Metric space model

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.