Mann Iterates And Ishikawa Iterates In Banach Space

It is primarily studied in Banach space that some kinds of mappings converge to these mappings'fixed point in sense of Mann iterations and Ishikawa iterations. First, the mappings'convergence problems in compac-tive space is extended in this paper In this paper. Because there is at least one subsequence in any numerous sequences in compactive space, the question mentioned above can be proved easily. In addition, it continues to discuss the mappings'convergence problems in non-compactive space. Since the condition of compact is not available, condition I is introduced. The result of the above is still correct.First, the history of fixed points is represented. Lots of theories and defi-nitions are quoted. We can know something about the fixed point development.Second, it is primarily studied nonexpansive mappings'convergence pro-blems in sense of Mann iteration or Ishikawa iteration. At first, it can be known some definitions related with this paper. Such as nonexpansive mapping, Mann iteration and Ishikawa iteration, Hausdoff metric and some related theorems. In addition, it is proved that in compactive space, the mapping in sense of Mann iteration or Ishikawa iteration converges to the fixed point of the mapping.Finally, it is proved that these mappings converge in non-compactive space. Since the compactive space is not available, the result is still true by assumption of condition I. A new Mann iteraton is introduced. Then this paper gives us the results that the mapping converges to its fixed point which can be nonexpansive mapping, quasi-nonexpansive mapping or generalized nonexpansive mapping.