The fixed point theorem of contraction maps is established,an example about that contraction maps have not fixed point in a completment metric space is given,and approach theorem of fixed point of contraction maps in compact metric spaces is obtained.

The present paper proves that the continuous map of a compact metric space is the sufficient condition of distributional chaos in a squence.

Fixed point theorem in compact matric space;

Closed images of locally compact metric spaces;

Transitivity and chaos of a sequence of maps in a compact metric space;

A further fixed point theorem was proved for mappings satisfying implicit relation functions on two compact metric spaces.

Finally,as to an equicontinuous homeomorphism on a compact metric space,from the minimality and compactness of the space,a finite subcover is acquired,which is formed by open neighborhoods centered at the finite-lengthed orbits of some finite points with radius ε2.

Here, we provide two new fixed point theorems in complete and compact metric spaces into which the concept of a w-distance was introduced.