Research On The Convergence And Divergence Of Sequence Based On Fixed Point Theorem

The fixed point theorem is an important theorem in functional analysis.It has been widely used in solving the limit of sequence,the unique solution of differential equations and the integral mean value theorem.In this paper,we use the fixed point theorem to study the convergence and divergence of recursive sequence.Firstly,we define the uniformly contraction sequence and power contraction sequence in the real number space,so that we can use the fixed point theorem to get the limit of the recursive sequence.Based on the concept of contraction coefficient,we define the divergent contraction coefficient and the divergent fixed point,making it easier to judge the diverging speed.We define the feature principal part and the feature term,which helps us to solve the speed of a class of series' s divergence.Secondly,based on the properties of the cone metric space,the fixed point theorem and the common fixed point theorem,we generalize the uniformly contraction theorem and the power contraction theorem.What's more,we can solve the common convergence of two recursive sequences.Finally,based on the concept of the cone metric spaces with Banach algebras and the properties of c-distance,using the fixed point theorem of the quasi-contractions,we get the uniformly quasi-contraction theorem and the power quasi-contraction theorem.