Fuzzy Metric Space

We generalize the definition of the first order Sobolev spaces with zero boundary values to an arbitrary metric space endowed with a Borel regular measure. We show that many classical results extend to the metric setting. These include completeness, lattice properties and removable sets.

In this paper we study on contribution of fixed point theorem in Metric spaces and Quasi Metric spaces. Definition: 1 (Metric Space) Let X be a non-empty set-A function XxX →R (the set of reals) such that p:XxX→R is called a metric or distance function (if ad only if) p satisfies the following conditions. (i) p (x,y) ≥ 0 for all x, y x (ii) p (x,y)= 0 if x=y (iii) p (x,y) = p (y,x) for all x,y x (iv) p (x,y) ≥ p (x,z)+ p (z ,y) for all x,y,z X If p is a metric for X, then the pair (X, p) is called a metric space. Definition: 2 (Cauchy sequence) Let (X,p) be a metric space .Then a sequence { xn} of points of X is said to be a Cauchy sequence if for each >0,there exists a positive integer n0 such that m,nn0 implies p(xm,xn) < . Definition: 3 (Completeness): A metric space(X,p) is said to be complete if every Cauchy sequence in X converges to point of X. Definition: 4 (Contraction mapping) Let (X, p) be a complete metric space. A mapping T:XX is said to be a contraction mapping if there exists a real number  with 0 << 1 such that, p{T(x) ,T(y)})  p (x,y) <p(x,y)  x, y,  X i.e In a contraction mapping, the distance between the images of any two points is less than the distance between the points.