Approximation Of Variational Inequalities And Nonlinear Operator Equations

In this thesis, several new(systems of) variational inequality problems, fixed point problems,nonlinear operator equation problems and the problems of their common elements are introduced and studied in the setting of infinite dimension real Hilbert spaces. Using the projection operator technique, systems of Wiener-Hopf equations technique, auxiliary principle technique and the strongly positive linear bounded operator technique respectively, we modify Mann iteration method, extragradient method, hybrid method, proximal point method and viscosity approximation method, construct several new iterative algorithms, and then prove the convergence of the generated sequences. The results in this paper can be viewed as the improvement, extension and supplementation of the corresponding results announced by many others. This thesis consists of five chapters.In Chapter 1, we state the research background of the variational inequality theory and nonlinear operator equation, recall some basic concepts and theories, and also briefly introduce the main work and the structure arrangement of this thesis.In chapter 2, we introduce and study a class of new system of generalized set-valued nonlinear quasi-variational inequalities problem. Firstly, the senses of equivalences between the system of generalized set-valued nonlinear quasi-variational inequalities, the corresponding fixed point problems and the system of Wiener-Hopf equations are proved. Secondly, by using the projection operator technique and system of Wiener-Hopf equations technique respectively, we suggest two new iterative algorithms to find the approximate solutions. Finally, both the existence of solutions and the convergence of different types of generated sequences are proved. It is the first time to use the system of Wiener-Hopf equations technique to solve the system of variational inequalities problems. Through comparing the two methods, it is easy to see that, the Wiener-Hopf equation technique is more general than the projection operator technique.In chapter 3, based on chapter 2, the auxiliary principle technique is extended to study a system of generalized set-valued strongly nonlinear mixed implicit quasi-variational-like inequalities problem. Firstly, we establish the existence of solutions of the corresponding system of auxiliary variational inequalities problem. Then, using the existence result, we construct a new iterative algorithm. Finally, both the existence of solutions of the original problem and the convergence of the generated sequences are proved. We give an affirmative answer to the open problem raised by Noor et al.(Korean J. Comput. Appl. Math. 1998, 1: 73-89; J. Comput. Appl. Math. 1993, 47:285-312).In chapter 4, we introduce and analyze a new general hybrid iterative algorithm for the problem of finding a common element of the set of common zeros of two families of finite maximal monotone mappings, the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping. The algorithm is based on four well-known methods: Mann’s iteration method, composite method, outerapproximation method and extragradient method. We prove the strong convergence theorem for the proposed algorithm and the existence of the common element. The results greatly extend and improve the corresponding results of [Wei L., Tan R.L. Fixed Point Theory and Applications.2014, 77(1)].In chapter 5, based on chapter 4, by the strongly positive linear bounded operator technique,a new generalized Mann-type hybrid composite extragradient CQ iterative algorithm is first constructed. Then using the algorithm, we find a common element of the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping, the set of zeros of two families of finite maximal monotone mappings and the set of fixed points of an asymptoticallyκ-strict pseudocontractive mappings in the intermediate sense. Finally, both the existence of the common element and the strong convergence of the generated sequences are proved.