非线性映射的不动点定理与遍历理论

【作者】 顾文颖；

【导师】 庄亚栋； 李刚；

【作者基本信息】 扬州大学 ， 基础数学， 2001， 硕士

【摘要】 非线性算子半群的遍历理论的研究开始于二十世纪七十年代中期，随后由于被广泛应用于微分方程的数值解，正解的存在性理论，控制论，最优化等问题中而得到了很大发展。J.B.Baillon[1]在1975年首先提出了Hilbert空间的凸闭集上非扩张映射的遍历收敛定理。Takahashi ＆ Zhang [4]，Tan ＆ Xu ［5］分别将Baillon的定理推广到渐近非扩张和渐近非扩张型半群。1997年，Li［6］，Li＆Ma[7]在Hilbert空间上成功的去掉了凸、闭等条件，在一般半群上得到了遍历压缩定理，极大地推广了遍历定理的应用范围。Rouhani [8]首先引入了殆非扩张曲线的概念，它包含了非扩张半群的殆轨道，并且给出了殆非扩张曲线的一些渐近性态。接着Li ＆ Ma [9]引入了渐近殆非扩张曲线（AANC）的概念，它比殆非扩张曲线要广得多，并且包含了所有渐近非扩张型半群的殆轨道。他们讨论了渐近殆非扩张曲线的渐近性态及遍历定理，并给出了当C非凸闭集时渐近非扩张半群的渐近性态及遍历定理。本文第一章首先给出了了在一般拓扑半群上渐近非扩张型半群(?)中任意函数的遍历压缩定理，它包含了以往一大批文章的结果。接着本文引入了渐近非扩张型曲线的概念，它包含了所有渐近非扩张型半群的殆轨道。进一步，本文还证明了渐近非扩张型曲线的遍历定理。 1965年，Kirk [20]证明了若C是具正规结构Banach空间的弱紧凸子集，则C上非扩张映射T有不动点。Goebel & Kirk [14]给出了一致凸Banach空间，渐近非扩张映射的不动点定理。Kirk [21]将其推广到渐近非扩张型半群。［17］，[13]，[19]，[16]，[18]又将其推广到更广的空间。Li ＆ Sims[25]解决了正规结构的Banach空间中，渐近非扩张型映照是否存在不动点这一多年未知的问题。本文第二章仍是继续这方面的工作。首先给出了判定不动点存在的充分及充要条件，从而简化了一些已知结果的证明。并且本文还将[18]，[19]推广到了k-Lipschitzian半群的情形。更多还原

【Abstract】 2 Abstract The research of nonlinear ergodic theory began in the mid-seventies. Consequently, it got great development because it was widely used in many questions such as the numerical solution of differential equation, the existence theory of positive solution, control theory, optimization. Baillon proved the first nonlinear ergodic theorem for nonexpansive mappings in the framework of Hilbert space. A series extensions of Baillon抯 result have been given, e.g., Takahashi and Zhang [4] , Tan and Xu [5] proved the ergodic theorem for asymptotically nonexpansive and asymptotically nonexpansive type semigroup respectively. Futher, Li [6], Li and Ma [7] indicated that some key conditions of previous results, such as C is convex and closed, are not necessary. Rouhani [8] first introduced the notion of almost nonexpansive curves, which contains all the almost-orbits of nonexpansive semigroups. He got some asymptotic behaviours of almost nonexpansive curves. In [9], Li and Ma introduced the concept of asymptotically almost nonexpansive curves(AANC) on [0, + a3], which is much more general than almost nonexpansive curves, and contains all the almost-orbits of asymptotically nonexpansive type semigroups. By discussing the asymptotically behavior and ergodic theorem of AANC, they got the corresponding results of asymptotically nonexpansive type semigroups when C is not necessary convex and closed. In the present paper, we first prove the edgodic retraction theorem for any functions on 3. This result contains many previous work. After giving the definition of asymptotically nonexpansive type curves, which contains all the almost-orbits of asymptotically nonexpansive type semigroups, we will prove the ergodic theorem of this curves. In 1965 , Kirk [20] proved that if C is a weakly compat convex subset of a Banach space with normal structure, then every nonexpansive self-mapping T of C has a fixed poind. Seven years late, in 1972, Goebel and Kirk [5] proved that if the space X is assumed to be uniformly convex, then every asymptotically nonexpansive self-mapping T of C has a fixed point. This was extended to mappings of asymptotically nonexpansvie type by Kirk in [12]. More recently these results have been extended to wider classes of spaces, see for example [8], [4], [10], [7] and [9]. Li and Sims [25] solve the open question that whether normal structure implies the existence of fixed points for mappings of asymptotically nonexpansive type. This present paper point out the conditions when fixed points exist and so is a futher step toward answering the above question.. We also represents an extension of the results of [10] and [9] to uniformly k-Lipsehitzian semigroup. 3更多还原