Let X be a complete distance space and F be a contractive mapping from X to X, then F must have only one fixed point in X and start from any point of X. Let's start with the sequence x 1=f(x0), x2=f(x 1), ..., xn = f (xn-/klk).
Where f is a compressed map if it compresses the distance between every two points in x by at least k times, where k is a constant less than 1, that is, the distance d (f(x), f(y)) between the images of x and y at every two points in x is not more than k times the distance d(x, y) between x and y.
Contraction mapping method is a common method in fixed point method. Barnach gave the contraction mapping in 1922. This idea can be traced back to Picard's successive approximation method for solving ordinary differential equations.
This method can provide the existence and uniqueness of various equations and solutions, as long as the solution of the equation can be transformed into a fixed point of contractive mapping. This method has been extended to many aspects, such as nonexpansive mapping, mapping family, set-valued mapping, probability metric space and so on.