Some Convergence Problem Of Iterative Sequences In Banach Space

In recent years, the problem of iterative approximation of fixed point for non-self mappings becomes a very popular subject. Among many directions of the fixed point researches, it becomes a main problem that the convergence of making approximating fixed point sequence and its application in control, nonlinear operator and derivative equation etc. Its research will play an important role in its application in reality. In this paper, we mainly studied fixed point iterative sequences of some kinds of non-self mappings in Banach spaces.First, the history of iterative scheme is represented. Lots of theories and definitions are quoted. We can know something about the iterative scheme development. As the same time, we show the fixed point development and the theorems related with Banach spaces.Second, it is primarily studied the equilibrium problem and iterative scheme about the fixed point in Hilbert spaces. In this paper, we introduce an equilibrium problem and definitions related with this paper firstly. In addition, it is proved some qualities in Hilbert spaces. Secondly, we propose an iterative scheme of an equilibrium problem and establish some weak and strong convergence theorems of the sequences generated by our proposed scheme of a strict pseudo-contraction mapping and an asymptotically pseudo-contraction mapping in Hilbert spaces.Finally, we discuss convergence problem of three step iterations for certain asymptotically nonexpansive ? -pseudo-contractive mapping in a real Banach space. Then it is provided that the convergences of three kinds of iterations for a Lipschitz operator are equivalent. Finally, the strong convergence of the scheme to a fixed point is shown in a Banach space with uniformly Ga?teaux differenti-able norm.