The Iterative Algorithm Of Fixed Point Problem And Split Feasibility Problem Based On Implicit Midpoint Formula

The theories and methods of fixed points is a branch of nonlinear functional analysis,and they are also play an important role in the theories of existence and uniqueness of solutions of ordinary differential equations.It also promotes the development of iterative algorithms for numerical solutions of ordinary differential equations.In the last century,the main research of mathematicians on the fixed point problem have changed from the analysis of its existence to the exploration of iterative methods.For various iterative algorithms,we not only study the existence and uniqueness of fixed points,but also construct an iterative program of fixed points problem which have an arbitrary degree of accuracy by contraction mapping.In order to approaching the fixed point of a nonlinear operator in the history,there have been constructed many alternative formats,such as Picard iteration,Mann iteration,Ishikawa iteration,etc.Iterative algorithms for numerical solutions of differential equations are very efficient algorithm systems.It is important to apply these algorithms to the iterative process of fixed points which can broaden the iterative algorithm system for solving fixed points.In this thesis we constructed a new iterative algorithm based on the iterative algorithm of the numerical solution of the differential equation and the iterative algorithms of the fixed point.The main content of the study is divided into three parts:In the first part,we constructed a fixed point iterative algorithm based on Simpson method and studied its generalization algorithm.Based on the existing fixed point iterative algorithm of trapezoidal formula and we constructed a iterative algorithm of fixed point based on the implicit Simpson method using the Euler formula,and proved the weak convergence and related properties of the algorithm.This is the same idea that under the high-order numerical format of the differential equation we continued to promote the iterative algorithm on the basis of the previous algorithm,and finally we got the fixed point iterative algorithm in the form of convex combination of Mann iterative algorithm,and proved the weak convergence and related properties of the algorithm.In the second part,we constructed a fixed point iterative algorithm based on implicit Runge-Kutta method and studied its generalization algorithm.Based on the existing fixed point iterative algorithm of the implicit midpoint formula and combined with the Euler formula,we constructed a fixed point iterative algorithm of the implicit Runge-Kutta method and proved the weak convergence and related properties of the algorithm.This is the same idea that under the high-order numerical format of the differential equation we continued to promote the iterative algorithm on the basis of the previous algorithm,and finally we got the fixed point iterative algorithm in the form of convex combination of Mann iterative algorithm,and proved the weak convergence and related properties of the algorithm.In the three part,this kind of iterative algorithm is applied to solve the Split feasibility problem and multi-set split feasibility problem.Then we gave a new iterative algorithms format and new parameter range and changed the iterative calculation process of the iterative format.