy metric spaces by using the fuzzy Banach contraction theorem. In addition to that, we discuss some results on fuzzy fractals such as Collage Theorem and Falling Leaves Theorem in the standard fuzzy metric spaces with respect to the standard Hausdorff fuzzy metrics.

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.