The Theorems On Existence Of Fixed Point And Its Strong Convergence For Nonlinear Operator

The fixed point theory of nonlinear operators is an important part of nonlinear function analysis. Meanwhile, the problem of approximating to solutions of nonlinear operator equations becomes the topic that people study in the recently years. In this paper, we investigate the existence of fixed point for two classes of nonlinear operators and the existence of solution for nonlinear operator equations. We also propose a new composite iterative method for nonexpansive mappings. Then we obtain some strong convergence theorems using the new composite iterative method. Results presented in this paper improve, extend and unify many authors's recent results. This paper includes four chapters.Chapter 1, we recall the history and present situation of the fixed point theory for nonlinear operators, and we also give a summary of this work.Chapter 2, we obtain some new fixed point theorems of 1-set-contractive operators in Banach spaces by using topological degree methods. In addition,we investigate several existence theorems of solutions for the equation Ax=μx(μ≥1).Chapter 3, by means of nonlinear alternative for nonexpansive mappings and property of strictly convex(concave) functions, some new fixed point theorems of nonexpansive mappings are obtained in Hilbert spaces. The results improve and extend the corresponding results.Chapter 4, we propose a new composite iterative method for computing common fixed points of nonexpansive mappings in Banach space. Then we prove that such a sequence converges strongly to a common fixed point of nonexpansive mappings. We also investigate the strong convergence of the viscosity approximate methods with the weakly contractive mapping. Moreover, we apply our results to the problem of finding a common fixed point of a countable family of nonexpansive mappings and the problem of finding a zero of an accretive operator.