Some Results On Fixed Point Theory

The theory of fixed points is an important topic of morden 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.The first chapter, prepare knowledge. Introduce some basic knowledge and clas-sical conclusion in the theory of Banach spaces. Also, we give a summary of this work.The second chapter, introduce a new iterative scheme to approximate com-mon fixed points of two asymptotically nonexpansive mappings. Also, we prove the weak and strong convergence theorems when the iterative coeffcients satisfy certain conditions.The third chapter, motivated by the viscosity approximation methods for find-ing zeros of accretive operators, we concern about the strong convergence theorems of a new iterative scheme of pseudocontractive mappings in Banach spaces.The fourth chapter, under the framework of uniformly convex Banach spaces which admit a weakly sequentially continuous duality mapping, we use viscosity approximation methods for finding the fixed point of nonexpansive mapping and obtain the strong convergence theorems.The fifth chapter, summarize the mainly results of this thesis and raise some problems which need to be solved.