The Fixed Point Theory Of Some Nonlinear Operator

Fixed point theory is an important part of the nonlinear functional analysis theory. Banach contraction principle was proposed and proved by Banach in 1920's, fixed point theory was caused widespread concern and in-depth study. Since then, a series of new concepts of contractive mappings and new fixed point theorems of contractive mappings were proposed, some of these results have been successfully applied to the research of the existence and uniqueness of solution to equation, and some of these results have been applied to random operator theory and financial mathematics, and so on. Now fixed point theory is a perfect system. The fixed point theory of the nonlinear operator was concerning by some scholars.Fixed point problems and convergence problems of several typical nonlinear operators are studied in this thesis. In particular, fixed point problems of asymptotically nonexpansive mappings in product space are discussed, the convergence problems of the new interative sequence for nonexpansive mappings under specific conditions are discussed in this thesis. The thesis mainly consists of three parts.Firstly, the history of fixed point theory and the background of nonlinear operator theory are introduced, the relevant literature are summarized.Secondly, the convergence problems of new iterative sequence for non-expansive mappings in reflexive Banach space are considered. Under appropriate conditions, the iterative sequence converges strongly to a fixed point of nonexpansive mapping was proved, and a partial answer to Halpern's open question is given. The results presented in this paper extend and improve some recent results.Finally, some fixed point theorems for asymptotically nonexpansive mappings in product spaces are proved. The conditions of asymptotically nonexpansive mappings which have fixed points are found. The results presented in this paper also extend and improve some recent results.