New fixed point theorems in partial fuzzy metric spaces

Abstract

The notion of a fuzzy set is generally very useful in today s time. Lofti Zadeh, devised it in the 1960s as an approach to deal with uncertainty caused by inaccuracy or ambiguity instead of randomness. A quotset of real numbers significantly greater than 1quot is a good illustration of what it means. Further, distance is a quantity which is used in day to day life. In mathematics, metric is a function which defines distance between two points of a set. In a particular set, if the metric operator satisfies some fundamental postulates then such set is known as a metric space. The most common metric space is the n-dimensional Euclidean space. On the same tune, partial metric space has been introduced by Matthews (1994) as a generalization of a metric space, having self-distance that is not be zero. Under this condition the self-distance of every point does not have to be zero, the necessary basic notions to generalize a fuzzy metric space and their topological properties into a partial fuzzy metric space is the main component of this research work. Zadeh (1965) proposed fuzzy set theory to deal with instances of ambiguous situations, while the concept of a fuzzy metric space proposed by Kramosil and Michalek (1975), which is closely associated to the probabilistic category of metric spaces. As it is known that a fuzzy metric space deals with instances where there is probability of a random occurrence of an event. As a consequence, fuzzy metric space theory is more extensive than theory of a metric space. Also, the theory of fixed points which is a part of non-linear analysis is very useful to find the conditions in which single valued or multi-valued mappings have solutions. The concept of metric spaces as introduced by Frechet (1906) further helped in the development of fixed point theory. Banach (1922) established the fixed point theorem in metric spaces, which serves as an important tool in solving the problems of finding fixed points, which is indeed an important component for the analysis of non-linear systems. In this