Realization-based Method Of Successive Projection And Boundary Point Algorithms And Their Applications

This thesis studies two kinds of new algorithms for fixed point problems of nonexpansive mappings. Firstly a new algorithm is presented for modifying the hybrid method, called Realization-based method of successive projection for nonexpansive mappings and nonexpansive semigroups. This algorithm avoids the use of projection operators on complex closed convex sets. In addition, another new algorithm is proposed: Generalized viscous boundary point algorithm for nonexpansive mappings, which covers Halpern iterative algorithm, Viscosity iterative algorithm and Boundary point algorithm. After that, two new algorithm is introduced for modifying Mann iteration process by boundary point method and their strong convergence is proved.The specific content of this paper is organized as follows:Firstly, Realization-based method of successive projection for nonexpansive mappings and nonexpansive semigroups is proposed and the strong convergence theorem is proved. In addition, its data results is given. Finally, Proximal point algorithm is improved to a more simple and easier to implement algorithm..Secondly, Generalized viscous boundary points is given and its strong convergence is proved. If the nonexpansive mapping is replaced with the sequence of average mappings, the same convergence result holds in the case of weakened conditions. At last, an algorithm is considered for finding a common fixed point of a family of finite nonexpansive mappings.Lastly, two new modified Mann iteration by boundary point method are proposed and the strong convergence theorems are proved. This iteration process can directly solve the minimum norm fixed point.