【作者】 张国辉；

【导师】 崔云安；

【作者基本信息】 哈尔滨理工大学 ， 基础数学， 2010， 硕士

【摘要】 非自映射不动点的迭代逼近问题已成为近年来学术界研究的活跃课题。在不动点问题研究的众多方向中,关于构造渐近不动点序列的迭代收敛问题以及其在控制、非线性算子和微分方程等方面的理论结合及应用成为研究的主流问题,并在实际运用中起到至关重要的作用。本文主要研究了Banach空间上的几类非扩张映射下迭代序列的收敛性问题。首先,我们讲述了迭代序列的发展概况。通过引用大量前人的定义和定理,使我们对迭代序列的发展史有了一定程度的认识。同时,对于不动点的发展史和Banach空间相关知识也有一定程度的了解。其次,我们主要研究Hilbert空间中均衡问题和不动点问题的迭代解。我们先提出均衡问题,给出与定理相关的定义,同时给出Hilbert空间的一些特性。接着给出一个关于均衡问题的迭代,讨论了Hilbert空间中在严格伪压缩映射与渐近伪压缩映射下的该迭代的弱收敛性和强收敛性问题。最后,在Banach空间中,我们讨论了一类渐近非扩张? -伪压缩映射下的三重迭代序列的收敛性和Lipschitz映射下三种迭代(改进的Mann迭代,改进的Ishikawa迭代和改进的三重迭代)收敛性的等价性;然后讨论具有一致Ga?teaux可微范数的一致凸Banach空间中迭代序列的强收敛问题。更多还原

【Abstract】 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.更多还原