Iteration Algorithms For Approximating To Fixed Points Of Nonexpansive Mappings And Solutions Of The System Of Variational Inequalities

This paper discusses the problem on approximating to the fixed points of nonexpansive mappings and solutions of the system of variational inequalities(SNVI). The paper have four chapters.In the first chapter, we introduce the background and simple history of nonlinear operators and iterative methods, and the information of implicit iteration processes.In the second chapter, we discuss two general composite implicit iteration sequences with errors. The schemes presented in this paper included various previous concrete iteration schemes. Hence, the results presented in this paper extend, generalize and improve the results of Mann, Ishikawa, Xu and Osilike, Zhou and Chang, and other authors.In the third chapter, we apply the monotone CQ iteration method which different from the previous CQ iteration methods to obtain strong convergence. In the proved process of this paper, Cauchy sequences method is used, so we complete the proof without using the demi-closedness principle, Opial's condition or other about weak topological technologies.In the fourth chapter, we introduce and consider a new system of relaxed cocoercive generalized nonlinear variational inequalities involving a notion of projective nonexpansive.Based on the projection technique,we suggest and analyze new iterative methods for this system of generalized nonlinear variational inequalities.We also study the convergence analysis of the new iterative methods under mild condition.