Viscosity Approximation Methods For Two Classes Of Fixed Point Problems Of Nonlinear Operators

The fixed point theory of nonlinear operators is an important part of nonlinear functional analysis and is one of the emphasis problems of people's regard. Especially, the problem of approximating to solutions of nonlinear operator equations becomes the topic that people study in the recently years, and the corresponding research has gained many great achievements. Among many directions of the fixed point researches, it becomes main problem that the convergence problem about making approximating fixed point sequences and its application in control, nonlinear operator and derivative equation etc. The research of this problem will play an important role in its application in reality.In this paper, we are concerned about viscosity approximation methods for two classes of fixed point problems of nonlinear operators which are very interesting and quite important problems in the study of nonlinear approximation theory. So one topic of this thesis is to use viscosity methods to approximate zero point of a family of m-accretive mappings in reflexive Banach spaces. Another topic of thesis is to use viscosity methods for equilibrium problems and fixed point problems in Hilbert space. Results presented in this paper improve, extend and unify many authors' recent results.The main results obtained in this dissertation may be summarized as follows.Chapter 1 recalls the history and present situation of iterative algorithms for solutions of nonlinear operator equations. The iterative algorithm for fixed point problems of nonlinear operators is an important part of fixed point theory and is one of the emphasis problems of people's regard. There are many iterative algorithms for the solution of fixed point of mappings. And we also give a summary of this work.In Chapter 2 , a general iterative algorithm by viscosity method to approximate a common point of a finite family of m-accretive mappings in a reflexive Banach space which has a weakly continuous duality mapping are studied. Furthermore , two classes of new and general iterative algorithms are suggested and the convergence of sequences generated by two class of algorithms are proved.In Chapter 3 , we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of k—strictly pseudo-contractive mappings in a Hilbert space. Then, we prove two strong convergence theorems .