Iterative Approximation On Fixed Points Of Nolinear Operators And Solutions Of Related Problems

The theory of nonlinear operators is theoretical basis and basic tools of nonlinear science,it has already been an important branch of modern mathematics and plays an important role inthe other branches. The fixed point theory of nonlinear operators is an constituent importantpart of nonlinear functional analysis, especially, the problem of approximating to solutionsof nonlinear operator equations （systems） becomes the active topic that people study in therecently years.The fixed points of nonlinear operators are closely related to the equilibrium prob-lems、variational inequalities、zero points of nonlinear operators, they can be also convertedto one another. Based on the transformation of the relationship between them, a new nonlin-ear operator and its iteration algorithm are constructed, through approximating fixed pointsof the new nonlinear operator, the equilibrium problems or variational inequalities are solved,and the strong （weak） convergence theorems are obtained. The nonlinear operators theoryare studied mainly by generalizing the space, improving the iterative algorithm and reducingthe restrictions of coefficient or the constraint of operators, then the more meaningful and themore generalized results are obtained.In this paper，we mainly study the fixed points, the equilibrium problems, variationalinequalities, zero points of nonlinear operators and the split feasibility problems, and so on.By using the different methods respectively, the solutions of the problem are iteratively ap-proximated, some strong convergence theorems are obtained, and some numerical examplesare presented supporting the theorems. Simultaneously, the comparison for the convergencerate of some classical iteration algorithms are considered and the comparison criteria is given,and the theoretical results are gotten, and some numerical examples are given to support ourresults. The results presented in this paper improve, extend and unify many authors’s recentresults. This paper includes six chapters. Now we will describe them briefly one by one.Chapter1principally states that the research background and present situation of the the-ory of nonlinear operators in Hilbert spaces and Banach spaces, and we also briefly introducethe main work and the structure arrangement of this work.Chapter2mainly constructs three different iterative algorithms, in certain conditions, toapproximate the common solutions of the fixed points of nonexpansive mappings, the varia-tional inequalities and the equilibrium problems, the the common solutions of the fixed pointsof infinite nonexpansive mappings and the equilibrium problems, the common solutionsof the fixed points of infinite nonexpansive mappings, the variational inequalities and the equilibrium problems, respectively. And some strong convergence theorems are obtained.Chapter3present the relations between the H-accretive operators （H-monotone oper- ators） and the-accretive operators （maximal monotone operators）, mainly discusses aboutiterative approximation problems for the zero points of the H-monotone operators and theH-accretive operators in Hilbert spaces and Banach spaces, respectively. Using the classicaliteration algorithms, some strong （weak） convergence theorems are obtained.Chapter4mainly pays attention to iterative approximation problems for the fixed pointsof the multi-valued nonexpansive mappings in Banach spaces. The two implicit iterationalgorithms are given, in certain conditions, some strong （weak） convergence theorems are ob-tained. In addition, we give comments for the gaps existed in the D.R.Sahu [47]�'�B.Panyanak[81].Chapter5mainly constructs three general iteration algorithms and proves the sequencesgenerated by the iteration algorithms converge to the common fixed point of the finite opera-tors {T₁, T₂,...,T_N} in Banach spaces. Applying the results in Hilbert spaces, the multiple-setsplit feasibility problem are solved.The last Chapter mainly considers the comparison for the convergence rate of some clas-sical iteration algorithms. In Hilbert space, the comparison criteria of convergence rate isgiven, the convergence rate of the Mann, Ishikawa and Noor iteration scheme for Lipschitzcontinuous and strongly monotone mapping are compared, the theoretical results are obtained,and some numerical examples are given to support our results.