Fuzzy Soft Metric Space PDF Download

The various uncertainties arise in complicated problems in Economics, Engineering, Environmental Science, Medical Science and Social Science. The methods of classical Mathematics may not be successfully used to solve them. Mathematical theories such as probability theory, fuzzy set theory and rough set theory were established by researchers to model uncertainties appearing in the above fields. But all these theories have their own difficulties. To overcome these difficulties, In 1999 Molodstov[7] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties. As the problem of setting the membership function does not arise in soft set theory, it can be easily applied to many different fields. In 2003, Maji.et.al.[5] studied some operations on the soft set theory. In 2009, M.I.Ali et.al.[1] studied some new operations on soft sets and its applications. In 2013, Sujoy Das et.al.[11] proposed soft metric space. In 2015, Thangaraj Beaula et.al.,[12] established the fuzzy soft metric spaces. In chapter 1, the basic definitions, examples, properties and theorems are given which are used for throughout the dissertation. In chapter 2, we defined Fuzzy soft metric space with suitable illustrations. We proved arbitrary union of fuzzy soft open set is fuzzy soft open set and the intersection of finite number of fuzzy soft open set is fuzzy soft open set. In chapter 3, Cauchy sequence are defined. First category, second category, dense, nowhere dense are all defined with suitable illustrations. We established Cantor intersection theorem on complete fuzzy soft metric space and also we proved Baires category theorem on fuzzy soft metric space. In chapter 4, fuzzy soft open cover, fuzzy soft compact set and fuzzy soft totally bounded set are defined. We proved some important theorems. Also we defined Bolzano Weirstress property and based on this we proved theorem namely fuzzy soft metric space becomes fuzzy soft sequentially compact if and only if fuzzy soft metric space has the property Bolzano Weirstrass. In chapter 5, we defined convex fuzzy soft metric space. Also we defined self mapping, fixed point and convergence of convex fuzzy soft metric space. Using these all we proved fixed point theorem on convex fuzzy soft metric space.