Viscosity Approximation Methods For Nonexpansive Mappings Of Banach Space

Fixed point theory is an important component of the recent rapid development of non-linear functional analysis theory, and has closed contact with many branches of modern mathematics, such as topology theory, modern analysis, operator theory, space agency theory and so on, especially plays an important role in the problem of the existence and uniqueness of the solution to various types of equations having been established. It is widely used, and is an important tool to study the nonlinear equations, such as deterministic or stochastic differential equations, integral equations, functional differential equations, functional equations and a variety of operator equations.Among the fixed point problem of the many directions, it becomes an important task on the iterative convergence problem of the fixed-point sequence and its application in modern variational problems, optimization problems, economic equilibrium problems. Nonself mapping fixed-point theory is an important component of the fixed-point theory, particularly nonself mapping fixed-point iterative approximation problem has become an active research topic of many scholars at home and abroad.In this paper, the problem on approximating to the nonexpansive fixed points is discussed. The paper includes two parts as follows:First, by using of the viscosity approxiamtion methods for nonexpansive mappings, the strong convergence theorem of non-self nonexpansive mappings is obtained. And it not only weakens the condition of the space, but also a new iterative sequence is given, and the necessary and sufficient conditions for its convergence are proved.Second, based on the previous results, in the context of Banach space with a uniform G a∧teaux differentiable norm, the strong convergence problem of non-self nonexpansive mappings and with errors of non-self nonexpansive mappings in Reich-Takahashi iterative algorithm are discussed. It provides the basis for looking for more general conditions to the approximate problem of Reich-Takahashi iterative sequence.