Fixed Point Theorem Forα-β-nonexpansive Mappings And Convergence Theorems For The Implicit Iteration Process With Errors

This paper studies the existence of fixed points for a new nonexpansive mapping and strong convergence of the iterative sequence in Banach space; The strong con-vergence of the iteration sequence in a real uniformly convex and uniformly smooth Banach space and the existence and uniqueness of fixed points for implicit iteration in convex metric spaces are proved. The paper contains four parts as following:In chapter one. we will discuss the background of fixed point theory, the main contents and significance of this paper.In chapter two. a class of new nonexpansive mapping is introducted in Banach spaces, a necessary and sufficient condition of existence of fixed points is got in a u-niformly convex Banach space, under certain conditions, Ishikawa iterative sequence of the nonexpansion mapping converges to its fixed point is proved, and finally the sequence converges strongly to a fixed point of the nonexpansive mapping is given. The results extend many maps and some of most general fixed point theorems for iteration sequence.In chapter three,we will introduce a class of quasi-Φ-asymptotically nonexpan-sive nonself mappings. to study the modify Halpern and Ishkawa type iterative algorithm. and to prove iterative sequence generated by Cauchy sequences under a certain the conditions, and to prove a strong convergence of the sequences. The results extend some fixed point theorem of iterative sequence in Banach spaces. In chapter four, we consider an implicit iterative sequence with errors of a finite asymptotic nonexpansive mapping, we prove that the sequence is a Cauchy sequence under certain conditions, we get the sequence converges asymptotically to a common fixed point of two finite families of asymptotically quasi-nonexpansive mappings in a complete convex metric spaces. As a consequence of our result, we obtain some related convergence theorems of iterative sequence.