Iterative Approximation Of A Class Of Nonlinear Problem

In this dissertation,we study the fixed point problems of nonlinear operators and the variational inequalities problems in the setting of infinite-dimensional Hilbert spaces or Banach spaces by using demi-closedness principle,fixed point technique,projection operator technique and geometric properties of Banach spaces.We modify viscosity method and extragradient method in the references therein,and prove the convergence of these modified algorithms.The results in this dissertation improve and extend the corresponding results announced by many others.This dissertation consists of five chapters.In Chapter 1,we state the research background and present situation of the variational inequality problems and fixed point 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,we study variational inequality and fixed point problems for quasi-pseudo-contractive mappings with and without Lipschitz assumption in Hilbert s-paces,respectively.It is proven that the sequences generated by the proposed itera-tive algorithms converge strongly to the common solution of the variational inequality and fixed point problems.Numerical example illustrates the theoretical result.In Chapter 4,we discuss a,class of generalized variational inequality problems involved in strictly pseudo-contractive mapping in Banach spaces by using some new algorithms which couple modified extra-gradient method with extended Mann itera-tion.It is proven that the sequences generated by the proposed iterative algorithms converge strongly to the solution of the generalized variational inequality problem.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,we summarize the main work and discuss the future research of this dissertation.