Some Convergence Properties Of Fixed Point Iterative Sequences For Some Mappings

The problem of fixed point is one of the more important problems that people regard, and the corresponding research has gained many great achievements. Among many directions of the fixed point researches, it becomes main problem that the convergence problem about making approximating fixed point sequences and its application in control, nonlinear operator and derivative equation etc. The research of this problem will play an important role in its application in reality. In this paper, the Ishikawa, Mann and three-step iterative sequences are discussed and some properties of the convergence of the Ishikawa, Mann and three-step iterative processes with errors and their detail conclusion under some mappings are introduced.The thesis mainly consists of three parts. the first is the convergence problems of fixed point about uniformly ( L-α)- Lipschitz asymptotically nonexpansive mapping; the second is the convergence problem of uniformly L-Lipschitz asymptotically nonexpansive type mappings; the last is the converg-ence theorems of a strictly pseudo-contractive map.We first discuss the convergence problems of fixed point about uniformly ( L -α)-Lipschitz asymptotically nonexpansive mapping. The results are given by the convergence problem of Ishikawa, Mann and three-step iterative processes with errors, and we prove it. By using a new method, the iterative approximation problems of fixed points for uniformly ( L -α)- Lipschitzasymptotically nonexpansive mapping in Banach spaces were studied, and the bounded assumption of domain and range was dropped.Second, we study the convergence problem of three-step iterative sequence approximation to fixed points of uniformly L-Lipschitz asymptotically nonexpansive type mappings with error in uniformly convex Banach space, and we prove the convergence theorems of three-step iterative sequence with error approximation asymptotically nonexpansive type mapping, The fixed point properties of Ishikawa and Mann iterative sequences are extended to three-step iterative sequence. and we improve some recent corresponding results.At last, we get some convergence theorems for Ishikawa and Mann iterative sequences with errors of a strictly pseudo-contractive map in a nonempty bounded closed convex subset of a Hilbert space. The fixed point properties of nonexpansive mapping are extended to strictly pseudo-contractive map. The results improved and extended some recently results.