Iterative Approximation Of Solutions For Fixed Point Problems Of Nonlinear Operators And Split Feasibility Problems

In this dissertation,we study the variational inequalities problems,the fixed point problems of nonlinear operators and the split feasibility problems in the setting of infinite-dimensional Hilbert spaces or Banach spaces.In order to solve these problems,we modify extragradient method,Bregman projection method,shrinking projection method,damped method,hybrid method and viscosity approximation method,and prove the convergence of these modified algorithms by using projection operator technique and demi-closedness principle.The results in this dissertation can be viewed as the improvement,extension and supplementation of the corresponding results announced by many others.This dissertation consists of six chapters.In Chapter 1,we state the research background and present situation of the split feasibility problems,and also the main work and the structure of this dissertation.In Chapter 2,we recall some basic concepts and preliminaries.In Chapter 3,the purpose of this chapter is to study the extra-gradient methods for solving split feasibility and fixed point problems involved in pseudo-contractive mappings in real Hilbert spaces.We propose an Ishikawa-type extra-gradient iterative algorithm for finding a solution of the split feasibility and fixed point problems involved in pseudo-contractive mappings with Lipschitz assumption.Moreover,we also suggest a Mann-type extra-gradient iterative algorithm for finding a solution of the split feasibility and fixed point problems involved in pseudo-contractive mappings without Lipschitz assumption.We prove that the sequences generated by the proposed iterative algorithms converge weakly to a solution of the split feasibility and fixed point problems.The results presented in this paper extend and improve some well-known results in the literature.Numerical example illustrates the theoretical result.In Chapter 4,based on the study of the definition of Bregman quasi-strictly pseudo-contractive mappings and hybrid projection,we propose a new hybrid projection algorithm in the setting of p-uniformly convex real Banach spaces which are also uniformly smooth for approximating the common solution of split feasibility and fixed point problems.It is proven that the sequences generated by the proposed iterative algorithm converge strongly to the common solution of split feasibility and fixed point problems.The results presented in this paper extend and improve some well-known results in the literature.Numerical example illustrates the theoretical result.In Chapter 5,the purpose of this chapter is to study the shrinking projection method for finding a common solution of proximal split feasibility problem,fixed point problem and variational inequality problem.We construct the appropriate iterative algorithm to approximate the common solution of the proximal split feasibility problems,fixed point problems and variational inequality problems by shrinking projection method.The strong convergence of the proposed algorithm is proved.In Chapter 6,the aim of the chapter is to study an iterative algorithm for finding the minimum norm solution of the proximal split feasibility problem.We present a damped algorithm from which strong convergence theorem is obtained in Hilbert space.Numerical example illustrates the theoretical result.