Mathematica Eterna
Open Access

Abstract

Frequently Convergent Properties of Solutions for a Discrete Dynamical System

Fanqiang Bu, Hui Li and Yuanhong Tao

The classic concept of limit is not enough to accurately describe the property of convergent sequence, however the definition of frequent convergence of sequence, defined by the concept of frequent measure, can get the better details of divergent sequence than the classic concept of convergence. In this thesis, using the definition and properties of frequent measure and frequent convergence, we study the frequently convergent properties of difference equations xn+k = 1 − x 2 n . We first present a fixed point theorem and then define a polynomial function, which are both closely related to the above diffrence equations. Through different monotonic properties of the above polynomial function on a different intervals, we detailed disscuss the solution of the above diffrence equation as k = 2, that is xn+2 = 1 − x 2 n , when initial values in diffrent intervals, and then we generalize the conclusion to the case k being any positive integer.