Some Problems On Fixed Point Theory

The theory of fixed points is an important topic of mordern mathematics and it is widely used in many subjects such as control and optimization theory, nonlinear operator etc. In this thesis, we discuss the problem on approximating fixed points. The thesis is divided into the following parts.Firstly, we recall present situation of the research on fixed point problems, and we also give a summary of this work.In Chapter 1, we start with the related preliminary and some basic results. Also, we introduce some conceptions and notations needed in the following chapters.In Chapter 2, motivated by the viscosity approximation methods for finding zeros of accretive operators, we mainly concern about necessary and sufficient conditions for strong convergence of iterative sequence of pseudo-contraction mapping.In Chapter 3, under the framework of strictly convex reflexive Banach space with uni-formly Gateaux differentiale norm, we use viscosity approximation methods for finding the common fixed point of nonexpansive nonself mappings. Aslo, we prove the iterative scheme strongly converges to the common fixed point of nonexpansive nonself mappings when the iterative coefficients satisfy certain conditions.In Chapter 4, we introduce a new iteartive scheme to approximate the solutions of generalized equilibrium problems and common fixed point of an infinite family of nonex-pansive mappings in Hilbert space. The strong convergence theorems of iterative algo-rithms are obtained by using viscosity approximation methods.In Chapter 5, under the framework of Hilbert space, we concern about the solutions of equilibrium problems and variational inequality problems, the common fixed points of an infinite family of nonexpansive mappings. We use viscosity approximation method to construct iterative sequence. Aslo, we prove the iterative scheme strongly converges to the solutions of equilibrium problems and variational inequality problems, the common fixed points of nonexpansive mappings.